Closed subsets and continuity of sheaf cohomology

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

A section along a closed set can be represented on one neighborhood of that set. The same principle holds for cohomology, provided the ambient space has the separation and refinement properties needed for soft sheaves. A single injective model makes the comparison concrete in every degree and preserves its maps. Compactness then gives a second continuity theorem for decreasing families of closed sets.

Let \(X\) be paracompact Hausdorff and \(\mathcal R\) a sheaf of unital rings. Modules are left modules, and the rings need not be commutative. All indices are sets in fixed universes, with choice. Restriction to a subset means inverse-image restriction of sheaves, with all ambient germs retained. Every derived complex below is bounded below.

Use Soft sheaves on paracompact Hausdorff spaces, Theorem 2.2 and Sections 3–5, for the closed-neighborhood section theorem and closed restrictions of soft sheaves; Godement resolutions and flabby sheaf cohomology, Sections 1–6, for enough injectives, their flabbiness and their injective restrictions to opens; and Universal cohomology and acyclic resolutions, Theorem 4.1, for the full normalized acyclic comparison. Closed-restriction maps are constructed in Finite closed covers and sheaf cohomology. The examples use the contractible-space theorem of Interval cohomology and homotopy invariance for sheaves and the diagram description in Finite-poset sheaves and canonical homological resolutions. Basic references are the Stacks project's compact-neighborhood theorem [S] and Demailly's discussion of soft sheaves [D].

1. The neighborhood functor and its scalars

For a closed subset \(C\subseteq X\), let \(\mathcal N(C)\) be the open neighborhoods of \(C\), ordered by shrinking: there is an arrow \(U\to V\) when \(V\subseteq U\). It is filtered, since \(U\cap V\) is a common later neighborhood. For \(C=\varnothing\), its empty neighborhood is terminal. Put

The closed-neighborhood section theorem identifies \(N_C(\mathcal R)\) with \(R_C\) as a ring, and identifies \(N_C(\mathcal F)\) with \(\Gamma(C,i_C^{-1}\mathcal F)\) as a left \(R_C\)-module. To see the action directly, represent a scalar on \(U\) and a section on \(V\), restrict both to \(U\cap V\), and multiply there. Refining again shows independence of representatives. The same construction verifies the unit, addition and associativity laws, in the original left-action order.

Lemma 1.1. Filtered colimits of abelian groups are exact and commute with cohomology of diagrams of complexes. Consequently \(N_C\) is an additive left exact functor to \(\operatorname{Mod}(R_C)\).

Proof. An element of a filtered colimit has a representative at one stage. It is zero precisely when its representative becomes zero at a later stage: the finite chain of identifications defining equality can be put at a common stage, with any parallel arrows equalized there. For a stagewise short exact row, a kernel element therefore lifts into the stage kernel after moving far enough; a target element lifts at its representative stage; and a source element that becomes zero is already zero at some later stage. These are exactness at all three positions. The construction uses compatible representatives to preserve addition. Applied to cycles, boundaries and their quotient, exactness gives

The map takes a represented cocycle class to that same cocycle in the colimit. Each section functor is left exact, and the filtered colimit preserves that exactness. Thus \(N_C\) is left exact on underlying groups, hence on the modules in (1.1). Its maps are \(R_C\)-linear by the common-neighborhood multiplication rule. \(\square\)

A cofinal neighborhood basis here means a family \(\mathcal B\subseteq\mathcal N(C)\) such that every neighborhood contains a member of \(\mathcal B\). It is directed under shrinking: given two members, choose a member inside their intersection. Restricting (1.1) to \(\mathcal B\) does not change it. Indeed a representative moves to a member inside its domain, and any equality moves to a member inside the neighborhood witnessing the equality. This proves both surjectivity and injectivity of the basis comparison, including its operations.

2. Restrict one injective model

Write \(\mathcal F_C=i_C^{-1}\mathcal F\). A closed subset \(C\) is paracompact Hausdorff, by the refinement theorem in the soft-sheaf prerequisite. An injective \(\mathcal I\) on \(X\) is soft. Its restriction \(\mathcal I_C\) is soft on \(C\), and therefore \(\Gamma(C,-)\)-acyclic there. Restriction to an open \(U\), on the other hand, preserves injectives: its left adjoint, extension by zero, is exact, as proved in the Godement prerequisite.

These facts give two usable restrictions of one global injective resolution. The open neighborhoods need not themselves be paracompact; their injective restrictions compute cohomology without that hypothesis.

Theorem 2.1 (closed-neighborhood cohomology). For every module sheaf \(\mathcal F\), every closed \(C\subseteq X\), and every \(q\geq0\), the actual restriction maps induce an isomorphism

It is natural in coefficients, respects nested closed-subset restrictions, and commutes with all connecting maps of short exact coefficient rows. A closed subset with this property is called taut for sheaf cohomology.

Proof. Choose an injective resolution \(\mathcal F\to\mathcal I^\bullet\) on \(X\). For every open \(U\), its restricted resolution is injective, so its actual section complex computes \(H^q(U,\mathcal F|_U)\). On \(C\), inverse image is exact and every \(\mathcal I_C^n\) is soft and acyclic. The full acyclic-resolution comparison therefore identifies the cohomology of \(\Gamma(C,\mathcal I_C^\bullet)\) with that of \(\mathcal F_C\), with its given augmentation. In every degree the closed-neighborhood section theorem gives a natural isomorphism, and those isomorphisms commute with the differentials:

Take cohomology and apply Lemma 1.1. The resulting isomorphism is (2.1). Its contribution from a fixed neighborhood is the map restricting that very injective section complex to its closed inverse image, followed by the normalized acyclic comparison. This is the usual derived restriction: the chain map starts with the actual restriction on \(\mathcal F\), and the acyclic comparison respects that augmentation. Thus (2.1) is the asserted canonical map. Section 3 proves the full derived naturality, including its connecting maps. \(\square\)

The colimit on the left of (2.1) is naturally an \(R_C\)-module. Represent a scalar by a section on a neighborhood \(V\), and a cohomology class on \(U\); restrict the class to \(U\cap V\) and multiply there. The cochain action commutes with the differential. Common refinements prove independence, and (2.2) proves that (2.1) respects this action. No commutation of ring elements is needed.

3. The full bounded-below comparison and all its maps

For a bounded-below complex \(K\), choose a bounded-below injective model \(K\to I\). Define the concrete neighborhood complex

It computes \(RN_C(K)\). This definition uses an actual diagram of complexes with restriction maps. In particular, no colimit of an unspecified diagram in the ordinary derived category is being presumed.

Theorem 3.1. There is a natural normalized isomorphism of functors on \(D^+(\operatorname{Mod}(\mathcal R))\), with values in \(D^+(\operatorname{Mod}(R_C))\):

Its representative on the model \(I\) is the degreewise isomorphism \(\mathsf N_C(K;I)\to\Gamma(C,i_C^{-1}I)\) from (2.2), followed by the canonical acyclic comparison. It preserves the degree-zero identification (1.1), all derived morphisms, shifts, triangles and connecting maps. For every integer \(q\),

Proof. Each \(I^n\) is injective on \(X\). Thus \(I_C^n\) is soft on \(C\), and the full bounded-below acyclic-computation theorem identifies \(\Gamma(C,I_C)\) with \(R\Gamma(C,i_C^{-1}K)\), with the actual map from \(K\). Applying (2.2) termwise gives (3.2).

Here are its independence and morphism checks. A map between bounded-below injective models restricts on every open and on \(C\); section restriction commutes with it. Hence the degreewise isomorphism is a natural map of complexes. A chain homotopy remains a chain homotopy after either restriction, sections or filtered colimit, since these operations are additive and preserve its defining equation. Comparison maps of injective models and their inverse homotopy classes therefore give the same derived isomorphism. Every derived map is represented by the corresponding homotopy class between such injective models, by the injective comparison theorem in the prerequisites. This proves naturality for all those maps.

Finite direct sums of injectives are injective. Consequently the cone of a map between these injective models is again an injective-term bounded-below complex. The degreewise map (2.2) commutes with shifts and with the standard signed cone differential, because it is additive and commutes with each component map. The canonical acyclic comparison has the same cone and shift compatibility. Thus (3.2) is a comparison of exact functors and respects all distinguished triangles. In particular it respects the triangle of a short exact sheaf row, including every connecting map. On a sheaf in degree zero, the augmentation makes its zeroth map precisely (1.1); this verifies the normalization.

Finally, \(I|_U\) is a bounded-below injective model of \(K|_U\) for each open \(U\). Lemma 1.1 applied to (3.1) now gives (3.3), with exactly these restriction maps. \(\square\)

If \(D\subseteq C\) are closed, the map \(N_C(\mathcal F)\to N_D(\mathcal F)\) regards a section near \(C\) as one near \(D\). On (2.2) it is exactly restriction from \(C\) to \(D\). The same is true on the injective-model complexes in Theorem 3.1. For a further closed subset, both composites are the same section restriction, so the derived comparisons compose coherently. These maps are semilinear for the natural ring map \(R_C\to R_D\). Restricting to any cofinal neighborhood basis gives the same comparisons by the representative argument in Section 1.

4. Continuity for decreasing closed families

Let \(T\) be a nonempty directed partially ordered set. Suppose \((C_t)_{t\in T}\) is a family of closed subsets of \(X\), with \(C_s\subseteq C_t\) for \(t\leq s\), and with at least one \(C_{t_0}\) compact. Put \(C_\infty=\bigcap_tC_t\).

Lemma 4.1. Every open neighborhood \(U\) of \(C_\infty\) contains some \(C_s\). One can require \(s\) to lie beyond any specified finite set of indices.

Proof. Pass to the cofinal tail \(t\geq t_0\). If no \(C_t\) on this tail lay in \(U\), the sets \(C_t\cap(X\setminus U)\) would be nonempty closed subsets of the compact space \(C_{t_0}\). Every finite intersection contains one of them, by directedness. They therefore have nonempty total intersection: otherwise their complements would be an open cover of the compact space with a finite subcover, contrary to the finite-intersection property. But the total intersection is \(C_\infty\setminus U=\varnothing\). This contradiction proves the assertion. To impose finitely many lower bounds on \(s\), choose a common upper bound with the index already found. Its closed set is smaller. The proof also covers an empty intersection: taking \(U=\varnothing\) then forces some \(C_s\) to be empty. \(\square\)

Lemma 4.2. For every sheaf \(\mathcal F\), the actual closed restrictions induce a bijection

Proof. A section on \(C_\infty\) is represented on an open neighborhood \(U\), by the closed-neighborhood theorem. Choose \(C_s\subseteq U\) by Lemma 4.1 and restrict that representative to \(C_s\). This proves surjectivity.

For injectivity, represent a section on \(C_t\) by a section \(f\) on an open neighborhood \(V\) of \(C_t\). If its restriction to \(C_\infty\) is zero, the zero-germ locus of \(f\) inside \(V\) is an open neighborhood \(W\) of \(C_\infty\). Lemma 4.1 gives \(s\geq t\) with \(C_s\subseteq W\). The restriction of the original section to \(C_s\) is therefore zero. This is precisely the zero criterion in the filtered colimit. Apply the argument to the difference for two module sections; for sheaves of sets use their open equality locus instead. The actual restriction maps prove naturality. \(\square\)

Theorem 4.3. Under these hypotheses, for every bounded-below \(K\) and integer \(q\), the canonical map is an isomorphism

It has a concrete complex representative, is natural in all derived coefficient maps, and respects the restriction maps and connecting maps. For a sheaf in degree zero this gives the same formula for ordinary sheaf cohomology.

Proof. Choose one global bounded-below injective model \(K\to I\). Every \(C_t\), and also \(C_\infty\), is closed in \(X\). The restricted terms of \(I\) are therefore soft and acyclic on each of these spaces, regardless of whether a particular earlier \(C_t\) was compact. They compute all the displayed hypercohomology groups. Lemma 4.2 gives a natural degreewise isomorphism

Apply Lemma 1.1 and the normalized acyclic comparisons to get (4.2). The injective-model homotopy, shift and cone argument of Section 3 applies to (4.3) as well, proving all the stated map compatibilities. These complexes can be regarded as modules over \(\Gamma(X,\mathcal R)\). More precisely, Lemma 4.2 applied to the ring sheaf gives \(\operatorname{colim}_tR_{C_t}\simeq R_{C_\infty}\); the common-stage multiplication rule equips (4.2) and (4.3) with that full left action. \(\square\)

There is no countability assumption on \(T\) and no bound on the cardinality of a closed set. Compactness supplies eventual containment of a whole closed set in a prescribed neighborhood. The hypotheses do not assert an analogous containment for decreasing noncompact sets.

5. Four exercises with complete solutions

Exercise 1 (introductory: separate neighborhood radii). Take \(C=\mathbb Z\subset\mathbb R\), an abelian group \(A\), and the constant sheaf \(A_{\mathbb R}\). Compute (2.1) in every degree. Show that the neighborhoods

are cofinal, but the family with a single constant radius is not cofinal.

Solution. An arbitrary neighborhood of \(\mathbb Z\) contains an interval of some positive radius about each integer. Shrink those radii below \(1/3\); their union in (5.1) is still a neighborhood and is contained in the prescribed one. Thus this family is cofinal. Its intervals are disjoint. Each is contractible, so the interval-homotopy prerequisite gives \(H^0=A\) and \(H^{q>0}=0\) on each component. An injective section complex on the disjoint union is the product of its component section complexes. Products of abelian groups are exact by coordinatewise lifts with choice, so their cohomology is the product of the component cohomologies. Hence \(H^0(U_{\boldsymbol\varepsilon},A)=\prod_{n\in\mathbb Z}A\) and the positive groups vanish. The restrictions between smaller such neighborhoods identify the same integer coordinates.

The closed space \(\mathbb Z\) is discrete. Sheaves on it are families of stalk groups, and sections are their product. This product functor is exact, proving \(H^0(\mathbb Z,A)=\prod_nA\) and all positive groups zero, directly as well. The map (2.1) sends each component value to its germ at that integer and is this product identification.

For noncofinality of a common radius, use the neighborhood with \(\varepsilon_n=1/(10+|n|)\). For any constant radius \(\varepsilon>0\), set \(d=\min(\varepsilon/2,1/4)\) and choose \(n\) so large that \(\varepsilon_n<d\). Then \(n+d\) is in the constant-radius neighborhood but outside (5.1): it is outside the interval about \(n\), and its distance from every other integer is at least \(3/4\). Thus no constant-radius member lies in this neighborhood. The closed set is noncompact, and separate radii are essential. \(\square\)

Exercise 2 (intermediate: the Cantor set). Let \(C_n\subset[0,1]\) be the union of the \(2^n\) closed intervals remaining after \(n\) middle-third removals, with \(C_0=[0,1]\), and let \(K=\bigcap_nC_n\). Compute all \(H^q(K,A_K)\), including the actual transition maps. Explain why degree zero is generally smaller than the group of all functions \(K\to A\).

Solution. The ambient interval is compact Hausdorff and paracompact, the latter also following from its closed inclusion in the real line. Each \(C_n\) is a finite disjoint union of contractible closed intervals. Its component sheaf categories and section complexes split as finite products. The interval-homotopy theorem therefore gives \(H^0(C_n,A)=A^{2^n}\) and zero positive groups. Restriction to \(C_{n+1}\) sends the value on a parent interval to that same value on each of its two children. Theorem 4.3 yields

Here \(w\) records the successive left or right choices. To identify the first group concretely, the intersections of \(K\) with the level-\(n\) intervals are clopen in \(K\): the finitely many complementary level-\(n\) pieces are closed. These cylinders have diameter \(3^{-n}\) and form a neighborhood basis. Indeed, a sufficiently small cylinder containing a point lies inside any prescribed relative open neighborhood of that point. A locally constant map \(K\to A\), with \(A\) discrete, is constant on a cylinder around every point. Compactness of \(K\) gives finitely many such cylinders covering it. Refine to the largest of their levels; the function is constant on each cylinder at that level. Conversely every level assignment gives a locally constant function. These are exactly the representatives and identifications in (5.2).

If \(A\ne0\), choose \(a\ne0\) and a point \(x\in K\). Every cylinder containing \(x\) has another point: continue its left/right choices in two different ways, whose nested intervals of lengths tending to zero determine distinct points. Thus the function taking value \(a\) just at \(x\) and zero elsewhere is not locally constant. It is an element of the product \(\prod_{x\in K}A\), but not of the group in (5.2). The cohomology comparison retains local constancy. \(\square\)

Exercise 3 (intermediate: restriction and support give different degrees). Let \(X=\mathbb R\), \(C=\{0\}\), and let \(A\) be an abelian group. Compute the neighborhood-cohomology colimit for \(A_X\). Compare it with the right derived groups of supported sections

Solution. Theorem 2.1 identifies the neighborhood colimit with \(H^q(\{0\},A)=A\) in degree zero and zero otherwise. It uses closed inverse image, which retains the stalk at zero.

For support, take an injective resolution \(A_X\to I\). Injective sheaves are flabby, so in every degree restriction from \(X\) to \(X\setminus C\) is onto. Thus there is a short exact row of complexes

By definition the first complex computes \(R\Gamma_C\); open restriction preserves injectives, so the others compute the indicated ordinary cohomology. The real line is contractible, and its punctured complement is two disjoint contractible half-lines. The homotopy theorem and the finite-product calculation give \(A\) and \(A\oplus A\) in degree zero, respectively, and zero positive groups. Their restriction map is the diagonal \(a\mapsto(a,a)\). The exact cohomology sequence of (5.4) therefore gives \(H_C^0(X,A_X)=0\), \(H_C^1(X,A_X)\simeq(A\oplus A)/\Delta A\simeq A\), and zero in every degree \(q\geq2\). An explicit quotient isomorphism is \([(u,v)]\mapsto v-u\); its kernel is exactly the diagonal and it is onto. With this isomorphism the connecting map sends a pair of half-line values to their right-minus-left difference. This computes the actual boundary normalization. \(\square\)

Exercise 4 (advanced: two hypotheses that prevent escape). Give a decreasing noncompact closed family for which (4.2) fails. Then let \(Y=\{a,b,c\}\) have the upper-set topology for \(a<c\), \(b<c\), and let \(D=\{a,b\}\). For a nonzero abelian group \(A\), compute the degree-zero neighborhood comparison for the constant sheaf \(A_Y\). Explain the topological hypotheses that fail in these examples.

Solution. In \(\mathbb R\), take \(C_n=[n,\infty)\), \(n\geq1\). These are decreasing noncompact closed sets with empty intersection. Each is contractible, so its zeroth constant-sheaf cohomology is \(A\), and restriction to a later tail is the identity under the constant-value identification. The colimit is therefore \(A\), whereas the cohomology of the empty intersection is zero. No stage is compact. The sets keep escaping toward infinity instead of entering the empty neighborhood of their intersection.

For \(Y\), the complement \(\{c\}\) is open, so \(D\) is closed. Every open containing \(a\) contains \(c\), and the same is true for \(b\). Therefore the only open neighborhood of all of \(D\) is \(Y\). The finite-poset sheaf description identifies \(A_Y\) with the diagram \(A\xrightarrow{1}A\xleftarrow{1}A\), whose entries are at \(a,c,b\). Its global sections are triples with the two endpoint values both equal to the value at \(c\), hence form one copy of \(A\). The subspace \(D\) is discrete: intersect \(\{a,c\}\) or \(\{b,c\}\) with it. Its restricted sheaf has global sections \(A\oplus A\). The neighborhood comparison is thus the diagonal

which is not onto when \(A\ne0\). This finite space is paracompact: every open cover has a finite subcover, and a finite family is locally finite. It is not Hausdorff, since any neighborhoods of \(a\) and \(b\) meet at \(c\). Thus closedness and paracompactness alone do not supply the degree-zero comparison used throughout the lesson. \(\square\)

References

• [S] The Stacks Project Authors, Cohomology on Hausdorff quasi-compact spaces, Lemma 20.16.3. This gives the compact-subset neighborhood-cohomology theorem with its ambient separation condition. The arguments above use the linked soft-sheaf results for general closed subsets of paracompact Hausdorff spaces.
• [D] Jean-Pierre Demailly, Complex Analytic and Differential Geometry, Chapter IV, Section 4.B, for soft sheaves and their acyclicity. Freely accessible author-hosted text.