# SH02-FSB — Formal stabilization over arbitrary neighborhood sets Original programme text: CC0 1.0 Universal. This reading proves the displayed formal and derived comparison statements used in local sheaf duality. Basic category and derived-functor foundations remain explicit prerequisites; this scoped source repair does not certify all surrounding SH-01 foundations. A neighborhood calculation often becomes constant only after transition maps have discarded temporary terms. The right notion of stabilization records those maps. This lesson first constructs that formal notion, then separates it from the derived inverse limit of an actual module diagram. The distinction allows the neighborhood set to be arbitrarily large and directed. ## SH02-FSB-SETUP — Indices, categories, and the two contracts All indexing sets are small, nonempty directed partially ordered sets. An inverse system in a locally small category $\mathcal C$ is a functor $A:I^{\mathrm{op}}\to\mathcal C$; write $a_{ji}:A_j\to A_i$ for $j\geq i$. A direct system is a functor $I\to\mathcal C$. We use a fixed universe for these sets and a larger universe when forming functor categories. The categories needed below are $\operatorname{Mod}(k)$ and the ordinary derived category $D(k)$, where $k$ is a unital commutative ring. The categorical and inverse-limit results in this lesson do not require finite global dimension. The formal calculus through SH02-FSB-FUNCTORS supports [SH02-CB-IMP-PRO-CALCULUS](cohomological-biduality.md#SH02-CB-IMP-PRO-CALCULUS). The derived module-diagram calculation in SH02-FSB-COFINALITY and SH02-FSB-PRISM supports [SH02-CB-IMP-DERIVED-COFINALITY](cohomological-biduality.md#SH02-CB-IMP-DERIVED-COFINALITY). Basic categories, complexes, projective or injective resolutions, and the derived functors they define remain prerequisite material. The proofs below identify the resolutions and comparison maps needed here. ## SH02-FSB-HOM — Formal maps and constant objects For inverse systems $A:I^{\mathrm{op}}\to\mathcal C$ and $B:J^{\mathrm{op}}\to\mathcal C$, define $$ \operatorname{Hom}_{\operatorname{Pro}(\mathcal C)}(A,B) =\varprojlim_{j\in J}\varinjlim_{i\in I} \operatorname{Hom}_{\mathcal C}(A_i,B_j). $$ The inner colimit uses precomposition by $a_{i'i}$ when $i'\geq i$. Consequently a component at target $j$ is represented by a map $A_i\to B_j$, and two representatives agree precisely when their composites from some common later $A_{i'}$ agree. A formal map is a family of such classes compatible with every target transition. A finite number of equalities can always be witnessed after one further source index, by directedness. Here is a construction that verifies composition rather than presuming it. Associate to $A$ the covariant functor $$ L_A(W)=\varinjlim_i\operatorname{Hom}_{\mathcal C}(A_i,W). $$ A formal map $f:A\to B$ defines a natural transformation $L_B\to L_A$: represent an element of $L_B(W)$ by $B_j\to W$, choose a representative $A_i\to B_j$ of $f_j$, and compose. Refining either representative gives the same class, by the compatibility just stated. Conversely, evaluate a natural transformation at the class of $\mathrm{id}_{B_j}$ in $L_B(B_j)$. These operations are inverse. Composition and identities of natural transformations therefore make the displayed Hom sets into a category. This also proves associativity without selecting simultaneous representatives for infinitely many components. For direct systems, the dual construction gives $$ \operatorname{Hom}_{\operatorname{Ind}(\mathcal C)}(A,B) =\varprojlim_i\varinjlim_j \operatorname{Hom}_{\mathcal C}(A_i,B_j), \qquad \operatorname{Ind}(\mathcal C) =\operatorname{Pro}(\mathcal C^{\mathrm{op}})^{\mathrm{op}}. $$ The constant-system functor $c:\mathcal C\to\operatorname{Pro}(\mathcal C)$ is fully faithful: substituting two one-object systems in the formula gives $\operatorname{Hom}(cP,cQ)=\operatorname{Hom}_{\mathcal C}(P,Q)$. The same is true for ind-objects. These are the conventions of [Stacks, Tag 05PW](https://stacks.math.columbia.edu/tag/05PW) and [Tag 05PX](https://stacks.math.columbia.edu/tag/05PX); the construction above supplies the composition details used here. ## SH02-FSB-REPRESENTED — What a representative actually supplies A pro-system $A$ is represented by $Q\in\mathcal C$ when there is a specified isomorphism $cQ\simeq A$. It is equivalent to give a compatible cone $p_i:Q\to A_i$, an index $i_0$, and a map $q:A_{i_0}\to Q$ such that $$ q p_{i_0}=\mathrm{id}_Q, \qquad a_{\ell j}=p_jq a_{\ell i_0} \quad\text{for some }\ell\geq j,i_0\text{ for each }j. $$ Indeed, the cone is precisely a map $p:cQ\to A$, and a map $A\to cQ$ is represented by one such $q$. The first identity says $qp=\mathrm{id}_{cQ}$; the eventual identities say $pq=\mathrm{id}_A$ by the Hom formula. This proves both directions of the criterion. Taking opposites gives its ind version, with a cocone and eventual equalities after advancing the target index. For any $T\in\mathcal C$, applying $\operatorname{Hom}_{\operatorname{Pro}(\mathcal C)}(cT,-)$ to this isomorphism gives $$ \operatorname{Hom}_{\mathcal C}(T,Q) \simeq\varprojlim_i\operatorname{Hom}_{\mathcal C}(T,A_i). $$ Thus $Q$ is also the ordinary categorical limit with the displayed cone. Dually an ind representative is the ordinary categorical colimit. The converse is false: merely being an ordinary limit does not produce the eventual inverse $q$. The corresponding representative characterizations are [Stacks, Tag 05PZ](https://stacks.math.columbia.edu/tag/05PZ) and its [ind version, Tag 05PY](https://stacks.math.columbia.edu/tag/05PY). If $\mathcal C$ has a zero object, then $$ A\simeq0\text{ in }\operatorname{Pro}(\mathcal C) \quad\Longleftrightarrow\quad \text{for every }i\text{ some }j\geq i\text{ has }a_{ji}=0. $$ The identity of $A$ is zero exactly when its component at each $A_i$ becomes zero in the inner colimit. This is the displayed criterion. A zero identity makes the unique maps to and from the zero object inverse, proving the assertion. In the ind case the criterion is that every source term is killed by some later transition. ## SH02-FSB-REINDEX — Cofinal changes preserve the formal object Call an order-preserving map $\phi:J\to I$ cofinal here when, for every $i\in I$, the upper comma set $$ J_i=\{j\in J:i\leq\phi(j)\} $$ is nonempty and directed. For maps between directed sets, nonemptiness already implies the directed condition: $J_i$ is upward closed in $J$. We retain both words to identify the exact indexing condition used later in the derived proof. The transition maps give a canonical formal isomorphism between $A$ and its restriction $\phi^*A$. To check it, for each $i$ choose $j\in J_i$ and use $A_{\phi(j)}\to A_i$. Different choices become equal at a common upper index. This defines $\phi^*A\to A$. In the other direction, the component at target $j$ is represented by $\mathrm{id}_{A_{\phi(j)}}$. Both composites equal the identity after a further transition, so they are inverse formal maps. Their definition also shows naturality in $A$ and compatibility with successive reindexings. This is the directed-set instance of the ordinary cofinality statement [Stacks, Tag 04E7](https://stacks.math.columbia.edu/tag/04E7), applied also to opposite categories. No derived inverse-limit assertion has yet been used. ## SH02-FSB-STRICT — Finite acyclic diagrams can be made levelwise **Proposition.** Let $D$ be a finite category admitting a degree function on its objects that strictly increases along every nonidentity arrow. Suppose a diagram $D\to\operatorname{Pro}(\mathcal C)$ is given, with each vertex $v$ represented by a specified directed system $A^v:I_v^{\mathrm{op}}\to\mathcal C$. There is a directed set $R$, cofinal maps $r_v:R\to I_v$, and a diagram $D\to\mathcal C^{R^{\mathrm{op}}}$ whose formal image is the given diagram. In particular this applies to a single morphism, composable morphisms, parallel maps with a prescribed equality, and finite commutative squares. **Proof.** A level consists of indices $i_v\in I_v$ and, for every arrow $\alpha:v\to w$, a map $$ f_\alpha:A^v_{i_v}\longrightarrow A^w_{i_w} $$ representing the prescribed formal component at target $i_w$. Require identity and composition equations at that level. Levels exist with arbitrarily prescribed lower bounds on their indices. Choose the vertices in decreasing degree. When choosing $i_v$, all targets of its nonidentity arrows have already been fixed. Each of the finitely many formal components has a representative, and directedness supplies a common source index above their indices and the prescribed bound. For each composable pair starting at $v$, the proposed composite and the proposed map for the composite arrow represent the same formal component. A further source index makes that equality hold. There are finitely many such equations. This finishes the construction at $v$ without changing its already chosen targets. Order levels by coordinatewise increase, requiring in addition that every square made from an arrow $f_\alpha$ and the original transition maps commute. This is a partial order. Any two levels have an upper level: repeat the decreasing-degree construction above, demanding indices above those of both old levels and demanding commutativity with both old arrow maps. Each new demand is equality between two representatives of the same formal component at a fixed old target. It is therefore achieved by a further source index. There are finitely many demands at each vertex. The same construction works while imposing an additional lower bound at any vertex. Let $R$ be this set of levels. It is small because the index sets and the relevant Hom sets are small, and it is directed by the preceding construction. Each projection $r_v$ is order preserving; its upper comma set above any prescribed $i_v$ is nonempty and directed by the same construction. Hence $r_v$ is cofinal. The transition squares built into the order make every $f_\alpha$ a natural transformation of the reindexed systems. The required equations already hold at every level. This proves the proposition. $\square$ To impose equality between two given parallel formal maps, apply the construction to the diagram in which those arrows are identified; finite lists of compatible composition equations are handled the same way. Taking opposites proves the ind version. With $\mathcal C=D(k)$, the level maps and equations lie in $D(k)$. This does **not** assert that arbitrary diagrams in $D(k)$ lift to strictly commuting chain maps. In the application below, applying $H^q$ gives actual module maps and actual equations, which is all that is needed. The acyclicity condition on $D$ matters. Let every term of an inverse sequence be a nonzero module $M$, with zero transition maps between distinct indices. This system is formally zero. Its formal isomorphism to the constant zero system has inverse equations, but no cofinal reindexing can turn its terms into modules isomorphic to zero. Thus one cannot require arbitrary inverse equations to hold levelwise. The next argument uses their correct eventual form. ## SH02-FSB-INVERSE — Kernels and cokernels of a pro-isomorphism Let $f:A\to B$ be a strict morphism of inverse module systems on one directed set $I$, and suppose its formal pro-morphism is invertible. For each $i$ there are $j\geq i$ and a map $g:B_j\to A_i$ satisfying $$ g f_j=a_{ji},\qquad f_i g=b_{ji}. $$ To prove this, represent the component of the formal inverse at $A_i$ by $g_0:B_\ell\to A_i$. The equation $fg=\mathrm{id}_B$, at target $B_i$, holds after restricting $B_\ell$ to some later term. The equation $gf=\mathrm{id}_A$, at target $A_i$, similarly holds after restricting the composite $A_\ell\xrightarrow{f_\ell}B_\ell\xrightarrow{g_0}A_i$. A common later index $j\geq i,\ell$ makes both equalities hold, and $g=g_0b_{j\ell}$ has the required properties. Now $a_{ji}$ kills $\ker(f_j)$, because it factors through $f_j$. Also $b_{ji}$ induces zero on $\operatorname{coker}(f_j)\to\operatorname{coker}(f_i)$, because it factors through $f_i$. The levelwise kernel and cokernel systems are therefore pro-zero. This proof uses kernels and cokernels in modules, not an assertion about kernels in a derived category. For clarity, levelwise kernels and cokernels of any strict module-system morphism also have the formal universal properties. For an arbitrary pro-system $Z$, the Hom formula gives $$ \begin{aligned} \operatorname{Hom}_{\mathrm{Pro}}(Z,\{\ker f_i\}) &=\ker\bigl(\operatorname{Hom}_{\mathrm{Pro}}(Z,A) \longrightarrow\operatorname{Hom}_{\mathrm{Pro}}(Z,B)\bigr),\\ \operatorname{Hom}_{\mathrm{Pro}}(\{\operatorname{coker}f_i\},Z) &=\ker\bigl(\operatorname{Hom}_{\mathrm{Pro}}(B,Z) \longrightarrow\operatorname{Hom}_{\mathrm{Pro}}(A,Z)\bigr). \end{aligned} $$ To verify the first equality, apply $\operatorname{Hom}(Z_j,-)$ to each kernel and then take $\varprojlim_i\varinjlim_j$. A filtered colimit commutes with this kernel: a class mapping to zero is represented by a map whose composite becomes zero at some later index, and two resulting kernel representatives become equal after a further index. Ordinary limits preserve kernels because the compatibility and zero equations can be imposed together. For the second equality use $\operatorname{Hom}(\operatorname{coker}f_i,Z_j)=\ker(\operatorname{Hom}(B_i,Z_j)\to\operatorname{Hom}(A_i,Z_j))$ and take $\varprojlim_j\varinjlim_i$. The same representative argument applies. These natural identities are exactly the kernel and cokernel universal properties; they do not presume an abelian-category theorem about all pro-objects. ## SH02-FSB-FUNCTORS — Applying a functor to the formal system Every covariant functor $T:\mathcal C\to\mathcal E$ induces functors on ind- and pro-categories by applying $T$ to all terms and all representatives of morphisms. Eventual equality remains equality after applying $T$, and the description of composition in SH02-FSB-HOM shows functoriality. Natural transformations of functors extend termwise and preserve the same compositions. In particular, a specified formal isomorphism $A\simeq cQ$ is sent to a specified formal isomorphism $TA\simeq c(TQ)$. This is also [Stacks, Tag 05SH](https://stacks.math.columbia.edu/tag/05SH). A contravariant functor $T:\mathcal C^{\mathrm{op}}\to\mathcal E$ instead gives $$ \operatorname{Pro}(\mathcal C)^{\mathrm{op}} \longrightarrow\operatorname{Ind}(\mathcal E), \qquad \operatorname{Ind}(\mathcal C)^{\mathrm{op}} \longrightarrow\operatorname{Pro}(\mathcal E). $$ Indeed, a representative $A_i\to B_j$ becomes $T(B_j)\to T(A_i)$, and the Hom formulas have exactly these reversed source and target indices. After the dual-sections adjunction identifies the terms, this explains why $R\operatorname{Hom}_k(-,k)$ converts a represented compact-support pro-system into a represented ordinary-section ind-system. That sheaf-theoretic identification is [SH02-EX-DUAL-SECTIONS](exceptional-operations.md#SH02-EX-DUAL-SECTIONS). The formal argument makes no claim that this functor commutes with an arbitrary ordinary or derived inverse limit. ## SH02-FSB-COFINALITY — A resolution for the derived comparison **Theorem.** Let $A:I^{\mathrm{op}}\to\operatorname{Mod}(k)$ be an inverse module system and let $\phi:J\to I$ be order preserving. If every $J_i=\{j:i\leq\phi(j)\}$ is nonempty and directed, restriction induces a natural isomorphism $$ r_\phi(A):R\varprojlim_I A \xrightarrow{\sim}R\varprojlim_J\phi^*A. $$ This isomorphism is the derived restriction of compatible families. It respects composition of indexing maps. **Proof.** Work in the abelian diagram category $\mathcal A_I=\operatorname{Fun}(I^{\mathrm{op}},\operatorname{Mod}(k))$, whose exact sequences are objectwise exact. For $i\in I$ define the free representable diagram $$ P_i(h)=k[\operatorname{Hom}_I(h,i)] =\begin{cases}k&h\leq i,\\0&h\nleq i.\end{cases} $$ Its transition maps between nonzero terms are identities. The map sending a natural transformation to its value on $1\in P_i(i)$ gives $\operatorname{Hom}_{\mathcal A_I}(P_i,A)=A_i$. Hence $P_i$ is projective. Direct sums of these objects are projective, because products of surjections of modules are surjective. We use the usual axiom of choice here. For completeness this diagram category has enough injectives. The right adjoint $E_i$ to evaluation at $i$ sends a module $M$ to the diagram with value $M$ at $h$ when $i\leq h$, and zero otherwise. Embed each $A_i$ into an injective module $Q_i$. Adjunction gives a monomorphism $$ A\longrightarrow\prod_i E_i(Q_i): $$ at $h$ its $i=h$ component is the chosen embedding of $A_h$. Each $E_i(Q_i)$ is injective because evaluation is exact, and a product of injectives is injective because its Hom functor is a product of exact functors. This constructs the resolutions defining the right derived inverse limit. Consider the augmented chain complex in $\mathcal A_I$ $$ P^I_n=\bigoplus_{i_0\leq\cdots\leq i_n}P_{i_0} \quad(n\geq0), \qquad P^I_0\longrightarrow c k. $$ Repeated indices are allowed. The boundary is the alternating sum of vertex deletions. Deleting the first vertex uses $P_{i_0}\to P_{i_1}$; the other faces keep the coefficient diagram. The augmentation sends each generator to $1$. The face identities give $\partial^2=0$. At $h$, this is the free augmented chain complex of the poset $\{i:h\leq i\}$. Prepending its initial element $h$ gives a contraction, including the augmentation degree. Thus $P^I_\bullet\to c k$ is a projective resolution. Since $\varprojlim_I A=\operatorname{Hom}_{\mathcal A_I}(c k,A)$, this resolution computes the derived limit. Explicitly, for an injective resolution $A\to Q^\bullet$, the first-quadrant bicomplex $\operatorname{Hom}(P^I_p,Q^q)$ computes $\operatorname{Hom}(P^I_\bullet,A)$ by projectivity and $\varprojlim_I Q^\bullet$ by injectivity. There are finitely many terms in each total degree, so the two calculations identify the same derived object. The resulting cochain model is $$ \begin{aligned} C^n(I,A)&=\prod_{i_0\leq\cdots\leq i_n}A_{i_0},\\ (dc)_{i_0,\ldots,i_{n+1}} &=a_{i_1i_0}c_{i_1,\ldots,i_{n+1}} +\sum_{r=1}^{n+1}(-1)^r c_{i_0,\ldots,\widehat{i_r},\ldots,i_{n+1}}. \end{aligned} $$ In degree zero its cycles are the compatible families. This agrees with the product construction in [Stacks, Tag 08RZ](https://stacks.math.columbia.edu/tag/08RZ); the module Ext version in [Tag 08S0](https://stacks.math.columbia.edu/tag/08S0) applies with the first coefficient diagram equal to the objectwise projective module $k$. The construction here proves the required case and identifies its maps. Now form another augmented complex in $\mathcal A_I$: $$ Q^\phi_n=\bigoplus_{j_0\leq\cdots\leq j_n}P_{\phi(j_0)}. $$ At $i$ its augmented chain complex is the nerve complex of $J_i$. To prove exactness, an augmented cycle involves only finitely many vertices. Choose $b\in J_i$ above those vertices. On chains using vertices at most $b$, appending $b$ with sign $(-1)^{n+1}$ in degree $n$, and sending the augmentation generator to $[b]$, gives $\partial h+h\partial=\mathrm{id}$. Therefore every such cycle bounds. Nonemptiness gives surjectivity of the augmentation. This finite-support argument proves exactness even when there is no single upper bound for all vertices. Thus $Q^\phi_\bullet\to c k$ is another projective resolution. There is a specified augmented chain map $$ F_\phi:Q^\phi_\bullet\longrightarrow P^I_\bullet, \qquad [j_0,\ldots,j_n]\longmapsto [\phi(j_0),\ldots,\phi(j_n)], $$ using the identity on $P_{\phi(j_0)}$. Allowing repeated vertices makes the formula valid for any order-preserving $\phi$. It commutes with every face and induces the identity on $c k$. An augmented map between projective resolutions that induces the identity is a chain homotopy equivalence. Here are the lifting details. Lift a map in degree zero through the other augmentation. Once degrees below $n$ have been constructed, the required degree-$n$ boundary lands in the cycles at degree $n-1$; exactness and projectivity lift it through the next boundary. This constructs a reverse chain map. Two lifts of the same augmentation are homotopic: their difference in degree zero lifts through the degree-one boundary; at degree $n$, subtract the already constructed homotopy term, observe that the remainder lands in cycles, and lift through degree $n+1$. Applying this to both composites produces the two homotopies. Applying $\operatorname{Hom}_{\mathcal A_I}(-,A)$ to $F_\phi$ therefore gives a cochain homotopy equivalence. Under the displayed cochain models it is precisely $$ (r_\phi c)_{j_0,\ldots,j_n} =c_{\phi(j_0),\ldots,\phi(j_n)}. $$ This is natural in $A$, restricts compatible families in degree zero, and satisfies $r_{\phi\psi}=r_\psi r_\phi$ on the complexes themselves. It is the claimed natural derived comparison. $\square$ The ordinary initiality theorem, [Stacks, Tag 002R](https://stacks.math.columbia.edu/tag/002R), concerns ordinary limits. The two projective resolutions above supply the additional derived assertion under the stated upper-comma condition. ## SH02-FSB-PRISM — The comparison for a natural change of index Let $u,v:J\to I$ be order preserving and suppose $u(j)\leq v(j)$ for every $j$. Transitions define a map of diagrams $$ t:v^*A\longrightarrow u^*A, \qquad t_j=a_{v(j),u(j)}. $$ Then, without any cofinality assumption on $u$ or $v$, $$ R\varprojlim_J(t)\,r_v=r_u. $$ To prove the equality for the actual comparison maps, define, for $n\geq1$, $$ (H^nc)_{j_0,\ldots,j_{n-1}} =\sum_{r=0}^{n-1}(-1)^r c_{u(j_0),\ldots,u(j_r),v(j_r),\ldots,v(j_{n-1})}, \qquad H^0=0. $$ Every index string is increasing, and every term lies in $A_{u(j_0)}$. This defines a homotopy with $$ dH+Hd=C(t)r_v-r_u. $$ The verification is the prism boundary calculation, which can be seen before applying Hom. A string $[j_0,\ldots,j_n]$ with coefficient $P_{u(j_0)}$ is sent to $$ \sum_{r=0}^n(-1)^r [u(j_0),\ldots,u(j_r),v(j_r),\ldots,v(j_n)]. $$ A deletion strictly before or after the change from $u$ to $v$ cancels with the term obtained by first deleting the corresponding source vertex. The two faces at adjacent change positions cancel each other. The only outer faces left are the all-$v$ string, with coefficient map $P_{u(j_0)}\to P_{v(j_0)}$, and minus the all-$u$ string. Applying Hom gives the displayed formula. In degree zero it reads $$ (H^1dc)_j=a_{v(j),u(j)}c_{v(j)}-c_{u(j)}, $$ which fixes the sign and the direction of $t$. This proves the identity in $D(k)$ as well as its cochain realization. In particular, if $s:J\to J$ satisfies $\mathrm{id}\leq s$ and $z:s^*B\to B$ is the transition map, then $C(z)r_s$ is homotopic to the identity. If every component of $z$ is zero, the homotopy shows $$ d(-H)+(-H)d=\mathrm{id}_{C(J,B)}. $$ Thus this particular derived-limit complex is contractible. This conclusion needs no separate cofinality assertion about $s$. ## SH02-FSB-NET — Applying the bridge to a pro-zero net **Theorem.** If an inverse module system $A$ is pro-zero, then $R^p\varprojlim_I A=0$ for every $p\geq0$. A formal pro-isomorphism of inverse module systems consequently induces an isomorphism on all derived inverse limits. **Proof.** Let $$ J=\{(S,n):S\subset I\text{ finite},\ n\in\mathbb N\}, \qquad (S,n)\leq(T,m)\Longleftrightarrow S\subseteq T, n\leq m, $$ with $0\in\mathbb N$. This is directed. The point $(S,n)$ has at most $2^{|S|}(n+1)$ predecessors, and every strict predecessor has smaller $|S|+n$. Construct $\phi:J\to I$ by induction on that nonnegative integer. At a point $p=(S,n)$, choose, for each strict predecessor $q$, an index $\kappa_q\geq\phi(q)$ that kills the map into $A_{\phi(q)}$. Choose $\phi(p)$ above the finite set consisting of $S$ and all these $\kappa_q$. At $(\varnothing,0)$ choose any element of $I$. Choices at the same rank do not depend on one another. The map $\phi$ is order preserving, since the killing indices dominate the previous chosen indices. Moreover every transition $A_{\phi(p)}\to A_{\phi(q)}$ for $q