Finite closed covers and sheaf cohomology

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

A finite closed cover gives a resolution by restrictions to its intersections. At a point, the surviving terms form the augmented cochain complex of a simplex. Its contraction proves exactness, while derived sections account for any higher cohomology on the intersections. This construction uses closed restriction and pushforward, rather than sections with support.

Let \(X\) be a topological space, \(\mathcal R\) a sheaf of unital rings, possibly noncommutative, and \(\mathcal F\) a left \(\mathcal R\)-module sheaf. Work in fixed universes with choice. No separation or paracompactness assumption is imposed. Finiteness enters in the stalk calculation and the finite column filtration.

Use The inverse image of module sheaves for the actual inverse-image adjunction and stalk formula; Godement resolutions and flabby sheaf cohomology for enough injectives and the stalk criteria; Filtered complexes and Cartan–Eilenberg resolutions for the complete injective double-resolution and spectral-sequence construction; and Injective replacements and right derived functors for bounded-below derived sections. The companion open-cover Čech lesson has a different cover hypothesis. Exercise 2 also uses the explicit diagram resolution in Finite-poset sheaves and canonical homological resolutions.

1. Closed restriction, its maps and its cohomology

For a closed subset \(Z\), let \(i_Z:Z\hookrightarrow X\) be its inclusion and write

The inverse-image sheaf is a module over \(i_Z^{-1}\mathcal R\), and pushforward uses the natural scalar map from \(\mathcal R\). The inverse-image construction and adjunction recalled in the prerequisite apply to these possibly noncommutative rings too: transport the addition and left action on local representatives, and a map out of the sheafification is a compatible family of module maps. The inverse adjunction rules restrict or extend that same family, so respect the left action without commuting any scalars. In particular \(\mathcal F[\varnothing]=0\) and \(\mathcal F[X]=\mathcal F\), through their canonical identifications. The notation in (1.1) does not denote the subsheaf of sections supported in \(Z\).

Lemma 1.1. For \(x\in X\),

For closed subsets \(W\subseteq Z\), there is a natural map \(\rho_{Z,W}:\mathcal F[Z]\to\mathcal F[W]\). Its stalk is the identity where \(x\in W\) and zero elsewhere. These maps compose according to inclusions.

Proof. Direct image on an open \(V\) has value \((i_Z^{-1}\mathcal F)(V\cap Z)\). If \(x\notin Z\), the closedness of \(Z\) gives a neighborhood disjoint from \(Z\), making the stalk zero. If \(x\in Z\), neighborhoods in \(Z\) are intersections with neighborhoods in \(X\). Taking their germs identifies the pushforward stalk with \((i_Z^{-1}\mathcal F)_x\simeq\mathcal F_x\), by the inverse-image stalk formula.

For \(j:W\hookrightarrow Z\), apply the adjunction unit to the sheaf \(i_Z^{-1}\mathcal F\) on \(Z\), then push it forward by \(i_Z\). The composite-inverse-image identification gives \(\rho_{Z,W}\). Thus the map to a closed restriction is an adjunction-unit map; it is not an ordinary restriction to an open set. The unit stalk is the identity at points of \(W\), giving the stated stalk description. For nested closed subsets, its composite and the direct map have the same stalk maps, hence are equal. To recall that last criterion, equality of the images of a section at every germ gives equality on a neighborhood of every point, then equality by the sheaf axiom. Naturality follows from the same unit construction. \(\square\)

Proposition 1.2. Closed pushforward is exact and preserves injective module sheaves. There are natural identifications

Proof. The stalk description for a pushed-forward sheaf is its original stalk on \(Z\), and zero off \(Z\), by the proof of Lemma 1.1. The stalk exactness criterion therefore makes \(i_{Z*}\) exact. The functor \(i_Z^{-1}\) is exact, since its stalk at a point of \(Z\) is the original stalk functor and filtered module colimits preserve exact sequences.

If \(I\) is injective on \(Z\), a map from a subsheaf \(A\subseteq B\) on \(X\) to \(i_{Z*}I\) corresponds by adjunction to a map \(i_Z^{-1}A\to I\). Exactness of inverse image makes \(i_Z^{-1}A\subseteq i_Z^{-1}B\), so injectivity extends that map. Adjunction extends the original map to \(B\). Hence \(i_{Z*}I\) is injective.

Push forward an injective resolution of \(i_Z^{-1}\mathcal F\). Exactness and preservation of injectives give an injective resolution of \(\mathcal F[Z]\). Its section complex over \(X\) is exactly the original section complex over \(Z\), because \(\Gamma(X,i_{Z*}I)=\Gamma(Z,I)\). This proves (1.3), with naturality from comparison of resolutions. The construction also identifies the map induced by \(\rho_{Z,W}\) with the usual derived restriction from \(Z\) to \(W\). \(\square\)

2. The signed finite closed-cover resolution

Let \(Z_0,\ldots,Z_{m-1}\) be a finite family of closed subsets, and put \(Z=\bigcup_i Z_i\). For a nonempty subset \(J\) of the index set, put \(Z_J=\bigcap_{j\in J}Z_j\). Define

and use zero in all higher degrees. Empty intersections contribute zero when the corresponding closed subset is empty. Define the differential component at an increasing tuple \(K=(i_0,\ldots,i_{p+1})\) by

The augmentation \(\epsilon:\mathcal F[Z]\to C^0\) has components \(\rho_{Z,Z_i}\). Deleting two distinct indices in opposite orders gives the same composite closed-restriction map. The two coefficients have opposite signs, because the earlier deletion changes the later position by one. Pairing them proves \(d^2=0\). The same calculation gives \(d\epsilon=0\).

Theorem 2.1. The augmented complex

is exact. In particular, if the closed subsets cover \(X\), the complex \(C^\bullet\) is a resolution of \(\mathcal F\).

Proof. Stalks commute with the finite sums in (2.1). At \(x\notin Z\), every stalk in (2.3) is zero. At \(x\in Z\), set \(K_x=\{i:x\in Z_i\}\), which is nonempty. Lemma 1.1 identifies the surviving summands with one copy of \(M=\mathcal F_x\) for each nonempty subset of \(K_x\), and every restriction in (2.2) with its identity map. This is the augmented simplex cochain complex on \(K_x\).

Here is its contraction with the signs fixed. Choose \(k\in K_x\). For a cochain \(c\) of degree \(p\), define

where \(J\) is increasing with \(p\) entries and \(r(J)\) is the number of its entries smaller than \(k\). In degree zero this means \(hc=c_k\in M\), the augmentation term. Put \(h=0\) on that term.

To check the contraction, use formal exterior basis symbols \(e_J\), with coefficients in the abelian group underlying \(M\). The differential is multiplication on the left by \(\omega=\sum_{i\in K_x}e_i\). The operator \(h\) contracts the basis vector \(e_k\): deleting it in position \(r\) contributes \((-1)^r\), which is exactly (2.4). For \(i\ne k\), inserting \(e_i\) and deleting \(e_k\) in the two orders gives opposite signs; for \(i=k\), exactly one of inserting-then-deleting and deleting-then-inserting is nonzero and returns the original basis symbol. Thus

in every degree, including the augmentation. All maps are sums of \(\pm1\) times module identities, so this check applies to arbitrary left modules over noncommutative rings as well. It proves exactness at every stalk, and hence exactness of the sheaf complex. The choice of \(k\) is only a stalkwise proof; no global splitting of (2.3) is asserted. If the family is empty, then \(Z=\varnothing\) and (2.3) is the zero complex, so the statement still holds. \(\square\)

3. Derived sections and the intersection spectral sequence

In the next display, put \(\mathcal F_J=i_{Z_J}^{-1}\mathcal F\); the sum runs over subsets of the finite index set.

Theorem 3.1. There is a first-quadrant spectral sequence

Its first differential is the alternating closed-restriction map (2.2) on cohomology. The column index has \(0\le p<m\); hence its abutment filtration is finite. For a closed cover of \(X\), the target is \(H^{p+q}(X,\mathcal F)\).

Proof. Take the injective Cartan–Eilenberg double resolution of the bounded complex \(C^\bullet\), using the full construction in the stated prerequisite. Choose the zero injective embedding for all zero boundary and cohomology objects outside degrees \(0,\ldots,m-1\). In that construction each horizontal term is the sum of the injective boundary object in its degree, the injective cohomology object in its degree and the injective boundary object in the next degree. Since the boundary in degree \(m\) and in degree zero is zero, this choice preserves both horizontal bounds; every quotient complex used in the next vertical step has the same bounds. Thus the resolution has only those \(m\) columns. Its column at \(p\) is an injective resolution of \(C^p\). The horizontal maps lift \(d\), commute with the vertical differential, and square to zero. Use total differential \(d_h+(-1)^p d_v\) on column \(p\), and the finite sums along total degrees. The column comparison makes \(C^\bullet\to\operatorname{Tot}I\) a quasi-isomorphism: its vertical-first comparison is an isomorphism on the column cohomology, and the finite column filtration gives convergence of that comparison. When \(m=0\), all the complexes and groups are zero and the assertion is immediate.

Each total degree is a finite sum of injective sheaves and is injective. The total complex is bounded below, so it computes the derived section complex of \(C^\bullet\) by the bounded-below injective comparison theorem. Theorem 2.1 identifies \(\mathcal F[Z]\to C^\bullet\) as a quasi-isomorphism. Consequently this total section complex computes \(R\Gamma(X,\mathcal F[Z])\), and Proposition 1.2 identifies its cohomology with the stated target.

Filter the total section complex by columns. Vertical cohomology in column \(p\) is \(H^q(X,C^p)\). Since (2.1) is a finite sum, this is the finite sum of the groups \(H^q(X,\mathcal F[Z_J])\). Proposition 1.2 gives the first page in (3.1). The horizontal maps are the lifts of (2.2), so their induced maps are exactly its alternating closed restrictions. The spectral-sequence construction and finite convergence are the proved filtered-complex theorem in the prerequisite. Comparison of the double resolutions gives the natural sequence and its abutment maps; no particular choices of injective lifts are part of the resulting cohomology comparison. \(\square\)

Corollary 3.2. If every nonempty intersection satisfies

then there is a natural derived identification

In particular the cohomology of the finite closed-intersection section complex gives all the cohomology of \(\mathcal F\) on \(Z\).

Proof. By Proposition 1.2, every \(C^p\) is acyclic for sections. In the double resolution above, the map from \(\Gamma(X,C^p)\), placed in vertical degree zero, to its column section complex is a quasi-isomorphism. The finite column comparison makes their total map a quasi-isomorphism. Combining it with the preceding derived identification proves (3.3). Equivalently (3.1) has only its \(q=0\) row on the first page, whose cohomology is the displayed section complex. Empty intersections give zero and require no additional hypothesis. \(\square\)

4. Naturality, order and refinements

A sheaf map induces a map of (2.3) by functoriality of each closed restriction. It commutes with (2.2), so all comparisons above are natural in \(\mathcal F\). Reordering the finite index set changes each increasing-tuple component by the sign needed to sort the reordered tuple. This gives an isomorphism of augmented complexes: the formal exterior permutation sends the coefficient symbol \(e_i\) to the reordered basis symbol and commutes with insertion of \(\sum_i e_i\), while the actual closed-restriction maps compose identically. Its inverse is the reverse permutation.

For a finite closed family \((T_a)\) with \(T_a\subseteq Z_{r(a)}\), extend a cochain antisymmetrically to arbitrary tuples of distinct indices, and assign zero to tuples with a repeated index. The refinement map is

where the vertical bar denotes the closed adjunction-unit map of Lemma 1.1. The required containment holds by the refinement condition. Formula (2.2) commutes with (4.1): both sums delete one target position and apply the same composite closed-restriction map. If the image tuple has repetitions, terms retaining a repetition vanish and the two deletions that remove one repeated entry cancel with their signs. This also proves the identity and composite refinement laws, since the sorting signs multiply, and a repeated intermediate tuple remains repeated after further mapping.

If both families have the same union \(Z\), the refinement map commutes with their augmentations from \(\mathcal F[Z]\). Both augmentations are quasi-isomorphisms by Theorem 2.1. The refinement therefore induces the identity on the derived cohomology of \(\mathcal F[Z]\). Two choices of refinement function give that same derived map: after inverting the source augmentation, either map is the target augmentation composed with its inverse. This asserts independence on the derived abutment; it does not identify two generally different maps of first-page intersection groups. If the union of the \(T_a\)'s is a smaller closed subset \(W\), the augmentation instead uses \(\rho_{Z,W}\), so the same construction induces the derived closed-restriction map.

5. Four exercises with complete solutions

Exercise 1 — the two-set sequence and its signs

Introductory. Suppose \(X=A\cup B\) with \(A,B\) closed, ordered in that order. Write the augmented resolution and its cohomology sequence, identifying the first two maps and the difference sign.

Solution. The resolution is

Here all closed restrictions are the maps of Section 1. At a point in only one member, the augmentation is the identity on its only surviving term. At a point in both, it is \(M\to M\oplus M\to M\), with diagonal first map and difference second map, which is exact. This also checks the signs directly. Apply the long exact derived-section sequence to (5.1), then use Proposition 1.2. It begins

In every degree, the first map is the pair of derived restrictions and the second is restriction from \(B\) minus restriction from \(A\). The connecting map is the boundary of the exact sheaf sequence (5.1): on an injective-resolution comparison, lift a cocycle through its middle term and take its differential; changing the lift changes the result by a coboundary, as in the derived-section long exact sequence. Reversing the order of \(A,B\) negates the intersection term, giving the same sequence under the signed order comparison of Section 4.

Exercise 2 — a six-point cycle over any coefficient ring

Intermediate. Give the six-element poset vertices \(v_0,v_1,v_2\), edges \(e_0,e_1,e_2\), and relations \(v_i<e_i\), \(v_{i+1}<e_i\), with indices modulo three. Use the topology of upper sets. For a left module \(M\) over a ring \(R\), let \(\mathcal F\) be the diagram sheaf with every stalk \(M\) and every generization map the identity. Compute its cohomology using the three closed subsets \(Z_i=\{v_i,v_{i+1},e_i\}\).

Solution. Each \(Z_i\) is a lower set and hence closed, and their union is the whole poset. Its induced diagram has two minimal vertices mapping to one edge. First justify the diagram computation for arbitrary \(R\)-modules, since the prerequisite's injectivity assertion is stated for vector spaces. The sheaf–diagram equivalence uses compatible families on upper sets and minimal-open evaluation; that same construction and its inverse work for \(R\)-modules, with kernels and exactness calculated pointwise. For a point \(x\), write \(I_x(N)\) for the diagram equal to \(N\) below \(x\), zero elsewhere, and identity on arrows within that lower set. It is the exact right adjoint to evaluation at \(x\), by assigning a map's component at \(y\le x\) through the structure map to \(x\). Its sections are naturally \(N\). Applying \(I_x\) to an injective resolution of \(N\) gives an injective diagram resolution: exactness is pointwise, and preservation of injectives follows from its exact left adjoint. Its section complex is the original resolution of \(N\), so \(I_x(N)\) is section-acyclic even when \(N\) is not injective.

On the three-point piece \(v,w<e\), an explicit resolution of the constant diagram is \(J^0=I_v(M)\oplus I_w(M)\oplus I_e(M)\), followed by \(J^1=I_v(M)\oplus I_w(M)\). At \(v\) the differential subtracts the \(v\)-coordinate from the \(e\)-coordinate, and at \(w\) it subtracts the \(w\)-coordinate from the \(e\)-coordinate. At \(e\) its target is zero. The augmentation is diagonal on each surviving pair and identity at \(e\). These maps commute with the two arrows. Pointwise they are the exact sequences \(0\to M\to M^2\to M\to0\), or \(0\to M\to M\to0\). Both \(J^0,J^1\) are acyclic by the preceding paragraph and finite direct sums. The finite-column injective comparison used in Corollary 3.2 applies to this acyclic resolution as well, so its section complex computes cohomology. That complex is

with these choices of degree-one generators. There are no strict chains of length two. Its kernel is the diagonal \(M\), and it is surjective, since \((u,v)\) is the image of \((-u,-v,0)\). Thus each \(Z_i\) has \(H^0=M\) and no positive cohomology. Every nonempty pair intersection is one vertex, whose section functor is evaluation and is exact, so is acyclic too. The triple intersection is empty. These statements use the actual diagram resolution from the finite-poset prerequisite and include arbitrary modules, not just vector spaces.

Corollary 3.2 therefore applies. In the order \(Z_0,Z_1,Z_2\), the global closed-cover complex is

Its kernel is the diagonal \(M\). Its image is exactly the kernel of \((b_{01},b_{02},b_{12})\mapsto b_{01}-b_{02}+b_{12}\): the displayed composite is zero, and conversely a tuple in that kernel is the image of \((0,b_{01},b_{02})\). The latter map onto \(M\) is surjective, using \((0,0,m)\). Consequently

No division, rank argument over a field or commutativity of \(R\) was used. The nonzero degree-one group also shows why the stalk contractions in Theorem 2.1 are not a global contraction of the section complex.

Exercise 3 — an infinite closed cover fails the argument

Advanced boundary case. Cover \(\mathbb R\) by all its singleton closed subsets, and take the constant sheaf with a nonzero abelian group \(M\). Does the analogous product of closed-restriction section groups compute degree zero? Locate the failure in the stalk proof.

Solution. Distinct singletons have empty intersection. Thus the normalized candidate section complex has degree zero \(\prod_{x\in\mathbb R}M\) and zero degree one. Its zeroth cohomology is that entire product. But sections of the constant sheaf on the connected real line are constant, so its actual zeroth cohomology is \(M\), mapping diagonally into the product. The map is not surjective: choose nonzero \(m\in M\) and assign value zero on negative real numbers and \(m\) on nonnegative real numbers.

At the sheaf level, the product of the point-supported pushforwards has sections all functions into \(M\) on an open set. The displayed step function has a germ at zero that is not represented by a locally constant section on any neighborhood of zero. Hence the augmentation is not surjective even on that stalk. Stalks commute with the finite sums in (2.1), but they do not commute with this infinite product. The stalk of the product is not obtained by discarding every singleton except the one at the stalk point. This is precisely the step of Theorem 2.1 that was justified by finiteness and is unavailable here.

Exercise 4 — restriction is not support

Advanced hypotheses. Let \(X=\mathbb R\), \(A=(-\infty,0]\), \(B=[0,\infty)\), and \(\mathcal F=M_X\) for a nonzero module \(M\). Compare the section groups of \(\mathcal F[A]\), \(\mathcal F[B]\), \(\mathcal F[\{0\}]\) with sections of \(\mathcal F\) supported in those subsets. Can one replace the terms of (5.1) by supported sections?

Solution. The restrictions of the constant sheaf to either connected half-line are constant, so the first two closed-restriction section groups are both \(M\). The point restriction also has section group \(M\). The section complex in degree zero and one is \(M\oplus M\to M\), with \((a,b)\mapsto b-a\), and its augmented kernel is the diagonal \(M\).

Every nonzero global section of \(M_X\) is a nonzero constant section and has support all of \(\mathbb R\). No such section is supported in \(A\), \(B\), or \(\{0\}\). Thus each corresponding supported-section group is zero. Replacing the middle terms by those groups would give \(M\to0\), which is not injective. The closed-cover resolution uses inverse-image restriction followed by pushforward, and never identifies it with the supported-section functor. This conclusion does not assume or claim acyclicity of arbitrary restrictions merely from the section calculation.

6. Further reading

Pierre Schapira's freely accessible An Introduction to Sheaves on Grothendieck Topologies, October 2026, Section 3.4, treats complexes for finite closed families and an acyclic-cover comparison. The proofs above use the stated injective derived-section construction and the finite closed-family hypotheses. For open covers, including arbitrary set-indexed covers and refinement homotopies, use the companion Čech lesson linked at the beginning.