Local module categories and Morita equivalence

Written by GPT-6.1 Sol (OpenAI), reasoning effort Ultra, September 2026. Self-checked; no independent review. Public domain (CC0).

A sheaf of rings is one way to describe local linear algebra. It is not the only way. Different sheaves of rings can have equivalent categories of modules, compatibly on every smaller region. For example, a matrix algebra and its scalar ring describe the same module theory. This lesson identifies the object that implements such an equivalence: a bimodule with a tensor inverse. This will let us glue module categories even when the rings themselves do not glue.

We assume categories, adjunctions, sheaves on a small Grothendieck site, and tensor products of modules. For topological spaces, the module-sheaf foundations are explained in Sheaves of modules on a ringed space, Section 2, Theorem 2.1. For a general site, use The abelian category of sheaves of modules for the corresponding commutative foundations. The arguments below use sheafification and local sections, rather than points of the site. They allow noncommutative algebra sheaves.

Basic references are [D'Agnolo–Polesello], [Stacks], and [Schapira, Sheaves]. The tensor description of functors is the sheaf version of the Eilenberg–Watts argument; the equivalence theorem is Morita theory.

1. Keep track of both actions

Let \(\mathcal C\) be a small site. Write \(X\) for the terminal sheaf of its topos. Let \(\mathcal O\) be a sheaf of commutative unital rings, and let \(A,B\) be sheaves of associative unital \(\mathcal O\)-algebras. The image of \(\mathcal O\) is central in each algebra. All modules are left modules unless a right action is written explicitly. The same arguments apply over the sheaf of integers if no scalar ring is specified.

Write \(\mathsf{Mod}(A)(U)\) for the category of \(A|_U\)-module sheaves on the slice site \(\mathcal C/U\). These categories, with restriction and its canonical comparison isomorphisms, form the stack \(\mathsf{Mod}(A)\). A functor between such stacks includes coherent isomorphisms expressing its compatibility with restriction. An \(\mathcal O\)-linear functor acts linearly on morphism sheaves. Global modules are the compatible families of these slice modules: evaluate a family on each test object to recover its sheaf, and restrict a sheaf to obtain the inverse construction. Thus a stack equivalence also induces an equivalence on global modules. The notation \(F_X\) denotes this induced equivalence. This interpretation applies on every slice topos, even when its terminal sheaf is not represented by an object of the original site.

An \((A,B)\)-bimodule \(P\) has a left \(A\)-action and a commuting right \(B\)-action, with the same \(\mathcal O\)-action on both sides. Equivalently it is a left module over \(A\otimes_{\mathcal O}B^{\mathrm{op}}\). Its tensor functor has the direction

For an \(A\)-module \(M\), the sheaf \(\mathcal Hom_A(P,M)\) is a left \(B\)-module by

The ordinary tensor–Hom adjunction, on each slice, is

To verify it, a map on the left sends \(n\) to the map \(p\mapsto f(p\otimes n)\). Balancing gives \(q_{bn}(p)=q_n(pb)\), which is exactly \(B\)-linearity. Conversely evaluate such a family at \((p,n)\). These constructions commute with restriction and are inverse.

Set \(P^\vee=\mathcal Hom_A(P,A)\). It is a \((B,A)\)-bimodule, where \((qa)(p)=q(p)a\). The evaluation map has a fixed direction:

It is balanced because \(q(pb)=(bq)(p)\), and it respects both \(A\)-actions.

2. A local finite presentation produces a dual basis

An \(A\)-module \(P\) is flat if \(-\otimes_A P\), on right \(A\)-module sheaves, is exact. It is faithfully flat if that functor is also faithful. These conditions are required on every slice. An object is locally finitely presented if a covering gives presentations \(A^m\to A^n\to P\to0\) with finite \(m,n\), allowed to vary between covering pieces.

Lemma 2.1. If \(P\) is flat and locally finitely presented, then locally it is a direct summand of a finite free left \(A\)-module. Moreover, for every left \(A\)-module \(M\), the map

is an isomorphism of sheaves of abelian groups.

Proof. Work on a piece with finite presentation \(A^m\to A^n\to P\to0\). Applying \(\mathcal Hom_A(-,A)\) gives a left exact sequence of right modules. Tensoring it with the flat left module \(P\) preserves exactness. Compare it with the left exact sequence obtained by applying \(\mathcal Hom_A(-,P)\). For a finite free module \(L\), the map

is an isomorphism: both sides are a finite product of copies of \(P\), and evaluation identifies their coordinates. Taking kernels in the two sequences shows that

The identity endomorphism is therefore locally the image of a finite sum \(\sum_{i=1}^r q_i\otimes p_i\). A section of a sheaf tensor product has such a representation after a covering; further refine if necessary. We obtain

The maps \(P\to A^r\), \(p\mapsto(q_i(p))_i\), and \(A^r\to P\), \((a_i)\mapsto\sum_i a_i p_i\), have composite the identity on \(P\). This proves the direct-summand assertion.

For finite free \(P\), (2.1) is an isomorphism by the same coordinate calculation. Both sides preserve finite direct-summand decompositions in \(P\), so it is an isomorphism for a summand of \(A^r\). The conclusion is local and hence holds on the site. \(\square\)

No commutativity of \(A\) was used. In (2.2), each coefficient \(q_i(p)\) acts on the left. Moving it to the right would change the assertion.

3. The sheaf Morita theorem

Theorem 3.1. For an \((A,B)\)-bimodule \(P\), the following conditions are equivalent.

1. There is a \((B,A)\)-bimodule \(Q\) and bimodule isomorphisms \[ P\otimes_B Q\simeq A,\qquad Q\otimes_A P\simeq B. \]
2. The functor \(P\otimes_B-\) is an equivalence of \(\mathcal O\)-linear stacks of modules.
3. As a left \(A\)-module, \(P\) is faithfully flat and locally finitely presented, and right multiplication induces an isomorphism \[ B^{\mathrm{op}}\simeq\mathcal End_A(P). \]

Under these conditions, one may take \(Q=P^\vee\). Evaluation (1.2) is an isomorphism. The inverse equivalence is

Such a bimodule \(P\) is called invertible.

There is also a symmetric criterion: \(P\) is faithfully flat and locally finitely presented as a right \(B\)-module, and \(A\simeq\mathcal End_{B^{\mathrm{op}}}(P)\). Equivalently, one can require the left-module Hom functor in (3.1) to be an equivalence, the right-module tensor functor \(-\otimes_A P\) to be an equivalence, or its right adjoint \(\mathcal Hom_{B^{\mathrm{op}}}(P,-)\) to be an equivalence. These descriptions impose the same condition: an adjoint of an equivalence is its inverse, by uniqueness of adjoints.

Proof. Condition 1 implies condition 2 by associativity of tensor products. Tensoring first with \(P\), then with \(Q\), gives the identity functor up to the stated isomorphisms; reversing their order gives the other identity. This is valid on every slice and commutes with restriction.

Condition 2 also implies condition 1. Indeed, the tensor reconstruction argument in Section 4 uses only presentations, restriction adjunctions and equivalences; it does not use this theorem. Apply that argument to an inverse equivalence \(G\), and put \(Q=G(A)\). It gives \(G\simeq Q\otimes_A-\). Evaluating the two composite equivalences at the regular modules gives the isomorphisms in condition 1. Naturality with respect to right multiplication makes these isomorphisms bimodule maps.

Under conditions 1 and 2, the right adjoint in (1.1) is an inverse equivalence. Its counit is the evaluation isomorphism

Its unit, at \(B\), identifies \(B\) with \(\mathcal Hom_A(P,P)\), carrying \(b\) to right multiplication by \(b\). As a ring of endomorphisms this is \(B^{\mathrm{op}}\): the composite of right multiplications by \(b\) and \(c\), in that order of composition, is right multiplication by \(cb\).

The inverse \(Q\otimes_A-\) and this right adjoint are naturally isomorphic. Evaluating at \(A\), and using naturality for its right multiplications, identifies \(Q\) with \(P^\vee\) as a bimodule. Choose the resulting unit and counit of the adjunction. Locally the image of \(1_B\) under the unit \(B\to P^\vee\otimes_A P\) has a finite expression \(\sum_i q_i\otimes p_i\). The triangle identity at \(P\) says precisely

As in Lemma 2.1, this splits \(P\) off a finite free left \(A\)-module. Hence \(P\) is locally finite projective, flat and finitely presented. Formula (2.1) now gives \(\mathcal Hom_A(P,M)=P^\vee\otimes_A M\). The counit at \(A\) and unit at \(B\) are the required tensor-inverse maps. Finally, if a right \(A\)-module sheaf \(N\) satisfies \(N\otimes_A P=0\), then

Since \(-\otimes_A P\) is exact, it is faithful: a morphism sent to zero has image sent to zero, so its image is zero. This proves condition 3 and condition 1.

Now assume condition 3. Lemma 2.1 gives an isomorphism

It sends \(q\otimes p'\) to the unique \(b\) such that \(x b=q(x)p'\) for all local \(x\in P\). The opposite ring in the hypothesis ensures that \(\delta\) is a \((B,B)\)-bimodule map. For example, replacing \(q\) by \(b_0q\) replaces \(b\) by \(b_0b\); replacing \(p'\) by \(p'b_1\) replaces it by \(bb_1\).

Tensor evaluation \(\varepsilon\) on the right with \(P\). Its resulting map

is the isomorphism induced by \(\delta\), because both maps send \(p\otimes q\otimes p'\) to \(q(p)p'=p\delta(q\otimes p')\). Faithful flatness of the left \(A\)-module \(P\) reflects isomorphisms of right \(A\)-modules: exactness takes kernel and cokernel to those of the tensor map, and faithfulness detects their vanishing. Hence \(\varepsilon\) is an isomorphism. This proves condition 1.

Apply the result just proved to \(B^{\mathrm{op}},A^{\mathrm{op}}\) and the same underlying bimodule. It gives the symmetric criterion. Formula (3.1) follows from Lemma 2.1 and (1.1). \(\square\)

4. Recover the bimodule from an equivalence

We now show that Theorem 3.1 describes every linear equivalence of module stacks. The ability to restrict the equivalence is essential.

For an object \(U\) of the site, define \(A[U]\) by sheafifying

Yoneda and sheafification give

Indeed a module map is determined by the section at the summand labeled by \(\mathrm{id}_U\). For \(s\in M(U)\), send \(b\) in the summand labeled by \(a\) to \(b\,s|_a\). This gives its inverse. Taking a coproduct over all pairs \((U,s)\) gives an epimorphism onto any \(M\), since each local section is in its image. Applying the same construction to its kernel gives a presentation by such coproducts.

Restriction \(j_U^*\) to \(\mathcal C/U\) has a left adjoint \(j_{U!}\). On a module sheaf \(N\) of the slice it is the sheafification of

A map from this presheaf to \(M\) is precisely a map \(N\to j_U^*M\); this proves the adjunction after sheafification. In particular \(j_{U!}A|_U=A[U]\). Tensoring with a fixed bimodule commutes with this construction: tensor products distribute over the displayed sums and commute with sheafification.

Theorem 4.1. Let \(F:\mathsf{Mod}(B)\to\mathsf{Mod}(A)\) be an \(\mathcal O\)-linear equivalence of stacks. Then \(P=F(B)\) has a canonical \((A,B)\)-bimodule structure, and there is a natural isomorphism of stack functors

The bimodule \(P\) satisfies Theorem 3.1.

Proof. On a slice, \(b\in B(U)\) gives the left \(B\)-linear map \(r_b:B|_U\to B|_U\), \(x\mapsto xb\). Define \(pb=F(r_b)(p)\). Since \(r_c\circ r_b=r_{bc}\), this is a right action. It commutes with the left \(A\)-action, and \(\mathcal O\)-linearity identifies the two scalar actions. Restriction compatibility makes the actions into sheaf actions.

For \(m\in M(U)\), let \(\ell_m:B|_U\to M|_U\) send \(b\) to \(bm\). Define a map locally by

It is additive in both variables. It is balanced because \(\ell_{bm}=\ell_m\circ r_b\). It is natural in \(M\), compatible with restriction, and left \(A\)-linear. Thus it defines \(\theta_M:P\otimes_B M\to F(M)\).

Choose an inverse equivalence \(G\). The restriction isomorphisms for \(F,G\), together with the adjunction for \(j_{U!}\), give

For clarity, maps from the left side to \(L\) correspond successively to maps from \(j_{U!}N\) to \(G_XL\), from \(N\) to \(j_U^*G_XL\), from \(N\) to \(G_Uj_U^*L\), and from \(j_{U!}F_UN\) to \(L\). This proves the isomorphism and its naturality.

Taking \(N=B|_U\), both sides of \(\theta_{B[U]}\) identify with \(j_{U!}P|_U\). Under these identifications (4.3) is the identity: on the universal section labeled by \(\mathrm{id}_U\), it is \(F(\mathrm{id}_B)\). Hence it is an isomorphism on these generators and their coproducts. Both functors preserve coproducts and cokernels, the first by tensor adjunction and the second because it is an equivalence. A presentation following (4.1) therefore proves that \(\theta_M\) is an isomorphism for every \(M\). The same construction on slices proves (4.2) as a statement about stacks. Theorem 3.1 applies. \(\square\)

5. Matrices and line bundles

Let \(R\) be a commutative ring sheaf and \(A=M_n(R)\), with \(n\geq1\). Take \(P=R^n\) as column vectors, an \((A,R)\)-bimodule, and \(Q=(R^n)^*\) as row vectors. Column times row induces

The elementary columns and rows map to the matrix units, proving this isomorphism. Row times column induces \(Q\otimes_A P\simeq R\). In that tensor product, write every row and column using matrix units. Balancing reduces every tensor to a scalar multiple of \(e_1^*\otimes e_1\), and its pairing is that scalar. Thus the second map also is an isomorphism. Theorem 3.1 gives \(\mathsf{Mod}(M_n(R))\simeq\mathsf{Mod}(R)\). The rings need not be isomorphic.

For another example, let \(L\) be a locally free \(R\)-module of rank one. Then \(L^*=\mathcal Hom_R(L,R)\) is its tensor inverse. The functor \(L\otimes_R-\) is a linear autoequivalence. The line bundle is allowed to be globally nontrivial.

Corollary 5.1. On a locally ringed topological space with nonzero commutative stalk rings, every \(R\)-linear autoequivalence of \(\mathsf{Mod}(R)\) is tensoring with a line bundle, unique up to isomorphism.

Proof. By Theorem 4.1 it is tensoring with a bimodule \(P\). The two \(R\)-actions agree by linearity. Locally \(P\) is finite projective. A finite projective module over a commutative local ring is free: lift a basis of its reduction modulo the maximal ideal, use Nakayama to get a surjection from a finite free module, split it, and apply Nakayama to the finitely generated complementary summand. If its rank were at least two, its matrix endomorphism ring would be noncommutative. Rank zero is excluded by faithful flatness. Since \(\mathcal End_R(P)\simeq R\), its rank is one. Evaluation at \(R\) recovers \(P\), proving uniqueness. \(\square\)

To pass from a stalk basis to a neighborhood basis, use a finite splitting \(P\subset R^n\). Lift the finitely many entries of a basis map and its inverse at the stalk to a neighborhood. Their composites are the identities at the stalk; shrink so the finitely many matrix entries of these identities agree on that neighborhood. This gives a basis there. On general ringed sites, the corresponding local-freeness statement requires a precise covering condition; Stacks, Invertible modules on locally ringed topoi gives that formulation.

6. Exercises with solutions

Exercise 6.1 (foundation). For left \(B\)-modules, determine the ring structure of \(\operatorname{End}_B(B)\). Explain why \(B\), rather than \(B^{\mathrm{op}}\), would give the wrong action in Theorem 3.1.

Solution. A left-linear map has the form \(x\mapsto xb\), where \(b\) is its value at \(1\). The composite \(r_b\circ r_c\) has value \(cb\). Hence the endomorphism ring is \(B^{\mathrm{op}}\). On \(P\), a left \(B\)-action obtained from these maps would reverse the multiplication law. A right \(B\)-action has exactly the required law \((pb)c=p(bc)\).

Exercise 6.2 (calculation). For \(A=M_2(R)\), compute the module corresponding to an \(A\)-module \(M\). Construct both maps of the equivalence without choosing generators for \(M\).

Solution. Put \(e=E_{11}\). The inverse sends \(M\) to \(eM\). The forward map \(R^2\otimes_R eM\to M\) sends \(e_i\otimes m\) to \(E_{i1}m\). Its inverse sends \(m\) to \(e_1\otimes E_{11}m+e_2\otimes E_{12}m\). Each second factor lies in \(eM\). Their composite on \(M\) is \((E_{11}+E_{21}E_{12})m=(E_{11}+E_{22})m=m\). On \(e_i\otimes m\), the reverse composite uses \(E_{1j}E_{i1}=\delta_{ji}E_{11}\), so it is the identity as well. These formulas respect restriction.

Exercise 6.3 (hypotheses). Over a field \(k\), the module \(P=k^2\) is faithfully flat and finitely presented. Does \(P\otimes_k-\) define an autoequivalence of \(k\)-modules?

Solution. No. The scalar map \(k\to\operatorname{End}_k(k^2)=M_2(k)\) is not onto; for instance \(E_{12}\) is not a scalar matrix. Thus condition 3 fails. Directly, the induced map from \(\operatorname{End}_k(k)\) to \(\operatorname{End}_k(P)\) is not onto, so the tensor functor is not full. Faithful flatness alone does not imply Morita equivalence.

Exercise 6.4 (synthesis). Suppose \(P\) implements an equivalence from \(B\)-modules to \(A\)-modules and \(T\) one from \(C\)-modules to \(B\)-modules. Describe the bimodule and inverse for the composite.

Solution. The bimodule is \(P\otimes_B T\), with its left \(A\)- and right \(C\)-actions. If \(Q\) and \(S\) are inverses of \(P\) and \(T\), its inverse is \(S\otimes_B Q\). Associativity gives \[ (P\otimes_B T)\otimes_C(S\otimes_B Q) \simeq P\otimes_B B\otimes_B Q\simeq A. \] Reversing the order gives \(C\). Every map respects the outside actions. The composite equivalence follows from Theorem 3.1.

References

• [D'Agnolo–Polesello] Andrea D'Agnolo and Pietro Polesello, Stacks of twisted modules and integral transforms, in Geometric Aspects of Dwork Theory, Walter de Gruyter, 2004. Open preprint.
• [Stacks] The Stacks Project Authors, The Stacks Project: Modules on Sites, Stacks, and sheaves of modules on ringed spaces.
• [Schapira, Sheaves] Pierre Schapira, An Introduction to Sheaves on Grothendieck Topologies, lecture notes, version of 1 August 2026, Sections 2.5–2.6, for modules over sheaves of rings and ringed sites.