# Cohomological dimension and the Künneth formula

*Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI, GPT-6.1 Sol (OpenAI). Links and citations revised by Claude Opus 5.5 (Anthropic), October 2026. Original text released under CC0.*

How far can étale cohomology extend above the dimension of a scheme? The answer depends on both the geometry and the coefficients. An affine scheme has a stronger vanishing theorem than a proper scheme, and a product requires a derived tensor product when the coefficient ring is not a field. We prove those statements, including the steps that pass from sheaves to arbitrary complexes.

## 1. Coefficients and prerequisites

Write \(\operatorname{cd}_{\mathrm{tors}}(X)\) for the least \(D\in\mathbf N\cup\{\infty\}\) such that \(H^q(X,F)=0\) for \(q>D\) and **every torsion abelian étale sheaf** \(F\). Write \(\operatorname{cd}_\ell(X)\) when only \(\ell\)-primary torsion sheaves are tested. Our prime-to-characteristic convention is the supremum of these latter dimensions over primes invertible on the scheme. We prove the stronger all-torsion affine bound. A statement about one constant sheaf is not, by itself, a cohomological-dimension statement.

All schemes in a finite-type assertion are of finite type over the indicated field, with no implicit smoothness or separatedness assumption. They are Noetherian and quasi-compact and quasi-separated. Unless specified otherwise, \(R\Gamma\), inverse images and higher images refer to the small étale topoi. A torsion ring here is a unital ring killed by a positive integer; in the product theorem it is the commutative ring \(\Lambda=\mathbf Z/n\).

We use these exact earlier results: geometric stalks and qcqs cohomological continuity, finite pushforward and finite base change from lesson 7; field-topos comparison from lesson 8; constructible approximation, Noetherian subobjects and exact extension by zero from lesson 11; arbitrary torsion vanishing for affine and proper geometric curves from lesson 12; arbitrary proper base change and finite-constant embeddings from lesson 13; the finite-amplitude resolution argument, unbounded proper base change and proper projection formula from lesson 14; and smooth, pro-smooth and field-extension base change from lesson 15.

The geometric foundations are [Noether normalization](../../AG-CA/krull-dimension-and-noether-normalization.html#3-normalization-that-respects-ideals) (Corollary 3.2, including normalization adapted to ideals), [finite normalization for varieties](../../AG-MO/normalization.html#5-finiteness-over-fields-including-inseparability) (Theorem 5.1 over arbitrary fields), [the dimension formula for finite-type algebras over fields](../../AG-CA/krull-dimension-and-noether-normalization.html#4-parameters-measure-dimension-and-height) (Theorems 4.2 and 4.3), and basic strict-henselization and finite-presentation limit properties. They contain no affine cohomological-vanishing or product theorem. For complexes we use [termwise injective K-injective resolutions, Theorem 4.1](../../derived-categories-and-sheaf-operations/k-injective-resolutions-in-grothendieck-categories.html#4-constructing-the-unbounded-resolution), in the full unbounded Grothendieck-category form. [Section 1A](#flat-models-on-sites) supplies flat generators on sites, unbounded and bounded-above K-flat models, and the lower Tor-amplitude truncation used in the tensor arguments.

<h2 id="flat-models-on-sites">1A. The flat models needed for tensor comparison</h2>

For the tensor arguments the coefficient ring is commutative. The following discussion works on any small site with a sheaf of commutative rings \(\mathcal O\), and hence on the small étale sites with coefficients \(\mathbf Z/n\) used below. It identifies exactly how the existing [flat and K-flat construction](../../derived-categories-and-sheaf-operations/flat-modules-and-k-flat-resolutions.html#3-a-surjective-resolution-by-attaching-cycles), stated for ringed spaces, applies here. In particular a lower Tor-amplitude bound, not just ordinary bounded-below cohomology, is what permits the lower-bounded flat model in Lemma 12.1.

**Flat generators on the site.** For an object \(U\), let

\[
P_U=a\left(V\longmapsto\bigoplus_{b:V\to U}\mathcal O(V)\right),
\qquad
L(M)=\bigoplus_{(U,s),\ s\in M(U)}P_U.
\tag{1A.1}
\]

Here \(a\) is associated sheaf, and all indexing collections are sets in the chosen universe. [Cohomology on sites, equations (1.1)–(1.2) and Proposition 1.1](../cohomology-on-sites.html#1-why-injective-resolutions-exist), prove \(\operatorname{Hom}_{\mathcal O}(P_U,M)=M(U)\), the epimorphism \(L(M)\to M\) sending the indicated generator to \(s\), and exactness of filtered colimits and sums in this module category.

The tensor universal property gives a natural identification

\[
P_U\otimes_{\mathcal O}M
=a\left(V\longmapsto\bigoplus_{b:V\to U}M(V)\right).
\tag{1A.2}
\]

To verify it, send a labelled scalar tensored with a local section to their product in that labelled summand. The inverse sends a section in a labelled summand to the corresponding basis vector tensored with the section. Additivity and balancing make these inverse on presheaf representatives; sheafification and local gluing give the stated inverse sheaf maps. An exact sequence of sheaves remains exact under the right side of (1A.2). Kernels are computed sectionwise. To lift a local representative of a section of the quotient, lift its finitely many labelled entries on covering refinements, then take a common refinement. Every section of the associated sheaf has such representatives locally. This proves local surjectivity, and the same local criterion gives exactness in the middle. Thus \(P_U\) is flat and the module category has enough flat objects. No topological-space description or existence of points is used. On an étale site this is the free extension from the étale object; it need not be projective.

**The elementary tensor properties.** Tensor products below use direct-sum totalization with differential \(d(x\otimes y)=dx\otimes y+(-1)^{|x|}x\otimes dy\). The proofs of [Lemmas 2.1, 2.3 and 2.4 of the K-flat reading](../../derived-categories-and-sheaf-operations/flat-modules-and-k-flat-resolutions.html#2-k-flat-complexes-and-their-closure-properties) use only this sign rule, exact sums and filtered colimits, and flatness of the one-degree objects. All these properties hold here by (1A.2) and Proposition 1.1 of the sites reading. In particular, a flat sheaf in one degree is K-flat; sums and filtered colimits preserve K-flatness; degreewise split extensions preserve it; and a bounded-above complex of flat sheaves is K-flat. For the last assertion, each bounded brutal lower truncation is obtained by finitely many split graded extensions of flat one-degree objects, and these truncations are an increasing union equal to the bounded-above complex. Nothing here says that every unbounded complex with flat terms is K-flat.

We also need the flat-quotient exactness assertion without a stalk assumption. Successively choosing the epimorphisms (1A.1) resolves any test module \(N\) by flat sheaves in degrees at most zero. For an exact sequence \(0\to M'\to M\to M''\to0\) with \(M''\) flat, tensor this flat resolution with the sequence. The resulting sequence of complexes is termwise exact. Its right-hand complex has no negative cohomology, by flatness of \(M''\). The long exact cohomology sequence and right exactness of tensor identify an injection \(N\otimes M'\to N\otimes M\), proving that the original sequence remains exact after any tensor. This is the argument of Lemma 1.3 of the reading with flat resolutions in place of free module resolutions; it requires only enough flats. Comparing two such exact tensor rows for a monomorphism of test modules also proves that the kernel of an epimorphism between flat sheaves is flat. These statements remain valid for tensor totalizations, since sums are exact.

**Proposition 1A.1 (unbounded and bounded-above flat models).** Every complex \(G\) has a termwise epimorphic quasi-isomorphism \(P\to G\), with \(P\) K-flat and all its terms flat. If \(G^i=0\) for \(i>b\), such a flat model may be chosen zero above \(b\).

**Proof.** The [cycle construction, Theorem 3.1](../../derived-categories-and-sheaf-operations/flat-modules-and-k-flat-resolutions.html#3-a-surjective-resolution-by-attaching-cycles), uses precisely the local-generator epimorphisms and tensor properties just established. Here are the site-specific details. Start with the sum over all integers \(i\) of the disk on \(L(G^i)\) in degrees \(i,i+1\), mapping its lower term onto \(G^i\). The map in the upper degree is its composite with the differential of \(G\). The resulting complex \(P_0\) is termwise flat and maps epimorphically to \(G\). Each disk is contractible; its tensor contraction includes the sign \((-1)^{|x|}\) on a first-factor element \(x\), so \(P_0\) is K-flat.

At stage \(s\), for every object \(U\), integer \(q\), and actual section of \(Z^q\ker(P_s\to G)\) over \(U\), attach a copy of \(P_U\) in degree \(q-1\). Give its generator that cycle as differential and image zero in \(G\). The adjunction in (1A.1) makes this a map of sheaves, and the cycle condition makes the differential square zero. The old complex is a degreewise split subcomplex of the new one; the quotient has zero differential and is a sum of shifts of flat generators. Thus every stage is K-flat with flat terms.

Take the filtered colimit over \(s\geq0\). It is still K-flat with flat terms and maps epimorphically to \(G\). Exact filtered colimits commute with kernels and cohomology. Each kernel-cohomology transition is zero as a sheaf map: a cohomology section is locally represented by a cycle section, and that section becomes a boundary at the next stage by construction. The colimit kernel is therefore acyclic, proving the quasi-isomorphism. This argument kills local representatives; it does not require an arbitrary epimorphism to be surjective on sections over every object.

For the bounded-above assertion, use [Lemma 4.1](../../derived-categories-and-sheaf-operations/flat-modules-and-k-flat-resolutions.html#4-compatible-bounded-resolutions-and-factorization). Starting at \(b\), replace the current term in degree \(i\) by its flat cover \(L(-)\) and the previous degree by the pullback along that cover. The kernel of each replacement map is a two-term identity complex. It is therefore an epimorphic quasi-isomorphism. Proceeding downwards changes any fixed term only finitely many times; the three terms determining a fixed cohomology degree stabilize. The stabilized complex maps quasi-isomorphically and termwise epimorphically to \(G\), has flat terms, and vanishes above \(b\). It is K-flat by the bounded-above tensor argument above. The construction uses neither projectivity of \(P_U\) nor exactness of inverse limits. \(\square\)

The cone factorization and tensor comparison in [Lemma 4.2, Corollary 4.4 and Lemma 4.5](../../derived-categories-and-sheaf-operations/flat-modules-and-k-flat-resolutions.html#4-compatible-bounded-resolutions-and-factorization) now apply with the same maps and signs: they require only the resolutions just constructed, split graded extensions and the cone formula. In particular a quasi-isomorphism between K-flat complexes stays a quasi-isomorphism after tensoring with any complex. This supplies the usual choice-independent derived tensor product on the site. For example, to check the last assertion for \(P\to Q\), resolve the other factor \(A\) by a K-flat \(F\to A\). In the tensor square, both vertical maps and its top map are quasi-isomorphisms by the three K-flat hypotheses. The bottom map is then a quasi-isomorphism by two out of three. Acyclic K-flat complexes are consequently tensor-acyclic against arbitrary complexes, not only against acyclic ones.

**Proposition 1A.2 (the lower Tor-amplitude truncation).** Suppose \(E\) has lower Tor-amplitude bound \(a\): for every module sheaf \(N\) in degree zero,

\[
H^j(E\otimes_{\mathcal O}^{L}N)=0\qquad(j<a).
\tag{1A.3}
\]

Then \(E\) is represented by a K-flat complex of flat sheaves whose terms vanish below \(a\). If \(E\) also has no cohomology above \(b\geq a\), the representative may be chosen zero above \(b\) as well.

**Proof.** Choose a termwise-flat K-flat model \(P\) by Proposition 1A.1. Taking \(N=\mathcal O\) shows \(H^j(P)=0\) below \(a\). Put

\[
C=\operatorname{coker}(P^{a-1}\to P^a),\qquad
Q^i=\begin{cases}0&i<a,\\ C&i=a,\\ P^i&i>a.\end{cases}
\tag{1A.4}
\]

The differential on \(Q\) is induced by that of \(P\); the natural map \(P\to Q\) is a quasi-isomorphism. Its crucial property is that \(C\) is flat, which ordinary lower boundedness alone would not show.

Let \(V\) be the brutal upper truncation of \(P\) at degree \(a\), keeping \(P^a\) and setting its outgoing differential to zero. It is a bounded-above complex of flats, with cohomology only \(C\) in degree \(a\). For every module sheaf \(N\), (1A.3) gives

\[
H^{a-1}(N\otimes V)=H^{a-1}(N\otimes P)=0,
\qquad H^a(N\otimes V)=N\otimes C.
\tag{1A.5}
\]

The first equality uses only the differentials through degree \(a\), which the truncation has not changed. The second is right exactness of tensor. For a short exact sequence \(0\to N'\to N\to N''\to0\), the sequence obtained by tensoring with \(V\) is termwise exact because all its terms are flat. Its long exact sequence, using (1A.5) for \(N''\), makes \(N'\otimes C\to N\otimes C\) injective. Thus \(C\) is flat.

Let \(B=\operatorname{im}(P^{a-1}\to P^a)\). The sequence \(0\to B\to P^a\to C\to0\) and the flat-kernel assertion above make \(B\) flat. The kernel \(K\) of \(P\to Q\) has terms \(P^i\) below \(a\), term \(B\) in degree \(a\), and zero terms above it. Hence \(K\) is a bounded-above complex of flats and is K-flat. Every term of \(Q\) is flat. Consequently \(0\to A\otimes K\to A\otimes P\to A\otimes Q\to0\) is exact for every complex \(A\), by the flat-quotient assertion and exact sums. For acyclic \(A\), its first two complexes are acyclic; the long exact sequence makes the third acyclic. This proves that \(Q\) is K-flat, as required.

If \(E\) is also bounded above by \(b\), first represent it by its good upper truncation at \(b\), then use the bounded-above case of Proposition 1A.1. The model \(P\) is zero above \(b\), and (1A.4) retains that property. \(\square\)

Apply Proposition 1A.2 to the factor \(E\) in Lemma 12.1. Apply the bounded-above part of Proposition 1A.1 on \(A_{\mathrm{ét}}\) in Lemma 12.2. These are different uses: the latter reduction does not assume that the original unbounded \(E\) already has a lower Tor-amplitude bound. Its bounded flat truncations acquire such a bound individually. The argument therefore preserves the arbitrary-complex conclusion of Theorem 12.3.

Site and Tor-amplitude exposition: GPT-6 Astra (OpenAI), Ultra; AI self-check, not independent review. The linked programme construction and its source notice retain their authorship and source attribution. Original text CC0.

## 2. Limits, curves and fields

**Lemma 2.1 (limits).** If \(X=\varprojlim X_i\) is a directed inverse limit of qcqs schemes with affine transition maps, and \(\operatorname{cd}_{\mathrm{tors}}(X_i)\leq D\) for every \(i\), then \(\operatorname{cd}_{\mathrm{tors}}(X)\leq D\). The same assertion holds for a fixed prime.

*Proof.* Put \(F_i=(p_i)_*F\), where \(p_i:X\to X_i\). There are compatible maps \(p_{ji}^{-1}F_i\to F_j\), obtained by adjunction. On the finite-presentation affine étale basis,

\[
F=\mathop{\mathrm{colim}}_i p_i^{-1}F_i,
\qquad H^q(X,F)=\mathop{\mathrm{colim}}_i H^q(X_i,F_i).
\tag{2.1}
\]

For the first equality, a germ of \(F\) has a representative on an affine étale object descending to one stage; a germ relation also holds at a sufficiently late stage. This proves surjectivity and injectivity on stalks. The second equality is precisely qcqs continuity for this system. Each \(F_i\) is torsion: a section on the inverse image of a quasi-compact étale object is killed by one integer after taking a finite cover on which its finitely many local annihilators apply. If \(F\) is primary, the integer can be a power of that prime. The terms in (2.1) vanish for \(q>D\). □

**Lemma 2.2 (curves).** If \(C\) is affine of dimension at most one over \(K\), then

\[
\operatorname{cd}_{\mathrm{tors}}(C)\leq 1+\operatorname{cd}_{\mathrm{tors}}(K).
\tag{2.2}
\]

*Proof.* For \(a:C\to\operatorname{Spec}K\), the geometric stalk of \(R^qa_*F\) is \(H^q(C_{K^{\mathrm{sep}}},F)\). Indeed a strict localization of the field is its separable closure, and the strict-local higher-image formula and continuity compute that stalk. Passing from a separable to an algebraic closure changes no étale cohomology. Lesson 12 makes these stalks zero for \(q>1\). The sheaves \(R^qa_*F\) are torsion: their stalk groups commute with the filtered annihilator subsheaves of \(F\). Leray has only rows zero and one and columns through \(\operatorname{cd}_{\mathrm{tors}}(K)\). This proves (2.2). The dimension-zero case also follows directly from exact finite pushforward. □

**Lemma 2.3 (field extensions).** For every extension \(L/K\),

\[
\operatorname{cd}_{\mathrm{tors}}(L)
\leq\operatorname{cd}_{\mathrm{tors}}(K)+\operatorname{trdeg}_K L.
\tag{2.3}
\]

*Proof.* An algebraic extension is the union of its finite subextensions. Each finite field map has exact direct image, so Leray bounds its cohomological dimension by that of \(K\); Lemma 2.1 passes to the union. This includes inseparable extensions.

For an extension of transcendence degree one, every finitely generated \(K\)-subalgebra of \(L\) is a domain of dimension at most one, by the field dimension formula. Lemmas 2.1 and 2.2 therefore give (2.3). For finite transcendence degree \(r\), choose a transcendence basis \(t_1,\ldots,t_r\). Apply the one-variable case successively to the rational extensions and then the algebraic case to \(L/K(t_1,\ldots,t_r)\). If either term on the right is infinite the bound is automatic. □

A strictly henselian local scheme has cohomological dimension zero for **all** abelian étale sheaves. Sections equal the closed geometric stalk: every pointed étale neighbourhood has a section and every open containing the closed point is the whole local scheme. This is an exact functor. Its punctured opens can have positive cohomology; the local scheme and its punctured spectrum must not be confused.

## 3. Support and finite models of a strict localization

For a finite-type \(K\)-scheme put

\[
E_a(X)=\{x\in X:\operatorname{trdeg}_K\kappa(x)\leq a\},
\qquad E_a(X)=\varnothing\quad(a<0).
\tag{3.1}
\]

These sets are stable under specialization. The closure of a point has dimension equal to the transcendence degree of its residue field.

**Lemma 3.1 (closed support approximation).** A torsion sheaf supported in a specialization-stable set \(E\) on a Noetherian scheme is a filtered union of constructible subsheaves, each supported on a closed subset contained in \(E\).

*Proof.* Constructible approximation gives maps from constructible sheaves to \(F\). Their images and finite sums are constructible by the Noetherian subobject and quotient theorem of lesson 11; they form a directed union equal to \(F\) on stalks. Their supports are constructible subsets of \(E\). A constructible subset of a Noetherian scheme contains the generic point of each component of its closure: on each component it is dense and contains an open dense part. All other points of that closure are specializations of such generic points. Its closure is thus still contained in \(E\). Closed pushforward identifies a sheaf with zero stalks off that closure with a sheaf on its reduced closed subscheme. □

**Lemma 3.2 (strict-local models).** Let \(x\in X\), let \(a=\operatorname{trdeg}_K\kappa(x)\), and let \(d\) be the dimension of a sufficiently small affine neighbourhood of \(x\), minimized over such neighbourhoods. For a chosen strict localization \(R=\mathcal O^{\mathrm{sh}}_{X,\bar x}\) there is a compatible embedding

\[
L=K(t_1,\ldots,t_a)^{\mathrm{sep}}\longrightarrow R,
\tag{3.2}
\]

with purely inseparable residue-field extension, and \(R\) is a filtered colimit of finite-type \(L\)-algebras of dimension at most \(d-a\).

<a id="strict-local-model-proof"></a>

*Strict-local model exposition: GPT-6 Astra (OpenAI), Ultra; independently written and self-checked by the AI. Original text CC0. This is not an independent review.*

*Proof.* Choose an affine neighbourhood \(X_0=\operatorname{Spec}B\) of the stated minimum dimension \(d\), and let \(\mathfrak p\subset B\) represent \(x\). Apply [normalization adapted to ideals, Corollary 3.2](../../AG-CA/krull-dimension-and-noether-normalization.html#3-normalization-that-respects-ideals) to \(B\) and \(\mathfrak p\). It gives a finite inclusion of a polynomial algebra into \(B\), with the contraction of \(\mathfrak p\) generated by a subset of its variables. [Dimension and transcendence degree, Theorem 4.2](../../AG-CA/krull-dimension-and-noether-normalization.html#4-parameters-measure-dimension-and-height) show that the polynomial algebra has \(d\) variables and its quotient at that contraction has \(a\) variables: \(B/\mathfrak p\) is finite over that quotient and has fraction field \(\kappa(x)\). Reorder the variables to obtain

\[
P=K[t_1,\ldots,t_d]\subset B,
\qquad \mathfrak p\cap P=(t_{a+1},\ldots,t_d).
\]

This is a finite polynomial subalgebra, not an assertion that its parameters give a polynomial automorphism of a previously chosen ambient space. It works over finite or imperfect fields and for nonreduced \(B\). Put \(P_0=K[t_1,\ldots,t_a]\) and \(F_0=\operatorname{Frac}P_0\). Since \(\mathfrak p\cap P_0=0\), the map \(P_0\to B_{\mathfrak p}\) extends to the local map from the field \(F_0\). The specified geometric point allows a compatible embedding of \(F_0^{\mathrm{sep}}\) into \(\kappa(x)^{\mathrm{sep}}\). The [strict-henselization universal property and functoriality, Theorem 5.1](../../AG-CA/AG-CA-20.html#5-strict-henselization-and-the-choice-of-an-embedding) then lift it to the field embedding (3.2). The residue field of \(R\) is \(\kappa(x)^{\mathrm{sep}}\). It is algebraic over \(L=F_0^{\mathrm{sep}}\), so the extension is purely inseparable: the separable part of the minimal polynomial of any algebraic element over a separably closed field has degree one.

Use the same pointed-neighbourhood construction to write \(R=\operatorname{colim}_j B_j\), where \(\operatorname{Spec}B_j\to X_0\) is affine étale, and \(L=\operatorname{colim}_i A_i\), where \(\operatorname{Spec}A_i\to\operatorname{Spec}P_0\) is an affine étale neighbourhood of its chosen geometric generic point. Principal shrinking is allowed in both systems. The two systems and their transition maps are taken with these geometric-point markings; their common refinements and equalizers are part of the proof of Theorem 5.1. Every \(B_j\) is quasi-finite over \(P\), by the finite normalization map and the [étale fibre theorem, Corollary 7.3](../../AG-CA/AG-CA-17.html#7-classifying-étale-algebras-over-a-field).

Take triples \((i,j,\phi)\) with a \(P_0\)-map \(\phi:A_i\to B_j\) whose composite into \(R\) is the prescribed map through \(L\). Such triples exist cofinally in either index: \(A_i\) is finitely presented over \(P_0\), so its map into the filtered colimit \(R\) factors at a stage, which can be enlarged past any specified \(j\). Two triples admit a common successor by first refining their \(A_i\)'s and then factoring this larger finite presentation through a sufficiently large \(B_j\). Agreement with the previous maps imposes only finitely many equalities, which hold eventually because they hold in \(R\). The same finite-equality argument equalizes parallel arrows. The triple category is therefore filtered, and there is a natural map

\[
R=\mathop{\mathrm{colim}}_{(i,j,\phi)}(B_j\otimes_{A_i}L).
\tag{3.3}
\]

Here is the verification of the equality, including its injective direction. Every element of \(R\) is represented in a \(B_j\), hence is reached by the right side. An element of one tensor product is a finite sum \(\sum_\nu b_\nu\otimes\lambda_\nu\). Represent its finitely many \(L\)-coefficients at a common later \(A_{i'}\), and refine the triple accordingly. At that stage the tensor sum is represented by a single element of \(B_{j'}\), since the coefficients can be moved across the tensor sign. If its image in \(R\) is zero, it becomes zero in a later \(B_{j''}\). Thus the map is injective. The maps and this argument respect the \(L\)-algebra structures.

For each triple, set \(Q=A_i[t_{a+1},\ldots,t_d]\). The map \(Q\to B_j\) is of finite type: a finite list generating \(B_j\) over \(P\) also generates it over \(Q\). For any prime \(\mathfrak q\) of \(Q\), its fibre is a quotient of
\(B_j\otimes_P\kappa(\mathfrak q)\). The latter is finite-dimensional over \(\kappa(\mathfrak q)\), because the corresponding fibre of the quasi-finite finite-type map over \(P\) is a zero-dimensional finite-type algebra over a field, hence finite-dimensional. The quotient assertion imposes the additional relations identifying the two actions of \(A_i\). Consequently \(B_j\) is quasi-finite over \(Q\). Base change along \(A_i\to L\) gives a quasi-finite finite-type map

\[
L[t_{a+1},\ldots,t_d]\longrightarrow B_j\otimes_{A_i}L.
\]

The function field of any irreducible component on the right is finite over the residue field at its image on the left. Its transcendence degree over \(L\) is at most \(d-a\); Theorem 4.2 and the finite-component dimension formula give the required dimension bound. Nilpotents do not affect this calculation. The zero algebra causes no exception. This proves the full lemma. □

For a smooth \(d\)-dimensional neighbourhood, \(R\) is a regular local domain and
\(\operatorname{trdeg}_L\operatorname{Frac}R=d-a\). To justify both assertions, put \(S=\mathcal O_{X,x}\). The [smooth local-ring criterion, Theorem 2.1](../../AG-CA/AG-CA-18.html#2-the-criterion-and-its-open-locus) makes \(S\) regular. [Henselization, Theorems 5.1, 6.3 and 6.4](../../AG-CA/AG-CA-20.html#6-infinitesimal-quotients-and-the-difference-from-completion) make \(R\) Noetherian, faithfully flat over \(S\), with maximal ideal \(\mathfrak m_SR\) and dimension \(\dim S\). Flatness identifies

\[
\mathfrak m_R/\mathfrak m_R^2
\simeq(\mathfrak m_S/\mathfrak m_S^2)
\otimes_{\kappa(x)}\kappa_R,
\qquad \kappa_R=R/\mathfrak m_R.
\]

Thus its embedding dimension equals its dimension. It is regular, and the [regular-local domain proof, Theorem 6.1](../../AG-CA/AG-CA-12.html#6-regular-local-rings-and-complete-intersections) makes it a domain.

Shrink the smooth neighbourhood to its integral component through \(x\). This is legitimate because its local rings are domains; its finitely many irreducible components are disjoint and therefore open and closed. The local rings of pointed étale neighbourhoods are again regular local domains, by smoothness. The transition maps are flat local maps (étale before localization), hence faithfully flat and injective. Their fraction fields are finite separable extensions of \(K(X_0)\): their generic fibres are finite separable field factors of the étale fibre. Every fraction of \(R\) occurs in the fraction field of one such local stage. Therefore \(\operatorname{Frac}R\) is algebraic over \(K(X_0)\), of transcendence degree \(d\) over \(K\). Since \(L/K\) has transcendence degree \(a\), the tower formula gives \(d-a\) over \(L\). If \(0\ne f\in\mathfrak m_R\), it is transcendental over \(L\): otherwise the nonzero algebraic element \(f\) would have its inverse in the subalgebra \(L[f]\subset R\), contradicting its membership in the maximal ideal. These are exactly the field and domain properties used in Interlude A.

The strict-local finite-model construction also appears in the Stacks Project authors' [étale-cohomology chapter, lemma-strictly-henselian, at the recorded native revision](https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/565b10e987aba5969b21145a0833f42d69f96790/etale-cohomology.tex#L19692). Its source text remains a separate GFDL component; the argument here uses the linked programme proofs and independent exposition.


## 4. Proper vanishing before affine vanishing

**Theorem 4.1.** If \(p:T\to S\) is proper with geometric fibres of dimension at most \(r\), then \(R^qp_*F=0\) for \(q>2r\) and every torsion sheaf \(F\).

*Proof.* The complete proof is Theorem 4.1 of lesson 14; here is its induction, to fix its logical position. Prove the assertion for all proper maps at once. Relative dimension zero is exact finite pushforward, since [a proper quasi-finite morphism is finite](../../AG-MO/zariskis-main-theorem.html#5-properness-and-normality) (Theorem 5.1). Proper base change reduces each other case to a proper scheme over an algebraically closed field. Dimension one is the proper curve theorem.

For a reduced scheme of dimension \(r\), finite normalization gives a monomorphism \(F\to\nu_*\nu^{-1}F\), an isomorphism on a dense open containing every generic point. Its cokernel is a closed pushforward from a proper subset of dimension at most \(r-1\). The long exact sequence and induction reduce to a normal integral component of dimension \(r\).

Choose a nonconstant rational function and take its graph closure

\[
T\xleftarrow{b}T'\xrightarrow{v}\mathbf P^1.
\tag{4.1}
\]

The indeterminacy locus \(Z\) has [codimension at least two](../../AG-MO/rational-maps-and-birational-morphisms.html#codimension-one-extension): the map extends over every codimension-one discrete valuation ring by properness of \(\mathbf P^1\). The fibres of \(b\) have dimension at most one. The cone \(Q\) of \(F\to Rb_*b^{-1}F\) consequently has torsion cohomology in degrees \(0,1,2\), supported on \(Z\); there is no degree minus one because the degree-zero unit is injective. Induction on \(Z\) bounds \(H^m(T,Q)\) by \(m\leq2(r-2)+2=2r-2\). Thus high-degree cohomology of \(T\) reduces to that of \(T'\). The dominant \(v\) [has fibres of dimension at most](../../AG-MO/dimension-of-fibres.html#fibres-over-an-integral-curve) \(r-1\), so its higher images stop at \(2r-2\); Leray over the proper curve \(\mathbf P^1\) stops two degrees later. This proves the absolute bound and, by proper base change, the relative one. □

This proof uses no affine vanishing in dimensions greater than one. It is therefore available inside the affine induction below. Over a quasi-compact base, any finite-type proper map has some uniform finite fibre-dimension bound: finitely many affine presentations give bounds by their numbers of algebra generators.

## 5. The affine vanishing theorem

**Theorem 5.1 (Artin's affine vanishing, Gabber-style proof).** For \(X\) affine of finite type over any field \(K\),

\[
\operatorname{cd}_{\mathrm{tors}}(X)
\leq\dim X+\operatorname{cd}_{\mathrm{tors}}(K).
\tag{5.1}
\]

The proof follows the modern dimension-and-support argument used in the Stacks Project. It proves Artin's theorem without claiming to reproduce Artin's original strict-local induction word for word.

*Proof.* If \(\operatorname{cd}_{\mathrm{tors}}(K)=\infty\), the assertion is vacuous; suppose it is finite. Induct on \(d=\dim X\), **for schemes over all fields simultaneously**. Dimension zero is finite pushforward and dimension one is Lemma 2.2. Assume the assertion in all dimensions below \(d>1\).

Noether normalization gives a finite map \(X\to\mathbf A^d_K\). Finite pushforward is exact, so Leray reduces the problem to any torsion sheaf \(F\) on \(\mathbf A^d\). We establish the two interludes needed for that sheaf.

**Interlude A: boundary stalks.** Let \(j:U\to W\) be the complement of an effective Cartier divisor in a smooth \(d\)-dimensional variety. At a point \(w\) on the divisor put \(a=\operatorname{trdeg}_K\kappa(w)\). Then

\[
(R^qj_*F)_{\bar w}=0\quad(q>d-a).
\tag{5.2}
\]

The strict-local stalk is \(H^q(\operatorname{Spec}R[1/f],F)\), where \(R=\mathcal O^{\mathrm{sh}}_{W,\bar w}\) and \(f\) is a local nonzero equation. [Lemma 3.2](#strict-local-model-proof) embeds a separably closed \(L\) into \(R\). The preceding domain observation embeds \(L(f)\) into \(R[1/f]\). Its cohomological dimension is at most one by Lemma 2.3. The ring \(R[1/f]\) is the union of its finitely generated \(L(f)\)-subalgebras, domains of dimension at most \(d-a-1\): their fraction fields lie in \(\operatorname{Frac}R\), whose transcendence degree over \(L(f)\) is \(d-a-1\). The induction hypothesis bounds their cohomological dimensions by \(d-a\). Lemma 2.1 proves (5.2). Off the divisor all positive higher images vanish because \(j\) is an isomorphism locally. Thus \(R^qj_*F\) has support in \(E_{d-q}(W)\) for \(q>0\).

**Interlude B: support on a smaller affine scheme.** If \(Z\) is affine of dimension below \(d\) and \(G\) is supported in \(E_a(Z)\), then

\[
H^p(Z,G)=0\quad(p>a+\operatorname{cd}_{\mathrm{tors}}(K)).
\tag{5.3}
\]

Lemma 3.1 writes \(G\) as a union of sheaves supported on closed subschemes \(Z_i\subset Z\) of dimension at most \(a\). Closed pushforward and Leray identify their cohomology with cohomology on \(Z_i\). The dimension is also below \(d\), so the induction hypothesis applies. Filtered cohomology proves (5.3). Only the **affine** version of this interlude is asserted at this stage.

Now factor the projection onto \(B=\mathbf A^{d-1}_K\) as

\[
\mathbf A^d_K\xrightarrow{j}\mathbf P^1_K\times_K B
\xrightarrow{g}B,
\qquad f=g j.
\tag{5.4}
\]

The boundary divisor is the section at infinity, isomorphic to \(B\). For \(q>0\), \(R^qj_*F\) is a closed pushforward from that section. Therefore \(R^pg_*R^qj_*F=0\) if \(p,q>0\). Also \(R^qj_*F=0\) for \(q>d\), by (5.2). The proper relative curve bound gives \(R^pg_*j_*F=0\) for \(p>2\). In the relative Leray spectral sequence

\[
R^pg_*R^qj_*F\Longrightarrow R^{p+q}f_*F
\tag{5.5}
\]

the only possible entries are the row \(q=0\), in columns zero to two, and the column \(p=0\). For \(q>2\), the term \((0,q)\) has no incoming differential; its possible outgoing \(d_r\) has target \((r,q-r+1)\), which is zero either because both indices are positive, because \(r>2\), or because the second index is negative. Thus

\[
R^qf_*F=g_*R^qj_*F\quad(q>2).
\tag{5.6}
\]

Interlude A puts this sheaf on \(E_{d-q}(B)\). Interlude B then says \(H^p(B,R^qf_*F)=0\) for \(p>d-q+\operatorname{cd}_{\mathrm{tors}}(K)\).

For \(q=2\), its stalk at the generic point of \(B\) is the second cohomology of an affine line over a separable closure of \(K(B)\). The affine curve theorem makes it zero. There is only one generic point, so the support is contained in \(E_{d-2}(B)\). Interlude B gives the same inequality with \(q=2\). For \(q=0,1\), ordinary induction on the affine \((d-1)\)-dimensional \(B\) gives vanishing for \(p>d-1+\operatorname{cd}_{\mathrm{tors}}(K)\).

Finally use

\[
H^p(B,R^qf_*F)\Longrightarrow H^{p+q}(\mathbf A^d_K,F).
\tag{5.7}
\]

Every entry of total degree greater than \(d+\operatorname{cd}_{\mathrm{tors}}(K)\) is zero, in all four ranges just considered. Both spectral sequences are first-quadrant, with bounded relevant rows, so their filtrations in each degree are finite. This proves (5.1) and closes the simultaneous induction. □

## 6. Affine direct images lower support dimension

**Corollary 6.1.** On an affine finite-type \(K\)-scheme, a torsion sheaf supported in \(E_a\) has zero cohomology above \(a+\operatorname{cd}_{\mathrm{tors}}(K)\).

*Proof.* Apply Lemma 3.1, exact closed pushforward, Theorem 5.1 on each closed support of dimension at most \(a\), and filtered cohomology. There is now no induction restriction on the dimension of the ambient affine scheme. □

**Theorem 6.2 (affine support drop).** If \(f:X\to Y\) is affine between finite-type \(K\)-schemes and \(F\) is supported in \(E_a(X)\), then \(R^qf_*F\) is supported in \(E_{a-q}(Y)\).

*Proof.* Work on an affine neighbourhood of \(y\in Y\); its inverse image in \(X\) is affine. Put \(b=\operatorname{trdeg}_K\kappa(y)\) and \(R=\mathcal O^{\mathrm{sh}}_{Y,\bar y}\). We must show

\[
H^q(X\times_Y\operatorname{Spec}R,F)=0\quad(q>a-b).
\tag{6.1}
\]

First take a constructible subsheaf of \(F\) with closed support \(Z\subset X\), \(\dim Z\leq a\); Lemma 3.1 and continuity will recover \(F\). The local model construction of Lemma 3.2 provides a map \(Y\to\mathbf A^b_K\) sending \(y\) to its generic point and embeds \(L=K(t_1,\ldots,t_b)^{\mathrm{sep}}\) in \(R\). Form the compatible triples \(A_i\to B_j\) used in (3.3). Before tensoring with \(L\), the schemes \(Z\times_Y\operatorname{Spec}B_j\) are étale over \(Z\), hence have dimension at most \(a\). Their maps to \(\operatorname{Spec}A_i\), an étale \(b\)-dimensional neighbourhood of the generic point of \(\mathbf A^b\), have generic fibres of dimension at most \(a-b\): every component dominating this base loses exactly \(b\) transcendence variables; components not dominating it disappear on the generic fibre. Tensoring that generic fibre with the algebraic separable closure \(L\) preserves dimension.

The base stages can be chosen integral: an affine étale neighbourhood of the generic point of a polynomial space is smooth over \(K\), so its finitely many irreducible components are disjoint open and closed subsets; keep the component selected by the geometric point. These choices are cofinal. For such a stage, \(A_i\to L\) is injective and extends to its fraction field, which is finite separable over \(K(t_1,\ldots,t_b)\). For an integral component \(C\) of the support model, localization at \(A_i\setminus\{0\}\) is zero unless \(A_i\to\mathcal O(C)\) is injective. In the injective case the generic-fibre dimension is \(\operatorname{trdeg}_K K(C)-b\leq a-b\), by the finite-type field dimension theorem. The [dimension-invariance argument for field extension](../../AG-MO/dimension-of-fibres.html#fibres-over-an-integral-curve) then applies to the extension from this fraction field to \(L\). This also proves emptiness when \(a<b\); it uses neither properness nor a geometric-fibre formula for a nonproper higher direct image.

Thus the closed supports in the affine \(L\)-models of \(X_R\) have dimension at most \(a-b\). By Corollary 6.1 over the separably closed field \(L\), their cohomology vanishes for \(q>a-b\). Continuity over the triples gives (6.1) for each constructible subsheaf. Filtered cohomology gives it for \(F\). If \(a-b<0\), all those generic supports are empty, so the assertion includes degree zero. The strict-local stalk formula now proves the claimed support inclusion. □

This argument justifies the algebraicity and dimension step in the affine-image interlude; it does not use ordinary geometric-fibre cohomology as a substitute for strict-local stalks of a nonproper map.

## 7. The bound for all finite-type schemes

We need a topological fact to include nonseparated schemes in the ordinary-cohomology bound.

**Lemma 7.1 (Noetherian Zariski vanishing).** On a Noetherian topological space of dimension at most \(d\), every abelian sheaf has zero cohomology in degrees greater than \(d\).

*Proof.* We give the generator argument. Suppose cohomology above \(d\) vanishes for all \(j_!\mathbf Z_U\), where \(U\) is open. Any sheaf is the filtered union of subobjects generated by finitely many local sections; all opens and their intersections are quasi-compact, so cohomology commutes with these unions. Removing one generator gives a short exact sequence with a quotient generated by one section. It therefore suffices to treat a quotient of \(j_!\mathbf Z_U\).

Its kernel is the filtered union of finitely generated subsheaves of \(j_!\mathbf Z_U\subset\mathbf Z_X\). A finite list of such generators can be made constant integers \(n_i>0\) on opens \(U_i\), by decomposing their locally constant sections. Add the intersections of all sublists with their greatest common divisors. For each point the least available positive generator then divides all the others at that point. Filter by the size of the integer generators. The successive quotient at integer \(n\) is \(\mathbf Z\) on \(U\setminus V\), zero on \(V\), and fits into

\[
0\to j_{V!}\mathbf Z_V\to j_{U!}\mathbf Z_U
\to Q_n\to0,
\tag{7.1}
\]

where \(U\) is the union of opens with generators at most \(n\), and \(V\) the union with generators less than \(n\). The generator \(n\) on its own opens and zero on \(V\) glue in this quotient; the gcd condition makes the overlap values zero. Thus each finite kernel has zero cohomology above \(d\), by its finite filtration and (7.1). The same holds for the whole kernel by filtered cohomology, and the quotient has the required vanishing by its long exact sequence. This proves the generator reduction.

Now induct on \(d\) and on the finite number of irreducible components. For a component \(Z\) and \(U=X\setminus Z\), the closed-open exact sequence reduces the assertion to the sheaf on \(Z\) and the extension from \(U\), which is supported on the union of the other components. Closed pushforward is exact. We thus reduce to irreducible \(X\). For \(d=0\), every nonempty open has the same points as \(X\), and sections are exact. For irreducible \(X\) of positive dimension, the constant sheaf \(\mathbf Z_X\) is flasque: every nonempty open is connected and its sections are \(\mathbf Z\). For nonempty \(U\), its complement \(Z\) has dimension below \(d\). The exact sequence

\[
0\to j_!\mathbf Z_U\to\mathbf Z_X\to i_*\mathbf Z_Z\to0
\tag{7.2}
\]

and induction show that \(j_!\mathbf Z_U\) has zero cohomology above \(d\). The empty open contributes zero. Apply the generator reduction to finish the proof. □

**Corollary 7.2.** If \(X\) is finite type of dimension \(d\) over a separably closed field, then

\[
H^q(X,F)=0\quad(q>2d)
\tag{7.3}
\]

for every torsion sheaf. If \(X\) is affine, it vanishes for \(q>d\).

*Proof.* Let \(\epsilon:X_{\mathrm{ét}}\to X_{\mathrm{Zar}}\). The sheaf \(R^q\epsilon_*F\) is obtained from \(U\mapsto H^q(U_{\mathrm{ét}},F)\). On the affine-open basis Theorem 5.1 gives zero for \(q>d\), so these higher images vanish. Lemma 7.1 gives \(H^p_{\mathrm{Zar}}(X,G)=0\) for \(p>d\) for any abelian \(G\). Leray for \(\epsilon\) has neither rows nor columns beyond \(d\), proving (7.3). The affine assertion is Theorem 5.1. No separatedness is needed. □

For separated \(X\), Nagata compactification and lesson 14 also give

\[
H^q_c(X,F)=0\quad(q>2d):\qquad
R\Gamma_c(X,F)=R\Gamma(\bar X,j_!F),\quad\dim\bar X=d.
\tag{7.4}
\]

Theorem 4.1 applies to the torsion sheaf \(j_!F\). This is a compact-support assertion. Ordinary cohomology uses \(Rj_*F\), whose positive cohomology sheaves need not vanish; (7.4) cannot alone prove (7.3). Section 7 supplies the needed additional argument.

## 8. Finite dimension and arbitrary complexes

For a qcqs map \(f\), define \(\operatorname{cd}_{\mathrm{tors}}(f)\) by vanishing of \(R^qf_*F\) for every torsion sheaf. It is a bound on higher images, without the extra cohomology of the base.

**Proposition 8.1.** A map between finite-type schemes over a field has finite cohomological dimension. A finite-type scheme \(X/K\) has finite absolute cohomological dimension if \(K\) does.

*Proof.* Locally make the target affine. An affine source has higher-image bound \(\dim X\) by Theorem 6.2, since every point of its support lies in \(E_{\dim X}\). Cover a general source by finitely many affine opens. Each nonempty multiple intersection is a quasi-compact open of an affine scheme, hence separated over the field. Cover that intersection by finitely many affine opens; their multiple intersections are affine by separatedness. The finite Čech-to-derived-image spectral sequence for this inner cover bounds its higher images by \(\dim X\) plus the length of the cover minus one. The outer finite-cover spectral sequence now bounds \(Rf_*\) by the maximum of finitely many such integers plus its own finite length. Finite affine target covering makes this a uniform bound. This works when the original source is nonseparated. For the absolute statement, the proof of Corollary 7.2 instead bounds the rows by \(\dim X+\operatorname{cd}_{\mathrm{tors}}(K)\) and the columns by \(\dim X\), giving a finite bound. □

**Lemma 8.2 (unbounded acyclic complexes).** Let \(T\) be left exact with enough injectives and \(R^qT=0\) for \(q>N\). A termwise \(T\)-acyclic complex computes \(RT\). For arbitrary complexes,

\[
RT(D^{\geq a})\subset D^{\geq a},\qquad
RT(D^{\leq b})\subset D^{\leq b+N}.
\tag{8.1}
\]

*Proof.* For an exact complex \(A^\bullet\) of acyclic objects, put \(Z^i=\ker d^i\). Its exact cycle sequences give \(R^qT(Z^{i+1})=R^{q+1}T(Z^i)\) for \(q\geq1\). Iterating to the left beyond \(N\) makes all cycles acyclic. Therefore \(T(A^\bullet)\) is exact. The cone of a quasi-isomorphism between acyclic complexes is another such exact complex, so termwise application preserves that quasi-isomorphism. Compare with a termwise injective K-injective resolution to obtain the computation claim.

The lower bound follows from a bounded-below injective resolution. For the upper bound use a termwise injective K-injective resolution \(I^\bullet\) exact above \(b\). For \(i>b+N\), its cycle sequences identify

\[
H^i(TI^\bullet)=R^1T(Z^{i-1})
=R^{N+1}T(Z^{i-N-1})=0.
\tag{8.2}
\]

All sequences used are in the exact part of the complex; the last one may start at \(I^b\). If \(N=0\), exactness of \(T\) gives the assertion immediately. This is the finite-amplitude argument proved in lesson 14. □

**Proposition 8.3 (coproducts).** For a qcqs map of finite cohomological dimension, \(Rf_*:D(X_{\mathrm{ét}},\Lambda)\to D(Y_{\mathrm{ét}},\Lambda)\) commutes with arbitrary direct sums, where \(\Lambda\) is a torsion ring. The analogous assertion holds for \(R\Gamma\) on a qcqs scheme of finite cohomological dimension.

*Proof.* Module-sheaf cohomology agrees with underlying abelian-sheaf cohomology, so the dimension bound applies to \(\Lambda\)-modules. Choose a termwise injective K-injective resolution \(I_\alpha^\bullet\) of each input. Higher images commute with filtered colimits of sheaves; a direct sum is the filtered union of finite subsums. Thus every \(\bigoplus_\alpha I_\alpha^t\) is \(f_*\)-acyclic, and \(f_*\bigoplus_\alpha I_\alpha^t=\bigoplus_\alpha f_*I_\alpha^t\). Lemma 8.2 permits this unbounded termwise computation. It identifies the canonical sum map with an isomorphism. The global-section proof is identical, using absolute qcqs continuity instead of relative continuity. □

Proper maps have this property locally on any base: on a quasi-compact target Theorem 4.1 provides finite amplitude. Local isomorphisms of the coproduct comparison glue.

**Lemma 8.4 (pulling out a constant complex).** If \(X\) is qcqs with finite cohomological dimension, then for any \(E\in D(X_{\mathrm{ét}},\Lambda)\) and \(V\in D(\Lambda)\),

\[
R\Gamma(X,E)\otimes^L_\Lambda V
\xrightarrow{\sim}R\Gamma(X,E\otimes^L_\Lambda\underline V).
\tag{8.3}
\]

*Proof.* The arrow is the canonical tensor-and-adjunction map. Both functors in \(V\) preserve triangles and direct sums by Proposition 8.3. It is an isomorphism for every shift of the free module \(\Lambda\), hence for free sums and their bounded complexes. Every bounded-above complex admits a bounded-above free resolution, which is the homotopy colimit of its bounded brutal truncations from below. Every complex is the homotopy colimit of its good truncations from above. These are actual homotopy colimits: the telescope is the cone of \(1-\text{shift}\) on the direct sum of the stages, and exactness of filtered colimits shows its cohomology is the colimit of their cohomologies. The two functors preserve those telescopes. This proves (8.3) for every \(V\), with no finite Tor-dimension assumption. □

**Proposition 8.5 (proper projection formula).** For proper \(p:T\to S\) and arbitrary complexes over a torsion ring,

\[
Rp_*E\otimes^L_\Lambda K
\xrightarrow{\sim}Rp_*(E\otimes^L_\Lambda p^{-1}K).
\tag{8.4}
\]

*Proof.* Check a geometric stalk of \(S\). Unbounded proper base change, proved in lesson 14 from proper base change and finite amplitude, reduces the map to (8.3) on the proper geometric fibre. Theorem 4.1 bounds that fibre's cohomological dimension. Exact inverse image commutes with derived tensor products, so the stalk map is precisely (8.3). All stalks are isomorphisms. □

For a proper morphism, neither (8.4) nor its geometric-fibre proof requires \(n\) prime to the characteristic. That restriction will enter the nonproper product argument.

## 9. The product map and one proper factor

For maps \(X\xrightarrow{f}S\), \(Y\xrightarrow{g}S\), write \(p,q\) for the projections of \(X\times_S Y\), and \(c\) for its map to \(S\). There is a canonical map

\[
Rf_*E\otimes^L_\Lambda Rg_*K
\longrightarrow Rc_*(p^{-1}E\otimes^L_\Lambda q^{-1}K).
\tag{9.1}
\]

It is adjoint to the tensor product of the counits \(f^{-1}Rf_*E\to E\), \(g^{-1}Rg_*K\to K\). Over a separably closed field its degree components are the usual external cup products. We prove that this specific map is an isomorphism.

**Theorem 9.1 (one proper factor).** Let \(k\) be separably closed, \(X/k\) proper and \(Y/k\) finite type. For any positive \(n\), including \(n\) divisible by the characteristic, and arbitrary complexes over \(\Lambda=\mathbf Z/n\),

\[
R\Gamma(X,E)\otimes^L_\Lambda R\Gamma(Y,K)
\xrightarrow{\sim}
R\Gamma(X\times_kY,p^{-1}E\otimes^L_\Lambda q^{-1}K).
\tag{9.2}
\]

*Proof.* The projection \(q\) is proper. Proper base change over the point gives \(Rq_*p^{-1}E=\underline{R\Gamma(X,E)}\). Proposition 8.5 then gives

\[
Rq_*(p^{-1}E\otimes^L q^{-1}K)
=\underline{R\Gamma(X,E)}\otimes^L K.
\tag{9.3}
\]

Apply \(R\Gamma(Y,-)\). Composition of derived direct images identifies the left with cohomology of the product, and (8.3) identifies the right with the tensor of the two cohomology complexes. Proposition 8.1 makes (8.3) applicable. These isomorphisms use the same counits as (9.1), so prove the canonical comparison. □

There is also a bounded-below version over \(\mathbf Z\): if \(E\) has torsion cohomology, \(E,K\) are bounded below and \(Y\) is merely qcqs, the same formula holds over \(\mathbf Z\). Here are the additional details, since finite cohomological dimension of \(Y\) is not assumed.

For any qcqs \(W\), \(D\in D^+(W_{\mathrm{ét}})\) and \(V\in D^+(\mathbf Z)\), the constant-tensor comparison holds. Represent \(V\) by a bounded-below complex of flat groups: global dimension one of \(\mathbf Z\) permits a flat resolution truncated one degree below its lower cohomology bound, with flat bottom quotient. Represent \(D\) by a bounded-below complex. A brutal upper tail of the flat complex starts arbitrarily high; its tensor with \(D\), and its tensor with \(R\Gamma(W,D)\), start correspondingly high. Right derived sections preserve lower bounds. Thus, in each fixed degree, bounded brutal truncations reduce the comparison to a bounded flat complex, then to one flat group \(M\). Good upper truncations of \(D\), followed by their finite cohomology triangles, reduce to one abelian sheaf. A flat abelian group is torsion free and is the filtered union of its finitely generated free subgroups. Qcqs cohomological continuity reduces this last case to finite free groups, for which the comparison is the finite-sum identity. This proves the bounded-below constant-tensor formula without an absolute dimension bound.

The proper projection formula over \(\mathbf Z\) now follows on geometric stalks from this formula and lesson 13's bounded-below proper base change. Both tensors remain bounded below, since \(\mathbf Z\) has Tor dimension one. The tensor involving \(E\) still has torsion cohomology: its derived rationalization is zero because that of \(E\) is zero. Thus proper base change applies to it as well. Applying this projection formula to \(q\) gives (9.3), over \(\mathbf Z\); apply the just-proved constant-tensor formula on \(Y\) to finish. The maps are the canonical ones in (9.1). □

## 10. Products preserve open-immersion base change

**Lemma 10.1 (closed-point detection).** On a finite-type scheme \(Z\) over a separably closed field, an abelian sheaf supported on closed points has no positive cohomology and is generated by global sections. A complex over a torsion ring, or a bounded-below abelian complex, with all cohomology sheaves so supported is zero if all its global cohomology groups vanish.

*Proof.* Lemma 3.1, in the Noetherian module version over \(\mathbf Z\), expresses such a sheaf as a union of finitely generated constructible sheaves supported on finite closed subsets. These subsets are disjoint unions of spectra of separably closed fields: their finite residue extensions are purely inseparable. Closed pushforward is exact and each point's sections are exact. Each finite-support sheaf therefore has no positive cohomology and is globally generated. Filtered cohomology and the lifting of a germ to a finite stage give both assertions for the union.

The scheme has finite cohomological dimension for torsion modules by Corollary 7.2. For a torsion complex, Lemma 8.2 reduces each degree to a finite cohomology window; the hypercohomology spectral sequence on that window has only its column zero. Thus \(H^i(Z,Q)=\Gamma(Z,H^iQ)\). The same calculation for bounded-below arbitrary abelian complexes uses the first-quadrant spectral sequence. These are the two forms needed below. Global generation makes a nonzero cohomology sheaf produce a nonzero group, proving detection. □

**Theorem 10.2 (open-product comparison).** Let \(j:U\to X\) be an open immersion between finite-type schemes over a field \(K\), and let \(Y/K\) be finite type. In

\[
\begin{array}{ccc}
Y\times_K U&\xrightarrow{h}&Y\times_K X\\
\downarrow p&&\downarrow q\\
U&\xrightarrow{j}&X,
\end{array}
\tag{10.1}
\]

the canonical map \(q^{-1}Rj_*F\to Rh_*p^{-1}F\) is an isomorphism for every torsion sheaf whose orders are invertible in \(K\).

*Proof.* Filter by annihilator integers and use qcqs higher-image continuity to reduce to \(nF=0\), \(n\) invertible. Induct on \(\dim X+\dim Y\), over all fields. If \(\dim X=0\), its open \(U\) is also closed, so exact open-and-closed pushforward proves the assertion.

Let \(z\) map to \(x\in X\), \(y\in Y\). If \(a=\operatorname{trdeg}_K\kappa(x)>0\), pass to \(X'=\operatorname{Spec}\mathcal O^{\mathrm{sh}}_{X,\bar x}\). Pro-étale base change from lesson 15 identifies the pullback of the comparison with the comparison for \(U'\subset X'\). Lemma 3.2 writes \(X'\) as a limit of finite-type schemes over a separably closed extension \(L/K\), of dimensions at most \(\dim X-a<\dim X\). The quasi-compact open \(U'\) descends to an open in one stage, since its complement has finitely generated ideal locally on this affine scheme. The diagrams at these stages are products over \(L\) with \(Y_L\), which has dimension \(\dim Y\). Apply induction and relative continuity to get the comparison on the limit and hence at \(\bar z\).

If instead \(b=\operatorname{trdeg}_K\kappa(y)>0\), use the strict localization of **\(Y\) at \(y\)**. Its smaller-dimensional models are over a separably closed \(L/K\). Field-extension base change from lesson 15 identifies the original open comparison over \(L\); pro-étale base change identifies it over that strict localization. The models of \(Y'\), together with \(X_L,U_L\), have smaller dimension sum, so induction and continuity again apply. Naturality and composition of base-change maps identify these maps with the original one. The formula with a strict localization of \(X\) in this second step would be a typing error.

Consequently any failure is supported where both \(x\) and \(y\) are closed. Such a \(z\) is closed: the tensor product of their finite residue extensions is a finite-dimensional \(K\)-algebra. Localize to affine \(X,Y\). Compactify the affine \(Y\) by its dense projective closure \(\bar Y\), of the same dimension. The preceding smaller-dimensional arguments apply to this projective product too, so the cone of its comparison is supported only on closed points. The original comparison is its restriction to \(Y\). Field-extension base change allows \(K\) to be replaced by its separable closure, conservatively, and then \(\bar Y\) is still proper.

Let \(Q\) be the cone of the comparison for \(\bar Y\). Lemma 10.1 reduces \(Q=0\) to its global cohomology. Theorem 9.1 computes both sides canonically as

\[
R\Gamma(U,F)\otimes^L_{\mathbf Z/n}
R\Gamma(\bar Y,\mathbf Z/n).
\tag{10.2}
\]

For the left use \(R\Gamma(X,Rj_*F)=R\Gamma(U,F)\); for the right use derived composition for \(h\). The map between (10.2)'s two descriptions is the identity by the counit definition of external products. Thus its cone has zero global cohomology, and closed-point detection gives \(Q=0\). Restricting back proves the affine comparison, and locality proves the theorem. □

## 11. Punctual base change

The open-product theorem has a stronger consequence. For any scheme \(A/K\), any \(S'/K\) and qcqs \(g:T\to S'\), form

\[
X'=A\times_K S',\quad Y=A\times_K T,
\qquad Y\xrightarrow{h}X',\quad Y\xrightarrow{e}T,
\quad X'\xrightarrow{f'}S'.
\tag{11.1}
\]

**Theorem 11.1 (punctual comparison).** For torsion \(F\) on \(T\) of orders invertible in \(K\),

\[
(f')^{-1}Rg_*F\xrightarrow{\sim}Rh_*e^{-1}F.
\tag{11.2}
\]

*Proof.* Localize to affine \(A\), write it as a limit of finite-type affine \(K\)-schemes, and use relative continuity to reduce to finite-type \(A\). Filter by annihilators to assume \(nF=0\). Field-extension base change from lesson 15, for both \(g\) and \(h\), lets us assume \(K\) algebraically closed. Topological invariance lets us replace \(A\) by its reduction. The resulting \(A\to\operatorname{Spec}K\) is flat, finitely presented and geometrically reduced.

Degree zero holds for any flat locally finitely presented map with geometrically reduced fibres, over an arbitrary base. [Section 11A](#connected-component-covers) constructs the required étale covering, proves the canonical degree-zero comparison on a covering basis, and supplies its generic-spreading argument in [Section 11B](#generic-geometric-connectedness-spreading). The sections lemma used there is Lemma 2.1 of lesson 15; no higher-cohomology assertion is used in this degree-zero step.

Suppose there is a least failing degree \(q>0\), with all comparisons in lower degrees valid for every square of the form (11.1). We give the reduction of a failure to a generic field; this is needed because \(g\) need not be finite type.

For an \(n\)-torsion sheaf \(M\), denote the degree-\(q\) comparison's cokernel by \(D_q(M)\). Embedding \(M\) into an injective \(I\), the two long exact sequences for \(0\to M\to I\to C\to0\) and the lower-degree comparisons show that the degree-\(q\) map is injective and

\[
D_q(M)\hookrightarrow R^qh_*e^{-1}I.
\tag{11.3}
\]

For \(M\subset N\), choose \(M\subset N\subset I\); the same embeddings in (11.3) show \(D_q(M)\hookrightarrow D_q(N)\). The diagram chase uses the identical cokernels of the two maps in degree \(q-1\); the source \(R^qg_*I\) is zero.

Choose a nonzero defect germ \(\xi\), on an affine base so \(T\) is qcqs. For a qcqs \(T'\to T\), let \(P(T')\) say that the image of this germ is still nonzero in the defect for the pulled sheaf. Nil-thickenings do not change \(P\). If \(T'\) is a limit of affine-transition schemes on all of which \(P\) holds, then \(P(T')\) holds: relative continuity and exact filtered colimits identify the defect groups, and an element is zero in such a colimit only if it becomes zero at a stage.

For a closed \(Z\subset T'\) with quasi-compact open complement \(V\), choose \(F|_V\hookrightarrow J\) with \(J\) injective. The stalk-exact monomorphism

\[
F|_{T'}\longrightarrow j_*J\oplus i_*F|_Z
\tag{11.4}
\]

preserves the defect by (11.3). If it remains nonzero in the closed summand, exact closed pushforward and its base change show \(P(Z)\). In the open summand, lower-degree comparison identifies the pulled \(j_*J\) with \(j'_*e_V^{-1}J\) and gives \(R^aj'_*e_V^{-1}J=0\) for \(1\leq a<q\). The relative Leray edge map

\[
R^qh'_*j'_*e_V^{-1}J\longrightarrow
R^q(h'j')_*e_V^{-1}J
\tag{11.5}
\]

is injective: its \((q,0)\) term has no outgoing differential, and every possible incoming term has second degree between one and \(q-1\). If \(P(V)\) failed, the restricted germ for \(F|_V\) would come from \(R^q(g'j)_*F|_V\); its image for \(J\) would be zero, since \(J\) is injective. This contradicts the nonzero image under (11.5). Hence \(P(V)\) holds. We have proved that a defect persists on either the closed or the open part.

Apply this to closed subschemes of \(T\). A descending chain has a closed inverse-limit intersection with affine transitions, on which \(P\) persists. Zorn's lemma gives a minimal closed subscheme with \(P\); replace it by its reduction. Every affine neighbourhood of any generic point in it has \(P\), because its proper closed complement cannot have \(P\). Their inverse limit is the spectrum of the residue field at that generic point. Indeed the local ring at a generic point of a reduced scheme is a field. Thus a failure exists with \(T=\operatorname{Spec}H\) a field.

Here are the integral replacements in this step, including their justification. An integral pushforward is exact, conservative, and commutes with arbitrary base change. To see exactness, on a strict-local target express the integral source as a limit of finite sources, and express an arbitrary sheaf as the colimit of its push-pull sheaves on those stages, as in (2.1). Continuity computes its higher cohomology by that on schemes finite over the strict-local target, which is zero by lesson 7. For conservativity, if the integral pushforward of \(M\) is zero, at every finite stage the finite pushforward of the stage sheaf \(p_{i*}M\) is zero. Finite pushforward is conservative by its product-of-stalks formula, so every stage sheaf is zero; (2.1) makes \(M=0\). Exactness then reflects kernels and cokernels. For base change, the same limit formula for degree-zero images, followed by finite base change at every stage, gives the canonical comparison. These arguments work for non-finite integral maps.

An integral surjection on the top allows replacement of \(F\) by its pulled sheaf: the unit \(F\to\pi_*\pi^{-1}F\) is injective (a geometric lift evaluates it as the identity), defects embed by (11.3), and exact integral pushforward and its comparison identify the new defect. Replace \(H\) by an algebraic closure. The sheaf then becomes a module over \(\mathbf Z/n\), a direct sum of finite cyclic groups \(\mathbf Z/d\), \(d\mid n\). Higher images commute with sums of sheaves, so one summand witnesses failure.

Replace the affine base by the normalization of the schematic closure of the image of \(\operatorname{Spec}H\), inside \(H\). This is an integral map; exact integral pushforward and conservativity transport the comparison and detect its failure. The resulting base is \(S'=\operatorname{Spec}B\), a normal domain whose fraction field \(L\) is algebraically closed. In fact it is the relative algebraic closure of the original fraction field inside \(H\); any algebraic element becomes integral after multiplication by a denominator. The extension \(H/L\) of separably closed fields has derived constant-sheaf unit an isomorphism by lesson 15. Derived composition therefore leaves the nonzero \(R^qh_*\mathbf Z/d\) unchanged if the top field is replaced by \(L\).

On a normal integral scheme with separably closed fraction field, separated étale charts split as disjoint unions of opens, by Lemma 6.1 of lesson 13. All local rings are strictly henselian, and on each irreducible open a constant sheaf is Zariski flasque. Étale and Zariski cohomology agree there, by the strict-local stalk argument. Hence

\[
R^q(\operatorname{Spec}L\to\operatorname{Spec}B)_*
\mathbf Z/d=0\quad(q>0).
\tag{11.6}
\]

Finally write \(B=\bigcup B_i\) as its finite-type \(K\)-subalgebras. Its fraction field is the colimit of the rings \((B_i)_b\), for nonzero \(b\in B_i\), ordered by enlargement and further inversion. The square for \(\operatorname{Spec}L\to\operatorname{Spec}B\) is the inverse limit of the squares for the open immersions \(\operatorname{Spec}(B_i)_b\to\operatorname{Spec}B_i\), with product factor \(A\). Theorem 10.2 proves comparison at every such stage. Relative continuity proves comparison on the limit. By (11.6) its positive target higher images must be zero, contradicting the surviving defect. There is no least failing degree. This proves (11.2). □

The theorem also holds for bounded-below complexes with admissible torsion cohomology: compare the hypercohomology spectral sequences, whose diagonals are finite. The unbounded tensor version below requires the separate finite-dimension argument.

<h2 id="connected-component-covers">11A. Connected-component covers and degree-zero base change</h2>

This argument concerns arbitrary bases. In particular, a geometrically reduced fibre need not be smooth, and the morphism need not be proper. The proof separates the construction of a cover from its sheaf-theoretic application.

<h3 id="component-a-1">11A.1. A specialization calculation with idempotents</h3>

Let \(R\) be a discrete valuation ring with uniformizer \(\pi\), and let \(Z\) be flat over \(R\). If the special fibre is reduced, restriction induces a bijection between idempotent sections on \(Z\) and on its generic fibre \(Z_\eta\).

Indeed, on an affine open \(W=\operatorname{Spec}B\), flatness makes multiplication by \(\pi\) injective. An idempotent on \(W_\eta\) can be written \(b/\pi^n\), with \(n\geq0\) minimal. Its equation in the localization gives

\[
b^2=\pi^n b
\]

in \(B\), since \(B\to B[1/\pi]\) is injective. If \(n>0\), the image of \(b\) in the reduced ring \(B/\pi B\) has square zero. Thus \(b\in\pi B\), contrary to minimality. The idempotent therefore extends to \(B\). These extensions agree on intersections: check on affine opens of an intersection and use the same injectivity. They glue uniquely, with their idempotent equations intact.

A nontrivial decomposition of a scheme into two open-and-closed subsets is equivalent to a nontrivial idempotent section. Consequently, if \(Z\) is connected, then \(Z_\eta\) is connected. No properness, finite-type hypothesis or reducedness of the total space was used in this calculation.

The reduced-special-fibre assumption matters. Over \(R=k[[t]]\), the algebra \(R[u]/(u(u-t))\) is free of rank two. Its generic fibre has two components, while its special fibre is \(k[u]/(u^2)\). The generic idempotent \(u/t\) does not extend. The total space is connected: its two irreducible components meet at the closed point. Thus flatness alone does not prevent two generic components from meeting.

<h3 id="component-a-2">11A.2. The component selected by a section</h3>

Suppose \(p:Z\to B\) is finitely presented and \(\sigma:B\to Z\) is a section. Write \(C_b\) for the connected component of \(Z_b\) containing \(\sigma(b)\), and let \(C\) be the union of these components as a subset of \(Z\).

Each \(C_b\) is geometrically connected. One way to see the field assertion is to pass to a separable closure. The finitely many connected components of a finite-type scheme descend to a finite separable extension. For a connected scheme their Galois action is transitive: an invariant nontrivial collection of components would descend to a nontrivial open-and-closed subset. A rational point belongs to a component fixed by every Galois automorphism. Transitivity therefore forces a single component. Passing from a separably closed field to any extension does not split connected components. This is the field-extension criterion for geometric connectedness.

It follows that this construction commutes with arbitrary base change. The inverse image of \(C_b\) in a fibre over a field extension is connected and contains the new section point, so it is contained in the new component. Conversely, the image of the new component is a connected subset containing the old section point and is therefore contained in \(C_b\).

Here is the constructibility input used below. For a finitely presented family with a section, \(C\) is locally constructible. Its proof can be organized as follows.

* Over an integral Noetherian base, the generic component containing the section is open and closed. Spread this finite open-and-closed decomposition after shrinking the base. One may do this by taking the closures of the two generic pieces and removing the images of their intersection and of the uncovered complement. Those images are constructible and omit the generic point, so they miss a dense open. Also shrink so that the section lies in the first piece.
* The first piece has geometrically connected generic fibre. The generic geometric-connectedness spreading theorem gives a further dense open over which all its fibres are geometrically connected. It is then exactly the component selected by the section.
* For an irreducible closed subset \(T\subset Z\) with generic point \(z\), apply this argument over the reduced closure of \(p(z)\). It makes \(C\cap T\) either empty or all of a nonempty open of \(T\). The Noetherian constructibility criterion applies to every such \(T\), not only to the irreducible components of \(Z\).
* Over an arbitrary affine base, descend the finite presentation and the section to a finite-type \(\mathbf Z\)-algebra. The construction commutes with base change, so the constructible description at that stage pulls back. These descriptions prove local constructibility.

The exact earlier constructibility criterion is AG-MO, *Quasi-finite morphisms and Chevalley*, Theorem 3.1. Finite-presentation and morphism descent are proved in AG-MO, *Limits and Noetherian approximation*, Theorems 3.2 and 4.1. The full generic geometric-connectedness spreading argument is given in [Section 11B](#generic-geometric-connectedness-spreading) below. It establishes the geometric-irreducibility and density steps explicitly; connectedness spreading is not being inferred from flatness alone.

Now assume \(p\) is flat and every fibre is geometrically reduced. We claim that \(C\) is open. First work over a Noetherian affine base, so that \(Z\) is Noetherian and \(C\) is constructible. It is enough to show that \(C\) is stable under generization. To justify this criterion, the complement is constructible. If a point of \(C\) lay in its closure, some locally closed piece of the complement would have an irreducible closure containing that point. The generic point of that closure belongs to the piece, contradicting stability under generization.

Let \(z'\leadsto z\) be a specialization with \(z\in C\), and suppose the points are distinct. Give \(\overline{\{z'\}}\) its reduced structure. The local ring at \(z\) is a Noetherian local domain which is not a field. AG-MO, *Valuation rings and separatedness*, Lemma 5.1 supplies a dominating DVR with the same fraction field. It gives a map

\[
\operatorname{Spec}R\longrightarrow Z
\]

whose generic and special points map to \(z'\) and \(z\). On \(Z_R\) there are two sections: the pullback of \(\sigma\), and the section \(\tau\) induced by this map. Their special points lie in the same connected component of the special fibre, by the base-change compatibility already proved.

Remove all other components of the special fibre. This removes a closed subset of \(Z_R\), since the fibre is Noetherian and has finitely many connected components. Both sections remain in the resulting open: an open of the spectrum of a DVR which contains its closed point is the whole spectrum. The remaining special fibre is connected. The total space is Noetherian, so its connected components are open. Retain the total-space component containing that special fibre. Both sections still remain, because their images are connected and meet that component. The total space is now connected, flat over \(R\), and has reduced special fibre. Section 11A.1 says its generic fibre is connected. Its two section points therefore belong to the same generic component, proving \(z'\in C\).

For a non-Noetherian base, fix a point \(b\). Descend \(p\), \(\sigma\), and flatness to a Noetherian affine stage; the flatness descent is AG-MO, *Limits and Noetherian approximation*, Theorem 4.3. If \(b_i\) is the image of \(b\), then the stage fibre at \(b_i\) is geometrically reduced, because its extension to \(\kappa(b)\) is geometrically reduced. The Noetherian argument at points of that fibre uses reducedness only at its special point: it proves that the selected component locus is a neighbourhood of the selected fibre component. Indeed, every generization of such a point belongs to the constructible component locus, so the complement-closure argument above shows that the point is interior. Pull this neighbourhood back. Varying \(b\) proves openness of \(C\). There is no assumption that every fibre of the chosen Noetherian stage is geometrically reduced.

If \(B\) is affine, then \(Z\) is qcqs. The already established locally constructible description makes the open \(C\) quasi-compact: on a finite affine cover it is an open constructible subset, hence a finite union of quasi-compact basic opens. Its inclusion into \(Z\) is a quasi-compact open immersion. Therefore \(C\to B\) is finitely presented, flat, and has nonempty geometrically connected, geometrically reduced fibres. Nonemptiness follows from the section.

<h3 id="component-a-3">11A.3. Building the cover without making the whole family smooth</h3>

**Component-cover theorem.** Let \(f:X\to S\) be flat and locally of finite presentation, with geometrically reduced fibres. There is an étale covering \(\{U_\lambda\to X\}\) and factorizations

\[
U_\lambda\xrightarrow{a_\lambda}V_\lambda\xrightarrow{v_\lambda}S
\]

such that \(v_\lambda\) is étale and \(a_\lambda\) is flat and finitely presented, with nonempty geometrically connected and geometrically reduced fibres. The \(V_\lambda\) can be affine. The same statement applies to each étale object over \(X\), so these factorizations provide a covering basis of the small étale site.

*Proof.* Fix \(x\in X\), with image \(s\). Restrict to affine neighbourhoods of these points; the restricted morphism is flat and finitely presented. Put \(k=\kappa(s)\), choose a separable closure \(k^s\), and lift \(x\) to a point \(\bar x\) of \(X_{k^s}\). Let \(D\) be its connected component. It is open and closed and is geometrically connected.

The geometrically reduced finite-type scheme \(D\) has a dense smooth open. In that open choose a closed point with separable residue field. Over \(k^s\) its residue field is \(k^s\) itself. For clarity, the two facts used here have the following algebraic content. A geometrically reduced function field is separably generated over the base field; a separating transcendence basis and a finite separable equation yield a smooth neighbourhood of the generic point after inverting finitely many coefficients and a nonzero derivative. Removing the pairwise intersections of irreducible components supplies such neighbourhoods on a dense open of a reduced scheme. A nonempty smooth scheme has an étale chart over affine space. Its image is open; over a separably closed field it contains a rational point, and a nonempty étale fibre over that point has a rational point. These arguments also show why reducedness over an imperfect field alone would not suffice.

Call the chosen smooth rational point \(d\). Its finitely many coordinates lie in a finite separable extension \(k'/k\) inside \(k^s\), so it descends to a \(k'\)-rational point \(d'\) of \(X_{k'}\). Smoothness at this point descends under the field extension. Realize \(k'\) as the residue field of a point \(s'\) in an étale neighbourhood \(S'\to S\). Concretely, lift a separable primitive polynomial to a monic polynomial over an affine neighbourhood of \(s\), and invert its derivative. The point selected by the residue-field quotient has residue field \(k'\).

The morphism \(X_{S'}\to S'\) is smooth near \(d'\). Here is the local algebra behind this assertion. Choose a finite polynomial presentation over the local base ring. The smooth fibre has, locally at \(d'\), a presentation by some of the defining equations with an invertible Jacobian minor. Lift these equations and invert that minor. They define a standard smooth algebra \(C\) mapping onto the original algebra \(B\), with a finitely generated kernel \(J\). The chosen fibre presentation identifies their fibre local rings. Because \(B\) is flat over the local base ring \(A\), tensoring \(0\to J\to C\to B\to0\) with its residue field stays injective on the left. At the selected prime this gives \(J/\mathfrak m_AJ=0\). Nakayama's lemma over that local ring of \(C\) gives \(J=0\) there. Finite generation allows a further neighbourhood on which \(J\) vanishes. Thus the original morphism has a standard smooth neighbourhood; this argument adds no Noetherian assumption on \(A\). A standard smooth presentation is étale over the affine space on its free coordinates, by the invertible Jacobian minor. We therefore have an étale chart

\[
W\longrightarrow\mathbf A^r_{S'}
\]

near \(d'\). Its image coordinates at \(d'\) belong to \(k'\). Lift them to functions on a neighbourhood of \(s'\), forming a section of the affine space. Pulling back the chart along this section gives an étale neighbourhood \(V\to S'\) with a point \(v\) of residue field \(k'\) and a map \(V\to X_{S'}\) taking \(v\) to \(d'\). Shrink \(V\) to an affine neighbourhood of \(v\).

This map supplies a section of \(X_V\to V\). Let \(U\subset X_V\) be its selected-component open from Section 11A.2. Then \(U\to V\) has exactly the asserted finiteness, flatness and fibre properties. The map \(U\to X\) is étale, since it is an open immersion followed by the base change of \(V\to S\).

It remains to check coverage at the original point, rather than only at the smooth point. Over the extension \(k'\subset k^s\), the selected component at \(v\) is the component containing \(d\), namely \(D\). It also contains \(\bar x\). Thus \(U\to X\) has a point over \(x\), even when \(x\) is singular. Repeating the construction for every \(x\) gives a covering.

If \(T\to X\) is étale, its composite with \(f\) is flat and locally finitely presented. Its geometric fibres are étale over geometrically reduced schemes and are therefore reduced. Apply the same construction to \(T\to S\). Covering arbitrary étale objects in this way proves the covering-basis assertion; a cover of \(X\) alone would not establish it. \(\square\)

<h3 id="component-a-4">11A.4. Degree-zero comparison on that basis</h3>

Consider a cartesian square with \(f:X\to S\) as in Section 11A.3 and an arbitrary morphism \(g:T\to S\). Write \(Y=X\times_S T\), \(h:Y\to X\), and \(e:Y\to T\). For every étale sheaf of sets \(F\) on \(T\), the canonical comparison

\[
f^{-1}g_*F\longrightarrow h_*e^{-1}F
\]

is an isomorphism. Neither torsion nor invertibility of a coefficient is needed in degree zero.

Take a basis object \(U\to X\) as above, factored through \(a:U\to V\) and an étale \(V\to S\). Restriction to \(U\) of the left side is \(a^{-1}((g_*F)|_V)\). Lemma 2.1 of *Smooth base change and local acyclicity* applies to \(a\), since it is flat, quasi-compact and has nonempty geometrically connected fibres. Apply the same lemma to its base change \(U\times_S T\to V\times_S T\). The resulting canonical identifications are

\[
\begin{aligned}
\Gamma(U,f^{-1}g_*F)
&=\Gamma(V,(g_*F)|_V)\\
&=\Gamma(V\times_S T,F)\\
&=\Gamma(U\times_S T,e^{-1}F)\\
&=\Gamma(U,h_*e^{-1}F).
\end{aligned}
\]

Here the middle sheaves mean their restrictions to the indicated étale objects. Every identification is induced by pullback or the definition of direct image; their composite is the canonical comparison, not merely some bijection of sets. The construction is natural under restriction of basis objects. Since every étale object is covered by these basis objects, the sheaf axiom makes the comparison an isomorphism on the whole site. This is precisely the degree-zero input in the punctual comparison argument, before its separate positive-degree induction.

<h3 id="component-a-5">11A.5. Source and provider locators</h3>

The geometric arguments are compared with *The Stacks Project*, by The Stacks Project authors, in the AI Integrated Stacks Project native edition at revision `565b10e987aba5969b21145a0833f42d69f96790`. The relevant labelled proofs in [more-morphisms.tex](https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/565b10e987aba5969b21145a0833f42d69f96790/more-morphisms.tex) are `lemma-connected-flat-over-dvr`, the `connected-along-section` lemmas, `lemma-geometrically-connected-generic-fibre`, `lemma-slice-smooth`, and `lemma-cover-by-geometrically-connected`. The field-level inputs occur in [varieties.tex](https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/565b10e987aba5969b21145a0833f42d69f96790/varieties.tex), especially `lemma-geometrically-reduced-dense-smooth-open` and `lemma-smooth-separable-closed-points-dense`. The source edition retains its GFDL terms; no native body is included here.

Exact earlier programme inputs are AG-MO *Valuation rings and separatedness*, Lemma 5.1; *Quasi-finite morphisms and Chevalley*, Theorem 3.1; *Limits and Noetherian approximation*, Theorems 3.2, 4.1 and 4.3; AG-CA *Formally smooth, unramified and étale ring maps*, Theorems 4.1 and 5.1, and *Smooth algebras over a field and the Jacobian criterion*, Lemma 1.1 and Theorem 3.2, for the field-level Jacobian and chart arguments; and AG-ET *Smooth base change and local acyclicity*, Lemma 2.1, for the sections comparison. Exact revisions and source/reader byte identities are recorded separately from this exposition.

The following earlier proofs supply the precise inputs used above:

- [AG-MO · Valuation rings and separatedness](../../AG-MO/valuation-rings-and-separatedness.html#5-the-noetherian-test-by-discrete-valuations) — Lemma5.1: DVR domination realizing Noetherian specializations.
- [AG-MO · Quasi finite morphisms and chevalley](../../AG-MO/quasi-finite-morphisms-and-chevalley.html#3-constructible-sets) — Theorem3.1, Theorem4.2, Theorem5.1: Noetherian constructibility criterion and finite-type constructible images.
- [AG-MO · Limits and noetherian approximation](../../AG-MO/limits-and-noetherian-approximation.html#3-finite-presentation-is-detected-by-limits) — Section3, Theorem4.3: Descent of finite presentations, sections, morphisms and eventual flatness.
- [AG-CA · Formally smooth unramified and etale ring maps](../../AG-CA/AG-CA-17.html#4-standard-smooth-presentations) — Theorem4.1, Theorem5.1, Proposition6.3, Theorem7.2: Standard smooth and etale presentations and finite separable residue extensions.
- [AG-CA · Smooth algebras over a field and the jacobian criterion](../../AG-CA/AG-CA-18.html#3-cotangent-spaces-and-separable-residue-fields) — Lemma1.1, Theorem3.2: Field-level smooth charts and Jacobian criterion at separably generated residue fields.
- [AG-CA · Krull dimension and noether normalization](../../AG-CA/krull-dimension-and-noether-normalization.html#3-normalization-that-respects-ideals) — Corollary3.2: Noether normalization for finite-type algebras over any field; weak Nullstellensatz input.
- [AG-FSE · Flat morphisms](../../AG-FSE/flat-morphisms.html#5-finding-a-flat-part-of-an-arbitrary-finite-type-family) — Lemma5.1, Corollary5.2: Generic freeness for finite modules over finite-type algebras; universal density of selected opens.
- [ag-etale-cohomology · Smooth base change and local acyclicity](../smooth-base-change-and-local-acyclicity.html#2-degree-zero-and-the-geometric-inputs) — Lemma2.1, Proposition2.2: Sections of pulled sheaves of sets along flat quasi-compact maps with nonempty geometrically connected fibres.

<h2 id="generic-geometric-connectedness-spreading">11B. Geometric connectedness near the generic fibre</h2>

The auxiliary results in this section are stated over integral Noetherian bases. They supply the Noetherian step of the selected-component argument; the subsequent approximation argument still gives the component-cover theorem over an arbitrary base. No flatness or properness is assumed in the spreading theorem itself.

<h3 id="component-b-1">11B.1. Keeping a polynomial absolutely irreducible</h3>

Let \(A\) be a domain, \(K\) its fraction field, and \(P\in A[x_1,\ldots,x_n]\) a polynomial which is irreducible over an algebraic closure of \(K\). There is a nonzero \(a\in A\) such that, for every homomorphism \(A_a\to E\) to a field, the specialized polynomial is irreducible over \(E\). The assertion includes further extensions of \(E\).

To prove this, choose positive integers \(w_i\) so that the weights

\[
\alpha_n+\sum_{i<n}w_i\alpha_i
\]

of the finitely many monomials in \(P\) are distinct. Powers of an integer larger than all the exponents suffice. Under the triangular automorphism \(x_i=y_i+y_n^{w_i}\) for \(i<n\), \(x_n=y_n\), the highest power of \(y_n\) has a nonzero coefficient in \(A\). Invert that coefficient and divide by it. We may now assume that \(P\) is monic of degree \(d>0\) in \(y_n\). These changes are invertible after every specialization of the localized coefficient ring.

If a specialized \(P\) factors nontrivially, both factors can be taken monic in \(y_n\): their leading coefficients multiply to \(1\) in \(E[y_1,\ldots,y_{n-1}]\), whose units are nonzero constants. Their degrees \(e_1,e_2\) in \(y_n\) must satisfy

\[
1\leq e_1,e_2,\qquad e_1+e_2=d.
\]

A monic factor of degree zero is exactly \(1\), so it is not a nontrivial factor. For each other variable, the degree of each factor is at most the corresponding degree of \(P\): degrees add when nonzero polynomials are multiplied over a domain.

For each of the finitely many positive-degree pairs, introduce variables for all coefficients of two monic polynomials within these bounds. Equating their product to \(P\) gives a finitely generated coefficient algebra \(D_{e_1,e_2}\) over \(A\). Its base change to \(\overline K\) has no \(\overline K\)-point. By the weak Nullstellensatz it is the zero algebra; faithful flatness of \(\overline K/K\) then gives \(D_{e_1,e_2}\otimes_AK=0\). Thus some nonzero element of \(A\) annihilates \(1\) in this algebra. Invert the product of these finitely many elements. Every nontrivial factorization after any field specialization would give a homomorphism from one of these zero algebras to a field, which is impossible. For \(d=1\), there are no positive-degree pairs, and this product is \(1\).

For completeness, the weak Nullstellensatz here follows from Noether normalization. A nonzero finite-type algebra over an algebraically closed field has a maximal quotient field \(L\), which is finite integral over a polynomial algebra in algebraically independent elements. A subring over which a field is integral is itself a field: an integral equation for the inverse of any nonzero subring element expresses that inverse in the subring. A polynomial algebra is a field only when there are no variables. Hence \(L\) is a finite algebraic extension of the algebraically closed coefficient field and equals that field. The required normalization proof is [AG-CA, Krull dimension and Noether normalization, Corollary 3.2](../../AG-CA/krull-dimension-and-noether-normalization.html#3-normalization-that-respects-ideals).

This proof also supplies geometric integrality of the specialized hypersurface: after any field extension, its coordinate ring is the quotient of a polynomial unique-factorization domain by an irreducible element and is therefore a domain.

<h3 id="component-b-2">11B.2. Spreading a dense open by generic freeness</h3>

Let \(A\) be a Noetherian domain with fraction field \(K\), let \(X\to\operatorname{Spec}A\) be of finite type, and let \(U\subset X\) be open. Suppose that \(X_K\) is reduced and \(U_K\) is dense in \(X_K\). After replacing \(A\) by \(A_a\) for one nonzero \(a\), the open \(U\) is dense in every fibre, and remains dense after every extension of its residue field.

Choose a finite affine cover \(\operatorname{Spec}B_i\) of \(X\). A chart with zero generic algebra disappears after inverting one nonzero element of \(A\): the equation \(1=0\) in \((B_i)_K\) already holds in some localization. On a remaining chart write its closed complement of \(U\) as \(V(I_i)\). The ideal \((I_i)_K\) is not contained in any minimal prime of \((B_i)_K\), because \(U_K\) contains the generic point of every component. Prime avoidance therefore supplies

\[
h_i\in I_i
\]

whose generic image lies in no minimal prime, after clearing a coefficient denominator. Multiplication by \(h_i\) on \((B_i)_K\) is injective: a reduced ring embeds in the product of its quotients by minimal primes, and each nonzero image of \(h_i\) acts injectively on the corresponding domain.

The kernel of multiplication by \(h_i\) on \(B_i\) is a finite \(B_i\)-module, since \(B_i\) is Noetherian. It vanishes after tensoring with \(K\); kill its finitely many generators by inverting another nonzero element of \(A\). Finally apply [AG-FSE, Flat morphisms, Lemma 5.1 and Corollary 5.2](../../AG-FSE/flat-morphisms.html#5-finding-a-flat-part-of-an-arbitrary-finite-type-family) to the finite \(B_i\)-module \(B_i/h_iB_i\). After further localization it is flat over \(A\). The exact sequence

\[
0\longrightarrow B_i\xrightarrow{h_i}B_i\longrightarrow B_i/h_iB_i\longrightarrow0
\]

therefore remains injective on the left after tensoring with any residue field and then with any field extension. In each resulting Noetherian ring, \(h_i\) belongs to no minimal prime. Indeed, localization at a minimal prime is a nonzero zero-dimensional Noetherian local ring, hence Artinian; an element in its maximal ideal is nilpotent and cannot act injectively. Thus \(D(h_i)\) is dense in each of these fibres. It is contained in the corresponding pullback of \(U\). A single product of the finitely many localization elements works for every chart and proves the assertion.

Only the quotient \(B_i/h_iB_i\), not a geometric dimension formula, was used to preserve this injectivity. Reducedness was needed on the generic fibre to choose \(h_i\); reducedness of all the specialized fibres was not assumed.

<h3 id="component-b-3">11B.3. Spreading geometric irreducibility</h3>

Let \(X\to\operatorname{Spec}A\) be of finite type, where \(A\) is a Noetherian domain. If its generic fibre is geometrically irreducible, there is a dense open of the base over which every fibre is geometrically irreducible.

We first spell out the field and finite-presentation steps. Over an algebraic closure \(\overline K\), give \(X_{\overline K}\) its reduced structure. This is an integral finite-type scheme. Its function field has a separating transcendence basis over \(\overline K\), followed by a finite separable simple extension. Consequently it is the fraction field of

\[
\overline K[t_1,\ldots,t_r,z]/(P),
\]

where \(P\) is irreducible. Take the primitive numerator of the minimal polynomial of the final generator, not an arbitrary polynomial multiple obtained by clearing denominators. Gauss's lemma gives irreducibility of that primitive numerator. If the function field is already \(\overline K(t_1,\ldots,t_r)\), use \(P=z\). If \(r=0\), the integral zero-dimensional scheme has function field \(\overline K\), and the same choice applies.

Here is a justification of the separating-basis assertion that also covers positive characteristic. Over a perfect field \(k\), choose a transcendence basis \(t_1,\ldots,t_r\) minimizing the inseparable degree of the finite extension \(L/k(t_1,\ldots,t_r)\). If that degree exceeds one, choose \(z\in L\) inseparable over this rational function field. The primitive irreducible polynomial relation \(Q(t_1,\ldots,t_r,z)=0\) has zero derivative in \(z\). Not all its derivatives can vanish: in characteristic \(p\), that would make it a polynomial in the \(p\)-th powers, and perfectness would make \(Q\) a \(p\)-th power, contradicting irreducibility. Choose \(t_j\) with nonzero derivative. The other \(r\) elements among \(t_1,\ldots,t_r,z\) are algebraically independent, since a relation among them would lie in the principal prime ideal \((Q)\) without involving \(t_j\). The field \(k(t_1,\ldots,t_r,z)\) is finite separable over the rational function field on these other elements. Multiplicativity of inseparable degrees now gives a smaller inseparable degree for \(L\), a contradiction. In characteristic zero any transcendence basis works. The finite separable primitive-element theorem then supplies the single final generator. These are also the field arguments in the admitted native algebra source, labels `lemma-characterize-separable-field-extensions` and `lemma-generating-finitely-generated-separable-field-extensions`.

The same separating-basis argument works for any finitely generated geometrically reduced field extension \(L/k\), including imperfect \(k\). In characteristic \(p\), geometric reducedness makes \(L\otimes_k k^{1/p}\) reduced. The map sending \(\alpha\otimes\beta\) to \(\alpha^p\beta^p\) in \(L\) is injective, since its value tensored with \(1\) is the \(p\)-th power of its input. Hence the \(p\)-th powers of any \(k\)-linearly independent elements of \(L\) are again \(k\)-linearly independent. Choose a polynomial relation of minimal total degree for the \(r+1\) generators in the preceding argument. If all its derivatives vanish, its monomials are \(p\)-th powers. Their roots have smaller total degree and are linearly independent by minimality, whereas their \(p\)-th powers satisfy the displayed relation. This is a contradiction. The same variable exchange decreases the inseparable degree. This general form supplies the separating coordinates at generic points of the geometrically reduced fibre in the smooth-anchor argument, without assuming that its residue field is perfect.

The primitive-element step can also be seen directly. For a finite separable extension of an infinite field, the finitely many embeddings into an algebraic closure differ on some algebra generator. Choose a linear combination of a finite generating list outside the finitely many hyperplanes where two embeddings agree. This element has as many distinct conjugates as the degree of the extension and therefore generates it. Over a finite field the extension is a finite field. Its multiplicative group is cyclic: in any finite subgroup of a field's multiplicative group, elements of the maximal prime-power orders can be multiplied to produce an element whose order is the group's exponent; every group element is a root of the polynomial of that degree, so the exponent equals the group's order. A generator of the multiplicative group also generates the field extension.

Two integral finite-type schemes with the same function field have isomorphic nonempty affine opens. On chosen affine opens, express finite algebra generators as fractions in the other algebra, and invert all the denominators in both directions. The two resulting subalgebras of the common function field are equal. Thus the reduced generic scheme above and the hypersurface have a common nonempty affine open.

All these data descend to a finite field extension \(K'/K\) inside \(\overline K\). To see this for the reduction as well as the equations, choose a finite affine cover of \(X_K\). Its nilradical after extension to \(\overline K\) is finitely generated on each chart, and is nilpotent. The generators, their nilpotence identities, overlap identifications, the coefficients of \(P\), and the finite data for the common open and its two inverse maps involve only finitely many algebraic coefficients. Enlarge \(K'\) to contain them. On this model the descended ideal is nilpotent and its quotient becomes the reduction after extension to \(\overline K\). Faithful flatness makes the quotient reduced already over \(K'\), so this ideal is exactly the nilradical. The resulting reduced generic scheme is geometrically integral. The polynomial \(P\) over \(K'\) is absolutely irreducible, because its extension to the algebraic closure is the chosen primitive polynomial. Finite-presentation descent, including maps and equalities, is given in [AG-MO, Limits and Noetherian approximation, Theorems 3.2 and 4.1](../../AG-MO/limits-and-noetherian-approximation.html#3-finite-presentation-is-detected-by-limits).

One can also verify the algebraically closed field criterion directly. A finite-type domain \(B\) over an algebraically closed field \(k\) remains a domain after any field extension \(E/k\). If two nonzero elements of \(B\otimes_k E\) had zero product, write them in a \(k\)-basis of \(B\). Put their finitely many coefficients in a finite-type subalgebra of \(E\), and invert one nonzero coefficient from each element. A \(k\)-point of this coefficient algebra, supplied by the weak Nullstellensatz, preserves these two nonzero coefficients and the zero-product identity. It would produce two nonzero elements of \(B\) with zero product. The contradiction proves the claim. For a general coefficient field whose algebraic-closure base change is a domain, extend any test field further to a common overfield containing that algebraic closure and use faithful flatness. This proves the geometric-integrality criterion just used.

After localizing \(A\), realize \(K'/K\) by a finite integral domain \(A'\) over \(A\): take finitely many generators of the finite field extension and invert the denominators in their monic algebraic equations. Then \(A'=A[\alpha_1,\ldots,\alpha_m]\) is finite, has fraction field \(K'\), and \(\operatorname{Spec}A'\to\operatorname{Spec}A\) is surjective. There is only one point over the generic point, since \(A'\otimes_AK=K'\).

Put \(X'=(X\times_A A')_{\mathrm{red}}\). Reduction commutes with localization at \(A'\setminus\{0\}\), so its generic fibre is the geometrically integral scheme just constructed. Clear the finitely many coefficient denominators of \(P\), and let \(H\) be the resulting hypersurface over a localization of \(A'\). Spread the two common affine opens and their isomorphism by clearing the denominators in the finite presentations, maps and inverse identities. We obtain an open \(W\subset X'\) identified with an open of \(H\); both generic fibres are nonempty and dense.

Section 11B.1 makes every fibre of \(H\) geometrically integral after a further localization. Section 11B.2 applied to \((X',W)\) and \((H,W)\) makes \(W\) universally dense in their fibres. In particular, in each geometric fibre it is a nonempty open of the integral hypersurface fibre. It is therefore irreducible, and its dense closure in the corresponding fibre of \(X'\) is irreducible. This proves geometric irreducibility for all fibres of \(X'\) over a dense open of \(\operatorname{Spec}A'\).

To return to \(A\), take the closed complement of this open in \(\operatorname{Spec}A'\) and remove its image in \(\operatorname{Spec}A\). The image is closed because the map is finite, and it omits the generic point because there is only one point above it. The remaining open is dense. For each of its points, choose a point above it in \(\operatorname{Spec}A'\). Reduction is a nilpotent closed immersion and remains a homeomorphism after every base change. Thus the just-proved geometric irreducibility of the reduced base change proves geometric irreducibility of the original fibre after extending its residue field. Geometric irreducibility descends along this field extension: extend further to a common algebraically closed field and use the surjectivity of field base change on spectra. This gives the asserted result over \(A\).

<h3 id="component-b-4">11B.4. Spreading geometric connectedness</h3>

Let \(X\to\operatorname{Spec}A\) be of finite type, with \(A\) a Noetherian domain, and suppose that \(X_K\) is nonempty and geometrically connected. There is a dense open of the base over which every fibre is nonempty and geometrically connected.

Over \(\overline K\), the generic fibre has finitely many irreducible components. Give them their reduced structures. Descend their defining ideals, their inclusions, and the finitely many identities for their union to a common finite field extension \(K'/K\). Each descended component is geometrically integral: its extension to the algebraic closure is integral, so the criterion in Section 11B.3 applies. They cover the generic fibre topologically by faithful flatness. As in Section 11B.3, extend this field extension to a finite surjective integral base change \(\operatorname{Spec}A'\to\operatorname{Spec}A\), with one point over the generic point.

Let \(Z_1,\ldots,Z_m\) be the reduced closures in \(X_{A'}\) of these generic components. Their union contains every point of the generic fibre. The complement is of finite type over the Noetherian base; its image is constructible by [AG-MO, Quasi-finite morphisms and Chevalley, Theorem 4.2](../../AG-MO/quasi-finite-morphisms-and-chevalley.html#4-why-a-dominant-finite-type-map-contains-an-open). A constructible subset of an irreducible Noetherian scheme which omits the generic point cannot be dense, so it misses a dense open. Shrink to this open; the \(Z_i\) now cover the whole family. Apply Section 11B.3 to each \(Z_i\), and shrink again, so all their fibres are geometrically irreducible, in particular nonempty.

Form the finite graph whose vertices are these generic components and whose edges record nonempty intersections over \(\overline K\). This graph is connected. Otherwise the unions belonging to two parts of the graph would be disjoint nonempty closed subsets covering the connected generic fibre; each would also be open. For each edge choose the scheme-theoretic intersection \(Z_i\cap Z_j\). Its generic fibre is nonempty. Its constructible image therefore contains a dense open of the base: a constructible set containing the generic point contains a nonempty open. Intersect these finitely many opens (there are no edges to impose if \(m=1\)). All those intersections now have nonempty fibres, and stay nonempty after every extension of the residue field, since field base change is faithfully flat and surjective on spectra.

Each geometric fibre of \(X_{A'}\) is thus a union of nonempty connected subsets indexed by this connected graph, with a nonempty intersection for every graph edge. Such a union is connected: a separation would put each connected subset on one side, and every edge would force its two endpoints onto the same side. It is also nonempty. Finally remove from \(\operatorname{Spec}A\) the finite image of the bad closed subset of \(\operatorname{Spec}A'\), as in Section 11B.3. Connectedness and nonemptiness of these geometric fibres descend under field extension. This proves the theorem.

The nonempty convention is harmless for the selected-component application, where a section is present. If the generic fibre is empty, a finite affine cover and the equation \(1=0\) in its generic algebras show that the entire family is empty over a dense open instead.

<h3 id="component-b-5">11B.5. Sources, scope and application</h3>

The argument above supplies the generic geometric-connectedness input in Section 11A.2 above. Together with its DVR specialization calculation, constructibility and finite-presentation approximation, it applies to the full flat, locally finitely presented, geometrically reduced family over an arbitrary base. The auxiliary Noetherian hypothesis is not a restriction on that final component-cover theorem.

The earlier programme proofs used here are AG-FSE *Flat morphisms*, Lemma 5.1 and Corollary 5.2; AG-CA *Krull dimension and Noether normalization*, Corollary 3.2; and AG-MO *Quasi-finite morphisms and Chevalley*, Theorems 3.1 and 4.2, and *Limits and Noetherian approximation*, Theorems 3.2 and 4.1. The field-theoretic comparison sources are The Stacks Project authors, with the AI Integrated Stacks Project, native revision `565b10e987aba5969b21145a0833f42d69f96790`, [algebra.tex](https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/565b10e987aba5969b21145a0833f42d69f96790/algebra.tex), labels `lemma-characterize-separable-field-extensions`, `lemma-mini-separability` and `lemma-generating-finitely-generated-separable-field-extensions`. The geometric comparison is [more-morphisms.tex](https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/565b10e987aba5969b21145a0833f42d69f96790/more-morphisms.tex), labels `lemma-irreducible-polynomial-over-domain`, `lemma-nowhere-dense-generic-fibre`, `lemma-geom-irreducible-generic-fibre` and `lemma-geometrically-connected-generic-fibre`.

Those native sources retain their own licence and attribution; their text is not included as a redistributed source component here. The dimension-theoretic density proof and finite-flat slicing construction remain useful alternative routes in that edition. The argument above uses generic freeness for density and a finite field extension for the generic irreducible components; that intermediate extension need not be separable or étale. The étale neighbourhood in the component-cover theorem is instead obtained from its separate smooth-anchor construction.

Component-cover and generic-spreading exposition: GPT-6 Astra (OpenAI), Ultra; AI self-check, not independent review. Earlier sources retain their stated authorship and licences.

## 12. Tensor comparison and the general Künneth formula

Use (11.1), and let \(p:X'\to A\). For \(n\) invertible in \(K\), the canonical tensor comparison is

\[
p^{-1}E\otimes^L_\Lambda(f')^{-1}Rg_*F
\longrightarrow
Rh_*(h^{-1}p^{-1}E\otimes^L_\Lambda e^{-1}F),
\qquad\Lambda=\mathbf Z/n.
\tag{12.1}
\]

**Lemma 12.1 (bounded-below tensor upgrade).** Map (12.1) is an isomorphism if \(E,F\) are bounded below and \(E\) has Tor amplitude bounded below.

*Proof.* First work over \(\mathbf F_\ell\), with both inputs sheaves and \(\ell\) invertible. For fixed \(F\), compare the degree-\(q\) functors

\[
A^q(E)=p^{-1}E\otimes(f')^{-1}R^qg_*F,
\quad
B^q(E)=R^qh_*(h^{-1}p^{-1}E\otimes e^{-1}F).
\tag{12.2}
\]

The \(A^q\) are exact in \(E\); the \(B^q\) form a cohomological sequence because tensor over a field is exact. Both commute with filtered colimits. Work locally on affine \(A\), approximate \(E\) by constructible sheaves, and use the finite-constant embedding of lesson 13:

\[
0\to E\to I=\bigoplus_i(\pi_i)_*\mathbf F_\ell\to C\to0,
\tag{12.3}
\]

where the \(\pi_i:A_i\to A\) are finite of finite presentation and \(C\) is constructible. For each summand, finite base change, finite exact pushforward and its projection formula identify (12.2) with the finite pushforward of the comparison (11.2) for \(A_i\). Thus it is an isomorphism in every degree for \(I\).

The following two-step induction fills in the dimension shifting. In degree zero, injectivity for every constructible \(E\) follows by embedding in \(I\) and using left exactness of \(B^0\). Once this also gives injectivity for \(C\), the exact source row and the left-exact target row in (12.3) give surjectivity for \(E\). Suppose all comparisons below \(q\) are isomorphisms. Since \(A^{q-1}(I)\to A^{q-1}(C)\) is surjective, the same is true for \(B^{q-1}\), and its connecting arrow is zero. Consequently \(B^q(E)\to B^q(I)\) is injective, which proves injectivity of the comparison for every \(E\). Apply that new injectivity to \(C\). An element of \(B^q(E)\), viewed in \(B^q(I)\), lifts through the comparison for \(I\); its image in \(A^q(C)\) is zero by injectivity for \(C\), hence it comes from \(A^q(E)\). This proves surjectivity. Filtered colimits recover all \(E\).

For \(\Lambda=\mathbf Z/n\), first take a flat sheaf \(E\). Filter a sheaf \(F\) by successive sheaves killed by prime factors \(\ell\) of \(n\): \(F[\ell]\) and \(F/F[\ell]\) have smaller annihilators when the original annihilator exceeds \(\ell\). In an \(\ell\)-killed case, flatness identifies

\[
E\otimes^L_\Lambda Rg_*F
=(E/\ell E)\otimes^L_{\mathbf F_\ell}Rg_*F,
\tag{12.4}
\]

and similarly on the target. Here derived pushforward of the \(\mathbf F_\ell\)-sheaf computes the same underlying cohomology; it may be regarded as a \(\Lambda\)-complex by restriction of scalars. The field-coefficient result applies. Long exact sequences and induction on the annihilator give the result for every sheaf \(F\).

Represent the given \(E\) by the bounded-below K-flat complex of flat sheaves supplied by [Proposition 1A.2](#flat-models-on-sites), using its lower Tor-amplitude bound. Let its terms start in degree \(a\), and let \(F\in D^{\geq b}\). Its brutal tail above \(N\), tensored with either \(Rg_*F\) or \(F\), starts in degree \(N+1+b\). Right derived images preserve lower bounds. Thus the tail cannot affect degrees below \(N+b\) of either side or their comparison. Bounded brutal truncations and finite triangles reduce to a single flat sheaf \(E\). In this case good upper truncations of \(F\) give a similar degreewise reduction to a bounded complex of sheaves, then to single sheaves. The already proved case identifies the map in every fixed degree, proving the lemma. □

**Lemma 12.2 (unbounded tensor upgrade).** If \(A,S',T\) are finite type over \(K\), (12.1) is an isomorphism for arbitrary \(E,F\in D(-,\mathbf Z/n)\).

*Proof.* Propositions 8.1 and 8.3 make \(Rg_*\) and \(Rh_*\) preserve direct sums. Derived inverse images and derived tensor preserve them too. Both sides of (12.1) therefore preserve triangles and homotopy colimits in each variable.

Write \(E\) as the homotopy colimit of \(\tau_{\leq N}E\), reducing to bounded above. Resolve this bounded-above object by the bounded-above flat construction of [Proposition 1A.1](#flat-models-on-sites) on **\(A\)**. Its bounded brutal truncations from below have a lower Tor-amplitude bound and have homotopy colimit equal to the resolution. Thus we may take \(E\) a bounded complex of flat sheaves. Independently write \(F\) as the homotopy colimit of its good upper truncations, represent a bounded-above stage by a bounded-above complex of sheaves, and take its bounded brutal truncations from below. We reduce to \(F\) bounded. Lemma 12.1 applies at each stage. Preserving the telescopes proves the arbitrary-complex assertion. Choosing a flat resolution only in \(F\) would not justify the Tor-amplitude requirement on \(E\); the factor choice above is deliberate. □

**Theorem 12.3 (general Künneth).** Let \(k\) be separably closed, \(X,Y\) finite type over \(k\), and \(n\) invertible in \(k\). For arbitrary \(E\in D(X_{\mathrm{ét}},\mathbf Z/n)\), \(K\in D(Y_{\mathrm{ét}},\mathbf Z/n)\), the external-product map is an isomorphism:

\[
R\Gamma(X,E)\otimes^L_{\mathbf Z/n}R\Gamma(Y,K)
\xrightarrow{\sim}
R\Gamma(X\times_kY,
\operatorname{pr}_1^{-1}E\otimes^L_{\mathbf Z/n}
\operatorname{pr}_2^{-1}K).
\tag{12.5}
\]

*Proof.* In Lemma 12.2 take \(A=X\), \(S'=\operatorname{Spec}k\), \(T=Y\), \(F=K\). Exact global sections on the separably closed field identify its direct image with the complex of groups \(R\Gamma(Y,K)\). The lemma gives

\[
R\operatorname{pr}_{1*}(\operatorname{pr}_1^{-1}E\otimes^L
\operatorname{pr}_2^{-1}K)
=E\otimes^L\underline{R\Gamma(Y,K)}.
\tag{12.6}
\]

Apply \(R\Gamma(X,-)\), use derived composition, and then Lemma 8.4, valid by Proposition 8.1. This gives (12.5). All arrows were defined by adjunction and tensor, so the comparison is (9.1), compatible with pullbacks, units and external cup products. Neither input has been assumed bounded, constructible or of finite Tor dimension. □

If \(\Lambda=\mathbf F_\ell\) and the inputs are sheaves, taking cohomology gives the familiar direct-sum formula

\[
H^r(X\times Y,E\boxtimes K)
=\bigoplus_{i+j=r}H^i(X,E)\otimes_{\mathbf F_\ell}H^j(Y,K).
\tag{12.7}
\]

Over \(\mathbf Z/n\), (12.5) is the assertion; Tor groups may contribute to its degree groups. Replacing its derived tensor by an ordinary tensor for arbitrary modules would lose those contributions.

## 13. Three examples, with sharp bounds

**Example 13.1 (affine and projective space).** Let \(k\) be separably closed and \(\ell\) invertible in \(k\). Then

\[
\operatorname{cd}_\ell(\mathbf A^r_k)=r,
\qquad\operatorname{cd}_\ell(\mathbf P^r_k)=2r.
\tag{13.1}
\]

The upper bounds are Theorems 5.1 and 4.1. Here is a proof that they are attained. Put \(j:U=\mathbf A^1\setminus\{0,1\}\to\mathbf A^1\) and \(Q=j_!\mathbf F_\ell\). The closed-open sequence gives

\[
H^1(\mathbf A^1,Q)=\operatorname{coker}
(\mathbf F_\ell\xrightarrow{\mathrm{diag}}\mathbf F_\ell^2)
=\mathbf F_\ell.
\tag{13.2}
\]

Indeed the constant affine-line cohomology has only degree zero by lesson 15, and each closed point is acyclic. The same sequence gives \(H^0(Q)=0\) and no higher groups. Apply Theorem 12.3 repeatedly to \(Q\boxtimes\cdots\boxtimes Q\) on \(\mathbf A^r\): its degree-\(r\) cohomology is \(\mathbf F_\ell\). Thus affine cohomological dimension is attained by a nonconstant sheaf, even though the constant sheaf has no positive cohomology.

For projective space, lesson 14 computed

\[
R\Gamma_c(\mathbf A^1,\mathbf F_\ell)
=\mathbf F_\ell(-1)[-2].
\tag{13.3}
\]

On the proper compactification \((\mathbf P^1)^r\), the external product of the \(j_!\mathbf F_\ell\) for \(j:\mathbf A^1\to\mathbf P^1\) is extension by zero from \(\mathbf A^r\): every stalk is constant if all its coordinates lie in the open, and zero otherwise. The proper-factor Künneth theorem gives

\[
R\Gamma_c(\mathbf A^r,\mathbf F_\ell)
=\mathbf F_\ell(-r)[-2r].
\tag{13.4}
\]

Use the closed-open triangle for \(\mathbf P^{r-1}\subset\mathbf P^r\), with complement \(\mathbf A^r\). The proper dimension bound kills \(H^{2r-1}\) and \(H^{2r}\) of \(\mathbf P^{r-1}\), so the triangle identifies \(H^{2r}(\mathbf P^r,\mathbf F_\ell)\) with (13.4)'s nonzero top group. This proves the second equality in (13.1). In fact induction in the same triangle gives

\[
H^{2i}(\mathbf P^r,\mathbf F_\ell)=\mathbf F_\ell(-i)
\ (0\leq i\leq r),\qquad H^{2i+1}=0.
\tag{13.5}
\]

The proof does not use a later purity or duality theorem. For \(r=0\), both equalities in (13.1) mean the acyclic separably closed point.

**Example 13.2 (a product of projective lines).** For \(k\) as above, Theorem 9.1 and the projective-line calculation give groups of ranks

\[
1,\ 0,\ 2,\ 0,\ 1
\tag{13.6}
\]

in degrees zero through four on \(\mathbf P^1\times\mathbf P^1\). In degree two the summands have twist \((-1)\), and in degree four the group has twist \((-2)\). Choose a primitive \(\ell\)-th root to trivialize twists. If \(a,b\) are the pullbacks of the projective-line degree-two generator, then external cup products give

\[
H^*(\mathbf P^1\times\mathbf P^1,\mathbf F_\ell)
=\mathbf F_\ell[a,b]/(a^2,b^2),\qquad\deg a=\deg b=2.
\tag{13.7}
\]

Both squares vanish by pullback from a curve with no degree-four cohomology; \(ab\) generates the one-dimensional degree-four group by Künneth. Together \(1,a,b,ab\) give the listed ranks, so there are no further relations. No choice of roots is needed for the version retaining the twists.

**Example 13.3 (all affine characteristic-\(p\) coefficients).** On **every affine scheme** of characteristic \(p\),

\[
\operatorname{cd}_p(X)\leq1.
\tag{13.8}
\]

We prove the arbitrary-sheaf assertion as well as its constant-coefficient special case. First let \(Y\) be affine Noetherian, \(v:V\to Y\) open, and let \(J\subset\mathcal O_Y\) be a coherent ideal defining the closed complement. There is an exact sequence of étale abelian sheaves

\[
0\to v_!\mathbf F_p\to J\xrightarrow{z\mapsto z^p-z}J\to0.
\tag{13.9}
\]

At a geometric point in \(V\), \(J=\mathcal O\) and this is Artin–Schreier. At a point on the complement use its strictly henselian local ring \(R\), with ideal \(I\). For \(c\in I\), the polynomial \(Z^p-Z-c\) has a unique root with closed residue zero. Reducing that root modulo \(I\) gives zero: in the local ring \(R/I\), all roots of \(Z^p-Z\) are the constant elements of \(\mathbf F_p\), and only zero has that closed residue. Hence the root lies in \(I\). This proves surjectivity on the stalk. A kernel root in \(I\) has the same property and is zero. Thus the kernel stalks are \(\mathbf F_p\) on \(V\) and zero off \(V\), exactly those of extension by zero, proving (13.9).

Quasi-coherent étale cohomology equals Zariski quasi-coherent cohomology, the descent comparison used for Artin–Schreier in lesson 12. Since \(Y\) is affine, \(J\) has no positive cohomology. Consequently

\[
H^q(Y,v_!\mathbf F_p)=0\quad(q\geq2).
\tag{13.10}
\]

Now let \(F\) be a constructible \(\mathbf F_p\)-sheaf on an affine Noetherian \(X\). We prove this same bound by induction on its finite locally constant stratification. Choose an open part \(U\) containing all generic points on which \(F\) is locally constant, and apply the closed-open sequence. The closed term is handled by induction on the proper closed remainder, which is still affine.

For a connected component of \(U\), lesson 11's Sylow reduction supplies a finite étale \(r:V\to U\) of degree prime to \(p\), such that \(r^{-1}F\) has a finite filtration with constant \(\mathbf F_p\) quotients. Trace makes \(F|_U\) a direct summand of \(r_*r^{-1}F\). Extend the quasi-finite separated \(V\to X\) by [Zariski's Main Theorem, Theorem 4.2](../../AG-MO/zariskis-main-theorem.html#4-the-separated-scheme-theorem) as \(V\xrightarrow{v}Y\xrightarrow{\pi}X\), with \(v\) open and \(\pi\) finite. Taking the schematic closure of \(V\) in \(Y\) ensures \(Y_U=V\): \(V\) is both open and closed in \(Y_U\), since it is finite over \(U\), and no extra part belongs to its closure. Finite base change and its zero boundary stalks therefore give

\[
j_!r_*r^{-1}F=\pi_*v_!r^{-1}F,
\tag{13.11}
\]

where \(j:U\to X\). The finite \(Y\) over affine \(X\) is affine. Exact \(v_!\) transports the filtration; (13.10) kills cohomology above one for each quotient, and hence for \(v_!r^{-1}F\). Exact finite pushforward and Leray give the same bound for (13.11). The trace splitting remains a splitting after exact \(j_!\), proving it for \(j_!F|_U\). Finitely many connected components cause only finite sums. The closed-open long exact sequence now proves \(H^q(X,F)=0\) for \(q\geq2\).

Constructible approximation and filtered cohomology recover every \(\mathbf F_p\)-sheaf on affine Noetherian \(X\). For \(p^mF=0\), the filtration by \(F[p]\) and successive quotients reduces to that case, with long exact sequences; for arbitrary primary torsion, take the filtered union of \(F[p^m]\). Finally an arbitrary affine \(\mathbf F_p\)-algebra is a filtered union of its finite-type subalgebras. The fixed-prime version of Lemma 2.1 passes this uniform bound to its spectrum. This proves (13.8) at the stated generality.

The bound is sharp for \(\mathbf A^r_k\), \(r\geq1\), in characteristic \(p\): the Artin–Schreier class of the first coordinate is nonzero. Restricting it to the first affine line reduces to the fact that a nonconstant \(b^p-b\) has degree divisible by \(p\), so cannot equal a coordinate. Thus \(\operatorname{cd}_p(\mathbf A^r_k)=1\), while \(\operatorname{cd}_\ell(\mathbf A^r_k)=r\) for \(\ell\ne p\).

The nonproper Künneth coefficient restriction is real. Over algebraically closed \(k\) of characteristic \(p\), the degree-one Artin–Schreier class of \(xy\) on \(\mathbf A^1\times\mathbf A^1\) is outside the sum of the two factors' classes. In the polynomial Artin–Schreier quotient, mixed monomials split into chains \((x^a y^b)^{p^i}\), with \(a,b>0\) and not both divisible by \(p\). If a polynomial \(c^p-c\) had mixed part equal to \(xy\), the highest occupied term of its chain through \(xy\) would contribute a still higher nonzero term under Frobenius, a contradiction. Pure-variable polynomials cannot cancel a mixed chain. Hence a characteristic-\(p\) version of (12.7) for these nonproper constant sheaves fails already in degree one.

## 14. Exercises

1. **Easy.** Over a separably closed \(k\), with \(\ell\ne\operatorname{char}k\), compute the groups, twists and cup-product algebra of \(\mathbf P^1\times\mathbf P^1\) with \(\mathbf F_\ell\)-coefficients.
2. **Medium.** Prove \(H^q(X,\mathbf Z/p)=0\) for \(q\geq2\) on every affine scheme of characteristic \(p\). Explain what additional argument extends this from the constant sheaf to \(\operatorname{cd}_p(X)\leq1\).
3. **Medium.** For separated finite-type \(X\) of dimension \(d\) over a separably closed field, use Nagata compactification and proper vanishing to prove \(H^q_c(X,F)=0\) for \(q>2d\). Then prove the requested ordinary bound \(H^q(X,F)=0\) for \(q>2d\), including nonseparated \(X\), and explain why compactification alone does not identify the two cohomologies.
4. **Medium.** For finite-type \(X\) over separably closed \(k\) and \(\ell\ne\operatorname{char}k\), prove that pullback gives \(H^*(X,\mathbf F_\ell)\simeq H^*(X\times\mathbf A^1,\mathbf F_\ell)\), compatibly with cup products.
5. **Hard.** Prove the affine curve bound \(\operatorname{cd}_{\mathrm{tors}}(C)\leq1+\operatorname{cd}_{\mathrm{tors}}(K)\) over arbitrary \(K\). Identify the geometric-stalk step and state why no prime-to-characteristic hypothesis is needed for this bound.

## 15. Full solutions

**Solution 1.** The projective-line theorem gives \(H^0=\mathbf F_\ell\), \(H^1=0\), \(H^2=\mathbf F_\ell(-1)\), with all later groups zero. Künneth over the field \(\mathbf F_\ell\) has no Tor terms. Degree zero is \(H^0\otimes H^0\); degree two has \(H^2\otimes H^0\) and \(H^0\otimes H^2\); degree four is \(H^2\otimes H^2\). These give ranks \(1,0,2,0,1\) and twists \(0,-1,-2\), with zero groups in every other degree.

Choose a twist trivialization and let \(a,b\) be the pullbacks of the degree-two generator on each factor. Their squares are pullbacks of zero degree-four groups on \(\mathbf P^1\). Their product is the external product of two generators, hence a generator in degree four. Graded commutativity has sign \((-1)^{2\cdot2}=1\), also for \(\ell=2\). The four basis elements \(1,a,b,ab\) establish \(\mathbf F_\ell[a,b]/(a^2,b^2)\). Retaining the twists avoids the choice of a root of unity. □

**Solution 2.** On the small étale site there is a short exact sequence

\[
0\to\mathbf Z/p\to\mathcal O_X
\xrightarrow{F-1}\mathcal O_X\to0.
\tag{15.1}
\]

Its kernel consists étale locally of the constant roots \(0,\ldots,p-1\). Surjectivity follows because adjoining a root of \(T^p-T-c\) gives a monic finite étale algebra with derivative \(-1\); its geometric fibres are nonempty, so it gives an étale cover. Quasi-coherent étale cohomology of \(\mathcal O_X\) is its Zariski cohomology, which is zero in positive degrees on affine \(X\). The long exact sequence makes every \(H^q(X,\mathbf Z/p)\), \(q\geq2\), zero and identifies

\[
H^1(X,\mathbf Z/p)=\Gamma(X,\mathcal O_X)/(F-1)\Gamma(X,\mathcal O_X).
\tag{15.2}
\]

This proves the exercise even for non-Noetherian affine rings. It does not alone test every primary torsion sheaf. For that assertion use the complete argument (13.9)–(13.11): apply Artin–Schreier to a coherent boundary ideal for open extension by zero, use a prime-to-\(p\) Sylow trace cover and its constant-quotient filtration, extend the cover into a finite affine scheme by [Zariski's Main Theorem, Theorem 4.2](../../AG-MO/zariskis-main-theorem.html#4-the-separated-scheme-theorem), then perform closed-open constructible dévissage. Primary-power filtrations and the limit theorem recover all affine schemes and all primary torsion sheaves. Each step was proved in Example 13.3. □

**Solution 3.** For separated \(X\), take a dense Nagata compactification \(j:X\to\bar X\), where \(\bar X\) is proper and has dimension \(d\); discard components not met by the open. Exact \(j_!\) preserves torsion. Thus \(H^q_c(X,F)=H^q(\bar X,j_!F)=0\) for \(q>2d\), by Theorem 4.1. This is the conclusion furnished by the stated compactification hint.

For ordinary cohomology, use \(\epsilon:X_{\mathrm{ét}}\to X_{\mathrm{Zar}}\). On every affine open, Theorem 5.1 over the separably closed field makes the cohomology presheaf zero above \(d\), so \(R^q\epsilon_*F=0\) for \(q>d\). Lemma 7.1 gives \(H^p_{\mathrm{Zar}}(X,G)=0\) for \(p>d\). The spectral sequence \(H^p_{\mathrm{Zar}}(X,R^q\epsilon_*F)\Rightarrow H^{p+q}(X,F)\) has no total degrees above \(2d\). It proves the ordinary bound even if \(X\) is nonseparated. Ordinary compactification instead gives \(R\Gamma(\bar X,Rj_*F)\), and positive \(R^qj_*F\) can occur. Substituting \(j_!F\) there would change the groups; for \(\mathbf A^1\), the constant ordinary group in degree zero is already different from its degree-zero compact-support group. □

**Solution 4.** Lesson 15 gives \(R\Gamma(\mathbf A^1,\mathbf F_\ell)=\mathbf F_\ell[0]\), with its degree-zero identification the constant-section map. Theorem 12.3 consequently identifies cohomology of \(X\times\mathbf A^1\) with \(R\Gamma(X,\mathbf F_\ell)\otimes^L\mathbf F_\ell=R\Gamma(X,\mathbf F_\ell)\). The canonical external-product definition sends \(u\) to \(\operatorname{pr}_1^*u\cup1\), so this is the actual projection pullback. Its inverse is restriction along the zero section, since that section followed by projection is the identity and pullback is already an isomorphism. Pullback preserves cup products, proving the ring assertion. Properness of \(X\) is unnecessary; invertibility of \(\ell\) is essential, as the characteristic-\(p\) example shows. □

**Solution 5.** Let \(f:C\to\operatorname{Spec}K\) and \(F\) be any torsion sheaf. Its higher-image geometric stalk is

\[
(R^qf_*F)_{\overline K}=H^q(C_{K^{\mathrm{sep}}},F|_{C_{K^{\mathrm{sep}}}}).
\tag{15.3}
\]

Use the strict-local stalk formula, expressing the separable closure as the limit of pointed field neighbourhoods; this is valid for a nonproper \(f\) because the base is a field. Purely inseparable passage to an algebraic closure preserves the étale topos. The arbitrary-torsion affine curve theorem of lesson 12 therefore kills (15.3) for \(q>1\), even for \(p\)-torsion in characteristic \(p\). Each remaining higher-image sheaf is torsion, by annihilator approximation and qcqs continuity. In Leray, \(H^i(K,R^qf_*F)\) vanishes for \(i>\operatorname{cd}_{\mathrm{tors}}(K)\), and rows \(q>1\) are zero. Thus \(H^m(C,F)=0\) for \(m>1+\operatorname{cd}_{\mathrm{tors}}(K)\). If the field dimension is infinite the inequality is automatic. This is the full coefficient scope of Lemma 2.2, rather than only a constant-sheaf computation. □

## 16. Sources, proof choices and corrections

- **[Modern dimension argument]** The Stacks Project Authors, [cohomological dimension, Tags 0F0P–0F0X](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/etale-cohomology.html#etale-cohomology-section-cd), and [proper dimension bound, Tag 095U](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/etale-cohomology.html#etale-cohomology-lemma-cohomological-dimension-proper). Sections 2–6 supply the limits, algebraic field-extension case, strict-local finite models, both interludes, all spectral-sequence ranges and the affine-image support bound. The induction's support interlude is stated on an affine scheme, where its use of affine induction is valid.
- **[Finite dimension and products]** [Finite cohomological dimension, Tags 0F0Y–0F0G](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/etale-cohomology.html#etale-cohomology-section-finite-cd), and [Künneth, Tags 0F13–0F1P](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/etale-cohomology.html#etale-cohomology-section-kunneth). Sections 8–12 prove the arbitrary-complex result in full. The inverse-limit reduction, closed-point detection, minimal defect argument, finite-constant dimension shifting and truncations are given explicitly, rather than replacing the general proof by its tag. The flat resolution in the unbounded tensor upgrade is taken in the factor required to have lower Tor amplitude.
- **[Auxiliary foundations]** [Noetherian topological vanishing](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/cohomology.html#cohomology-section-vanishing-Noetherian), [integral pushforward](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/etale-cohomology.html#etale-cohomology-lemma-what-integral), [finite-constant embeddings](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/etale-cohomology.html#etale-cohomology-lemma-constructible-maps-into-constant-general), [flat geometrically reduced connected-fibre covers](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/more-morphisms.html#more-morphisms-lemma-cover-by-geometrically-connected), and [finite Zariski main factorization](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/more-morphisms.html#more-morphisms-lemma-quasi-finite-separated-pass-through-finite). The latter geometric factorization is the explicit foundation in Example 13.3. Sections 7 and 11 give the topological and integral cohomological arguments; the finite-constant embedding is the proved earlier lesson 13 prerequisite.
- **[Artin–Schreier]** [The Artin–Schreier sequence](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/etale-cohomology.html#etale-cohomology-section-artin-schreier) gives the constant-coefficient affine computation. The stronger arbitrary-primary-sheaf assertion in Example 13.3 includes the boundary-ideal and Sylow trace steps.
- **[Historical affine theorem]** M. Artin, *[SGA 4](https://www.normalesup.org/~forgogozo/SGA4/)*, Exposé XIV, especially §§2–4, Théorème 3.1 and Corollaire 3.2. Its route uses strict-local dimension and a punctured strict-local model. The present proof follows the modern Gabber-style form named above. Exposé X, §5, contains the boundary-ideal Artin–Schreier mechanism used and proved in Example 13.3.
- **[Arcata statement]** P. Deligne, *[SGA 4½](https://publications.ias.edu/sites/default/files/Number32.pdf)*, “Cohomologie étale: les points de départ,” IV, 6.4: the affine-vanishing statement \(H^q(X,F)=0\) for \(q>\dim X\), proved here, and the Morse-theory analogy that follows it.

The examples distinguish cohomological dimension from constant cohomology, and the compactification exercise distinguishes ordinary from compact support. Finite \(n\) is invertible in the general nonproper Künneth theorem; one proper factor permits every \(n\). The linked AI Integrated Stacks Project reader contains AI-proposed corrections and additions and is not reviewed by the maintainers of the [official Stacks Project](https://stacks.math.columbia.edu/).