Locally constant module sheaves and transition maps

Written by GPT-6.1 Sol (OpenAI), October 2026. Original text: CC0.

For interval cohomology and homotopy invariance, use the complete interval and cylinder proof. For the global path-groupoid classification, its precise topology assumptions and all representation intertwiners, use Path transport and monodromy. A locally constant module sheaf has constant models on neighborhoods. A map between two such models need not itself be locally constant as a family of module maps. Finite generation of its source gives the common neighborhood needed for that conclusion. It also makes frames into torsors for the constant automorphism group, with genuinely nonabelian transition rules.

Let \(X\) be any topological space and \(R\) any unital ring, possibly noncommutative. All modules are left modules. Use fixed universes and small open covers. No local connectedness, separation or path hypothesis is imposed. Internal Hom below is a sheaf of abelian groups; for noncommutative \(R\), no scalar-module structure on those Hom groups is asserted.

We use Coefficient morphism sheaves and local isomorphisms, Sections 2–3 for the full sheaf of local maps; Twisted modules from coherent gluing, Theorem 2.1 for effective module descent; and The inverse image of module sheaves for exact stalks and the criterion for an isomorphism. The explicit ordinary descent formula below fixes the transition convention. Torsors, first cohomology and line bundles treats abelian torsors and their derived first cohomology. The automorphism groups here may be nonabelian. Local systems on finite posets has its separate finite-space and word-groupoid hypotheses.

1. Constant sheaves and their actual fibres

For an \(R\)-module \(M\), define \(M_X(U)\) to be the locally constant functions \(U\to M\), with \(M\) discrete. Restriction restricts functions. Functions agreeing on an open cover glue uniquely, and their glue is locally constant on the same neighborhoods. Thus this is a sheaf. Pointwise multiplication makes \(R_X\) a ring sheaf and \(M_X\) an \(R_X\)-module sheaf. The empty-open value is the zero module of empty functions.

The constant-presheaf map sends \(m\) to the constant function with value \(m\). Every locally constant function is locally of this form. Two such representatives have equal germs precisely when their values agree at that point. This proves that \(M_X\) is the associated sheaf of that presheaf: a presheaf map into any sheaf extends uniquely by applying it on constant-value neighborhoods and gluing, with agreement and uniqueness checked on their overlaps.

At each point there is a canonical identification

A representative is constant near \(x\), so evaluation is well defined and injective on germs; a constant function represents every value, so it is surjective. It respects the action through \((R_X)_x=R\). A module map \(a:M\to N\) gives \(a_X\) by composition with functions. Equation (1.1) makes all its stalk maps equal to \(a\). The stalk exactness criterion therefore proves that \(M\mapsto M_X\) is exact. In particular its kernels and cokernels are the constant sheaves of the algebraic kernels and cokernels, with their actual maps. This does not identify the cokernel with a sectionwise quotient on every open.

A module sheaf \(L\) is locally constant if every point has a neighborhood \(U\) and an isomorphism \(L|_U\simeq M_U\) for some \(R\)-module \(M\). The model may vary between neighborhoods. The qualification locally of finitely generated type means that these models can be chosen finitely generated. A form of \(M_X\) has such a cover with that same specified module \(M\) on every member. For a field, a finite-rank local system is locally constant with finite-dimensional models.

2. Open descent with all module maps

Take an open cover \((U_i)\), module sheaves \(F_i\) on its members and isomorphisms as below. Write \(U_{ij}=U_i\cap U_j\).

Every equation is on the appropriate triple overlap. Repeated indices give \(g_{ii}=1\) and \(g_{ji}=g_{ij}^{-1}\): an invertible idempotent is identity, and the repeated triple then gives both inverse equations. For an open \(V\), put \(V_i=V\cap U_i\) and \(V_{ij}=V\cap U_{ij}\). The descended sheaf is

Restrictions, addition and the \(R_X\)-action act componentwise. Compatible families on a cover of \(V\) glue in every \(F_i\); the overlap equations hold for the glued sections because they are local equalities. This proves the sheaf and module axioms. On \(V\subseteq U_i\), projection to \(s_i\) is an isomorphism with \(F_i(V)\). Its inverse sends \(s_i\) to the components \(g_{ji}(s_i|_{V\cap U_j})\). The cocycle law checks all equations, and the same equation shows that projection followed by this inverse recovers every family. These identifications are natural in \(V\).

For a second datum \((F'_i,g'_{ij})\), local module maps \(h_i:F_i\to F'_i\) descend exactly when

Under this condition, apply \(h_i\) to every component of (2.2). Conversely any global map has those local maps, and the global map is forced by them. These constructions preserve identity and composition. Restriction followed by (2.2) reconstructs an existing sheaf: its local sections agree under its restriction comparisons and glue uniquely. Conversely the projection isomorphisms just proved reconstruct the prescribed descent datum. Thus restriction is an equivalence with the category of these data, including every noninvertible morphism (2.3). Uniqueness of a descended object means uniqueness up to an isomorphism compatible with its given local identifications.

3. The finite-generation Hom comparison

Write \(\mathcal H(F,G)(U)=\operatorname{Hom}_{R_U}(F|_U,G|_U)\). A compatible family of local maps glues by applying it to each source section on smaller opens and gluing the target values. This is the morphism-sheaf construction in the prerequisite; it includes all further restrictions, and gives a sheaf of abelian groups. There is a natural map

For a locally constant family \(a_x\) of module maps, its value on a locally constant section \(s\) is the function \(x\mapsto a_x(s(x))\). Both factors are constant on a sufficiently small common neighborhood of each point, so this is a section of \(N_X\). The rule is linear and commutes with restrictions. It defines (3.1) and respects addition, identities and composition wherever defined.

Theorem 3.1. The map (3.1) is always a monomorphism. If \(M\) is finitely generated, it is an isomorphism for every \(N\).

Proof. A germ of its source is a single module map \(a:M\to N\), by (1.1). The induced map on the fibre at that point is exactly \(a\). A source germ mapping to zero must therefore be zero. This proves stalkwise injectivity and hence injectivity of sheaves.

Suppose \(m_1,\ldots,m_t\) generate \(M\), and let \(h\) be a sheaf map between the constant models on an open \(U\). Fix \(x\in U\). Each image \(h(m_j)\), where \(m_j\) is the constant section, is constant on a neighborhood of \(x\). Intersect these finitely many neighborhoods to obtain \(W\). Let \(a=h_x:M\to N\) be the map on fibres. On \(W\), the images of every generator agree with the constant sections \(a(m_j)\). Every element of \(M\) is a finite linear combination of those generators, so its constant section has the prescribed image \(a(m)\). Any locally constant section of \(M_W\) is locally one of these constant sections. Restriction and the sheaf axiom therefore give \(h|_W=a_W\) on all sections. This proves local surjectivity of (3.1), hence the isomorphism by its already-proved injectivity. For \(M=0\), the generating list can be empty and both Hom sheaves are zero. No finite presentation or condition on \(N\) was used. \(\square\)

Consequently, if \(F\) is locally constant of finitely generated type and \(G\) is locally constant, then \(\mathcal H(F,G)\) is locally constant. Choose simultaneous models near a point and apply Theorem 3.1 there. For a fixed finitely generated \(M\), composition in (3.1) identifies the endomorphism sheaf as a ring sheaf and its units as a group sheaf:

Indeed a local endomorphism is represented near each point by one actual endomorphism of \(M\). It is an isomorphism there exactly when that endomorphism is invertible; an inverse induces the inverse sheaf map. The same argument describes local isomorphisms between two different constant models with finitely generated source. Local constancy of these families does not mean that a whole disconnected overlap carries only one map.

4. Transition functions, gauge changes and refinements

Fix a finitely generated module \(M\), and put \(G=\operatorname{Aut}_R(M)\), with multiplication given by composition. Trivializations \(t_i:L|_{U_i}\to M_{U_i}\) for a form of \(M_X\) give \(g_{ij}=t_i t_j^{-1}\). Equation (3.2) makes this a locally constant \(G\)-valued function on \(U_{ij}\). Its order is fixed by its domain: it converts chart \(j\) coordinates to chart \(i\) coordinates. It satisfies (2.1).

Conversely any such \(G_X\)-valued cocycle glues a form \(L_g\) by (2.2), using the constant modules. Its chart projection is \(t_i\), and its transition is exactly \(g_{ij}\), by the inverse projection formula. Starting with \(L\), the map \(s\mapsto(t_i s)_i\) is an isomorphism with \(L_g\); its inverse glues the sections \(t_i^{-1}s_i\). These are actual inverse maps on every open.

A module map between two forms is a family of local maps satisfying (2.3); Theorem 3.1 expresses these as locally constant families of module maps. An isomorphism is precisely one whose local maps are invertible. Replacing coordinates by \(t'_i=a_i t_i\), with \(a_i\in G_X(U_i)\), gives

Conversely (4.1) makes \((s_i)\mapsto(a_i s_i)\) an isomorphism \(L_g\to L_{g'}\), with inverse components \(a_i^{-1}\). This proves both the gauge law and its complete isomorphism criterion on a fixed trivializing cover.

For a refining cover \((V_a)\), with \(V_a\subseteq U_{r(a)}\), the new transition is \(g_{r(a),r(b)}|_{V_a\cap V_b}\). Restricting the chart identifications supplies a canonical isomorphism between its glued object and \(L_g\). Identity and composite refinements give the same maps because they agree on every restricted chart. If \(r,s\) are two refinement choices, the gauge function on \(V_a\) is \(a_a=g_{s(a),r(a)}\). Its gauge equation is the product of three cocycle equations, explicitly

Thus two choices give isomorphic data with the actual coordinate comparison. Two covers have the common intersection refinement. Any isomorphism of their descended forms restricts there to invertible local coordinate maps, so yields a gauge relation. Conversely such a gauge relation descends to an isomorphism. Cocycles modulo gauge and common refinement therefore classify exactly the isomorphism classes of forms of \(M_X\). This is a pointed set with the constant form as neutral element; no abelian group or derived first-cohomology interpretation is asserted for a general \(G\).

5. Frames and nonabelian torsors

A right \(G_X\)-torsor is a sheaf of sets \(P\) with a right action, local sections and a simply transitive action locally: after choosing a section \(p\), the map \(g\mapsto pg\) is an isomorphism from \(G_X\) to \(P\). Equivalently the action-difference map \(P\times G_X\to P\times P\) is an isomorphism and sections exist locally. One implication uses these coordinate maps; in the other, that difference map gives the unique local coefficient relative to a chosen section. Those coefficients glue uniquely.

For a form \(L\), its frame sheaf is \(\operatorname{Fr}_M(L)=\operatorname{Isom}(M_X,L)\), with right action \(p\cdot g=p\circ g\). Maps and their inverses glue by the morphism-sheaf property, so this is a sheaf. Local trivializations supply frames. Relative to any frame, (3.2) identifies every other frame uniquely with \(pg\). Hence it is a \(G_X\)-torsor. This use of (3.2) is the finite-generation step.

Conversely take a torsor and local sections \(p_i\). On an overlap write

The coefficients are sections of \(G_X\), and associativity and uniqueness give \(g_{ij}g_{jk}=g_{ik}\). Glue constant \(M\)'s with this cocycle. The map from the torsor to its frame sheaf sends \(p_i g\) to \(t_i^{-1}\circ g\). On overlaps, \(p_j h=p_i g_{ij}h\), while \(t_j^{-1}=t_i^{-1}g_{ij}\); therefore the rule agrees there. It is locally bijective, equivariant and thus an isomorphism of sheaves. Starting with frames of \(L\), the same coordinates reconstruct \(L\) by Section 4. Changing the local origins or cover changes the coordinates by the gauge and refinement isomorphisms already proved.

Equivalently the reconstructed module is the associated sheaf of the sectionwise quotient of \(P\times M_X\) by \((pg,m)\sim(p,gm)\). On a chart with origin \(p_i\), the rule \([p_i g,m]\mapsto gm\) is well defined and bijective, with inverse \(m\mapsto[p_i,m]\); these identifications supply its local module operations and identify its transition with \(g_{ij}\). Sheafification glues them. For a frame torsor the map \([p,m]\mapsto p(m)\) identifies this module with \(L\), with both inverses checked in these charts.

An equivariant torsor map is an isomorphism: in local origins it is a translation, with its inverse translation. It induces the corresponding module isomorphism on the associated sheaves. Conversely a module isomorphism acts on frames by composition. These operations are inverse, as is visible in every frame. Thus the groupoid of forms with module isomorphisms is equivalent to the groupoid of right \(G_X\)-torsors. Noninvertible module maps remain present in (2.3); they are not asserted to give equivariant maps of frame torsors.

6. Finite-dimensional local systems form an exact category of sheaves

Let \(R=k\) be a field. Use the ambient abelian module-sheaf category proved in Injective objects in locally module stacks, Section 1. Kernels and cokernels of a morphism between finite-rank local systems are finite-rank local systems. To prove this at a point, choose constant finite-dimensional models for its source and target on a common neighborhood. Theorem 3.1 makes the morphism one fixed linear map on a smaller neighborhood. Exactness of the constant-sheaf functor then identifies its sheaf kernel and cokernel with the constant sheaves of that map's finite-dimensional kernel and cokernel. Finite sums are locally constant by taking a common finite intersection of trivializing neighborhoods. The full subcategory consequently has the inherited kernels, cokernels and finite biproducts. Its image and coimage coincide because they do in the ambient abelian sheaf category and all these constructions stay in the subcategory. It is therefore abelian, and its inclusion is exact.

It is also closed under extensions in all \(k_X\)-module sheaves. Given \(0\to L'\xrightarrow\iota E\xrightarrow\pi L''\to0\), trivialize \(L',L''\) near a point, with a finite basis for \(L''\). A sheaf epimorphism lifts each of these finitely many basis sections on a sufficiently small neighborhood of the point. Intersect those neighborhoods. Extending their lifts linearly defines a sheaf map \(\sigma:L''\to E\) there; constant basis expansions locally define it on all locally constant coefficient sections. The equation \(\pi\sigma=1\) holds on the basis and hence everywhere. The mutually inverse splitting maps are

The inverse \(\iota^{-1}\) in (6.1) is the inverse from the kernel subsheaf to \(L'\); its argument lies in that kernel. Substitution verifies both inverse identities. Thus \(E\) is constant of the sum rank on this neighborhood. The splitting is local; it need not extend over \(X\).

7. Four exercises with complete solutions

Exercise 1 — circle gluings over a ring with torsion

Introductory. Let \(X=\mathbb R/\mathbb Z\), \(U=X\setminus\{0\}\) and \(V=X\setminus\{1/2\}\). Their overlap has two interval components \(W_+,W_-\). For a finitely generated module \(M\) and \(T\in\operatorname{Aut}_R(M)\), glue the constant models with \(g_{UV}=1\) on \(W_+\) and \(g_{UV}=T\) on \(W_-\). Compute global sections. Compare two such gluings, and evaluate the example \(R=\mathbb Z\), \(M=\mathbb Z/6\), \(T=-1\).

Solution. Take the inverse reverse transition and identity self-transitions. With two charts every cocycle equation reduces to an identity or an inverse equation, so the data do glue. The two arcs are homeomorphic to open intervals and are connected. A locally constant function on a connected space is constant, because a nonempty fibre and its complement would otherwise separate it into open sets. For connectedness of an interval, suppose a nonempty proper subset is both open and closed. After reversing the order if needed, take a point a in it and a larger point b outside it. Its intersection with [a,b] has a supremum c belonging to that closed subset. Openness of the complement at b gives c<b, while openness of the subset at c supplies a larger point still in that subset and in [a,b], a contradiction. Thus sections on the two charts are values \(a,b\in M\). Their equations are \(a=b\) on \(W_+\) and \(a=Tb\) on \(W_-\), so

For a second gluing \(L_S\) with finitely generated model \(N\), Theorem 3.1 makes a local sheaf map on each connected chart a single module map \(h_U,h_V:M\to N\). Equation (2.3) on \(W_+\) says they are the same map \(h\), and on \(W_-\) it says \(hT=Sh\). Conversely these two equalities construct the descended map, so the Hom set is exactly the set of those intertwiners. An isomorphism is exactly an invertible intertwiner. For the same model this means \(S=hTh^{-1}\). This compares the explicitly constructed family; it does not invoke or prove a classification of all local systems by the topological fundamental group.

In the specified integer-module example, (7.1) consists of \(0,3\), hence is \(\mathbb Z/2\). The gluing is not the constant form: its automorphism group is abelian, so \(-1\ne1\) cannot be conjugate to \(1\). Thus a nonconstant form of a cyclic module can have nonzero global sections; multiplication by \(T-1\) need not be injective over these coefficients.

Exercise 2 — all endomorphisms and a nonsplit extension

Intermediate. Use the same circle and \(k=\mathbb F_2\). Take \(M=k^2\) and \(T=\begin{pmatrix}1&1\\0&1\end{pmatrix}\). Compute every endomorphism, the automorphism group and global sections of \(L_T\). Show that it is an extension of the constant rank-one sheaf by itself that does not split globally.

Solution. Write \(h=\begin{pmatrix}a&b\\c&d\end{pmatrix}\). Direct multiplication in \(hT=Th\) gives \(c=0\) and \(d=a\), with no condition on \(b\). Therefore the four endomorphisms are

Exactly the two with \(a=1\) are invertible. The other two are zero and the nonzero square-zero matrix with \(b=1\). Thus all maps, not just frame changes, matter. The fixed subspace of \(T\) is \(k e_1\), so (7.1) gives \(\Gamma(X,L_T)=k\).

The subspace \(k e_1\) and quotient by it both have identity transition, giving an exact sequence \(0\to k_X\to L_T\to k_X\to0\) in sheaves. A splitting would, by Exercise 1, be an intertwiner from the identity representation on \(k\) to \(T\), hence a column fixed by \(T\). Such a column has second coordinate zero. The quotient projection is precisely that second coordinate, so its composite cannot be identity. The sequence is not split globally. This is compatible with the local splittings of Section 6.

Exercise 3 — infinitely many generators need infinitely many neighborhoods

Advanced boundary case. Give \(X=\{\infty\}\cup\mathbb N_{>0}\) the topology in which every integer is isolated and neighborhoods of \(\infty\) contain all but finitely many integers. Let \(k\) be a field and \(M=\bigoplus_{n\ge1}k e_n\). Show that (3.1) for \(N=k\) is not surjective on the stalk at \(\infty\), and that its target Hom sheaf is not locally constant there. Explain the related failure for frames of an infinitely generated model.

Solution. At an even integer \(n\), use the coordinate map \(h_n:M\to k\), \(\sum a_j e_j\mapsto a_n\); at odd integers and at \(\infty\), use zero. For each fixed \(m\in M\), its image function \(x\mapsto h_x(m)\) has finite support, so is locally constant, including at \(\infty\). For a locally constant input section \(s\), the same conclusion holds locally: near \(\infty\), \(s\) is one fixed finite-support vector, and at an integer there is a singleton neighborhood. The pointwise rule therefore gives an actual sheaf morphism \(h:M_X\to k_X\), linear and compatible with every restriction.

Its map on the fibre at \(\infty\) is zero, but its germ as a local sheaf map is nonzero. Every neighborhood of \(\infty\) contains an even \(n\), where the constant section \(e_n\) has nonzero image on the singleton. A germ from the source of (3.1) would be represented by one constant map \(a:M\to k\); its fibre map at \(\infty\) is \(a\). To give the germ of \(h\), it would have to be zero, contradicting that nonzero germ. This proves failure of stalk surjectivity.

For any section of a locally constant sheaf of abelian groups, its nonzero-germ locus is open: in a constant chart the section is a locally constant function, and nonzero values give an open subset. The section \(h\) of the Hom sheaf has nonzero germ at \(\infty\) and zero germ at every odd integer. Its nonzero-germ locus contains no neighborhood of \(\infty\). The same is true after restriction to any such neighborhood, so the Hom sheaf cannot be locally constant there.

For frames, enlarge the model to \(M'=k e_0\oplus\bigoplus_{n\ge1}k e_n\). At even \(n\), let \(E_n(e_n)=e_0\), with all other basis vectors sent to zero. Set \(A_n=1+E_n\); at odd integers and at \(\infty\), set \(A_x=1\). Both this family and its pointwise inverse \(1-E_n\) act locally constantly on every fixed finite-support input, by the same argument. They define inverse automorphisms of \(M'_X\). Their family is not locally constant as a function to the discrete group \(\operatorname{Aut}_k(M')\): every neighborhood of \(\infty\) includes a nonidentity \(A_n\). It cannot come from that constant group sheaf, since its fibre at \(\infty\) would force the representing constant automorphism to be identity. Thus the finite-generation hypothesis in (3.2) and its frame-torsor identification cannot simply be dropped.

Exercise 4 — finitely generated models need not have finitely generated kernels

Advanced coefficient hypothesis. On a one-point space take \(R=k[x_1,x_2,\ldots]\), and let \(I=(x_1,x_2,\ldots)\). Both \(R\) and \(R/I\) are cyclic modules. Is the kernel of their quotient map finitely generated? What does this say about replacing the field in Section 6 by an arbitrary ring?

Solution. The kernel is \(I\). Suppose it had finitely many polynomial generators. Only finitely many variables occur in those polynomials. Choose \(x_j\) outside that finite set, and apply the ring map setting all variables in that set to zero while retaining \(x_j\) and the remaining variables. Every supposed generator maps to zero, since it has zero constant term and uses only the variables being set to zero. The entire generated ideal would therefore map to zero. But \(x_j\in I\) maps to the nonzero polynomial \(x_j\), a contradiction.

All sheaves on the point are constant. Thus this is a map between locally constant sheaves with finitely generated models whose kernel is locally constant but has no finitely generated model. The Hom comparison of Section 3 remains valid for the two cyclic modules. What fails is closure of the finitely generated coefficient class under kernels, which finite-dimensional vector spaces do satisfy. The general-ring Hom theorem alone does not supply that closure.

8. Further reading

Pierre Schapira's freely accessible An Introduction to Sheaves on Grothendieck Topologies, October 2026, Section 1.9, introduces locally constant sheaves and their construction by gluing. The precise finite-generation and coefficient boundaries used here are proved above. For finite-poset tensor duals, compactness, projectivity and the word-groupoid description, use the finite-poset lesson linked at the beginning. Topological path transport and homotopy invariance require their own statements and hypotheses.