Draft lesson.

Cohomological dimension of noetherian spaces

Written by GPT-6.1 Sol (OpenAI), October 2026. Public domain (CC0).

The dimension of a noetherian space bounds the degree in which any sheaf can have cohomology. A useful proof separates two phenomena: generic stalks give a flabby comparison object, while sheaves with zero generic stalks are assembled from smaller-dimensional closed pieces. The latter assembly is a filtered union. Zero generic stalks do not in general give one fixed proper closed support.

We use unital, possibly noncommutative ringed spaces and left module sheaves, fixed universes and choice. A noetherian space is one in which every open is quasi-compact. Its Krull dimension is the supremum of the lengths \(m\) of strict chains \(Z_0\subsetneq\cdots\subsetneq Z_m\) of nonempty irreducible closed subsets. The empty space has dimension \(-1\). The theorem below assumes a finite bound on this dimension, and imposes no separation or sobriety condition and no noetherianness condition on the coefficient rings.

Use Godement resolutions and flabby sheaf cohomology, Sections 1–9, for point pushforwards and flabby acyclicity, and Filtered colimits and sheaf cohomology on quasi-compact opens, Sections 1–6, for every filtered-colimit and cohomology comparison used here. Closed support and localization for locally module stacks, Sections 1–3, gives the supported-sheaf equivalence in its ordinary module specialization. The applications use Cech complexes, refinements and acyclic covers, Exercise 2, for the actual Mayer–Vietoris maps; Finite-poset sheaves and canonical homological resolutions, Sections 1–3, for exact least-point sections; Sheaf Ext and the local-to-global spectral sequence, Sections 4 and 6, for the Ext comparisons; and Polynomial rings, finite resolutions and derived duality, Section 2, for the full Koszul proofs.

1. Irreducible components and the dimension drop

A nonempty closed set is irreducible when it is not a union of two proper closed subsets. By induction on closed subsets of a noetherian space, every closed set is a finite union of irreducible closed subsets: a minimal counterexample would be reducible, with both proper closed pieces already finite unions. Remove contained members of such a finite union for \(X\). The remaining sets \(C_1,\ldots,C_t\) are precisely its maximal irreducible closed subsets. Indeed an irreducible set contained in their finite union is contained in one of them, by repeated use of irreducibility.

A generic point of \(C_i\) is a point \(\eta_i\) whose closure is \(C_i\). If all these points exist, no \(\eta_i\) belongs to another component. Otherwise its closure would be contained in that component. For nonempty \(X\), if a closed set \(Z\) avoids every \(\eta_i\) and \(\dim X\le d\), then

\dim Z\le d-1. \tag{1.1}

In fact the largest member of any irreducible closed chain in \(Z\) lies in some \(C_i\). It is strictly smaller than \(C_i\), since the latter contains \(\eta_i\notin Z\). Appending \(C_i\) adds one strict step to the chain in \(X\). Closed subsets inherit noetherianness, and their irreducible closed subsets are also closed in \(X\). These facts prove (1.1), including the conclusion that such \(Z\) is empty when \(d=0\).

2. Supply the generic points without changing sheaves

Here is the full reduction needed when generic points are absent. Let \(X^s\) be the set of all nonempty irreducible closed subsets of \(X\). For an open \(U\subseteq X\), put

U^s=\{C\in X^s:C\cap U\ne\varnothing\}. \tag{2.1}

These subsets form a topology. Arbitrary unions commute with the operation, and \((U\cap V)^s=U^s\cap V^s\): two nonempty opens in an irreducible space intersect. Also \(\varnothing^s=\varnothing\) and \(X^s\) is the whole space. The map \(x\mapsto\overline{\{x\}}\) pulls \(U^s\) back to \(U\). Indeed if the closure of \(x\) meets an open, that open contains \(x\). Thus \(U\mapsto U^s\) is a bijection of open sets, preserving inclusions, covers, arbitrary unions and finite intersections.

The closed subset corresponding to a closed \(A\subseteq X\) is

A^s_{\mathrm{cl}}=\{C\in X^s:C\subseteq A\}. \tag{2.2}

This is the complement of \((X\setminus A)^s\). The correspondence preserves finite unions and all intersections, and detects nonempty sets, since any \(x\in A\) gives \(\overline{\{x\}}\subseteq A\). It follows that the nonempty irreducible closed subsets on both sides correspond, with exactly the same strict inclusion chains. In particular noetherianness and dimension are preserved. For irreducible \(A\), the point \(A\in X^s\) has closure \(A^s_{\mathrm{cl}}\): a closed set (2.2) contains that point exactly when it contains \(A\). It is the unique such point, since distinct closed subsets give distinct sets (2.2). Hence \(X^s\) is sober, meaning that every nonempty irreducible closed subset has a unique generic point.

Transport coefficients by \(\mathcal O^s(U^s)=\mathcal O(U)\). This is a sheaf of rings, because the open correspondence preserves covers and their intersections. Sending a module sheaf \(F\) to \(F^s(U^s)=F(U)\), with precisely the same restriction and scalar maps, gives an equivalence of module-sheaf categories. Its inverse uses the inverse open correspondence. The two composites are identities on sections and maps. Both functors are additive and preserve kernels and cokernels, since they are inverse equivalences of abelian categories. The equivalence preserves injectives by transposing their extension property along the inverse equivalence. It preserves global sections as well. Applying it to an injective resolution therefore gives canonical isomorphisms

H^q(X,F)\simeq H^q(X^s,F^s) \tag{2.3}

in every degree, natural in all sheaf maps and compatible with the coefficient actions. Thus it suffices to prove the theorem for sober spaces. This construction is often called soberification; no assumption about generic points of the original space has been inserted.

3. Closed-supported sheaves and their cohomology

For closed \(i:Z\hookrightarrow X\), use the inverse-image ring on \(Z\). Recall the ordinary maps of the supported-sheaf prerequisite. The sheaf \(i_*N\) has sections \(N(U\cap Z)\). Its stalk is \(N_x\) on \(Z\) and zero off \(Z\). These formulas prove exactness of \(i_*\). Its exact left adjoint \(i^{-1}\) is inverse image with these coefficients; its counit \(i^{-1}i_*N\to N\) is the identity on germs. A sheaf zero off \(Z\) is identified with \(i_*i^{-1}F\) by the unit, an isomorphism at every stalk.

The adjunction with an exact left adjoint makes \(i_*\) preserve injectives: transpose a map from a subobject, extend into the injective target, and transpose back. As \(\Gamma(X,i_*N)=\Gamma(Z,N)\), applying \(i_*\) to an injective resolution gives

H^q(X,i_*N)\simeq H^q(Z,N). \tag{3.1}

These are the actual resolution-section identifications and are natural in \(N\), not just abstract isomorphisms of groups.

For a closed \(Z\) and its open complement \(j\), write

G_Z=\ker\bigl(G\to j_*j^{-1}G\bigr). \tag{3.2}

Its sections on \(U\) are exactly the sections of \(G(U)\) which vanish on \(U\setminus Z\). Thus it is a subsheaf supported on \(Z\), and if \(Z\subseteq Z'\) its canonical inclusion into \(G_{Z'}\) is the original subsection inclusion. We will use these maps in a filtered union.

4. Zero generic stalks give a filtered support decomposition

Now let \(X\) be sober and noetherian, with component generics \(\eta_1,\ldots,\eta_t\). Suppose \(G_{\eta_i}=0\) for every \(i\). Let \(\mathcal Z\) be the set of closed subsets avoiding all these points. It is directed by inclusion, since it contains the empty set and is closed under finite unions. Then the canonical map is an isomorphism

\mathop{\mathrm{colim}}_{Z\in\mathcal Z}G_Z\xrightarrow{\ \sim\ }G. \tag{4.1}

To check surjectivity on stalks, take a section \(g\in G(U)\) representing any germ. For each generic point belonging to \(U\), its zero germ lets us choose an open neighborhood \(W_i\subseteq U\) on which \(g\) vanishes. If \(\eta_i\notin U\), then \(U\cap C_i=\varnothing\): any nonempty open in \(C_i\) contains its generic point. In that case set \(W_i=X\setminus\bigcup_{j\ne i}C_j\). This is an open neighborhood of \(\eta_i\), contained in \(C_i\), so disjoint from \(U\). The union \(W=\bigcup_iW_i\) contains every generic point, and \(g\) vanishes on \(U\cap W\). Consequently \(g\in G_Z(U)\) for \(Z=X\setminus W\in\mathcal Z\). This proves the required germ lifting. Every map in the union is a subsection inclusion; exactness of filtered colimits, or the same germ test for zero, proves injectivity in (4.1).

The choice of \(Z\) can depend on the section. Nothing asserts that \(G\) vanishes on a single neighborhood of all generics. If vanishing above degree \(d-1\) has been established on all noetherian spaces of dimension at most \(d-1\), (1.1), (3.1) and (4.1) imply

H^q(X,G)=0\qquad(q\ge d). \tag{4.2}

Indeed each \(G_Z\) is a closed pushforward from such a space \(Z\). The filtered-colimit theorem applies on all of \(X\), since \(X\) is noetherian and hence quasi-compact. It identifies \(H^q(X,G)\) with \(\mathop{\mathrm{colim}}_Z H^q(X,G_Z)\) using precisely the inclusion maps in (3.2). All terms are zero for \(q\ge d\). For \(d=0\), all the \(Z\)'s are empty, and (4.1) says directly that \(G=0\).

5. The dimension induction

Theorem 5.1 (Grothendieck vanishing). If \(X\) is noetherian and \(\dim X\le d<\infty\), then for every module sheaf \(F\),

H^q(X,F)=0\qquad(q>d). \tag{5.1}

Proof. The empty space is immediate. Use Section 2 to pass to a sober space with the same dimension and cohomology. For each component generic let \(a_i:\{\eta_i\}\to X\) be the point map with its stalk ring, and form the canonical map

F\xrightarrow{u_F}E(F)=\prod_{i=1}^t a_{i,*}F_{\eta_i}. \tag{5.2}

On an open, its \(i\)-th component sends a section to its generic germ when that open contains \(\eta_i\), and to zero otherwise. These maps commute with restriction and all \(F\)-morphisms. Each point pushforward is flabby: its sections are the fixed module when the open contains the point and zero otherwise. The finite product \(E(F)\) is therefore flabby and has no positive cohomology.

At \(\eta_i\), the \(i\)-th factor of (5.2) is exactly \(F_{\eta_i}\). Every other factor vanishes on the neighborhood \(X\setminus\bigcup_{j\ne i}C_j\). Hence \(u_F\) is an isomorphism at every component generic. Put \(K=\ker u_F\), \(L=\operatorname{im}u_F\), and \(Q=\operatorname{coker}u_F\). Exact stalks give zero generic stalks for both \(K\) and \(Q\).

Induct on \(d\). In dimension zero, Section 4 gives \(K=Q=0\), so \(F\simeq E(F)\) is flabby and the conclusion follows. For \(d\ge1\), the induction hypothesis and (4.2) give \(H^q(X,K)=H^q(X,Q)=0\) for \(q\ge d\). Use the two actual short exact rows

0\to K\to F\to L\to0,\qquad 0\to L\to E(F)\to Q\to0. \tag{5.3}

For \(n>d\), we have \(n\ge2\). The connecting map of the second row identifies \(H^n(X,L)\) with \(H^{n-1}(X,Q)=0\), since the two intervening positive groups of \(E(F)\) vanish. The first row has \(H^n(X,K)=0\) and \(H^n(X,L)=0\), so exactness gives \(H^n(X,F)=0\). These are the canonical long-exact-sequence maps of (5.3); all kernel, image, quotient and generic-stalk maps are functorial. Finally (2.3) transfers the conclusion back to the original space. This proves the theorem without a sobriety assumption. \(\square\)

6. Local bounds and a global Ext bound

The same theorem applies on every open or closed subspace which is noetherian with the specified dimension bound. If \(f:X\to Y\) is a ringed-space map and \(Y\) has a basis of opens \(V\) such that \(f^{-1}V\) is noetherian of dimension at most \(d\), then

R^qf_*F=0\qquad(q>d). \tag{6.1}

The cohomology-presheaf description recalled in Section 6 of the filtered-colimit prerequisite identifies this sheaf with the associated presheaf of \(V\mapsto H^q(f^{-1}V,F)\). The theorem makes that presheaf zero on the stated basis, so every stalk is zero. This proves (6.1) for the actual direct-image functor and its coefficient actions. Neither properness nor an identification with the cohomology of a literal fiber has been used.

For the next statement assume \(\mathcal O\) commutative, \(X\) noetherian of dimension at most \(d\), and \(F\) locally equipped with an actual finite resolution of length at most \(r\) by finite projective module sheaves. Here finite projective means a local retract of a finite free module, and \(r\ge0\) is common to the covering. Then for every \(G\),

\operatorname{Ext}^n_X(F,G)=0\qquad(n>d+r). \tag{6.2}

Indeed those local resolutions compute internal derived Hom by the finite-resolution proof in the sheaf-Ext prerequisite; finite projective retracts preserve the exact Hom property used there. The local Hom complexes have terms only in degrees zero through \(r\), so \(\mathcal Ext^q(F,G)=0\) for \(q>r\). The local-to-global spectral sequence has \(E_2^{p,q}=H^p(X,\mathcal Ext^q(F,G))\), with \(p,q\ge0\). Theorem 5.1 kills \(p>d\), and the resolution bound kills \(q>r\). Every associated piece of its finite abutment filtration is therefore zero in total degrees greater than \(d+r\), proving (6.2). The finite local resolutions are essential hypotheses; freeness or projectivity of individual stalks is not a substitute.

7. Four exercises with full solutions

7.1 A sharp bound on a space with no generic point

Exercise 1 (basic). Let \(S\) be an infinite set with its cofinite topology, use \(\mathcal O=k_S\), and choose distinct \(x,y\in S\). Show that \(S\) is noetherian of dimension one but is not sober. For \(j:S\setminus\{x,y\}\hookrightarrow S\), compute all cohomology of \(j_!k\).

Solution. The proper closed sets are finite. A descending chain of closed sets stabilizes once it reaches a finite set, so the space is noetherian. Its irreducible closed subsets are the singletons and \(S\) itself: every finite set with at least two points is reducible, and any two nonempty opens of \(S\) intersect. Thus its dimension is one. Every point is closed, so none is generic for \(S\). The construction of Section 2 adds exactly one point \(\eta\), corresponding to \(S\); all nonempty cofinite opens become \(U\cup\{\eta\}\).

There is a short exact row

0\to j_!k\to k_S\longrightarrow k_x\oplus k_y\to0, \tag{7.1}

where the last map evaluates germs. At \(x\) or \(y\) it is the identity to the corresponding single factor; elsewhere its target stalk is zero. The kernel agrees with extension by zero from the complement, by the identity stalk maps on that complement and zero stalks at the two omitted points. These observations prove exactness and identify all arrows. The constant sheaf \(k_S\) is flabby, since every nonempty open is irreducible and has exactly the constant sections \(k\). Point sheaves are flabby as well. On global sections the last map is \(k\to k^2\), \(t\mapsto(t,t)\). The long exact sequence therefore gives \(H^0(S,j_!k)=0\), \(H^1(S,j_!k)=k^2/\Delta k\simeq k\) through \((a,b)\mapsto b-a\), and zero in higher degrees. The bound is sharp on this non-sober space.

7.2 A finite space sharp in every dimension

Exercise 2 (intermediate). Let \(P_d\) have two points in each level \(0,\ldots,d\), with each point below every point at a larger level and no relation between the two points at one level. Give it the upper-set topology. Determine its dimension and the cohomology of the constant sheaf with any abelian coefficient group \(A\).

Solution. A closed set in a finite ordered space is a lower set. Its maximal elements express it as a finite union of principal lower sets, so an irreducible closed set has exactly one maximal element and is principal. Strict chains of these sets are exactly strict point chains. Hence \(\dim P_d=d\); finiteness gives noetherianness.

For \(d=0\) the space is two discrete points, so \(H^0=A^2\) and higher groups vanish. For \(d\ge1\), cover \(P_d\) by the minimal opens of its two level-zero points. Sections on either open are exact evaluation at its least point, by the sheaf–diagram equivalence, with degree-zero constant group \(A\). The intersection is \(P_{d-1}\). The actual Mayer–Vietoris difference map in degree zero is the difference of these restrictions.

When \(d=1\), this map is \(A^2\to A^2\), \((s,t)\mapsto(t-s,t-s)\). Its kernel and cokernel are \(A\), giving \(H^0=H^1=A\). When \(d\ge2\), induction makes the intersection connected, with \(H^0=A\), and the difference map \(A^2\to A\) is surjective. Thus \(H^0(P_d,A)=A\), \(H^1(P_d,A)=0\), and the positive connecting maps identify \(H^n(P_d,A)\) with \(H^{n-1}(P_{d-1},A)\) for every \(n\ge2\). Induction gives, for \(d\ge1\), exactly \(A\) in degrees zero and \(d\), and zero in the other degrees. These computations use exact least-point sections and the full connecting maps, so they hold for arbitrary abelian \(A\), not only vector spaces. With \(A\ne0\) the dimension bound is sharp for every \(d\).

7.3 No single closed support for a zero generic stalk

Exercise 3 (advanced). On the soberification \(S^s=S\cup\{\eta\}\) of Exercise 1, form \(G=\bigoplus_{x\in S}k_x\). Show that \(G_\eta=0\), but \(G\) is not supported on any proper closed subset. Describe exactly the union (4.1) and all its cohomology.

Solution. The proper closed subsets of \(S^s\) are the finite subsets of \(S\). On a nonempty open \(U\cup\{\eta\}\), the filtered-colimit section theorem gives

G(U\cup\{\eta\})=\bigoplus_{x\in U}k. \tag{7.2}

Indeed the direct sum is the filtered colimit of its finite partial sums, whose sections have exactly those coordinates, and this space is noetherian. Each such section has finite support; removing that finite support gives a neighborhood of \(\eta\) on which it is zero. Thus its germ at \(\eta\) is zero. At every \(x\in S\), its stalk is \(k\): remove the other finitely many coordinates of any representative, retaining the \(x\)-coordinate. In particular \(G\) cannot be supported on any proper closed, finite set.

For a finite closed \(Z\subset S\), (3.2) and (7.2) give \(G_Z=\bigoplus_{x\in Z}k_x\). Its inclusion for \(Z\subset Z'\) is the ordinary zero extension of coordinates, and their filtered union is precisely \(G\). Restrictions in (7.2) are surjective coordinate projections, so \(G\) is flabby. Its cohomology is \(\bigoplus_{x\in S}k\) in degree zero and zero in all positive degrees. This example separates the section-dependent closed sets in (4.1) from the stronger, false assertion of one fixed proper closed support.

7.4 Sharp simultaneous space and resolution bounds

Exercise 4 (advanced). For \(d,r\ge0\), put \(R=k[t_1,\ldots,t_r]\), \(X=P_d\), \(\mathcal O=R_X\), \(F=k_X\) through the quotient by all the variables, and \(G=R_X\). Compute all global Ext groups and show that the bound \(d+r\) in (6.2) is sharp.

Solution. Use the ordered Koszul resolution of \(k\) over \(R\), and take its constant module-sheaf complex on \(P_d\). Its augmentation is exact at every stalk, with finite free terms in degrees \(-r,\ldots,0\), so it is an actual resolution of the required length. The polynomial prerequisite proves exactness by tensoring the one-variable contractions and proves, with the normalized Hom differential and its degree-sign correction, that its dual has only degree-\(r\) cohomology \(k\otimes_k\det(k^r)^*\). All these maps are coefficient maps, hence give the same constant-sheaf identifications here, including \(r=0\).

Thus \(\mathcal Ext^q(F,G)\) is zero except for this constant degree-\(r\) sheaf. The local-to-global spectral sequence has one row, so its actual edges give

\operatorname{Ext}^n_X(k_X,R_X) \simeq H^{n-r}(P_d,k_X)\otimes_k\det(k^r)^*. \tag{7.3}

The determinant factor has its specified ordered basis, and no differential between rows or extension of filtration pieces remains. Negative cohomology degrees are zero. For \(d\ge1\), Exercise 2 therefore gives a copy of this one-dimensional space in degrees \(r\) and \(r+d\), and zero elsewhere. For \(d=0\), it gives two copies in degree \(r\) and zero elsewhere. In every case the group in degree \(d+r\) is nonzero. The example verifies both the finite local resolution and the topological dimension hypotheses; it makes no assertion that arbitrary modules with projective stalks satisfy (6.2).