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Draft lesson.

Flat algebra epimorphisms and exact localization of module stacks

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

Inverting scalars can turn a module category into a full subcategory of itself. The same construction works for twisted module stacks, even when there is no global regular module. We construct the actual reflection, identify its exact quotient property and prove the bounded derived comparison. Flatness governs exactness; the epimorphism condition governs uniqueness of the added action.

1. The scalar multiplication test

All rings are unital, and zero rings are allowed. First let \(R\to S\) be a map of commutative rings. Write \[ m:S\otimes_R S\longrightarrow S,\qquad m(s\otimes t)=st. \tag{1.1} \]

Lemma 1.1. The ring map is an epimorphism in the category of commutative rings if and only if \(m\) is invertible. In that case \(s\otimes1=1\otimes s\) for every \(s\).

Proof. The two maps \(i_1,i_2:S\to S\otimes_R S\), inserting a unit in the other factor, agree on \(R\). If the given map is epic they agree everywhere. Hence \(s\otimes t=i_1(s)i_2(t)=i_1(st)\). Thus \(i_1m=1\), while \(mi_1=1\), proving invertibility.

Conversely, if \(m\) is invertible, both \(i_1,i_2\) are its inverse. For two ring maps \(f,g:S\to B\) agreeing on \(R\), the rule \(s\otimes t\mapsto f(s)g(t)\) is a ring map to the commutative ring \(B\). Its composites with \(i_1,i_2\) are \(f,g\), so they are equal. This is precisely epic cancellation. \(\square\)

For any \(S\)-module \(N\), the action map \[ S\otimes_R N\longrightarrow N \tag{1.2} \] has inverse \(n\mapsto1\otimes n\). To check the other composite, apply \(s\otimes t=st\otimes1\) to the balanced map \(S\otimes_R S\to S\otimes_R N\) sending \(s\otimes t\) to \(s\otimes tn\). It gives \(s\otimes n=1\otimes sn\). In particular the inverse in (1.2) is \(S\)-linear.

Restriction of scalars is therefore fully faithful. Indeed any \(R\)-linear map \(f:N\to N'\) respects these invertible units by naturality; conjugating \(S\otimes_R f\) by (1.2) shows \(f\) is \(S\)-linear. Conversely, full faithfulness makes the underlying \(R\)-linear unit \(S\to S\otimes_R S\) an \(S\)-linear map for the first-factor action. Evaluating its linearity at \(1\) gives \(1\otimes s=s\otimes1\), so Lemma 1.1 applies.

No flatness was used in these statements. We next retain them when module descriptions and their scalar actions must glue.

2. Scalar actions in a twisted module stack

Let \(X\) be a topological space, \(\mathcal R\) a commutative ring sheaf and \(\mathfrak M\) an \(\mathcal R\)-linear stack locally equivalent to \(\mathsf{Mod}(\mathcal A_i)\) on an open cover. Each \(\mathcal A_i\) is a unital algebra sheaf with central \(\mathcal R\)-action; it may be noncommutative. Stack morphisms and equivalences respect this action. Fix universes in which the cover, Hom sets and all diagrams used below are small.

Let \(\mathcal S\) be a commutative \(\mathcal R\)-algebra sheaf such that \[ m:\mathcal S\otimes_{\mathcal R}\mathcal S\xrightarrow{\sim}\mathcal S. \tag{2.1} \] Tensoring the underlying object with an \(\mathcal R\)-module sheaf is intrinsic. Locally, \(\mathcal S\otimes_{\mathcal R}K\) carries the action of \(\mathcal A_i\) on its \(K\)-factor. It represents the sheaf-valued functor \[ N\longmapsto \mathcal Hom_{\mathcal R} \bigl(\mathcal S,\mathcal Hom_{\mathfrak M}(K,N)\bigr). \tag{2.2} \] Currying is \(h\mapsto(s\mapsto(k\mapsto h(s\otimes k)))\). The balanced relation and centrality make it an \(\mathcal A_i\)-linear map precisely when each resulting map on \(K\) is \(\mathcal A_i\)-linear. Its inverse evaluates this formula on pure tensors. These rules commute with every restriction.

The universal-identification descent proof in Representing objects from local data, Sections 1 and 2 glues these local representations, with their maps. Thus no choice of a global regular module is needed. Tensor units, multiplication and associativity glue as well: their local maps represent the same natural transformations of (2.2), so uniqueness gives the comparisons and all their coherence equations. The explicit sheaf tensor and currying rules are also those of Bimodules and adjoints of module-stack functors, Section 2.

Define \(\mathfrak M_{\mathcal S}\) by equipping \(K\) with a unital associative action \(a:\mathcal S\otimes_{\mathcal R}K\to K\); its arrows respect this action. This is a stack. Underlying objects descend in \(\mathfrak M\); action maps descend as morphisms, and the unit and associativity equations are equalities checked on a cover. Morphisms likewise glue with their action equation. In the \(i\)-th chart it is exactly \(\mathsf{Mod}(\mathcal B_i)\), where \(\mathcal B_i=\mathcal A_i\otimes_{\mathcal R}\mathcal S\). Commuting \(\mathcal A_i\)- and \(\mathcal S\)-actions give the action \((a\otimes s)k=a(sk)\), and the two factor insertions recover them.

Write \[ \mathcal C=\mathfrak M(X),\quad \mathcal D=\mathfrak M_{\mathcal S}(X),\qquad F(K)=\mathcal S\otimes_{\mathcal R}K,\quad U:\mathcal D\to\mathcal C. \tag{2.3} \] In \(F(K)\) the \(\mathcal S\)-action is multiplication on its first factor. The adjunction \(F\dashv U\) has unit \(\eta_K(k)=1\otimes k\) and counit the specified action \(a\). The Hom rules send \(h:F(K)\to N\) to \(h\eta_K\), and \(f:K\to UN\) to \(s\otimes k\mapsto s f(k)\). They are inverse by the unit and action equations, natural, and commute with restriction; hence give one stack adjunction.

By (2.1), the monad \(UF\) has invertible multiplication. The calculation for an invertible-multiplication monad in Separable monads and the Karoubi completion of free modules, Exercise 8.1 proves that every action \(a\) is invertible, with inverse \(\eta_{UN}\). The same local calculation as (1.2) supplies this inverse here. An underlying map between two action objects preserves their units; composing with their inverse actions then proves its action equation. Thus \(U\) is fully faithful, and the counit \(FU\to1_{\mathcal D}\) is an isomorphism of stack functors.

3. Flatness makes the reflection exact

Now assume additionally that every \(\mathcal S_x\) is flat over \(\mathcal R_x\). Flat means that tensoring with it preserves short exact sequences. Tensor sheaves satisfy \[ (\mathcal S\otimes_{\mathcal R}M)_x \simeq\mathcal S_x\otimes_{\mathcal R_x}M_x. \tag{3.1} \] Indeed a finite sum of germ tensors has representatives on a common neighborhood, and any finite set of tensor relations holds on a smaller common neighborhood. These two facts construct the inverse of the displayed germ map and verify both composites. Sheafification changes no stalk: a locally represented section gives its original presheaf germ, and equality of the germs is witnessed after restriction. This proves (3.1) for the sheaf tensor construction.

In an \(\mathcal A_i\)-chart, restriction of an \(\mathcal B_i\)-module to \(\mathcal A_i\) preserves kernels and cokernels; both are computed on the same underlying sheaves with inherited action. Induction has underlying tensor \(\mathcal S\otimes_{\mathcal R}-\), which is exact by (3.1), flatness and the stalkwise exactness test. Exactness therefore holds globally for \(F,U\). The local-to-global abelian and exactness proof is Injective objects in locally module stacks, Section 1. Its Sections 2 and 5 also prove that both \(\mathcal C\) and \(\mathcal D\) have small colimits and enough injectives.

Put \(\mathcal K=\{K\in\mathcal C:F(K)=0\}\). This is a Serre subcategory: applying exact \(F\) to a short exact row shows that its middle object is killed exactly when both endpoints are killed. It is closed under small colimits, since \(F\), as a left adjoint, preserves colimits. For completeness, this follows by taking maps to any \(N\), applying the adjunction, then applying the colimit universal property in \(\mathcal C\); the resulting bijections identify the image cocone with a colimit cocone in \(\mathcal D\).

Proposition 3.1. For every \(K\), both \(\ker\eta_K\) and \(\operatorname{coker}\eta_K\) lie in \(\mathcal K\). An object is in the essential image of \(U\) exactly when \(\eta_K\) is invertible. There is at most one \(\mathcal S\)-action on any given object of that image.

Proof. The adjunction triangle gives \(\varepsilon_{FK}F\eta_K=1_{FK}\). Since \(\varepsilon\) is invertible, \(F\eta_K\) is its inverse. Exactness now kills the kernel and cokernel. If \(K=UN\), its unit is inverse to \(U\varepsilon_N\), hence invertible. Conversely an invertible unit identifies \(K\) with \(UFK\). Finally every possible action is inverse to this specified unit, proving uniqueness. \(\square\)

Call these image objects local. The unit need be neither monic nor epic in general; its two defects are exactly what the reflection kills.

4. The exact quotient property includes transformations

For an abelian category \(\mathcal E\), let \(\operatorname{Ex}(\mathcal C,\mathcal E)_{\mathcal K}\) consist of exact additive functors killing \(\mathcal K\), with all natural transformations between them.

Theorem 4.1. Precomposition with \(F\) is an equivalence \[ \operatorname{Ex}(\mathcal D,\mathcal E) \xrightarrow{\sim} \operatorname{Ex}(\mathcal C,\mathcal E)_{\mathcal K}. \tag{4.1} \] Consequently \(F\) realizes the exact Serre quotient of \(\mathcal C\) by \(\mathcal K\).

Proof. Its values kill \(\mathcal K\), and are exact because \(F\) is exact. Conversely, for an exact \(H\) killing \(\mathcal K\), put \(\overline H=HU\). This is exact because \(U\) is exact. Proposition 3.1 and exactness of \(H\) make \(H\eta:H\to HUF=\overline HF\) an isomorphism: its kernel and cokernel are the images of the two killed defects. This proves essential surjectivity.

For \(J,J':\mathcal D\to\mathcal E\) and \(\theta:JF\to J'F\), the unique recovered transformation has component \[ \overline\theta_N =J'(\varepsilon_N)\,\theta_{UN}\,J(\varepsilon_N)^{-1}. \tag{4.2} \] Naturality follows from that of \(\varepsilon,\theta\). At \(N=FK\), naturality of \(\theta\) on \(\eta_K\) and \(\varepsilon_{FK}F\eta_K=1\) show \(\overline\theta_{FK}=\theta_K\). Conversely any transformation on \(\mathcal D\) has (4.2) by naturality on \(\varepsilon_N\). This proves full faithfulness, including transformations. The construction respects identities and composition by cancellation of the intervening counit isomorphisms. Thus (4.1) has the claimed full universal property. \(\square\)

The quotient claim retains the functor \(F\) and its unit, rather than only an abstract equivalence of categories. No roof construction is needed to replace these explicit factorization maps.

5. Injectives and the bounded derived embedding

If \(I\) is injective in \(\mathcal D\), then \(UI\) is injective in \(\mathcal C\). Given a monomorphism \(K\to K'\) and a map \(K\to UI\), transpose it to \(FK\to I\). Exact \(F\) makes \(FK\to FK'\) monic, so injectivity extends the transposed map. Transpose back. This proves injective preservation on the actual global categories, not a stalkwise assertion of injectivity.

For bounded complexes, both exact functors act termwise and preserve quasi-isomorphisms. Their units and counits induce a derived adjunction, since they already obey both triangles as chain maps. A bounded complex \(N\) of \(\mathcal D\)-objects has a bounded-below injective replacement \(N\to I\). Restriction gives a bounded-below injective replacement \(UN\to UI\) in \(\mathcal C\). The existence and computing property of these replacements are proved in Derived Hom and bounded resolution pairs, Sections 2, 5 and 6.

For a bounded \(M\), full faithfulness of \(U\) identifies the complexes \[ \operatorname{Hom}_{\mathcal D}^{\bullet}(M,I) \xrightarrow{\sim} \operatorname{Hom}_{\mathcal C}^{\bullet}(UM,UI). \tag{5.1} \] In each degree the terms are the products of the termwise Hom groups; only finitely many factors occur because \(M\) is bounded. Restriction respects the differential \(d(f)=d_I f-(-1)^{|f|}f d_M\). The computing theorem therefore identifies (5.1) with the actual map induced by restriction on derived morphisms, naturally in both variables and shifts. It proves \[ D^b(\mathcal D)\longrightarrow D^b(\mathcal C) \quad\text{fully faithful}, \qquad \operatorname{Ext}_{\mathcal D}^{n}(M,N) \simeq\operatorname{Ext}_{\mathcal C}^{n}(UM,UN). \tag{5.2} \] The Ext formula here uses degree-zero objects and \(n\geq0\).

Proposition 5.1. The essential image consists exactly of bounded complexes whose cohomology objects are local.

Proof. Exactness of \(U\) makes the cohomology of \(UN\) local. Conversely, for a bounded complex \(K\), the termwise unit \(K\to UFK\) has cohomology maps \(\eta_{H^jK}\), since both functors are exact. If every cohomology object is local, all these maps are invertible by Proposition 3.1; hence this unit is a quasi-isomorphism. It identifies \(K\) with the restricted bounded complex \(FK\). \(\square\)

These are global bounded derived statements. They assert neither exactness of taking global sections nor a comparison for arbitrary unbounded complexes.

6. Orthogonality detects the local objects

Theorem 6.1. An object \(N\in\mathcal C\) is local if and only if \[ \operatorname{Hom}_{\mathcal C}(T,N)=0 =\operatorname{Ext}_{\mathcal C}^{1}(T,N) \quad\text{for every }T\in\mathcal K. \tag{6.1} \] For a local object all higher \(\operatorname{Ext}_{\mathcal C}^{n}(T,N)\) vanish as well.

Proof. If \(N=UN'\), choose an injective resolution of \(N'\) in \(\mathcal D\). Its restricted resolution computes Ext in \(\mathcal C\). In every degree the adjunction gives \(\operatorname{Hom}_{\mathcal C}(T,UI^j)=\operatorname{Hom}_{\mathcal D}(FT,I^j)=0\). Thus all the Ext groups, including Hom, vanish.

Conversely let (6.1) hold. The inclusion \(\ker\eta_N\to N\) is zero because its source is in \(\mathcal K\); being monic, it has zero source. Put \(C=\operatorname{coker}\eta_N\), also in \(\mathcal K\). The exact row \(0\to N\to UFN\to C\to0\) splits because its class in \(\operatorname{Ext}^1(C,N)\) vanishes. The exact class-to-splitting argument is Short exact extensions and the Baer group, Proposition 1.2. A section \(C\to UFN\) must be zero: its source is killed and its target is local, so the already proved forward implication gives zero Hom. Composing with the quotient then makes \(1_C=0\), whence \(C=0\). Both unit defects vanish, so \(\eta_N\) is invertible and \(N\) is local. \(\square\)

The degree-one condition is necessary even when there are no nonzero maps from killed objects. The nonflat example below also explains why an arbitrary ring epimorphism does not acquire (5.2).

7. Stalks, fractions and scalar base change

Condition (2.1) can be checked on stalks by (3.1) and Lemma 1.1. The same stalk test applies to the flatness hypothesis. These tests are made for the scalar sheaves, before choosing module charts; they consequently survive every change of chart.

For a global section \(f\) of \(\mathcal R\), the sheaf \(\mathcal S=\mathcal R[1/f]\) satisfies both hypotheses. On a stalk, its fractions have the relation \[ \frac{m}{f^a}=\frac{m'}{f^b} \quad\Longleftrightarrow\quad f^c(f^b m-f^a m')=0\text{ for some }c\geq0. \tag{7.1} \] This defines the localized module. The map \(\mathcal R_x[1/f]\otimes_{\mathcal R_x}M_x\to M_x[1/f]\) sends \((r/f^a)\otimes m\) to \(rm/f^a\); its inverse sends \(m/f^a\) to \((1/f^a)\otimes m\). Relation (7.1) verifies well-definedness, and the two rules are inverse on their generating elements.

Localization is exact: a zero fraction in the target of a localized injection is already killed by a power of \(f\) in its source; a kernel fraction has a power-of-\(f\) multiple represented in the original kernel; a quotient fraction lifts by lifting its numerator. These three tests prove preservation of every short exact row. Multiplication of two localized scalars is invertible as in Lemma 1.1, because both tensor factors have the same inverse to \(f\). Thus localization is a flat algebra epimorphism.

An object is killed precisely when, in each local module chart, every germ is annihilated by some power of the corresponding \(f_x\). An object is local precisely when multiplication by \(f\) is an isomorphism: fractions then act by the unique inverse powers, giving the action; conversely an action makes \(f\) invertible. Both statements follow from (7.1) on stalks and the sheaf exactness test. No uniform exponent on all points, or on all sections of an open set, is asserted.

More generally, for a commutative algebra sheaf \(\mathcal R'\), put \(\mathcal S'=\mathcal R'\otimes_{\mathcal R}\mathcal S\). It again has invertible multiplication: the two successive \(\mathcal S\)-factors contract by (2.1), leaving \(\mathcal R'\otimes\mathcal S\). It is flat over \(\mathcal R'\), because \(\mathcal S'\otimes_{\mathcal R'}M=\mathcal S\otimes_{\mathcal R}M\), and restriction of \(\mathcal R'\)-modules is exact. These are the actual associativity maps on pure tensors, so their comparisons commute with further scalar base change. This stability of the hypotheses makes no assertion that a nonflat change of scalars commutes with arbitrary derived Hom.

8. Four graded exercises with complete solutions

Exercise 8.1 — Arithmetic unit defects (intermediate)

Take \(R=\mathbb Z\), \(S=\mathbb Z[1/p]\) for a prime \(p\). Compute the unit on \(\mathbb Z\) and on \(\mathbb Z/n\), where \(n=p^a b>0\) and \(p\nmid b\). For \(b>1\), compute the self-Ext groups of \(\mathbb Z/b\) over both rings.

Solution. The unit on \(\mathbb Z\) is the inclusion into \(S\), with zero kernel and cokernel \(S/\mathbb Z\). Each fraction class is killed by some power of \(p\). This cokernel is nonzero, since \(1/p\) is not integral, so the unit need not be epic.

Since \(p\) is invertible in \(S\), the ideals \((n)\) and \((b)\) agree there. The quotient \(S/bS\) is \(\mathbb Z/b\): send \(r/p^j\) to \(r\) times the inverse of \(p^j\) modulo \(b\). Its inverse is induced by integers; both maps fix their generators, proving the isomorphism. Thus the unit on \(\mathbb Z/n\) is reduction modulo \(b\), with kernel the subgroup generated by \(b\), cyclic of order \(p^a\), and zero cokernel. This also covers \(b=1\), when the target is zero.

The free resolution \(0\to R\xrightarrow{b}R\to\mathbb Z/b\to0\) remains a free resolution over \(S\), using the quotient above and the injectivity of multiplication by \(b\) in these integral domains. Applying Hom into \(\mathbb Z/b\) gives a two-term complex with both terms \(\mathbb Z/b\) and zero differential. Over either ring, self-Ext is \(\mathbb Z/b\) in degrees zero and one, and zero in higher degrees. Restriction identifies the two Hom complexes by their values on \(1\), verifying the actual comparison in (5.2).

Exercise 8.2 — A flat quotient can forget a whole component (intermediate)

Let \(R=k\times k\) and let \(S=k\) by the first projection. Describe the reflection, its kernel and its quotient property, and compute its bounded derived essential image.

Solution. With \(e_1=(1,0),e_2=(0,1)\), every \(R\)-module is \(e_1M\oplus e_2M\), so is a pair \((M_1,M_2)\). Maps, kernels and cokernels are componentwise. As an \(R\)-module, \(S=Re_1\) is a direct summand of the free module \(R\), hence flat: tensoring with it is exactly projection onto the first component and is exact. The projection is surjective, hence epic, and directly \(S\otimes_R S\simeq S\).

The functors are \(F(M_1,M_2)=M_1\) and \(U(N)=(N,0)\). The unit \((M_1,M_2)\to(M_1,0)\) has kernel \((0,M_2)\); the counit is the identity. Killed objects are exactly the second-component pairs. Any exact functor killing them sends the split decomposition to its first-component value, and the inclusion/projection maps give the natural identification with its factor through \(F\), including transformations.

On bounded complexes the reflection is still first-component projection. Its essential image comprises exactly the pairs \((K_1,K_2)\) with acyclic \(K_2\), since they are quasi-isomorphic to \((K_1,0)\). This matches Proposition 5.1. The example shows that a flat epimorphism need not be faithfully flat or injective.

Exercise 8.3 — An epic nonflat scalar map loses derived morphisms (advanced)

For \(R=\mathbb Z\to S=\mathbb F_p\), prove that restriction is fully faithful on modules but fails to be fully faithful on bounded derived categories.

Solution. The map is surjective, hence epic, and \(S\otimes_R S\simeq S\). Every additive map of \(\mathbb F_p\)-vector spaces is \(\mathbb F_p\)-linear, since its scalars are residues of integers; this verifies ordinary full faithfulness directly.

Flatness fails: tensor the injection \(\mathbb Z\xrightarrow{p}\mathbb Z\) with \(S\). Its image is the zero map \(S\to S\), which is not monic. Thus induction is not exact and cannot be used as the exact functor of Sections 3–6.

Over \(\mathbb Z\), the free resolution \(0\to\mathbb Z\xrightarrow{p}\mathbb Z\to S\to0\) computes \(\operatorname{Ext}^1_{\mathbb Z}(S,S)=S\), because the induced Hom differential is zero. Over \(S\), the module \(S\) is free, so \(\operatorname{Ext}^1_S(S,S)=0\). Therefore restriction misses degree-one derived morphisms. Concretely \(0\to S\to\mathbb Z/p^2\to S\to0\), with inclusion \(1\mapsto p\), is nonsplit: a section would lift \(1\) to an element congruent to \(1\) modulo \(p\) killed by \(p\), and no such element exists. Its nonzero extension class cannot come from an \(S\)-linear extension, all of which split as vector spaces.

Exercise 8.4 — Stalkwise torsion has no global bound (advanced)

Let \(X=\{1,2,3,\ldots\}\) be discrete, let \(\mathcal R\) be the constant ring sheaf \(\mathbb Z\), and let \(\mathcal S=\mathcal R[1/p]\). Let \(M\) be the sheaf with stalk \(M_n=\mathbb Z/p^n\). Compare \(\Gamma(X,\mathcal S\otimes_{\mathcal R}M)\) with the ordinary \(\mathbb Z\)-module localization \(\mathbb Z[1/p]\otimes_{\mathbb Z}\Gamma(X,M)\).

Solution. A sheaf on a discrete space has sections over a subset equal to the product of its stalks: singleton sections glue uniquely, since their distinct overlaps are empty. Hence \(\Gamma(X,M)=\prod_{n\geq1}\mathbb Z/p^n\). Each stalk of the localized sheaf vanishes, since its elements are killed by \(p^n\), so the sheaf is zero and its global sections are zero.

In contrast, the element \(x=(1\bmod p^n)_n\) in that ordinary product is killed by no fixed power of \(p\): for a proposed exponent \(j\), its coordinate at \(n>j\) remains nonzero. By the zero-fraction criterion (7.1), \(x/1\) is nonzero in its ordinary localization. Thus the second displayed module is nonzero.

The comparison from the second module to the first is well-defined: global scalar fractions act on the localized sheaf and the unit gives the map on global sections. Here it sends every element to zero. This computes a failure of commuting scalar localization with global sections; it does not replace sheaf localization by tensoring over the ring of global scalar sections. All stalks are killed even though one global section has no uniform annihilating power.

Freely accessible references