Subanalytic chains and closed cycle supports

A subanalytic chain records an orientation and a coefficient on each smooth piece of a given dimension. Its support includes limiting points of those pieces. A cycle satisfies an additional compatibility condition at those limiting points. The dualizing object expresses that condition even at a branch or a singularity, where an orientation line on a manifold would be insufficient.

Original lesson text and solutions: CC0 1.0 Universal. Human mathematical sources retain their named credit and their own terms.

We use compatible subanalytic triangulations, the locally closed support and composition maps, and manifold cohomological dimension and orientation. These supply the precise geometric and sheaf operations used below. The dualizing complex from oriented simplices fixes the incidence signs used in the examples. Subanalytic chains were introduced for this purpose by M. Kashiwara, Index theorem for constructible sheaves, Astérisque 130 (1985), §1. Proper images, products, the local contraction proving the full chain resolution, coefficient flatness and intersections are treated in the following lessons.

Orientations give a sheaf of chains

Let $A$ be a commutative ring of finite global dimension. Let $X$ be a real analytic manifold, Hausdorff and countable at infinity, of dimension $n$; for several components use a uniform finite dimension bound and work component by component. Neither characteristic zero nor a field nor a global orientation is assumed. All tensor products in the initial chain definitions are ordinary sheaf tensor products over $A_X$.

For $p\geq0$, form an $A$-module from oriented subanalytic $p$-dimensional submanifolds $S\subset X$, with three rules:

[S_1\amalg S_2]=[S_1]+[S_2],\qquad [S]=[V]\quad(V\subset S\text{ open and dense}),\qquad [S^{\mathrm{op}}]=-[S]. \qquad\text{(1)}

The orientations on the first two expressions are inherited from the indicated pieces. The dense subset is subanalytic. The disjoint union rule concerns unions that are again submanifolds of the indicated dimension; a compatible finite decomposition and dense deletion handle the general local subdivision. A lower-dimensional piece does not become a $p$-chain. Set the module to zero for $p<0$.

Restriction to open sets gives a presheaf. Denote its sheafification by $\mathcal C_p$. For an arbitrary sheaf $F$ of $A$-modules put

\mathcal C_p(F)=\mathcal C_p\otimes_A F. \qquad\text{(2)}

Sheafification is essential: a global section may require an unbounded number of different coefficients on a locally finite family, although each germ has a finite description. Exercise 5 constructs one.

For a $p$-manifold $S$, let $\operatorname{or}_S$ be its integral orientation sign line tensored with $A$. An integral orientation chooses a generator of that line. Reversal multiplies it by $-1$, including when these become the same scalar in characteristic two. The manifold-duality normalization is

\omega_S\simeq\operatorname{or}_S[p], \qquad H^{-p}(\omega_S)=\operatorname{or}_S. \qquad\text{(3)}

Thus a degree-$p$ geometric piece will occupy cohomological degree $-p$ in the chain complex.

The lowest dualizing degree on a singular support

For any locally closed subanalytic subset $S$ let $j_S:S\to X$ be its inclusion. Its dualizing object is defined by exceptional composition, $\omega_S=j_S^!\omega_X$. If $\dim S\leq p$, then

H^q(S;F)=H_c^q(S;F)=0\quad(q>p),\qquad H^q(\omega_S)=0\quad(q<-p). \qquad\text{(4)}

Here the first assertion holds for every sheaf $F$ on $S$, with no coefficient finiteness condition.

Proof of the dimension bound. Triangulate the ambient analytic manifold compatibly with $S$. Intersect $S$ with each closed skeleton. These intersections are closed in $S$; their successive differences are disjoint unions of open simplices of the indicated dimension. A neighborhood of each point meets finitely many simplices. In a fixed-dimensional layer every simplex is open, since all its remaining faces have been removed. The layer is therefore a possibly disconnected manifold. There are only the dimension levels $0,\ldots,p$, regardless of the number of simplices globally. This constructs the required finite closed filtration.

The manifold dimension theorem (M2)–(M4) supplies a c-soft resolution of length $r$ for every sheaf on an $r$-dimensional layer. A locally closed subspace here is locally compact and second countable, hence countable at infinity. The same theorem proves that c-soft sheaves are soft and acyclic for ordinary sections on these spaces; it does not identify ordinary sections with compactly supported sections.

Open extension by zero preserves c-softness in this situation. To see the relevant extension property, a section of $j_!E$ on a compact subset has its nonzero germ support in a compact subset of the open stratum, away from its complement. Choose a compact neighborhood of that support inside the stratum. Prescribe the original values on its intersection with the given compact set, and zero on an outer compact boundary, where those prescriptions agree. C-soft extension then gives a section; it vanishes on a collar of that boundary, so its restriction extends by zero with compact support. Its extension by zero has all the desired values on the ambient compact set. This also applies to a disjoint union: a compactly supported section meets only finitely many members of a locally finite family.

Consequently the open-stratum term in $0\to j_!F|_U\to F\to i_*F|_{S\setminus U}\to0$ has ordinary and compact-support cohomology zero above $r$. For ordinary sections we use c-soft acyclicity on a countable-at-infinity locally compact space; for compact sections we use compact-support acyclicity. The closed term has the bound of the next filtration stage. Induction and the two long exact sequences give the first part of (4). This proof does not commute arbitrary ordinary global sections with a filtered colimit.

The dual-sections adjunction (EX.32)–(EX.33), together with open restriction of the dualizing object, gives on every relatively open neighborhood $V\subset S$,

R\Gamma(V;\omega_S|_V) \simeq R\operatorname{Hom}_A(R\Gamma_c(V;A),A). \qquad\text{(5)}

The complex inside the dual has no cohomology above $p$. Derived Hom into a module in degree zero therefore has no cohomology below $-p$. Taking stalks proves the second part of (4). The same bound also gives

H^{-p}(S;\omega_S) \simeq\Gamma(S;H^{-p}\omega_S) \simeq\operatorname{Hom}_A(H_c^p(S;A),A). \qquad\text{(6)}

Indeed separate $H_c^p(S;A)[-p]$ from the lower truncation in (5). The dual of the lower truncation begins in degree $1-p$; only the ordinary Hom of the top group contributes in degree $-p$. Likewise the sheaf-to-global spectral sequence has only its $H^0(S;H^{-p}\omega_S)$ term in that degree. No assertion that all higher Ext groups vanish is needed. $\square$

Write $T_S=j_{S*}H^{-p}(\omega_S). \qquad\text{(7)}$

The ordinary direct image in this formula retains boundary germs. It is not open extension by zero. For a locally closed subset $V\subset S$, support adjunction and the lower bound give

j_{V*}H^{-p}(\omega_V) \simeq H^0_V(T_S). \qquad\text{(8)}

For locally closed $V$, $H^0_V$ means degree zero of the sheaf operation $R\mathcal Hom(A_V,-)$, with $A_V$ extended by zero. In particular an open $V$ gives ordinary direct image after restriction. For closed $V$ it is the usual subsheaf of sections supported there.

To prove (8), first apply internal ordinary adjunction to the extension-by-zero sheaf $A_V$. Its restriction to $S$ is the extension-by-zero constant sheaf of $V\subset S$. Factoring that locally closed inclusion into an open and a closed inclusion, support adjunction gives $R\Gamma_V(Rj_{S*}\omega_S)\simeq Rj_{V*}\omega_V$. Exceptional composition supplies the dualizing object on $V$; the ordinary direct images compose. These are the locally closed support comparisons (EX.13)–(EX.16), with their natural restriction and closed-trace maps.

Both derived support and derived ordinary image preserve the lower bound $-p$. Apply either functor to the truncation triangle separating the sheaf $H^{-p}(\omega_S)[p]$ from the part beginning in degree $1-p$. The latter contributes nothing in degree $-p$. The former contributes the degree-zero ordinary image or degree-zero support of the lowest sheaf. Consequently $H^{-p}(Rj_{S*}\omega_S)=T_S$, and taking degree $-p$ of the preceding natural comparison gives precisely (8). This proves the identification of its maps as well as its objects.

Two consequences control the singular points. If $V$ is closed in $S$ and $\dim(S\setminus V)<p$, the closed-support trace induces an isomorphism

j_{V*}H^{-p}\omega_V \xrightarrow{\sim} T_S. \qquad\text{(9)}

Use the localization triangle and (4) on the lower-dimensional open complement; its ordinary derived image starts in degree $1-p$. Let $S$ be closed, with $\dim S\leq p$, and let $U$ be a subanalytic open subset of $S$ contained in and dense in its $p$-dimensional regular part. Set $B=S\setminus U$. The lower-dimensional components of $S$ lie in $B$; the subanalytic frontier and regular-locus bounds give $\dim B<p$. Then

0\longrightarrow T_S \longrightarrow j_{U*}\operatorname{or}_U \longrightarrow j_{B*}H^{1-p}\omega_B \qquad\text{(10)}

is exact. This time use the closed-support triangle for $B$; $\dim B<p$ kills its degree-$-p$ term. The displayed sequence makes no claim of surjectivity on the right. The last arrow measures failure of the orientations and coefficients on the regular pieces to fit at the singular set.

Directed comparison of locally closed representatives

Let $\mathscr L_p$ be the family of locally closed subanalytic subsets of dimension at most $p$. Use a comparison $S_1\preceq S_2$ when there is a subanalytic $V\subset S_1\cap S_2$ such that

V\text{ is open in }S_1,\quad V\text{ is closed in }S_2,\quad \dim(S_1\setminus V)<p. \qquad\text{(11)}

This is a directed preorder; one may quotient mutually comparable objects to obtain an ordered index. Distinct subsets differing by a detached lower-dimensional piece can be mutually comparable. Nothing below uses literal antisymmetry of subsets.

The comparison gives

T_{S_1}\longrightarrow T_{S_2} \qquad\text{(12)}

by restricting to $V$, then using the closed-support trace from $V$ to $S_2$. These maps are independent of the witness. If $V'$ is another witness, $V\cap V'$ is open and closed in $V$, with lower-dimensional complement. Equation (9) makes passage to that intersection an isomorphism. Both maps then factor through the same trace and restriction. For composition, let $V_{12}$ witness $S_1\preceq S_2$ and $V_{23}$ witness $S_2\preceq S_3$. The intersection $V_{12}\cap V_{23}$ is open in $V_{12}$, hence in $S_1$, and closed in $V_{23}$, hence in $S_3$. Its complement in $S_1$ is the union of $S_1\setminus V_{12}$ and a subset of $S_2\setminus V_{23}$; both have dimension less than $p$. Thus it is an actual comparison witness. Restrict the closed-trace map for $V_{12}\subset S_2$ to the open $V_{23}\subset S_2$. The open base-change comparison identifies it with the trace from their intersection. Composition of closed traces then gives the same map as this composite witness. Reflexivity uses $V=S$. These facts prove the preorder and functoriality, rather than presupposing them.

Here is a common upper representative for $S_1,S_2$: $R=(S_1\setminus\partial S_2)\cup(S_2\setminus\partial S_1), \qquad \partial S=\overline S\setminus S. \qquad\text{(13)}$

The frontier of a locally closed subanalytic set is closed and has smaller dimension. Indeed $R=(\overline S_1\cup\overline S_2)\setminus(\partial S_1\cup\partial S_2)$, which is locally closed. The witness $V_i=S_i\setminus\partial S_j$ is open in $S_i$ and closed in $R$: a point of $R$ in $\overline{V_i}\setminus V_i$ would lie in the frontier of $S_i$, removed from the other piece, or in the frontier of $S_j$, removed from the first. Its complement in $S_i$ has dimension less than $p$. Hence $S_i\preceq R$.

These comparisons yield a concrete description of chains:

\mathcal C_p\simeq\underset{S\in\mathscr L_p}{\operatorname{colim}}T_S. \qquad\text{(14)}

Proof. If $\dim S<p$, (4) gives $T_S=0$, and the empty set is a comparison successor. Otherwise remove all lower-dimensional pieces and the singular set, leaving the $p$-dimensional regular part $U$. It is open in $S$, closed in itself and has lower-dimensional complement, so it is a comparison successor in (11). On $U$, (3) gives its orientation line. Thus it suffices to compare the chain presentation with orientation sections on smooth top-dimensional pieces. Work in a relatively compact subanalytic neighborhood. A finite compatible triangulation subdivides the relevant pieces into oriented open $p$-cells; after deletion of lower-dimensional faces these form a dense subset. A section of an orientation line has a locally constant coefficient on each such cell. This produces the map from (1) to (14).

Conversely those cell coefficients are represented by a finite sum of oriented symbols on a sufficiently small neighborhood. Two representations can be compared on a common finite subdivision. Their difference is zero precisely when every open $p$-cell coefficient is zero; lower-dimensional leftovers are deleted by (1). Orientation reversal gives exactly the opposite coefficient. This proves surjectivity and injectivity on germs. Compatible restriction then proves the sheaf isomorphism. Nonorientable pieces cause no obstruction to this argument: the cells trivialize their sign lines, and their transition signs are retained when the pieces are compared. $\square$

Closed supports define cycles

Let $\mathscr K_p$ consist of closed subanalytic subsets of dimension at most $p$, ordered by inclusion. Their unions provide common upper bounds. For $S\subset S'$, (8) identifies $T_S$ with the subsheaf of $T_{S'}$ supported on $S$. In particular the transition is injective. Define

\mathcal Z_p=\underset{S\in\mathscr K_p}{\operatorname{colim}}T_S. \qquad\text{(15)}

There is an injection $\mathcal Z_p\hookrightarrow\mathcal C_p$. To check it, represent a germ on a closed $S$. If a later chain comparison killed it, restriction to the common dense top-dimensional regular pieces would be zero. The injection in (10), followed by a compatible subdivision, then makes the original germ zero. Thus a coefficient compatible on a closed support is not lost in the larger locally closed colimit.

An open interval's chain illustrates the distinction. Its orientation contributes through ordinary direct image from the interval, with nonzero germs at both endpoints. The lowest dualizing sheaf of the closed interval has zero endpoint stalks, as the earlier cellular calculation proves. A global nonzero oriented interval therefore does not furnish a cycle on that closed interval.

For an arbitrary sheaf $F$ define $\mathcal Z_p(F)=\mathcal Z_p\otimes_A F$. At this stage we have proved the injection and the kernel statement below for the untensored sheaves. Exactness after arbitrary tensoring needs the flatness argument supplied by the full local chain resolution.

The boundary comes from the frontier triangle

Take $S\in\mathscr L_p$, and put $B=\overline S\setminus S$. The localization triangle on $\overline S$ is

\omega_B\longrightarrow\omega_{\overline S} \longrightarrow Rj_*\omega_S\longrightarrow\omega_B[1], \qquad\text{(16)}

where the first and last objects are pushed forward from the closed frontier as necessary. Since $\dim B<p$, its cohomology gives

0\longrightarrow T_{\overline S} \longrightarrow T_S \xrightarrow{\ b_S\ }T_B^{\,p-1}. \qquad\text{(17)}

The superscript reminds us to take $H^{-(p-1)}(\omega_B)$ in the last term. Map that term into $\mathcal Z_{p-1}$. The compatibility with (12) can be checked on the same finite compatible subdivisions used for (14). On an oriented open $p$-cell the connecting map in (16), localized at an open codimension-one face, is the dual of the compact-support boundary of a half-collar. The oriented-boundary computation (11)–(12) identifies it with outward-normal-first incidence. On an interval this is terminal endpoint minus initial endpoint. After subdivision, each artificial interior face occurs twice with opposite induced orientations; their two closed-face maps agree and their coefficients add to zero over every $A$. Faces of codimension at least two contribute no top $(p-1)$-coefficient. Injection (10), applied in degree $p-1$, shows that equality of these regular-face coefficients is equality of the resulting closed cycles, including their singular germs. Thus replacing a representative by its dense regular pieces, refining it, reversing an orientation or composing a comparison preserves this boundary map. For $p=0$ the target is zero and the frontier is empty. Therefore (14) yields

b_p:\mathcal C_p\longrightarrow\mathcal Z_{p-1}, \qquad \partial_p:\mathcal C_p\xrightarrow{b_p}\mathcal Z_{p-1} \hookrightarrow\mathcal C_{p-1}. \qquad\text{(18)}

Equations (15) and (17) prove $\ker\partial_p=\mathcal Z_p,\qquad \partial_{p-1}\partial_p=0. \qquad\text{(19)}$

For the kernel assertion, a germ killed by $b_p$ can be represented by $T_S$. Its boundary in $T_B^{p-1}$ cannot become zero merely by enlarging a closed support, because those transitions are injective. Exactness of (17) therefore lifts the germ to $T_{\overline S}$, which is a cycle representative. Conversely a closed cycle has empty frontier and zero boundary. The image of the first boundary is already in $\mathcal Z_{p-1}$, so the second boundary vanishes. This proof does not assert that $b_p$ is onto.

Define a cohomological sheaf complex by $\mathcal C^{-p}=\mathcal C_p,\quad d^{-p}=\partial_p. \qquad\text{(20)}$

It is bounded in degrees $[-n,0]$. Since $X$ itself is terminal among closed supports of dimension at most $n$, $\mathcal Z_n=H^{-n}\omega_X=\operatorname{or}_X$. Its inclusion as the degree-$-n$ kernel defines a morphism $\omega_X=\operatorname{or}_X[n]\longrightarrow\mathcal C. \qquad\text{(21)}$

It is canonical with the orientation and trace conventions fixed above. The local half-ray contraction proves that (21) is a quasi-isomorphism by killing the lower-degree local cycle classes. The kernel identity (19) gives the complex and its top class; that local contraction supplies the additional exactness.

All these constructions commute with restriction to an open subset. Indeed the support representatives and traces restrict, every local representative can be chosen in a relatively compact subanalytic neighborhood of the point under examination, and sheafification is determined by those germs. Hence both $\mathcal C_p^X|_U=\mathcal C_p^U$ and $\mathcal Z_p^X|_U=\mathcal Z_p^U$.

Cutting chains proves softness

Let $W\subset X$ be subanalytic and open. On each oriented generator define $P_W[S]=[S\cap W]$, taking the empty intersection to zero and retaining the induced orientation. Intersection preserves disjoint union, orientation reversal and dense deletion inside the surviving open part. It therefore respects all three relations (1), commutes with restriction and induces a sheaf endomorphism of $\mathcal C_p$. Under (14), this is restriction from $T_S$ to the ordinary image of the dualizing sheaf on $S\cap W$, followed by its canonical map to the chain colimit. It satisfies

P_W^2=P_W,\qquad P_W|_W=\mathrm{id},\qquad P_W|_{X\setminus\overline W}=0. \qquad\text{(22)}

The closure in the last expression is necessary: a cut interval has new endpoint germs on $\partial W$. Also $P_W$ need not commute with the boundary; cutting can create a boundary.

For every sheaf $F$, $\mathcal C_p(F)$ is soft. Here softness means extension of a section from every closed subset, with no constructibility, local freeness or coefficient flatness assumption on $F$.

Proof. First we give the needed neighborhood representation for a section of any sheaf on a closed $Z$. Represent it near each point by a section $s_i$ on an open $U_i$. Choose a locally finite cover of a neighborhood of $Z$ by smaller open sets $V_i$, with $\overline V_i\subset U_i$. This is obtained by the compact-exhaustion and finite-ball construction below, applied to the varying neighborhoods $U_i$. In $U_i\cap U_j$, the locus where the germs of $s_i,s_j$ disagree is closed. Intersect that locus with $\overline V_i\cap\overline V_j$; the result is closed in $X$ and misses $Z$. The family of these intersections is locally finite, so their union $D$ is closed and misses $Z$. On $U=(\bigcup_i V_i)\setminus D$, all the sections agree on overlaps and glue. This proves that the prescribed section extends to an actual open neighborhood of $Z$.

Now choose a subanalytic open $W$ with $Z\subset W\subset\overline W\subset U. \qquad\text{(23)}$ Such a $W$ exists even when $Z$ is not subanalytic. Take a compact exhaustion $K_r\subset\operatorname{int}K_{r+1}$, with interiors covering $X$, and take negatively indexed sets to be empty. Each compact shell $Z\cap(K_r\setminus\operatorname{int}K_{r-1})$ is covered by finitely many analytic coordinate balls with compact closures in $U\cap(\operatorname{int}K_{r+1}\setminus K_{r-2})$. At a shell point such a ball exists, since $K_{r-2}\subset\operatorname{int}K_{r-1}$. Choose finitely many to cover that shell. Their closures form a locally finite family: a neighborhood with compact closure in some $\operatorname{int}K_m$ misses all sufficiently late shells' balls. The balls cover $Z$. Let $W$ be their union. Each ball is semianalytic in its chart and empty near points outside its compact chart closure, so their locally finite union is subanalytic. The union of their compact closures is closed and lies in $U$, proving (23). The same construction works with any prescribed open neighborhood at each chosen centre, which justifies the earlier $V_i$ selection.

Apply $P_W\otimes\mathrm{id}_F$ to the neighborhood extension. By (22) its support is contained in $\overline W$, so it extends by zero from $U$ to $X$, and it still equals the prescribed section near $Z$. This is the required global extension. The argument uses an actual idempotent on chains, without assuming that tensor product preserves a prior exact sequence. $\square$

Together with the local contraction proving (21), softness supplies an acyclic chain model. Softness of chains does not assert softness of cycles or flabbiness of chains; Exercise 5 exhibits the failure of flabbiness. Exact coefficient kernels require the cycle-flatness proof, which uses the completed local resolution and chain-stalk flatness from the products lesson.

Exercises with complete solutions

Deleting a point keeps a chain and cancels its interior boundary

Difficulty: Introductory.

Orient the real line to the right. Compare the chain on $(-2,3)$ with the sum of its pieces $(-2,0)$ and $(0,3)$. Determine the germ of $\mathcal C_1$, its boundary and its cycle kernel at $0$. Then cut the whole-line chain by $W=(0,3)$.

Solution. Dense deletion gives $[(-2,3)]=[(-2,0)]+[(0,3)]$. Near $0$, every one-dimensional subanalytic piece is, after dense deletion and subdivision, a left or right interval germ. Thus $(\mathcal C_1)_0=A\oplus A$, with coordinates $a_-$ and $a_+$ using the positive line orientation. Also $(\mathcal C_0)_0=A$, generated by $[0]$. Terminal-minus-initial incidence gives $\partial(a_-,a_+)=a_- -a_+,\qquad (\mathcal Z_1)_0=\{(a,a):a\in A\}. \qquad\text{(24)}$ The original interval has germ $(1,1)$; the two pieces contribute $(1,0)$ and $(0,1)$. Their boundaries at zero are respectively $+[0]$ and $-[0]$, which cancel. The remaining global boundary is $[3]-[-2]$, so the interval is not a global cycle.

The whole-line cycle is sent by $P_W$ to $[(0,3)]$, whose germ at zero is $(0,1)$ and whose boundary is $[3]-[0]$. In particular the cutoff has a nonzero germ at a point outside $W$, but on $\overline W$. It is idempotent and is not a chain map. These calculations remain valid over characteristic two: subtraction becomes addition and the kernel is still the diagonal.

A singular vertex imposes a conservation law

Difficulty: Intermediate.

Let $S\subset\mathbb R^2$ be three closed rays meeting only at their common endpoint $v$, oriented away from $v$. Over $\mathbb Z$, find the compatibility condition for a chain with ray weights $(a,b,c)$ to be a cycle. Test $(2,-3,1)$ and $(2,3,1)$. Explain what changes in the lowest degree if an isolated point $w\notin S$ is adjoined.

Solution. Each outward ray has initial endpoint $v$, so its contribution there is the negative of its coefficient. There are no finite outer endpoints. Thus the boundary is $-(a+b+c)[v]. \qquad\text{(25)}$ The lowest dualizing stalk at $v$ is the kernel of $\mathbb Z^3\xrightarrow{-(1,1,1)}\mathbb Z$, hence is free of rank two. This is the cellular local-duality calculation, and also the kernel in (10): ordinary orientation sections on the three open rays have three independent germs, while the singular point records their sum. The first test has zero boundary and gives a global cycle on the closed rays. The second has boundary $-6[v]$.

For $S'=S\amalg\{w\}$, the closed inclusion $S\subset S'$ has a zero-dimensional complement. Equation (9) with $p=1$ identifies their $H^{-1}$ sheaves after direct image; the isolated point contributes no one-cycle. It does contribute $H^0(\omega_{S'})_w=\mathbb Z$. Removing lower-dimensional pieces preserves the lowest indexed degree, not the entire dualizing object.

Ordinary boundary germs and closed cycle germs differ

Difficulty: Intermediate.

Let $I=[0,1]$ and $J=(0,1)$ in $\mathbb R$, with $p=1$. Compute $T_I$ and $T_J$, their global sections and the comparison $T_I\to T_J$. Show separately that (11) is not an antisymmetric relation on literal subsets.

Solution. The closed interval's cellular complex has an edge in degree $-1$, endpoint coefficients in degree zero, and the terminal-minus-initial differential. At an endpoint that differential is an isomorphism, while in the interior it is zero. Consequently $T_I=j_{J!}A_J,\qquad T_J=j_{J*}A_J=A_I \quad\text{as sheaves on }\mathbb R. \qquad\text{(26)}$ The first expression has zero endpoint stalks; the second has an $A$ stalk at each endpoint. A section of $j_{J!}A_J$ over $\mathbb R$ would be constant on the connected $J$ and vanish near both endpoints, hence is zero. Therefore $\Gamma(\mathbb R;T_I)=0$, whereas $\Gamma(\mathbb R;T_J)=A$. The comparison is the natural injection that is the identity on $J$ and zero from each zero endpoint stalk. Its boundary sends $a$ to $a([1]-[0])$.

For the second assertion take $S_1=J$ and $S_2=J\amalg\{2\}$. The common witness $J$ is open in each and closed in each as required in the two directions; the extra point has dimension zero, less than $p$. Thus $S_1\preceq S_2$ and $S_2\preceq S_1$, although $S_1\ne S_2$. Their $T$ sheaves are isomorphic because the point has no degree-$-1$ contribution. The directed system uses this comparison equivalence, rather than a false antisymmetry assertion.

Arbitrary point coefficients survive the cutoff and the kernel

Difficulty: Intermediate.

On $\mathbb R$, let $F=i_*M$, where $i:\{0\}\to\mathbb R$ and $M$ is any $A$-module. Compute $\mathcal C_1(F)$, $\mathcal C_0(F)$, their boundary and its kernel. Calculate $P_{(0,1)}$ on this coefficient sheaf. Does this calculation require $M$ to be flat?

Solution. Tensor product is determined on stalks. The two sheaves are zero away from zero and have stalks $M^2$ and $M$ there. Equation (24) becomes $M^2\xrightarrow{(a,b)\mapsto a-b}M,\qquad \ker=\{(a,a):a\in M\}. \qquad\text{(27)}$ This sequence is split: $m\mapsto(m,0)$ is a section of the boundary, and $(a,b)\mapsto(b,b)$ projects onto the kernel. It therefore survives tensoring with every module; no general flatness theorem has been used. Locally $\mathcal Z_1=\operatorname{or}_{\mathbb R}=A$, so its tensor with $F$ is exactly this diagonal copy of $M$.

The cutoff acts by $(a,b)\mapsto(0,b)$. It is idempotent, has a potentially nonzero germ at zero, and takes the diagonal $(m,m)$ to a chain of boundary $-m$. All these assertions hold, for example, with $A=\mathbb Z$ and $M=\mathbb Z/3$. The sheaves here are skyscrapers and are soft: a section on a closed set containing zero is determined by its $M$-coordinates, which extend globally; a closed set missing zero has only the zero section. This particular split calculation does not establish tensor exactness for all singular chain systems.

Sheafification allows many weights, but not accumulating zero-dimensional support

Difficulty: Advanced.

Over $\mathbb Z$, construct the zero-chain $\alpha=\sum_{r\geq1}r[r]$ on $\mathbb R$. Show why it is a sheaf section but has no finite presentation by the presheaf generators in (1). On $U=(0,1)$, consider instead the zero-chain with coefficient one at every $1/r$, $r\geq2$. Can it extend to $\mathbb R$? Reconcile the answer with softness.

Solution. The positive integers are a closed locally finite subanalytic zero-dimensional subset. On every relatively compact neighborhood only finitely many of them occur. Their finite weighted presentations agree on overlaps, hence glue to a section $\alpha$ of $\mathcal C_0$, or equivalently to a section of $T_{\mathbb N}$.

A finite sum $\sum_{\ell=1}^m a_\ell[S_\ell]$ has, at any point, a coefficient formed from the finite set of $a_\ell$, with each term absent or present with one of its orientation signs. There are only finitely many such sums. The weights $1,2,3,\ldots$ cannot be represented this way. Thus sheafification supplies more global sections than a single finite presheaf expression.

The reciprocal set is locally finite and closed in $U$, so its coefficient-one chain is a section there. If it extended across zero, some neighborhood of zero would have a finite local subanalytic zero-chain presentation. A zero-dimensional subanalytic subset has no infinite accumulation in a sufficiently small relatively compact neighborhood: a finite local stratification has only finitely many zero-dimensional strata. Such a presentation cannot contain every $1/r$. The extension is impossible. Softness concerns sections on closed subsets of the ambient $\mathbb R$, whereas the reciprocal support is not closed there; extension from every open subset would be flabbiness. This example distinguishes the two properties.

Orientation cycles over the integers and in characteristic two

Difficulty: Advanced.

Take $X=\mathbb RP^2$, $p=n=2$. Compute $\Gamma(X;\mathcal Z_2)$ over $\mathbb Z$ and over $\mathbb F_2$. Check the result using (6), and identify the lower dualizing cohomology over $\mathbb Z$ that the top cycle calculation does not see.

Solution. Since $X$ is a closed support, $\mathcal Z_2=\operatorname{or}_X$. Transport around an orientation-reversing loop acts by $-1$. An integral global coefficient must satisfy $a=-a$, hence $2a=0$. Over $\mathbb Z$ this forces $a=0$; over $\mathbb F_2$ every coefficient satisfies it and the sign line is constant, giving a one-dimensional space of top cycles.

For an independent check, the usual one-cell-in-each-dimension decomposition has chain boundary $2:A\to A$ in dimensions two to one and zero in dimensions one to zero. The attaching loop of the two-cell traverses the one-cell twice in the same induced direction, giving that integer $2$. Its cochain complex is $A\xrightarrow{0}A\xrightarrow{2}A$, and compactness makes ordinary and compact cohomology agree. Thus $H_c^2(X;A)=A/2A$. Formula (6) gives $\Gamma(X;\mathcal Z_2) =\operatorname{Hom}_A(A/2A,A)=\{a\in A:2a=0\}, \qquad\text{(28)}$ which has exactly the two values above.

The finite cellular cochain complex is free. Applying its derived $A$-dual computes $R\Gamma(X;\omega_X)$ by (5), with $A\xrightarrow{2}A\xrightarrow{0}A$ in degrees $-2,-1,0$, up to the harmless consistent basis sign. Over $\mathbb Z$ it has $H^{-2}=0$, $H^{-1}=\mathbb Z/2$ and $H^0=\mathbb Z$. In particular the dualizing object is not zero just because there is no integral top cycle. No characteristic-zero assumption or division by two belongs in these constructions.

Sources and mathematical credit

Masaki Kashiwara, Index theorem for constructible sheaves, Astérisque 130 (1985), pp. 193–209, §1.3–1.5 (pp. 195–196), is the source for the oriented subanalytic chain model, the use of noncompact supports, and tensoring the chain sheaf with a coefficient sheaf. That short account states the orientation resolution and coefficient-cohomology comparison. The closed-support colimit construction, frontier maps, coefficient cutoff extension and worked singular and torsion examples are proved in the programme text here and in the linked contraction lesson.

Pierre Schapira, An Introduction to Sheaves on Grothendieck Topologies, edition dated 01/08/2026, §4.7 (pp. 97–98) gives duality between ordinary dualizing sections and compactly supported cohomology; §5.1, Lemma 5.1.1 and Proposition 5.1.2 (pp. 105–106), gives the manifold cohomological-dimension and soft-resolution results. The dimension filtration and lowest-degree arguments above apply those operations to singular carriers with the explicit programme proofs linked at each use. The finite-global-dimension ring convention, arbitrary coefficient sheaves and absence of a global orientation are retained throughout.