Draft lesson.

Filtered colimits and sheaf cohomology on quasi-compact opens

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

A section of an associated sheaf has representatives near each point. A filtered colimit lets finitely many representatives and equalities meet at one stage. Quasi-compactness makes this useful globally: it reduces the local data to finitely many choices. Higher cohomology requires one further step, an acyclicity argument on all the relevant opens. Neither an infinite product of sections nor preservation of injectives can be assumed in that step.

Let \((X,\mathcal O)\) be a ringed space, with unital, possibly noncommutative rings, and use left module sheaves. Fix universes and choice. Every filtered category below is small and nonempty. Our topological hypothesis is that \(X\) has a basis \(\mathcal B\) of quasi-compact opens stable under finite intersections; we allow the empty open. We do not assume that \(X\) itself is quasi-compact. All coefficient colimits are colimits of module sheaves.

Godement resolutions and flabby sheaf cohomology, Sections 1–9, supplies an exact functorial Godement resolution and all-open flabby acyclicity. Cech complexes, refinements and acyclic covers, Sections 1–7, supplies the actual comparison edges and finite spectral filtrations. Universal cohomology and acyclic resolutions, Section 4, supplies the canonical bounded-below acyclic-model maps. For direct images use Higher direct images and the Leray spectral sequence, Sections 1–3. The exercises also use Finite-poset sheaves and canonical homological resolutions and Sheaf Ext and the local-to-global spectral sequence, Exercise 1.

1. The finite amount of topology that is needed

An open \(U\) is quasi-compact when every open cover of \(U\) has a finite subcover. Finite unions of quasi-compact opens are quasi-compact: restrict a cover to each member and combine its finite subcovers. Under our basis hypothesis, every quasi-compact open is a finite union of basis members. Distributing two such finite unions and using the basis intersection property shows that intersections of quasi-compact opens are again quasi-compact. Thus all finite intersections in a finite cover by quasi-compact opens stay within the hypothesis.

A space is noetherian if its increasing chains of opens stabilize; equivalently every open is quasi-compact. For completeness, if a cover of an open has no finite subcover, successively choose a member enlarging the union of the previously chosen members. This constructs a strictly increasing chain. Conversely, an increasing chain covers its union; a finite subcover makes one member the entire union, so the chain stabilizes. The same argument handles any directed union of opens. In particular a noetherian space satisfies our basis hypothesis with all opens as its basis. No noetherianness of the coefficient rings has been imposed.

2. Sections of a filtered colimit

For a filtered diagram \((F_i)_{i\in I}\), put \(F=\mathop{\mathrm{colim}}_i F_i\). Its stalk at \(x\) is \(\mathop{\mathrm{colim}}_i(F_i)_x\): both operations are colimits and the associated-sheaf map preserves stalks. Filtered colimits of modules are exact, since a finite collection of representatives and equalities can be put at one stage. Consequently filtered colimits of module sheaves are exact, by the stalk criterion.

For every quasi-compact open \(U\), the comparison induced by the colimit legs is an isomorphism

First check injectivity, also for any other quasi-compact open. A representative section whose image is zero has, at each point, a zero germ at some later stage on some neighborhood. Shrink these neighborhoods inside the open and take a finite subcover. Filteredness puts their finitely many arrows at one stage with the same composite from the original object. At that stage the section vanishes on the cover and hence vanishes. Equality of two arbitrary representatives reduces to this assertion after taking their difference at a common stage.

For surjectivity, represent a given section of \(F(U)\) locally by sections of some \(F_i\), using the local description of associated sheaves. Refine to basis members and take a finite subcover \(U_1,\ldots,U_m\). Place the finitely many representatives at a common stage. On each \(U_a\cap U_b\) their difference has zero image in \(F\). These intersections are quasi-compact, so the injectivity argument kills each difference at a later stage. There are only finitely many pairs. A further common stage makes all equalities hold, and the representatives then glue in that one sheaf. This proves surjectivity and both inverse operations in (2.1).

For a general filtered category, each occurrence of a common stage means a cocone on the finite diagram of arrows already chosen. Parallel arrows are equalized by filteredness. Thus the proof applies to arbitrary filtered categories and does not require monic transition maps or an index which is a linearly ordered set. All operations respect restriction to quasi-compact subopens, diagram morphisms, coefficient scalars and the original colimit legs. The empty-open assertion is the zero-module identity.

3. A complete Cech acyclicity criterion

Suppose a sheaf \(E\) has the following property: for every quasi-compact open \(U\) and every finite cover \(\mathcal U\) of \(U\) by quasi-compact opens,

Then \(H^q(U,E)=0\) for every such \(U\) and every \(q>0\). Here Cech cochains use all ordered tuples, including repetitions, and the differential and augmentation are exactly those of the Cech prerequisite.

We prove this simultaneously for all quasi-compact opens by induction on \(q>0\). A positive-degree cohomology class is locally zero. Indeed represent it by a section cycle in an injective resolution. At each stalk the positive-degree augmented resolution is exact, so the cycle is a boundary on some neighborhood. Restriction of injectives to that neighborhood remains injective, making the restricted cohomology class zero. Refine these neighborhoods to the basis and choose a finite cover \(\mathcal U\) of \(U\) on which the original class \(\alpha\in H^q(U,E)\) restricts to zero.

The actual Cech spectral sequence for this cover has

Every intersection is quasi-compact. The induction hypothesis kills the rows \(0<r<q\). In total degree \(q\), all positions with \(p>0\) therefore vanish at the stable page: those with \(p<q\) vanish already at page one, and \((q,0)\) vanishes at page two by (3.1). There are no positions with \(p>q\). Finite convergence now gives \(F^1H^q(U,E)=0\). The edge into \(E_1^{0,q}\) is the collection of restriction maps, because the augmentation restricts the actual resolution cycle. Its kernel is \(F^1\): there is no incoming differential into column zero, so the stable column-zero term embeds in its page-one term. Our chosen class is in that kernel and is zero. For \(q=1\) the same argument has no intermediate rows. This proves the induction and the criterion in all degrees, with no equality between an arbitrary single-cover Cech group and sheaf cohomology assumed.

4. Filtered colimits of flabby sheaves are acyclic on these opens

Let all \(E_i\) be flabby, and put \(E=\mathop{\mathrm{colim}}_i E_i\). Fix a finite quasi-compact cover \(\mathcal U\) of a quasi-compact open. In degree \(p\), its cochain product has only \(m^{p+1}\) factors, each on a quasi-compact intersection. Formula (2.1) and commutation of filtered colimits with finite products give an isomorphism of actual complexes

The maps are induced by the colimit legs and commute with every restriction in the alternating differential. Each \(E_i\) is acyclic on every open by flabby acyclicity. The acyclic-cover theorem therefore identifies its positive Cech groups with its positive cohomology on the union, which is zero. Exact filtered colimits commute with cohomology of complexes: apply exactness to the cycle, boundary and cohomology short exact sequences. Equation (4.1) proves (3.1) for \(E\). The criterion just proved makes \(E\) acyclic on every quasi-compact open.

This assertion does not say that \(E\) is flabby. On a noetherian space it is flabby as well: every open is quasi-compact, so (2.1) identifies each restriction of \(E\) with the colimit of the surjective restrictions of the \(E_i\). Lifting a representative proves surjectivity. In either case acyclicity is sufficient for the next argument. Injectivity of the colimit is never required.

5. All cohomology degrees and the actual comparison maps

For every quasi-compact open \(U\) and every \(q\ge0\),

Take the canonical functorial Godement resolutions \(F_i\to G^\bullet(F_i)\) from the first prerequisite. Their terms are flabby, their maps are functorial in every diagram arrow, and their augmentations are exact on stalks. Exact filtered colimits give an augmented resolution

Section 4 makes every \(E^r\) acyclic for sections on \(U\). The complete bounded-below acyclic-model theorem computes \(H^q(U,F)\) by \(H^q\Gamma(U,E^\bullet)\). Term by term (2.1) identifies this complex with \(\mathop{\mathrm{colim}}_i\Gamma(U,G^\bullet(F_i))\). Exact filtered colimits of modules commute with its cohomology, giving (5.1).

The map from each original resolution into (5.2) extends the colimit leg \(F_i\to F\). Naturality of the canonical acyclic-model comparison therefore identifies the resulting map with \(H^q(U,F_i)\to H^q(U,F)\) induced by that leg. Thus the isomorphism is precisely the displayed comparison, including the degree-zero map (2.1). It commutes with every morphism of filtered diagrams, restriction between quasi-compact opens, and the scalar action of \(\mathcal O(U)\). Choice comparisons are the derived comparison maps of the prerequisite. A final reindexing gives the same map: its colimit identification commutes with all legs, which characterize (5.1).

6. Higher direct images and arbitrary direct sums

Let \(f:(X,\mathcal O_X)\to(Y,\mathcal O_Y)\) be a ringed-space map. Assume the basis hypothesis on \(X\), and assume that \(Y\) has a basis \(\mathcal V\) for which every \(f^{-1}V\), \(V\in\mathcal V\), is quasi-compact. The basis \(\mathcal V\) itself need not consist of quasi-compact opens. Then the canonical maps are isomorphisms of module sheaves

To prove this with its maps, let \(P_i(V)=H^q(f^{-1}V,F_i)\). The direct-image prerequisite identifies its associated sheaf with \(R^qf_*F_i\), naturally in all restrictions and coefficients. Associated-sheaf formation is a left adjoint, so the colimit of these associated sheaves is the associated sheaf of the presheaf colimit. One can check this directly at a stalk: the two colimits over \(i\) and neighborhoods of the point commute, by their universal properties. On every \(V\in\mathcal V\), (5.1) gives the required isomorphism from that presheaf colimit to \(H^q(f^{-1}V,F)\). Hence the associated-sheaf map is an isomorphism on every stalk. These are the actual maps of (6.1); their \(\mathcal O_Y\)-actions are those already verified for the cohomology presheaf. No injective-forgetful assertion is involved.

If \(X\) is noetherian, every inverse open is quasi-compact, so this conclusion holds for every ringed-space map out of \(X\). For global cohomology the same conclusion requires \(X\) quasi-compact, or more generally applies on the quasi-compact open where the cohomology is taken. Local noetherianness alone is not that global condition.

Both (5.1) and (6.1) also give commutation with arbitrary small direct sums under their respective hypotheses. A direct sum is the filtered colimit of its finite partial sums, indexed by finite subsets, including the empty subset. Derived sections and derived direct images preserve finite sums: finite sums of injective resolutions are injective resolutions, and the original functors are additive. The resulting isomorphisms with arbitrary sums are induced by the original summand inclusions.

7. Diagrams of complexes with a common lower bound

There is a useful bounded-below version with no unbounded replacement claim. Let \(K_i\) be an actual filtered diagram of cochain complexes, all zero below one integer \(a\). For every quasi-compact open \(U\),

where \(\mathbb H^n(U,K)=H^nR\Gamma(U,K)\), and the complex colimit is termwise. To construct its comparisons, use the double complex \(G^p(K_i^r)\), \(p\ge0,r\ge a\), with total differential \(d_G+(-1)^p d_K\). The Godement term functors are exact, so every augmented horizontal row is exact, and the augmentation from \(K_i\) to its total is a quasi-isomorphism. This follows either stalkwise by the row comparison or from the exact first-page filtration: there are only finitely many positions in each total degree after shifting \(r\) by \(a\). Each total term is a finite sum of flabby sheaves.

These totals are functorial in every chain map. Their filtered colimit is a bounded-below complex of section-acyclic sheaves on \(U\), by Section 4, and its augmentation from \(\mathop{\mathrm{colim}}K_i\) is a quasi-isomorphism by exactness of filtered colimits. Apply the same acyclic-model comparison as in Section 5, then (2.1) termwise and exactness on cohomology. This gives (7.1), with the prescribed augmentation maps and all chain-map naturalities. Replacing this whole diagram by a termwise quasi-isomorphic diagram with the same kind of common bound preserves the comparisons, by exactness of the colimit and the augmentation squares.

Under the hypotheses of Section 6, the same proof on inverse neighborhoods and then on stalks gives commutation for every cohomology sheaf of \(Rf_*K_i\). This assertion concerns the specified diagrams of uniformly bounded-below complexes; it does not define a colimit for an arbitrary abstract diagram in the derived category, or assert convergence for complexes whose lower bounds tend to minus infinity.

8. Four exercises with full solutions

8.1 Infinite discrete sections

Exercise 1 (basic). On \(X=\mathbb N\) with the discrete topology and \(\mathcal O=k_X\), let \(F_n\) have stalk \(k\) at the first \(n\) points and zero elsewhere, with the evident inclusions. Compute the degree-zero comparison on \(X\), and identify exactly which hypothesis fails.

Solution. A sheaf on a discrete space is a family of modules: sections on an open are their product, as follows by gluing the cover by singleton opens. The sheaf colimit has stalk \(k\) everywhere and is \(k_X\). Thus \(\mathop{\mathrm{colim}}_n\Gamma(X,F_n)=k^{(\mathbb N)}\), whereas \(\Gamma(X,k_X)=k^{\mathbb N}\). The comparison is the finite-support inclusion and misses the constant sequence one. Singleton opens form an intersection-stable quasi-compact basis, but \(X\) is not quasi-compact, since its singleton cover has no finite subcover. On every finite open the comparison is an isomorphism. Exactness of filtered colimits does not make them commute with the infinite product used by the global section functor here.

8.2 A compact space and a nonflabby sum of injectives

Exercise 2 (intermediate). Let \(X=\{\infty\}\cup\mathbb N\), with isolated integers and neighborhoods of \(\infty\) containing every sufficiently large integer. With \(\mathcal O=k_X\), let \(F_n\) be the finite sum of the point sheaves at \(1,\ldots,n\). Show that every \(F_n\) is injective, but their colimit \(M\) is not flabby. Compute all \(H^q(X,M)\), and explain why the cohomology theorem still applies on \(X\).

Solution. A vector space over \(k\) is injective by extending a linear map from a subspace along a basis. The point pushforward is right adjoint to the exact stalk functor, so it preserves injectives. Hence the finite sum \(F_n\) is injective. Its colimit has stalks \(k\) at the integers and zero at \(\infty\). A section of \(M\) on \(X\) is exactly a finitely supported sequence: its zero germ at \(\infty\) forces it to vanish on some tail, and every finite sequence comes from a finite sum. On the open \(\mathbb N\), however, its sections are all of \(k^{\mathbb N}\). The restriction is the nonsurjective inclusion \(k^{(\mathbb N)}\hookrightarrow k^{\mathbb N}\). Thus \(M\) is not flabby, and in particular not injective.

Every cover of \(X\) has a finite subcover: choose a member containing \(\infty\), and then cover the finitely many remaining points. A basis consists of the singleton integers and the tails \(U_N=\{\infty\}\cup\{n\ge N\}\), together with the empty open. These are quasi-compact, and their finite intersections have the same form. The theorem therefore applies to \(X\): \(H^0(X,M)=k^{(\mathbb N)}\) and \(H^{q>0}(X,M)=0\), because the \(F_n\) are injective. The open \(\mathbb N\) is not quasi-compact, so \(X\) is not noetherian. This is exactly why Section 4 proves acyclicity on the indicated opens without claiming general preservation of flabbiness.

8.3 Higher cohomology on infinitely many circles

Exercise 3 (advanced). Let \(X=\coprod_{n\ge1}C_n\), where every \(C_n\) is the four-point space \(u,v<a,b\) with upper-set topology. Use \(\mathcal O=k_X\), and let \(F_n\) be the constant sheaf on the first \(n\) components, extended by zero. Compute the comparison in degree one on \(X\), including its actual map.

Solution. On one component the canonical normalized injective resolution from the finite-poset prerequisite has only degrees zero and one. Its section complex is \(k^4\xrightarrow{d}k^4\), with coordinates at vertices \((u,v,a,b)\) and edges \((ua,ub,va,vb)\), and

Its kernel is the diagonal copy of \(k\). Its image is the kernel of the surjective functional \(\Lambda(y)=y_{ua}-y_{ub}-y_{va}+y_{vb}\): the equations are solved by choosing \(x_u=0,x_a=y_{ua},x_b=y_{ub},x_v=y_{ua}-y_{va}\), and the last equation is exactly \(\Lambda(y)=0\). Hence the cokernel is \(k\) through \(\Lambda\), and higher cohomology is zero.

A sheaf on the disjoint union is exactly a family of component sheaves, with all maps componentwise; both reconstructions follow by restriction and gluing on the component cover. An object is injective if each component is injective: extend a given map in each component and assemble it, using choice. We may therefore combine the displayed component resolutions into an injective resolution on \(X\). Its section complex is the product of the component section complexes. Products of module sequences are exact by choosing lifts coordinatewise, so its degree-one cohomology is \(\prod_n k\). The filtered sheaf colimit of the \(F_n\) is \(k_X\), and \(H^1(X,F_n)=k^n\), with transition the zero extension to the next component. Thus the comparison is precisely

the finite-support inclusion, and is not surjective. The componentwise functional \(\Lambda\) verifies the identification and its naturality under these inclusions. Every point has a finite noetherian open neighborhood and the basis hypothesis holds, but \(X\) is not quasi-compact: its component cover has no finite subcover. This proves failure in positive degree despite local noetherianness and monic transition maps.

8.4 Localization kills or preserves both torsion Ext degrees

Exercise 4 (advanced). On one four-point circle take \(\mathcal O=\mathbb Z_X\), a prime \(p\), and \(A=(\mathbb Z/p^r)_X\) with \(r\ge1\). Let \(G_i=\mathcal O\) with every transition multiplication by a prime \(\ell\). Compute \(\mathop{\mathrm{colim}}_i\operatorname{Ext}^m_X(A,G_i)\) and its comparison with \(\operatorname{Ext}^m_X(A,\mathbb Z[1/\ell]_X)\), for all \(m\). Track the structural maps.

Solution. The local free resolution \(\mathcal O\xrightarrow{p^r}\mathcal O\to A\) gives normalized internal Hom differential \(-p^r\) in degrees zero and one. For \(G_i=\mathcal O\), its only cohomology sheaf is \(\mathbb Z/p^r\) in degree one. The local-to-global calculation in the sheaf-Ext prerequisite therefore gives \(\operatorname{Ext}^m_X(A,G_i)=\mathbb Z/p^r\) for \(m=1,2\), and zero otherwise. Postcomposition by the scalar \(\ell\) multiplies both terms of that Hom complex by \(\ell\); it consequently multiplies both Ext groups by \(\ell\), including the global degree-two term.

The sheaf colimit of the \(G_i\) is \(\mathbb Z[1/\ell]_X\), with stage \(i\) map \(z\mapsto z/\ell^i\), starting at \(i=0\). The same local resolution into this target has zero kernel and cokernel \(T=\mathbb Z[1/\ell]/p^r\mathbb Z[1/\ell]\) in degree one. Its global Ext groups are \(H^{m-1}(X,T_X)\). To compute these for the possibly nonfield group \(T\), use the two minimal opens at \(u,v\). Sections on either are exact evaluation at its least point, and their intersection is the discrete pair \(a,b\), with exact sections as well. The proved Mayer–Vietoris sequence is therefore computed by \(T^2\to T^2\), \((s,t)\mapsto(t-s,t-s)\). Its kernel and cokernel are \(T\) and all higher cohomology vanishes. Thus global Ext is \(T\) in degrees one and two and zero otherwise.

If \(\ell=p\), then \(T=0\), and the filtered groups are zero too: any class is killed by \(r\) further multiplications by \(p\). If \(\ell\ne p\), multiplication by \(\ell\) is invertible modulo \(p^r\), and \(T\simeq\mathbb Z/p^r\) sends \(z/\ell^i\) to \(\ell^{-i}z\). Under these identifications both comparisons in degrees one and two send the stage-\(i\) class \(z\) to \(\ell^{-i}z\); the other degrees are zero. The finite free local resolution, finite-circle cochain complex and their scalar naturalities justify this Ext comparison. A noetherian topological space by itself would not make arbitrary internal Hom commute with colimits in its second argument.

Further reading

The Stacks Project, Cohomology and colimits, states the quasi-compact basis hypothesis and the resulting cohomology and higher-direct-image comparisons. The arguments above also specify the acyclicity criterion, flabby models, structural maps and common complex bounds used here.