SH02-SUP-EXH-001 — Recovering sheaf cohomology from closed pieces

Support item: SH02-SUP-EXH-001.

This note supplies the geometric comparison needed before applying a Mittag-Leffler argument. Its proof does not require the closed pieces to be compact. In particular it applies to strips with a noncompact horizontal base.

SH02-EXH-COMPARISON — The comparison

Let T be a topological space and k a fixed unital coefficient ring. Let

$$T_0\subset T_1\subset\cdots\subset T, \qquad T_n\text{ closed},\qquad T_n\subset\operatorname{Int}_T(T_{n+1}),\qquad \bigcup_nT_n=T.$$

Write $i_n:T_n\hookrightarrow T$. For $K$ in the bounded-below derived category of sheaves of left $k$-modules on $T$, put $K_n=i_{n*}i_n^{-1}K$. Restriction to the smaller closed piece gives the transition $K_{n+1}\to K_n$. Then the restriction maps induce an equivalence

$$K\simeq\operatorname*{holim}_n K_n. \qquad\text{(1)}$$

Here holim means the homotopy inverse limit, or derived inverse limit. It is not an ordinary categorical inverse limit in the triangulated derived category. One may work in the enhanced derived category. Concretely, represent $K$ by a bounded-below complex $C$ and use the strict tower $i_{n*}i_n^{-1}C$. For $h_n:T_n\hookrightarrow T_{n+1}$, the transition is $i_{n+1*}$ applied to the restriction unit into $h_{n*}h_n^{-1}i_{n+1}^{-1}C$. The resulting strict map from the constant tower defines the map in (1) after derived inverse limit.

For an explicit resolution, use the Grothendieck category of inverse sequences of sheaves. Evaluation at index n preserves injectives: its left adjoint places the given sheaf at indices at most n, zero at later indices, and has identity transitions where applicable; it is exact. A bounded-below injective resolution of the tower consequently evaluates to K-injective complexes $I_n$. The cone of $1-\mathrm{shift}$ on their product, shifted by minus one, computes the homotopy limit. Strict compatibility provides the zero null-homotopy needed for the comparison map. This agrees with the enhanced-derived-category construction, without asserting uniqueness of a cone in a bare triangulated category.

Proof. Fix an open inclusion $j:U\hookrightarrow T$. Restriction $j^{-1}$ is exact and has the exact extension-by-zero functor $j_!$ as a left adjoint. The adjunction therefore passes to derived categories, and $j^{-1}$ preserves derived products. It is exact on distinguished triangles as well. Apply it to the derived-product triangle defining the countable homotopy limit,

$$\operatorname*{holim}_n K_n\longrightarrow \prod_n^{\mathrm D} K_n \xrightarrow{\,1-\mathrm{shift}\,}\prod_n^{\mathrm D}K_n.$$

It follows that restriction commutes with this homotopy limit. The products here are derived products; termwise products of arbitrary complexes must not be substituted without justification.

The open sets $U_N=\operatorname{Int}_T(T_N)$ cover $T$: a point of $T_n$ lies in $U_{n+1}$. On $U_N$, every $K_n$ with $n\geq N$ restricts to $K|_{U_N}$, with identity transition maps under the restriction identifications. Discarding a finite initial segment does not change a homotopy inverse limit. The homotopy limit of this remaining constant tower is $K|_{U_N}$, and the map in (1) restricts to that equivalence. For clarity, the latter constant-tower computation is elementary: the map $x\mapsto(x_0,(1-\mathrm{shift})x)$ identifies $\prod L$ with $L\oplus\prod L$. Its inverse has nth component $a-\sum_{0\leq r<n}y_r$, a finite sum. Thus the homotopy fibre of the second projection is $L$, with the diagonal comparison. Finite initial factors are removed by the same finite triangular elimination. Equivalences of complexes of sheaves can be checked on an open cover, by their cohomology sheaves. This proves (1). No compactness or Hausdorff assumption was used. $\square$

SH02-EXH-MILNOR — The precise cohomological obstruction

Closed pushforward $i_{n*}$ is exact and is right adjoint to the exact inverse-image functor. Thus it preserves injectives, and

$$R\Gamma(T,K_n)\simeq R\Gamma(T_n,i_n^{-1}K).$$

The derived-sections functor commutes with homotopy inverse limits. Applying the Milnor exact sequence to (1) gives, for every integer q,

$$0\longrightarrow \lim{}^1_n H^{q-1}(T_n,i_n^{-1}K) \longrightarrow H^q(T,K) \longrightarrow \lim_n H^q(T_n,i_n^{-1}K) \longrightarrow0. \qquad\text{(2)}$$

Consequently, the inverse-limit description of $H^q(T,K)$ holds if the tower in degree $q-1$ is Mittag-Leffler: for each n, the images in its nth term from all sufficiently late terms stabilize. Surjective transition maps in that degree are one sufficient condition. The relevant condition is on degree $q-1$; Mittag-Leffler in degree $q$ alone does not remove the left term in (2). For all-degree continuity, check the required condition in every degree.

If every transition is an isomorphism in every degree, global cohomology is identified with that on every piece. If transitions are eventually isomorphisms in every degree, this conclusion holds in each degree on its stable tail. The stabilization index may depend on the degree; a common tail requires a uniform index. In degree q the obstruction also uses degree q-1, so take a tail beyond both stabilization indices. No identification with arbitrary early pieces follows.

SH02-EXH-TOWER-PROOFS — Tower resolutions, sections and the Milnor sequence

Here are proofs of the inverse-limit facts used above. They apply to the same bounded-below complexes and to left modules over any unital ring. Neither exactness of products of sheaves nor compactness of the closed pieces is assumed.

Resolving the whole tower

Let $\mathcal A$ be the category of sheaves of left $k$-modules on $T$, and let $\operatorname{Tow}(\mathcal A)$ consist of sequences $F_{n+1}\xrightarrow{u_n}F_n$, indexed by the nonnegative integers. Its morphisms commute with all transitions. Kernels, cokernels and colimits are computed at each index, so it is abelian and has exact filtered colimits. These sheaf-category facts are proved in Sheaves of modules on a ringed space, Theorems 2.1 and 3.1.

For an object $M$ of $\mathcal A$, define $L_nM$ to have value $M$ at indices $0,\ldots,n$, zero at later indices, and identity transitions between its nonzero terms. A tower map $L_nM\to F$ is determined by its component $M\to F_n$: earlier components are its composites with the transitions. Thus $L_n$ is an exact left adjoint of evaluation at $n$. If $U$ is a generator of $\mathcal A$, then $\bigoplus_{n\geq0}L_nU$ is a generator of the tower category: a nonzero tower morphism has a nonzero component, detected by a map from the corresponding $L_nU$. The sheaf generator and its detection property are proved in Proposition 1.1 of K-injective resolutions in Grothendieck abelian categories. Hom collections in the tower category are subsets of products of Hom sets, hence sets. Therefore it is a locally small Grothendieck abelian category.

The injective embedding construction, Theorem 2.4, applies to this category. The degree-by-degree resolution construction, Theorem 4.1, consequently resolves any bounded-below complex of towers by a bounded-below complex $J^\bullet$ of injective towers. In particular it applies to the strict tower $i_{n*}i_n^{-1}C$ above. Evaluation preserves injectives, because its left adjoint $L_n$ is exact: extending a map into an evaluated injective is the same as extending the adjoint map from the corresponding monomorphism of towers. Hence each evaluated complex $J_n^\bullet$ is a bounded-below injective resolution of $K_n$. All these resolutions have the same lower bound.

An injective tower $J$ has split-surjective transitions. Indeed, $L_n(J_n)\to L_{n+1}(J_n)$ is a monomorphism. The tower map from its source to $J$ corresponding to $1_{J_n}$ extends by injectivity. At index $n+1$, the extension is a map $\sigma_n:J_n\to J_{n+1}$ satisfying $u_n\sigma_n=1_{J_n}$. For any family of morphisms $y_n:M\to J_n$, set $x_0=0$ and recursively set $x_{n+1}=\sigma_n(x_n-y_n)$. This gives $x_n-u_nx_{n+1}=y_n$. Applied to the projections from the product, the construction gives a right inverse of

$$\delta:\prod_{n\geq0}J_n\longrightarrow\prod_{n\geq0}J_n, \qquad \delta(x)_n=x_n-u_nx_{n+1}. \qquad\text{(E3)}$$

Each coordinate of that right inverse uses only finitely many projections and splittings. It is a genuine morphism of sheaves, not an assumed ability to lift infinitely many local sections on a common neighbourhood. Its kernel is $\lim_n J_n$, by the defining compatibility equations. The limit is also injective: limit is right adjoint to the exact constant-tower functor, so the same extension argument proves preservation of injectives.

Apply these facts separately in every degree of $J^\bullet$. The splittings need not commute with its differential; they establish degreewise split exactness of

$$0\longrightarrow\lim_n J_n^\bullet \longrightarrow\prod_n J_n^\bullet \xrightarrow{\delta}\prod_n J_n^\bullet\longrightarrow0. \qquad\text{(E4)}$$

The kernel-to-fibre comparison is a quasi-isomorphism, by the cohomology sequence of this short exact sequence of complexes. In the cone convention $d(y,x)=(dy+\delta x,-dx)$, the fibre is $\operatorname{Cone}(\delta)[-1]$, its differential is $(y,x)\mapsto(-dy-\delta x,dx)$, and the comparison sends a kernel element $x$ to $(0,x)$. This checks the actual comparison map and its signs.

The termwise limit of $J^\bullet$ computes the right derived limit by the cited bounded-below derived-functor construction. Each product in (E4) represents the derived product of the $K_n$: products of K-injective complexes are K-injective and represent derived products, as proved in Proposition 5.4 of the K-injective lesson. Thus (E4) proves the homotopy-limit triangle used in the geometric comparison, not merely a formula for an ordinary inverse limit. Comparison maps between tower resolutions are unique up to homotopy by the same bounded-below comparison theorem. The construction is therefore independent of the chosen resolution and natural in maps of strict towers.

Restriction, finite initial segments and sections

For an open inclusion $j:V\hookrightarrow T$, restriction is exact and has the exact left adjoint $j_!$, proved in Lemma 5.1 of the module-sheaf lesson. It therefore preserves injectives and K-injectives by the adjunction argument. Restriction also commutes with ordinary products: both products have sections on an open of $V$ equal to the product of the same sections on that open in $T$. Apply this to the resolved products in (E4). It proves commutation of restriction with the derived-product fibre, even though arbitrary products of sheaves need not be exact.

To remove the first $N$ terms of a tower, split its resolved product into the first $N$ factors and the tail. In these coordinates $\delta$ is block upper triangular. Its head block is upper triangular with identity diagonal; backward substitution gives its inverse using only finite sums of transition maps. Eliminating this block identifies its cone with the cone of the tail block plus the contractible cone of an identity. Thus deleting the head preserves the homotopy fibre and its comparison maps. For a constant tail use one K-injective model $I$ of its value. The coordinate transformation

$$\prod_{n\geq0} I\longrightarrow I\oplus\prod_{n\geq0}I, \qquad x\longmapsto\bigl(x_0,(x_n-x_{n+1})_n\bigr) \qquad\text{(E5)}$$

is a chain isomorphism. Its inverse has coordinate $a-\sum_{r<n}y_r$. It changes $\delta$ into projection onto the second factor, whose homotopy fibre is $I$. The diagonal map gives this identification. These computations justify precisely the local constant-tail step in (1).

Sections commute with products by the module-sheaf limit construction. The complexes in (E4), and their fibre, are K-injective; alternatively its kernel is a bounded-below complex of injectives by the limit argument above. Applying ordinary sections to these models therefore computes derived sections. Applying sections also commutes with the cone, since it commutes with finite sums and the displayed differentials. Consequently

$$R\Gamma\bigl(T,\operatorname{holim}_nK_n\bigr) \simeq\operatorname{Cone}\!\left( \prod_n R\Gamma(T,K_n)\xrightarrow{1-\mathrm{shift}} \prod_n R\Gamma(T,K_n)\right)[-1]. \qquad\text{(E6)}$$

Here the products on the right are represented by complexes of $k$-modules. Products of modules are exact: kernels are coordinatewise, and a product of surjections is surjective by choosing a lift in each coordinate. Hence cohomology of these products is the product of their cohomologies. The long exact sequence of the fibre (E6) gives a natural short exact sequence with left term the cokernel of $1-\mathrm{shift}$ on the degree-$(q-1)$ product and right term its kernel on the degree-$q$ product. For the present closed embeddings, exactness of $i_{n*}$ follows on stalks: its stalks are the original stalks on $T_n$, and zero off the closed set. Its exact left adjoint $i_n^{-1}$ shows that it preserves injectives. The equality of ordinary sections then identifies $R\Gamma(T,K_n)$ with $R\Gamma(T_n,i_n^{-1}K)$, as used in (2).

For completeness, the cokernel just obtained is the first right derived limit of a module tower, not a new notation for a different obstruction. Resolve a module tower by injective towers and use (E4) in the module category. Exactness of products says that the products of the resolutions resolve the products of the original modules. Passing to their cones therefore identifies derived limit with the two-term complex $\prod M_n\xrightarrow{1-\mathrm{shift}}\prod M_n$ in degrees zero and one. Its degree-zero cohomology is $\lim M_n$, its degree-one cohomology is $\lim^1 M_n$, and its higher cohomology is zero. Substitution into the preceding fibre sequence proves (2), with exactly the degree-$(q-1)$ obstruction asserted there.

Why the Mittag–Leffler hypothesis kills that obstruction

Let $M=(M_n,u_n)$ be a countable inverse sequence of left $k$-modules with stabilizing images. Let $S_n\subset M_n$ be the eventual image in $M_n$. The transitions carry $S_{n+1}$ onto $S_n$: choose an index beyond stabilization at both positions; a lift from that index of an element of $S_n$ maps into $S_{n+1}$. Put $Q_n=M_n/S_n$. For every fixed $n$, the transition $Q_m\to Q_n$ is zero for all sufficiently large $m$, since the corresponding image in $M_n$ has become $S_n$.

The difference map $\delta_S$ on $\prod S_n$ is onto. For a prescribed $y$, start with $x_0=0$ and choose $x_{n+1}$ mapping to $x_n-y_n$, using the surjective transitions. The difference map on $\prod Q_n$ is invertible: if $v_{n,m}:Q_m\to Q_n$ denotes the composite transition and $v_{n,n}=1$, its inverse sends $y$ to

$$x_n=\sum_{m\geq n}v_{n,m}(y_m). \qquad\text{(E7)}$$

For each $n$ only finitely many summands can be nonzero, by the preceding eventual-zero property. Subtracting the shifted expression leaves $y_n$. Conversely a compatible family in $Q$ is zero because each coordinate is the image of an arbitrarily late coordinate and that transition eventually vanishes. This proves both inverse identities.

Products of the short exact sequences $0\to S_n\to M_n\to Q_n\to0$ remain exact. Given $y\in\prod M_n$, solve its image under $\delta_Q$, lift that solution to $x'\in\prod M_n$, and observe that $y-\delta_Mx'$ lies in $\prod S_n$. Surjectivity of $\delta_S$ supplies $z$ with this difference, and $x'+z$ solves $\delta_Mx=y$. Thus $\lim^1 M_n=\operatorname{coker}\delta_M=0$. This proof uses the same classical choice convention as the injective constructions. It does not require finite generation, commutativity, a field, or a uniform stabilization index. Applied to the degree-$(q-1)$ tower it gives exactly the continuity criterion following (2), including all noncompact-strip restrictions.

SH02-EXH-STRIPS — Noncompact-strip application

For an arbitrary topological space $V$, take $T=V\times\mathbb R$ and $T_n=V\times[-n,n]$ for positive integers n. These are closed, satisfy the interior condition, and cover T. Thus (1) and (2) apply without requiring V to be compact. Any claimed vanishing of the lim-one term still needs a separate proof for the actual sheaf or complex under consideration. The geometry of the exhaustion alone does not prove that condition.

SH02-EXH-COUNTEREXAMPLE — Why the interior condition is substantive

A closed increasing cover alone does not suffice. Let $T=[0,1]$, $T_n=\{0\}\cup[1/n,1]$ for $n\geq2$, and $K=\mathbb Z_T$. Each piece has two components, hence degree-zero sections $\mathbb Z^2$, and its restriction transitions are identities. Global sections on the connected interval are $\mathbb Z$, with diagonal restriction. The degree-zero limit comparison is therefore not an isomorphism. The interiors fail to cover zero, exactly where the local proof cannot apply.

SH02-EXH-DEPENDENCIES — Dependencies and attribution

This is a short independent bridge built from standard derived-sheaf facts, not a claim of a new research theorem. The exact Stacks inputs are 0D60 (derived sections and the Milnor sequence) and 02UV (closed pushforward and cohomology). The internal tower proof supplies the derived extension-by-zero/restriction comparison, the countable homotopy-limit triangle, invariance under removing a finite initial segment, and the countable Mittag-Leffler vanishing theorem. Their precise supporting tags are 00A7, 01AK, 01AX, 08TC, 07D9, 0BK7 and 07KW. The nonunique-cone caution is 0H9J.

For noncommutative $k$, the same additive arguments apply to left modules: no tensor-product exactness, commutativity or finite global dimension is used. For these coefficients, commutation of derived sections with derived inverse limits follows from 08U1. Exact products of modules then give the Milnor sequence by the same product-triangle argument underlying 0D60. The internal tower and module proofs above establish these facts with this coefficient interpretation; the links also provide source credit and further reading.

Original text is dedicated under CC0 1.0 Universal.