Soft sheaves on paracompact Hausdorff spaces

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

Flabbiness extends sections from open sets. Softness extends sections along closed sets, retaining their germs in the ambient space. These are different extension properties. On a paracompact Hausdorff space, softness is enough to compute derived global sections. The proof uses a locally finite closed refinement and successive corrections of local lifts; compactness and a countable exhaustion are unnecessary.

Let \(X\) be a paracompact Hausdorff space. Here paracompact means that every open cover has a locally finite open refinement. Fix universes and choice, including a well-ordering of each small index set. Let \(\mathcal R\) be a sheaf of unital rings, possibly noncommutative, and work with left module sheaves. The same arguments apply to abelian sheaves, by taking the constant integer ring sheaf. Cohomology means the right derived functors of \(\Gamma(X,-)\); complexes are bounded below wherever a resolution computation is asserted.

We use Finite closed covers and sheaf cohomology, Section 1, for exact closed inverse image and the closed-pushforward stalk formula. Godement resolutions and flabby sheaf cohomology, Sections 1–2 and 6–7 supplies the abelian category, enough injectives, flabby germ-family sheaves and their injective embeddings. Universal cohomology and acyclic resolutions, Sections 3–4 supplies dimension shifting and full bounded-below acyclic computation. Our closed-cover gluing lemma below extends the finite stalk argument to a locally finite family.

A freely accessible reference for soft sheaves is Jean-Pierre Demailly's Complex Analytic and Differential Geometry, Chapter IV, Section 4.B. For the underlying sheaf operations, another reference is Pierre Schapira's An Introduction to Sheaves on Grothendieck Topologies. The constructions and proofs needed here are given below and in the linked prerequisites.

1. The topological refinement that the proof needs

A family \((E_a)\) is locally finite if each point has an open neighborhood meeting only finitely many members. Any union of a locally finite family of closed sets is closed: near a point outside that union, only finitely many closed sets have to be avoided. A subfamily remains locally finite. The closures of a locally finite family of open sets remain locally finite, since an open set meeting the closure of another open set meets that open set itself.

Lemma 1.1. A paracompact Hausdorff space is regular: if \(x\notin A\) and \(A\) is closed, there is an open neighborhood \(N\) of \(x\) with \(\overline N\cap A=\varnothing\).

Proof. For each \(a\in A\), choose disjoint open neighborhoods \(U_a\) of \(x\) and \(V_a\) of \(a\), using Hausdorffness. Refine the cover consisting of \(X\setminus A\) and all \(V_a\) by a locally finite open cover \((W_t)\), recording a containing cover member for each \(W_t\). If \(W_t\) meets \(A\), that containing member must be some \(V_a\); then \(x\notin\overline{W_t}\), because \(U_a\) is disjoint from \(W_t\). The set

is closed and omits \(x\). Every point of \(A\) belongs to one of these \(W_t\), so \(A\subseteq\operatorname{int}B\). Put \(N=X\setminus B\). Its closure avoids \(\operatorname{int}B\), and hence avoids \(A\). The case \(A=\varnothing\) uses \(N=X\). \(\square\)

Lemma 1.2. Every open cover \((O_a)\) has a locally finite open refinement \((V_t)\) and an assignment \(a(t)\) such that

Its closures \(K_t=\overline{V_t}\) are a locally finite closed cover, and their interiors cover \(X\). Every closed subspace of \(X\) is paracompact Hausdorff.

Proof. For each point choose a containing \(O_a\), then an open neighborhood whose closure lies in it, by Lemma 1.1. Take a locally finite open refinement of these neighborhoods. The closure of each refining member lies in the chosen original \(O_a\), proving (1.2). Local finiteness of the closures was proved above, and each \(V_t\) lies in the interior of its closure. For a closed \(C\subseteq X\), extend any cover by relative opens \(C\cap O_a\) to the cover \((O_a)\) together with \(X\setminus C\). Restrict a locally finite open refinement to \(C\), discarding empty members. A member assigned to \(X\setminus C\) restricts to the empty set; the others refine the prescribed cover. Hausdorffness passes to subspaces. \(\square\)

The refinement can be uncountable. In particular, a union of any initial segment of a well-ordered family \((K_t)\) is closed, because every such segment is still locally finite.

2. Gluing on closed subsets and extending to a neighborhood

For \(C\subseteq X\) closed, write \(\mathcal F|_C=i_C^{-1}\mathcal F\). A section in \(\Gamma(C,\mathcal F|_C)\) consists locally of ambient section germs along \(C\). It is not in general determined just by ordinary function values on \(C\). Inverse image is exact and has the original stalk at every point of \(C\), by the prerequisites.

Lemma 2.1 (locally finite closed gluing). If \((C_t)\) is a locally finite closed cover of a space \(Y\), compatible sections of \(\mathcal F|_{C_t}\) glue uniquely to a section of \(\mathcal F\) on \(Y\). Compatibility means equality as sections of the restricted sheaf on every \(C_t\cap C_u\).

Proof. Use the closed pushforwards \(\mathcal F[C_t]=i_{t*}i_t^{-1}\mathcal F\). Consider

The two arrows restrict from the first and second indices. At a point, choose a neighborhood meeting only finitely many \(C_t\). All other factors restrict to the zero sheaf there. Thus the stalk calculation uses only finite products, for which stalks commute with products. Among the surviving factors, those not containing the point have zero stalk; every factor containing it has stalk \(\mathcal F_y\), and the comparison maps on pair intersections are identities. The kernel of the two arrows is exactly the diagonal copy of \(\mathcal F_y\). The cover guarantees at least one such factor. This proves exactness at the first two positions of (2.1). Apply the left exact global-section functor. Its equalizer is precisely the set of compatible families, proving both existence and uniqueness. For sheaves of sets the identical stalk and equalizer argument uses terminal factors in place of zero factors. \(\square\)

Theorem 2.2 (sections along a closed set). For every sheaf \(\mathcal F\) and every closed \(C\subseteq X\), the natural map is a bijection

The transition maps restrict to smaller neighborhoods. The comparison is natural in sheaf morphisms and in closed-subset restriction. For module sheaves it respects the induced module structures.

Proof. Represent a section along \(C\) by sections \(s_a\in\mathcal F(O_a)\) on an ambient open cover of \(C\). Their germs agree at every point of \(C\) at which both are defined. Add \(X\setminus C\) to the cover of \(X\), with an arbitrary local section there when available; for modules use zero. For general sheaves of sets, omit this extra section and below use only refinement members whose assigned original open meets \(C\).

Apply Lemma 1.2 and write \(V_t\) for the locally finite open refinement with closure in its assigned \(O_{a(t)}\); retain only the members assigned to section-bearing opens. They still cover \(C\). On \(O_{a(t)}\cap O_{a(u)}\), let \(E_{tu}\) be the open set where \(s_{a(t)}\) and \(s_{a(u)}\) have equal germs. Equality of germs is open by its definition. The sets

are closed in \(X\): their closed intersection lies inside both section domains. They omit \(C\), where the germs agree. The family of their possibly nonempty members is locally finite, since near a point only finitely many \(\overline{V_t}\) occur. Thus \(B=\bigcup_{t,u}B_{tu}\) is closed and disjoint from \(C\). On the open neighborhood

the sections on the \(W\cap V_t\) agree on every overlap and hence glue. Their germs along \(C\) are the prescribed section. This proves surjectivity, including sheaves of sets without a global section. If \(C\) is empty, use its empty neighborhood.

If two representatives have the same restricted section, the locus on their common domain where their germs agree is an open neighborhood of \(C\). Restricting to it makes the representatives equal. This is exactly injectivity in (2.2). All the maps used to define the comparison are actual restrictions. Their composition proves naturality, and the addition and scalar formulas on representatives prove the module assertion. \(\square\)

The closedness of \(C\) and the locally finite refinement supply the single neighborhood in this proof. A collection of separate neighborhoods of its points would not by itself be a representative in (2.2).

3. Softness and the supply of soft objects

A module sheaf \(\mathcal S\) is soft if for every closed \(C\subseteq X\), the restriction of ambient germs is onto:

If \(\mathcal S\) is soft, its restriction to a closed \(C\) is soft on \(C\). Indeed any subset closed in \(C\) is closed in \(X\); extend its section to \(X\) by (3.1), then restrict to \(C\). Composition of inverse images identifies the germs in this assertion. The empty closed set imposes no condition.

Proposition 3.1. Every flabby module sheaf on \(X\) is soft. Every injective module sheaf is soft, and every module sheaf embeds into a soft sheaf.

Proof. Theorem 2.2 extends a section along a closed set to one open neighborhood. Flabbiness then extends it to \(X\), proving softness. For the injective assertion, use the monic germ-family embedding into an injective sheaf of the form \(R(J)\) in the Godement lesson. This target is flabby. If the source is already injective, the embedding splits, so the source is a retract of a flabby sheaf. A retract is flabby: include a section into the larger sheaf, extend there, and project back. Thus an injective is soft. Enough injectives now embeds every sheaf into a soft one. \(\square\)

Softness of a restriction to a closed subspace uses a closedness hypothesis. We will apply that fact only on the closed subspaces specified below.

4. Lifting sections across a soft kernel

Theorem 4.1. If

is exact and \(\mathcal A\) is soft, then \(\Gamma(X,\mathcal B)\to\Gamma(X,\mathcal C)\) is surjective. Hence applying global sections to (4.1) gives a short exact sequence. This assertion also holds after restriction to any closed subspace.

Proof. Fix \(c\in\Gamma(X,\mathcal C)\). Sheaf epimorphy gives local lifts on an open cover. Choose a locally finite closed refinement \((K_\alpha)_{\alpha<\kappa}\) as in Lemma 1.2, with a chosen local lift \(b_\alpha\) on a neighborhood of each \(K_\alpha\). Well-order the index set and put

Every \(L_\alpha\) is closed. We construct compatible lifts \(g_\alpha\) of \(c|_{L_\alpha}\) as sections of \(\mathcal B|_{L_\alpha}\). Start with the unique section on the empty set. At a successor stage, the difference between \(b_\alpha\) and \(g_\alpha\) on \(D=K_\alpha\cap L_\alpha\) has zero image in \(\mathcal C|_D\). Exact inverse image and left exact sections identify it with a section \(a_D\) of \(\mathcal A|_D\). Since \(D\) is closed in \(X\), softness extends it to \(a\in\Gamma(X,\mathcal A)\). Replace the new local lift by \(b_\alpha-j(a)\). Its restriction to \(K_\alpha\) now agrees with \(g_\alpha\) on \(D\), and its image is still \(c\). Lemma 2.1 for the two closed pieces glues them into \(g_{\alpha+1}\).

At a limit ordinal \(\lambda\), use the already chosen restrictions on \(K_\beta\), \(\beta<\lambda\). They agree on all pair intersections, since each successor construction retained every earlier restriction. They are a locally finite closed cover of \(L_\lambda\). Lemma 2.1 glues them into a unique \(g_\lambda\), which retains all earlier choices. Transfinite recursion therefore reaches \(L_\kappa=X\), and \(g_\kappa\) is the required global lift. This uses the given small well-ordered index set, not a countable subcover.

The first two positions of the global row are exact by left exactness. Its maps are the given maps in (4.1), so the resulting exact sequence is natural for every morphism of such rows; no functorial choice of a lift is asserted. For a closed subspace \(D\), Lemma 1.2 makes \(D\) paracompact Hausdorff, exact inverse image restricts the row, and \(\mathcal A|_D\) is soft by Section 3. The same argument applies there. \(\square\)

Corollary 4.2. Soft sheaves are closed under extensions and cokernels of monomorphisms between soft sheaves.

Proof. In (4.1), first suppose \(\mathcal A,\mathcal B\) are soft. A section of \(\mathcal C|_D\), for closed \(D\), lifts to \(\mathcal B|_D\) by the closed-subspace part of Theorem 4.1. Extend that lift globally by softness of \(\mathcal B\), then apply \(p\). This proves softness of \(\mathcal C\).

Next suppose \(\mathcal A,\mathcal C\) are soft and take \(s\in\Gamma(D,\mathcal B|_D)\). Extend \(p(s)\) to a global section \(c\) of \(\mathcal C\). Theorem 4.1 gives a global lift \(b\) in \(\mathcal B\). The difference \(s-b|_D\) belongs to \(\Gamma(D,\mathcal A|_D)\); extend it globally by softness of \(\mathcal A\), and add its image to \(b\). This extends \(s\), proving softness of \(\mathcal B\). The zero sheaf is soft, and finite direct sums are soft by their component extension maps. \(\square\)

5. Higher acyclicity and complete soft resolutions

Theorem 5.1. A soft module sheaf \(\mathcal S\) satisfies

Its restriction to every closed \(D\) also satisfies \(H^{q>0}(D,\mathcal S|_D)=0\).

Proof. Embed \(\mathcal S\) into an injective \(\mathcal I\), and put \(\mathcal Q=\mathcal I/\mathcal S\). Proposition 3.1 makes \(\mathcal I\) soft, and Corollary 4.2 makes \(\mathcal Q\) soft. Theorem 4.1 makes \(\Gamma(X,\mathcal I)\to\Gamma(X,\mathcal Q)\) onto. The derived long sequence and acyclicity of the injective therefore give \(H^1(X,\mathcal S)=0\). In higher degrees the same long sequence gives

Induct on \(q\), simultaneously for all soft sheaves. The preceding first-degree vanishing starts the induction, and \(\mathcal Q\) is again soft. This proves (5.1) in every positive degree. A closed \(D\) is paracompact Hausdorff and \(\mathcal S|_D\) is soft, so the same theorem on \(D\) gives the last assertion. \(\square\)

Theorem 5.2. For every bounded-below complex \(\mathcal S^\bullet\) of soft sheaves, its canonical acyclic-computation map is invertible:

Thus every bounded-below soft resolution computes derived sections, with its actual augmentation and maps on all derived morphisms.

Proof. Theorem 5.1 makes every term \(\Gamma(X,-)\)-acyclic. Apply the full bounded-below acyclic-computation theorem from the prerequisites. It identifies (5.3) with the canonical replacement transformation, including its augmentation, shifts and naturality. A quasi-isomorphism between such acyclic-term complexes is sent by \(\Gamma\) to a quasi-isomorphism by that theorem's comparison square. Its localization on these complexes therefore computes derived maps as well, rather than only cohomology groups of one resolution. Enough injectives and Proposition 3.1 provide a bounded-below soft model for every bounded-below sheaf complex. \(\square\)

This class computes ordinary global sections on the specified paracompact Hausdorff space. The proof keeps the closed sets and the locally finite cover throughout; replacing them by arbitrary subsets or an arbitrary nonlocally finite closed family would remove the gluing argument.

6. Local tests, partitions and soft rings

Theorem 6.1 (closed-neighborhood locality). A sheaf is soft on \(X\) if and only if every point has a closed neighborhood \(N\) for which \(\mathcal S|_N\) is soft on \(N\).

Proof. The forward direction is closed-subspace restriction from Section 3; regularity supplies closed neighborhoods. Conversely, cover \(X\) by the interiors of the specified closed neighborhoods, and use Lemma 1.2 to obtain a locally finite closed cover \((K_\alpha)\) with each \(K_\alpha\) contained in one of those interiors. The restriction to each \(K_\alpha\) is soft, since \(K_\alpha\) is closed in its assigned neighborhood.

Given a section \(s\) along a closed \(C\), begin with it on \(C\). Well-order the cover. Suppose a compatible extension has been chosen on the closed set \(C\cup\bigcup_{\beta<\alpha}K_\beta\). Its restriction to the intersection with \(K_\alpha\) extends to all of \(K_\alpha\), by softness on that closed piece. Glue it to the earlier extension by Lemma 2.1. At a limit stage, the family consisting of \(C\) and the preceding \(K_\beta\) is locally finite; Lemma 2.1 again glues all the compatible restrictions. The final section is global and extends \(s\). This proves softness. The argument tests closed neighborhoods; it does not assume that restriction to every open subset preserves softness. \(\square\)

For a global section \(s\), its support is the set of points at which its germ is nonzero. This set is closed, since the zero-germ locus is open. A locally finite family of global sections has a sum: on a neighborhood meeting only finitely many supports take that finite sum, and glue the resulting compatible sections. These operations also define every partial sum of such a family.

Theorem 6.2 (partitions of a soft section). If \(\mathcal S\) is soft and \((O_a)\) is any open cover, every \(s\in\Gamma(X,\mathcal S)\) admits sections \(s_a\) such that

Proof. First choose a locally finite open refinement \((V_t)\) with \(\overline{V_t}\subseteq O_{a(t)}\). Apply Lemma 1.2 once more to obtain a closed cover \((K_t)\) with \(K_t\subseteq V_t\) and interiors covering \(X\), allowing some \(K_t\) to be empty. To justify this common indexing, let \((L_j)\) be a locally finite closed refinement subordinate to the \(V_t\), assign \(L_j\) to a containing \(V_t\), and take the union of all pieces assigned to each \(t\). Each such union is closed by local finiteness, lies in \(V_t\), and contains the interiors of its assigned pieces. The resulting \((K_t)\) is locally finite because \(K_t\subseteq V_t\).

Well-order the indices. Write \(L_t=\bigcup_{u<t}K_u\). Inductively construct \(f_t\) supported in \(V_t\), with the sum of the preceding \(f_u\) equal to \(s\) along \(L_t\), and with every later correction zero along each earlier closed piece. The partial sum is defined, since all supports lie in the fixed locally finite family \((V_t)\). Let \(r_t=s-\sum_{u<t}f_u\), which is zero along \(L_t\). On the closed set

prescribe zero on \(X\setminus V_t\) and \(L_t\), and prescribe \(r_t\) on \(K_t\). These sections are compatible: \(K_t\) misses \(X\setminus V_t\), and the residual vanishes on its intersection with \(L_t\). Finite closed gluing gives a section along \(D_t\), and softness extends it to \(f_t\) on \(X\). Its germs vanish on the closed complement of \(V_t\), so its support lies in \(V_t\). The new residual is zero along \(L_t\cup K_t\). At a limit stage the locally finite sum retains all previous equalities: later corrections have zero germs along every preceding piece. This proves the induction for the full well-ordered index set. The final residual is zero at every point because the \(K_t\) cover, so \(s=\sum_t f_t\).

Finally put \(s_a=\sum_{t:a(t)=a}f_t\). Each subfamily is locally finite, and the union of its closed supports is closed and contained in \(O_a\). Thus \(\operatorname{supp}(s_a)\subseteq O_a\). Near a point only finitely many \(V_t\) meet, so only finitely many of the grouped supports meet; grouping also retains the sum. This proves the stated assertion for the original cover. \(\square\)

Theorem 6.3 (modules over a soft ring sheaf). If the underlying additive sheaf of \(\mathcal R\) is soft, every left \(\mathcal R\)-module sheaf is soft. Consequently \(\Gamma(X,-)\) is exact on \(\operatorname{Mod}(\mathcal R)\), and every such module sheaf has zero positive global cohomology. This holds for noncommutative unital ring sheaves too.

Proof. Let a module section along closed \(C\) be represented by \(f\) on an open neighborhood \(U\), using Theorem 2.2. On the closed set \(C\cup(X\setminus U)\), prescribe the ring section one along \(C\) and zero along \(X\setminus U\). The two pieces are disjoint closed subsets of their union and hence are relatively open there, so these sections glue. Softness of the additive ring sheaf extends the resulting section to \(\rho\in\Gamma(X,\mathcal R)\). Its germ is one on \(C\) and zero on \(X\setminus U\). Hence \(\rho=0\) on an open neighborhood of that complement. Glue \(\rho f\) on \(U\) to zero on this latter neighborhood. The resulting global module section agrees with \(f\) on a neighborhood of \(C\), where \(\rho=1\), so it extends the required germs. This proves softness. Left multiplication uses the unit and requires no commutation of scalars.

Every kernel in a short exact module row is now soft, so Theorem 4.1 makes global sections exact. Theorem 5.1 gives the cohomological assertion. Applied to \(s=1\) in \(\mathcal R\), Theorem 6.2 also gives a locally finite partition of unity by ring sections subordinate to any specified open cover. \(\square\)

7. Four exercises with complete solutions

Exercise 1 (introductory: discrete and connected spaces). Prove that every module sheaf on a discrete space is soft and has zero higher global cohomology. Show that the constant integer sheaf on \(\mathbb R\) is not soft, using the closed subset \(\{0,1\}\).

Solution. The singleton cover gives \(\Gamma(U,\mathcal F)=\prod_{x\in U}\mathcal F_x\) on a discrete space, with coordinate restrictions. A section on any closed subset, which here is any subset, extends by zero at the missing points. This proves softness. The singleton cover refines every open cover and is locally finite, so the space is paracompact Hausdorff. Theorem 5.1 gives higher vanishing. One can also compute it directly: global sections are the product of component modules, and products of module epimorphisms are onto by choice, so this global-section functor is exact.

On \(\mathbb R\), the constant integer sheaf has locally constant integer-valued functions as sections. A global such function is constant because \(\mathbb R\) is connected. On the discrete two-point closed subspace its restricted sheaf has independent integer stalks, giving \(\mathbb Z^2\). The restriction map is the diagonal \(\mathbb Z\to\mathbb Z^2\); it misses \((0,1)\). This disproves softness. The restricted section specifies the two local integer germs, not a single global integer.

Exercise 2 (intermediate: soft but not flabby). Let \(\mathcal C\) be the sheaf of continuous real-valued functions on \(\mathbb R\). Prove that it is soft and not flabby. Give the closed-germ extension explicitly, rather than use extension of function values as a substitute. Show that the same cutoff argument works on any metrizable paracompact Hausdorff space.

Solution. First, \(\mathbb R\) is paracompact. Given an open cover, choose a finite subcover of each compact interval \([n,n+1]\), \(n\in\mathbb Z\). Intersect each chosen member with \((n-1,n+2)\). The resulting family refines the cover and covers \(\mathbb R\); near any point only finitely many integers \(n\) can contribute, and each contributes only finitely many sets. Thus it is locally finite.

Let \(A\) be closed and take a section of \(\mathcal C|_A\). Theorem 2.2 represents it by a continuous function \(f\) on an open neighborhood \(U\). If \(A\) is empty, use zero; if \(U\) is the whole space, use \(f\). Otherwise use a compatible metric and, for an open \(T\supseteq A\), put

For nonempty \(A\), this contains \(A\), and \(\overline{V(T)}\subseteq T\). Indeed at a point outside \(T\) the second distance is zero and the distance to the closed set \(A\) is positive, so the weak inequality for the closure cannot hold. Set \(V_2=V(U)\), \(V_1=V(V_2)\). Then \(A\subseteq V_1\), \(\overline{V_1}\subseteq V_2\), and \(\overline{V_2}\subseteq U\). The two nonempty disjoint closed sets \(P=\overline{V_1}\), \(Q=X\setminus V_2\) give

The denominator is positive everywhere. This continuous function is one on \(V_1\), zero outside \(V_2\), and has support in \(\overline{V_2}\subseteq U\). Define \(g=\chi f\) on \(U\) and zero outside \(U\). This is continuous: outside \(U\) each point has a neighborhood disjoint from \(\overline{V_2}\), where the function is zero. On \(V_1\) it equals \(f\), so its germs along \(A\) are the required ones. This proves softness and proves the asserted metric version. In particular, matching values only on \(A\) would have been insufficient; equality on \(V_1\) gives equality of germs.

The section \(t\mapsto1/t\) on \((0,1)\) has no continuous extension to \(\mathbb R\), by continuity at zero and its unbounded values on positive sequences approaching zero. Thus the sheaf is not flabby. Its higher global cohomology nevertheless vanishes by Theorem 5.1.

Exercise 3 (advanced: a soft quotient and a jump class). On \(\mathbb R\), let \(\mathcal D(U)\) be all real-valued functions on \(U\), let \(\mathcal C\) be the continuous-function subsheaf, and let \(\mathcal Q=\mathcal D/\mathcal C\) be the sheaf quotient. Compute \(\Gamma(\mathbb R,\mathcal Q)\) and its positive cohomology. For \(h(t)=0\) if \(t<0\) and \(h(t)=1\) if \(t\geq0\), show that its class gives a monomorphism \(i_*\mathbb R\to\mathcal Q\), where \(i:\{0\}\hookrightarrow\mathbb R\). Check linearity for the action of \(\mathcal C\), with evaluation at zero on the skyscraper.

Solution. Functions glue pointwise, so \(\mathcal D\) is a sheaf; extending by zero proves flabbiness. It is soft by Proposition 3.1. The continuous subsheaf is soft by Exercise 2's explicit cutoff proof. Corollary 4.2 makes the quotient soft. Theorem 4.1 and its left exact beginning give the actual identification

The vanishing is Theorem 5.1. The equality in (7.3) means that every global quotient section has a single global function representative; it is stronger than local representatives in the definition of a quotient sheaf.

For an open containing zero, send \(a\in\mathbb R\) to \([ah]\) on that open. On an open omitting zero, \(h\) is continuous, even if the open has components on both sides, so this class restricts to zero as required for a skyscraper map. A nonzero \(ah\) is discontinuous at zero on every neighborhood; its quotient germ is nonzero. Hence the map is injective at zero, and all its other source stalks are zero. The stalk criterion proves monicity. Finally, for continuous \(b\),

is continuous, since its two limiting values at zero are zero. Thus \([bah]=b(0)[ah]\). This is exactly the claimed module linearity, including the actual supported jump class.

Exercise 4 (advanced: the circle from a soft exponential resolution). On the circle \(S^1\), let \(\mathcal R\) be the additive sheaf of continuous real-valued functions and \(\mathcal T\) the multiplicative abelian sheaf of continuous circle-valued functions. Prove that both are soft, and use the exponential sequence to calculate \(H^q(S^1,\mathbb Z)\) in every degree. Identify the map whose cokernel gives degree one.

Solution. The circle is compact Hausdorff and hence paracompact: a finite subcover is already a locally finite refinement. Its usual metric and the cutoff construction of Exercise 2 prove softness of \(\mathcal R\).

For \(\mathcal T\), represent a section along a closed subset \(A\) by \(g:U\to S^1\) using Theorem 2.2. The empty subset is trivial. If \(A=S^1\), the section is already global. Otherwise compactness gives finitely many closed arcs lying in \(U\) whose interiors cover \(A\); their union \(K\) is a finite union of closed arcs. If \(K=S^1\), again \(g\) is already global. In the other case the complementary components have finitely many pairs of endpoints in \(K\). On each complementary closed arc choose a continuous path joining the two endpoint values of \(g\), for example by choosing real arguments of those two values and interpolating their arguments. These paths and \(g|_K\) agree at endpoints. Their pasting is continuous: the inverse image of any closed set is a finite union of subsets closed in the closed pieces, and hence is closed in the circle. The resulting global map equals \(g\) on the interior of \(K\), a neighborhood of \(A\), and therefore extends its germs. This proves softness of \(\mathcal T\).

The sequence of abelian sheaves

is exact. Its kernel is the sheaf of locally constant integer-valued functions. Every point of \(S^1\) has an arc with a continuous real argument: after rotation, on the arc with positive real part use the continuous function \(\arctan(\operatorname{Im}z/\operatorname{Re}z)\). Dividing by \(2\pi\) and adding the rotation argument gives a local inverse of \(E\). Pulling these arcs back along a circle-valued function gives local lifts, proving sheaf epimorphy. Its two nonzero resolution terms are soft, so Theorem 5.2 computes cohomology by the actual group map

Its kernel is \(\mathbb Z\), since an integer-valued continuous function on the connected circle is constant. To compute its cokernel, parametrize the circle by \(t\mapsto e^{2\pi i t}\), \(0\leq t\leq1\). Every continuous \(g:S^1\to S^1\) has a continuous real argument \(a(t)\) along this interval. Here is the finite subdivision needed to construct it. Cover the interval by inverse images of argument arcs. At each \(t\), choose \(\epsilon_t>0\) so that the interval of radius \(2\epsilon_t\) lies in one member of this cover, relatively to \([0,1]\). Finitely many intervals of radius \(\epsilon_t\) cover \([0,1]\). Take \(\delta\) smaller than all their radii. Any subinterval of length less than \(\delta\), by choosing one point in it and a covering half-radius interval, lies in the corresponding full-radius interval. A partition with mesh less than \(\delta\) therefore has each piece mapped into an argument arc. Choose a branch on the first piece and adjust each successive branch by an integer to match at the common endpoint. Gluing gives \(a\). Any two such arguments differ by a continuous integer-valued function on the connected interval, hence by a constant integer. Thus

is well defined. Adding arguments shows that this is a homomorphism for multiplication of functions. It is onto because \(g(z)=z^m\) has degree \(m\). It vanishes precisely on the image of (7.6): a global real argument has equal endpoint values, and conversely an interval argument with equal endpoints descends to a continuous real function on the circle. Therefore the cokernel of (7.6) is \(\mathbb Z\), with its specified degree map. We conclude

The connecting map from a global circle-valued function to \(H^1(S^1,\mathbb Z)\) is the quotient map induced by (7.6); using it to identify the cokernel gives exactly the degree just computed.

References

• Jean-Pierre Demailly, Complex Analytic and Differential Geometry, Chapter IV, Section 4.B, “Soft Sheaves over Paracompact Spaces.” Freely accessible author edition.
• Pierre Schapira, An Introduction to Sheaves on Grothendieck Topologies. Freely accessible notes.