\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

The Stacks project

20.39 Glueing complexes

We can glue complexes! More precisely, in certain circumstances we can glue locally given objects of the derived category to a global object. We first prove some easy cases and then we'll prove the very general [Theorem 3.2.4, BBD] in the setting of topological spaces and open coverings.

Lemma 20.39.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $X = U \cup V$ be the union of two open subspaces of $X$. Suppose given

  1. an object $A$ of $D(\mathcal{O}_ U)$,

  2. an object $B$ of $D(\mathcal{O}_ V)$, and

  3. an isomorphism $c : A|_{U \cap V} \to B|_{U \cap V}$.

Then there exists an object $F$ of $D(\mathcal{O}_ X)$ and isomorphisms $f : F|_ U \to A$, $g : F|_ V \to B$ such that $c = g|_{U \cap V} \circ f^{-1}|_{U \cap V}$. Moreover, given

  1. an object $E$ of $D(\mathcal{O}_ X)$,

  2. a morphism $a : A \to E|_ U$ of $D(\mathcal{O}_ U)$,

  3. a morphism $b : B \to E|_ V$ of $D(\mathcal{O}_ V)$,

such that

\[ a|_{U \cap V} = b|_{U \cap V} \circ c. \]

Then there exists a morphism $F \to E$ in $D(\mathcal{O}_ X)$ whose restriction to $U$ is $a \circ f$ and whose restriction to $V$ is $b \circ g$.

Proof. Denote $j_ U$, $j_ V$, $j_{U \cap V}$ the corresponding open immersions. Choose a distinguished triangle

\[ F \to Rj_{U, *}A \oplus Rj_{V, *}B \to Rj_{U \cap V, *}(B|_{U \cap V}) \to F[1] \]

where the map $Rj_{V, *}B \to Rj_{U \cap V, *}(B|_{U \cap V})$ is the obvious one and where $Rj_{U, *}A \to Rj_{U \cap V, *}(B|_{U \cap V})$ is the composition of $Rj_{U, *}A \to Rj_{U \cap V, *}(A|_{U \cap V})$ with $Rj_{U \cap V, *}c$. Restricting to $U$ we obtain

\[ F|_ U \to A \oplus (Rj_{V, *}B)|_ U \to (Rj_{U \cap V, *}(B|_{U \cap V}))|_ U \to F|_ U[1] \]

Denote $j : U \cap V \to U$. Compatibility of restriction to opens and cohomology shows that both $(Rj_{V, *}B)|_ U$ and $(Rj_{U \cap V, *}(B|_{U \cap V}))|_ U$ are canonically isomorphic to $Rj_*(B|_{U \cap V})$. Hence the second arrow of the last displayed diagram has a section, and we conclude that the morphism $F|_ U \to A$ is an isomorphism. Similarly, the morphism $F|_ V \to B$ is an isomorphism. The existence of the morphism $F \to E$ follows from the Mayer-Vietoris sequence for $\mathop{\mathrm{Hom}}\nolimits $, see Lemma 20.31.3. $\square$

Lemma 20.39.2. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. Let $\mathcal{B}$ be a basis for the topology on $Y$.

  1. Assume $K$ is in $D(\mathcal{O}_ X)$ such that for $V \in \mathcal{B}$ we have $H^ i(f^{-1}(V), K) = 0$ for $i < 0$. Then $Rf_*K$ has vanishing cohomology sheaves in negative degrees, $H^ i(f^{-1}(V), K) = 0$ for $i < 0$ for all opens $V \subset Y$, and the rule $V \mapsto H^0(f^{-1}V, K)$ is a sheaf on $Y$.

  2. Assume $K, L$ are in $D(\mathcal{O}_ X)$ such that for $V \in \mathcal{B}$ we have $\mathop{\mathrm{Ext}}\nolimits ^ i(K|_{f^{-1}V}, L|_{f^{-1}V}) = 0$ for $i < 0$. Then $\mathop{\mathrm{Ext}}\nolimits ^ i(K|_{f^{-1}V}, L|_{f^{-1}V}) = 0$ for $i < 0$ for all opens $V \subset Y$ and the rule $V \mapsto \mathop{\mathrm{Hom}}\nolimits (K|_{f^{-1}V}, L|_{f^{-1}V})$ is a sheaf on $Y$.

Proof. Lemma 20.30.6 tells us $H^ i(Rf_*K)$ is the sheaf associated to the presheaf $V \mapsto H^ i(f^{-1}(V), K) = H^ i(V, Rf_*K)$. The assumptions in (1) imply that $Rf_*K$ has vanishing cohomology sheaves in degrees $< 0$. We conclude that for any open $V \subset Y$ the cohomology group $H^ i(V, Rf_*K)$ is zero for $i < 0$ and is equal to $H^0(V, H^0(Rf_*K))$ for $i = 0$. This proves (1).

To prove (2) apply (1) to the complex $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)$ using Lemma 20.36.1 to do the translation. $\square$

Situation 20.39.3. Let $(X, \mathcal{O}_ X)$ be a ringed space. We are given

  1. a collection of opens $\mathcal{B}$ of $X$,

  2. for $U \in \mathcal{B}$ an object $K_ U$ in $D(\mathcal{O}_ U)$,

  3. for $V \subset U$ with $V, U \in \mathcal{B}$ an isomorphism $\rho ^ U_ V : K_ U|_ V \to K_ V$ in $D(\mathcal{O}_ V)$,

such that whenever we have $W \subset V \subset U$ with $U, V, W$ in $\mathcal{B}$, then $\rho ^ U_ W = \rho ^ V_ W \circ \rho ^ U_ V|_ W$.

We won't be able to prove anything about this without making more assumptions. An interesting case is where $\mathcal{B}$ is a basis for the topology on $X$. Another is the case where we have a morphism $f : X \to Y$ of topological spaces and the elements of $\mathcal{B}$ are the inverse images of the elements of a basis for the topology of $Y$.

In Situation 20.39.3 a solution will be a pair $(K, \rho _ U)$ where $K$ is an object of $D(\mathcal{O}_ X)$ and $\rho _ U : K|_ U \to K_ U$, $U \in \mathcal{B}$ are isomorphisms such that we have $\rho ^ U_ V \circ \rho _ U|_ V = \rho _ V$ for all $V \subset U$, $U, V \in \mathcal{B}$. In certain cases solutions are unique.

Lemma 20.39.4. In Situation 20.39.3 assume

  1. $X = \bigcup _{U \in \mathcal{B}} U$ and for $U, V \in \mathcal{B}$ we have $U \cap V = \bigcup _{W \in \mathcal{B}, W \subset U \cap V} W$,

  2. for any $U \in \mathcal{B}$ we have $\mathop{\mathrm{Ext}}\nolimits ^ i(K_ U, K_ U) = 0$ for $i < 0$.

If a solution $(K, \rho _ U)$ exists, then it is unique up to unique isomorphism and moreover $\mathop{\mathrm{Ext}}\nolimits ^ i(K, K) = 0$ for $i < 0$.

Proof. Let $(K, \rho _ U)$ and $(K', \rho '_ U)$ be a pair of solutions. Let $f : X \to Y$ be the continuous map constructed in Topology, Lemma 5.5.6. Set $\mathcal{O}_ Y = f_*\mathcal{O}_ X$. Then $K, K'$ and $\mathcal{B}$ are as in Lemma 20.39.2 part (2). Hence we obtain the vanishing of negative exts for $K$ and we see that the rule

\[ V \longmapsto \mathop{\mathrm{Hom}}\nolimits (K|_{f^{-1}V}, K'|_{f^{-1}V}) \]

is a sheaf on $Y$. As both $(K, \rho _ U)$ and $(K', \rho '_ U)$ are solutions the maps

\[ (\rho '_ U)^{-1} \circ \rho _ U : K|_ U \longrightarrow K'|_ U \]

over $U = f^{-1}(f(U))$ agree on overlaps. Hence we get a unique global section of the sheaf above which defines the desired isomorphism $K \to K'$ compatible with all structure available. $\square$

Remark 20.39.5. With notation and assumptions as in Lemma 20.39.4. Suppose that $U, V \in \mathcal{B}$. Let $\mathcal{B}'$ be the set of elements of $\mathcal{B}$ contained in $U \cap V$. Then

\[ (\{ K_{U'}\} _{U' \in \mathcal{B}'}, \{ \rho _{V'}^{U'}\} _{V' \subset U'\text{ with }U', V' \in \mathcal{B}'}) \]

is a system on the ringed space $U \cap V$ satisfying the assumptions of Lemma 20.39.4. Moreover, both $(K_ U|_{U \cap V}, \rho ^ U_{U'})$ and $(K_ V|_{U \cap V}, \rho ^ V_{U'})$ are solutions to this system. By the lemma we find a unique isomorphism

\[ \rho _{U, V} : K_ U|_{U \cap V} \longrightarrow K_ V|_{U \cap V} \]

such that for every $U' \subset U \cap V$, $U' \in \mathcal{B}$ the diagram

\[ \xymatrix{ K_ U|_{U'} \ar[rr]_{\rho _{U, V}|_{U'}} \ar[rd]_{\rho ^ U_{U'}} & & K_ V|_{U'} \ar[ld]^{\rho ^ V_{U'}} \\ & K_{U'} } \]

commutes. Pick a third element $W \in \mathcal{B}$. We obtain isomorphisms $\rho _{U, W} : K_ U|_{U \cap W} \to K_ W|_{U \cap W}$ and $\rho _{V, W} : K_ U|_{V \cap W} \to K_ W|_{V \cap W}$ satisfying similar properties to those of $\rho _{U, V}$. Finally, we have

\[ \rho _{U, W}|_{U \cap V \cap W} = \rho _{V, W}|_{U \cap V \cap W} \circ \rho _{U, V}|_{U \cap V \cap W} \]

This is true by the uniqueness in the lemma because both sides of the equality are the unique isomorphism compatible with the maps $\rho ^ U_{U''}$ and $\rho ^ W_{U''}$ for $U'' \subset U \cap V \cap W$, $U'' \in \mathcal{B}$. Some minor details omitted. The collection $(K_ U, \rho _{U, V})$ is a descent datum in the derived category for the open covering $\mathcal{U} : X = \bigcup _{U \in \mathcal{B}} U$ of $X$. In this language we are looking for “effectiveness of the descent datum” when we look for the existence of a solution.

Lemma 20.39.6. In Situation 20.39.3 assume

  1. $X = U_1 \cup \ldots \cup U_ n$ with $U_ i \in \mathcal{B}$,

  2. for $U, V \in \mathcal{B}$ we have $U \cap V = \bigcup _{W \in \mathcal{B}, W \subset U \cap V} W$,

  3. for any $U \in \mathcal{B}$ we have $\mathop{\mathrm{Ext}}\nolimits ^ i(K_ U, K_ U) = 0$ for $i < 0$.

Then a solution exists and is unique up to unique isomorphism.

Proof. Uniqueness was seen in Lemma 20.39.4. We may prove the lemma by induction on $n$. The case $n = 1$ is immediate.

The case $n = 2$. Consider the isomorphism $\rho _{U_1, U_2} : K_{U_1}|_{U_1 \cap U_2} \to K_{U_2}|_{U_1 \cap U_2}$ constructed in Remark 20.39.5. By Lemma 20.39.1 we obtain an object $K$ in $D(\mathcal{O}_ X)$ and isomorphisms $\rho _{U_1} : K|_{U_1} \to K_{U_1}$ and $\rho _{U_2} : K|_{U_2} \to K_{U_2}$ compatible with $\rho _{U_1, U_2}$. Take $U \in \mathcal{B}$. We will construct an isomorphism $\rho _ U : K|_ U \to K_ U$ and we will leave it to the reader to verify that $(K, \rho _ U)$ is a solution. Consider the set $\mathcal{B}'$ of elements of $\mathcal{B}$ contained in either $U \cap U_1$ or contained in $U \cap U_2$. Then $(K_ U, \rho ^ U_{U'})$ is a solution for the system $(\{ K_{U'}\} _{U' \in \mathcal{B}'}, \{ \rho _{V'}^{U'}\} _{V' \subset U'\text{ with }U', V' \in \mathcal{B}'})$ on the ringed space $U$. We claim that $(K|_ U, \tau _{U'})$ is another solution where $\tau _{U'}$ for $U' \in \mathcal{B}'$ is chosen as follows: if $U' \subset U_1$ then we take the composition

\[ K|_{U'} \xrightarrow {\rho _{U_1}|_{U'}} K_{U_1}|_{U'} \xrightarrow {\rho ^{U_1}_{U'}} K_{U'} \]

and if $U' \subset U_2$ then we take the composition

\[ K|_{U'} \xrightarrow {\rho _{U_2}|_{U'}} K_{U_2}|_{U'} \xrightarrow {\rho ^{U_2}_{U'}} K_{U'}. \]

To verify this is a solution use the property of the map $\rho _{U_1, U_2}$ described in Remark 20.39.5 and the compatibility of $\rho _{U_1}$ and $\rho _{U_2}$ with $\rho _{U_1, U_2}$. Having said this we apply Lemma 20.39.4 to see that we obtain a unique isomorphism $K|_{U'} \to K_{U'}$ compatible with the maps $\tau _{U'}$ and $\rho ^ U_{U'}$ for $U' \in \mathcal{B}'$.

The case $n > 2$. Consider the open subspace $X' = U_1 \cup \ldots \cup U_{n - 1}$ and let $\mathcal{B}'$ be the set of elements of $\mathcal{B}$ contained in $X'$. Then we find a system $(\{ K_ U\} _{U \in \mathcal{B}'}, \{ \rho _ V^ U\} _{U, V \in \mathcal{B}'})$ on the ringed space $X'$ to which we may apply our induction hypothesis. We find a solution $(K_{X'}, \rho ^{X'}_ U)$. Then we can consider the collection $\mathcal{B}^* = \mathcal{B} \cup \{ X'\} $ of opens of $X$ and we see that we obtain a system $(\{ K_ U\} _{U \in \mathcal{B}^*}, \{ \rho _ V^ U\} _{V \subset U\text{ with }U, V \in \mathcal{B}^*})$. Note that this new system also satisfies condition (3) by Lemma 20.39.4 applied to the solution $K_{X'}$. For this system we have $X = X' \cup U_ n$. This reduces us to the case $n = 2$ we worked out above. $\square$

Lemma 20.39.7. Let $X$ be a ringed space. Let $E$ be a well ordered set and let

\[ X = \bigcup \nolimits _{\alpha \in E} W_\alpha \]

be an open covering with $W_\alpha \subset W_{\alpha + 1}$ and $W_\alpha = \bigcup _{\beta < \alpha } W_\beta $ if $\alpha $ is not a successor. Let $K_\alpha $ be an object of $D(\mathcal{O}_{W_\alpha })$ with $\mathop{\mathrm{Ext}}\nolimits ^ i(K_\alpha , K_\alpha ) = 0$ for $i < 0$. Assume given isomorphisms $\rho _\beta ^\alpha : K_\alpha |_{W_\beta } \to K_\beta $ in $D(\mathcal{O}_{W_\beta })$ for all $\beta < \alpha $ with $\rho _\gamma ^\alpha = \rho _\gamma ^\beta \circ \rho ^\alpha _\beta |_{W_\gamma }$ for $\gamma < \beta < \alpha $. Then there exists an object $K$ in $D(\mathcal{O}_ X)$ and isomorphisms $K|_{W_\alpha } \to K_\alpha $ for $\alpha \in E$ compatible with the isomorphisms $\rho _\beta ^\alpha $.

Proof. In this proof $\alpha , \beta , \gamma , \ldots $ represent elements of $E$. Choose a K-injective complex $I_\alpha ^\bullet $ on $W_\alpha $ representing $K_\alpha $. For $\beta < \alpha $ denote $j_{\beta , \alpha } : W_\beta \to W_\alpha $ the inclusion morphism. By transfinite induction, we will construct for all $\beta < \alpha $ a map of complexes

\[ \tau _{\beta , \alpha } : (j_{\beta , \alpha })_!I_\beta ^\bullet \longrightarrow I_\alpha ^\bullet \]

representing the adjoint to the inverse of the isomorphism $\rho ^\alpha _\beta : K_\alpha |_{W_\beta } \to K_\beta $. Moreover, we will do this in such that for $\gamma < \beta < \alpha $ we have

\[ \tau _{\gamma , \alpha } = \tau _{\beta , \alpha } \circ (j_{\beta , \alpha })_!\tau _{\gamma , \beta } \]

as maps of complexes. Namely, suppose already given $\tau _{\gamma , \beta }$ composing correctly for all $\gamma < \beta < \alpha $. If $\alpha = \alpha ' + 1$ is a successor, then we choose any map of complexes

\[ (j_{\alpha ', \alpha })_!I_{\alpha '}^\bullet \to I_\alpha ^\bullet \]

which is adjoint to the inverse of the isomorphism $\rho ^\alpha _{\alpha '} : K_\alpha |_{W_{\alpha '}} \to K_{\alpha '}$ (possible because $I_\alpha ^\bullet $ is K-injective) and for any $\beta < \alpha '$ we set

\[ \tau _{\beta , \alpha } = \tau _{\alpha ', \alpha } \circ (j_{\alpha ', \alpha })_!\tau _{\beta , \alpha '} \]

If $\alpha $ is not a successor, then we can consider the complex on $W_\alpha $ given by

\[ C^\bullet = \mathop{\mathrm{colim}}\nolimits _{\beta < \alpha } (j_{\beta , \alpha })_!I_\beta ^\bullet \]

(termwise colimit) where the transition maps of the sequence are given by the maps $\tau _{\beta ', \beta }$ for $\beta ' < \beta < \alpha $. We claim that $C^\bullet $ represents $K_\alpha $. Namely, for $\beta < \alpha $ the restriction of the coprojection $(j_{\beta , \alpha })_!I_\beta ^\bullet \to C^\bullet $ gives a map

\[ \sigma _\beta : I_\beta ^\bullet \longrightarrow C^\bullet |_{W_\beta } \]

which is a quasi-isomorphism: if $x \in W_\beta $ then looking at stalks we get

\[ (C^\bullet )_ x = \mathop{\mathrm{colim}}\nolimits _{\beta ' < \alpha } \left((j_{\beta ', \alpha })_!I_{\beta '}^\bullet \right)_ x = \mathop{\mathrm{colim}}\nolimits _{\beta \leq \beta ' < \alpha } (I_{\beta '}^\bullet )_ x \longleftarrow (I_\beta ^\bullet )_ x \]

which is a quasi-isomorphism. Here we used that taking stalks commutes with colimits, that filtered colimits are exact, and that the maps $(I_\beta ^\bullet )_ x \to (I_{\beta '}^\bullet )_ x$ are quasi-isomorphisms for $\beta \leq \beta ' < \alpha $. Hence $(C^\bullet , \sigma _\beta ^{-1})$ is a solution to the system $(\{ K_\beta \} _{\beta < \alpha }, \{ \rho ^\beta _{\beta '}\} _{\beta ' < \beta < \alpha })$. Since $(K_\alpha , \rho ^\alpha _\beta )$ is another solution we obtain a unique isomorphism $\sigma : K_\alpha \to C^\bullet $ in $D(\mathcal{O}_{W_\alpha })$ compatible with all our maps, see Lemma 20.39.6 (this is where we use the vanishing of negative ext groups). Choose a morphism $\tau : C^\bullet \to I_\alpha ^\bullet $ of complexes representing $\sigma $. Then we set

\[ \tau _{\beta , \alpha } = \tau |_{W_\beta } \circ \sigma _\beta \]

to get the desired maps. Finally, we take $K$ to be the object of the derived category represented by the complex

\[ K^\bullet = \mathop{\mathrm{colim}}\nolimits _{\alpha \in E} (W_\alpha \to X)_!I_\alpha ^\bullet \]

where the transition maps are given by our carefully constructed maps $\tau _{\beta , \alpha }$ for $\beta < \alpha $. Arguing exactly as above we see that for all $\alpha $ the restriction of the coprojection determines an isomorphism

\[ K|_{W_\alpha } \longrightarrow K_\alpha \]

compatible with the given maps $\rho ^\alpha _\beta $. $\square$

Using transfinite induction we can prove the result in the general case.

reference

Theorem 20.39.8 (BBD gluing lemma). In Situation 20.39.3 assume

  1. $X = \bigcup _{U \in \mathcal{B}} U$,

  2. for $U, V \in \mathcal{B}$ we have $U \cap V = \bigcup _{W \in \mathcal{B}, W \subset U \cap V} W$,

  3. for any $U \in \mathcal{B}$ we have $\mathop{\mathrm{Ext}}\nolimits ^ i(K_ U, K_ U) = 0$ for $i < 0$.

Then there exists an object $K$ of $D(\mathcal{O}_ X)$ and isomorphisms $\rho _ U : K|_ U \to K_ U$ in $D(\mathcal{O}_ U)$ for $U \in \mathcal{B}$ such that $\rho ^ U_ V \circ \rho _ U|_ V = \rho _ V$ for all $V \subset U$ with $U, V \in \mathcal{B}$. The pair $(K, \rho _ U)$ is unique up to unique isomorphism.

Proof. A pair $(K, \rho _ U)$ is called a solution in the text above. The uniqueness follows from Lemma 20.39.4. If $X$ has a finite covering by elements of $\mathcal{B}$ (for example if $X$ is quasi-compact), then the theorem is a consequence of Lemma 20.39.6. In the general case we argue in exactly the same manner, using transfinite induction and Lemma 20.39.7.

First we use transfinite induction to choose opens $W_\alpha \subset X$ for any ordinal $\alpha $. Namely, we set $W_0 = \emptyset $. If $\alpha = \beta + 1$ is a sucessor, then either $W_\beta = X$ and we set $W_\alpha = X$ or $W_\beta \not= X$ and we set $W_\alpha = W_\beta \cup U_\alpha $ where $U_\alpha \in \mathcal{B}$ is not contained in $W_\beta $. If $\alpha $ is a limit ordinal we set $W_\alpha = \bigcup _{\beta < \alpha } W_\beta $. Then for large enough $\alpha $ we have $W_\alpha = X$. Observe that for every $\alpha $ the open $W_\alpha $ is a union of elements of $\mathcal{B}$. Hence if $\mathcal{B}_\alpha = \{ U \in \mathcal{B}, U \subset W_\alpha \} $, then

\[ S_\alpha = (\{ K_ U\} _{U \in \mathcal{B}_\alpha }, \{ \rho _ V^ U\} _{V \subset U\text{ with }U, V \in \mathcal{B}_\alpha }) \]

is a system as in Lemma 20.39.4 on the ringed space $W_\alpha $.

We will show by transfinite induction that for every $\alpha $ the system $S_\alpha $ has a solution. This will prove the theorem as this system is the system given in the theorem for large $\alpha $.

The case where $\alpha = \beta + 1$ is a successor ordinal. (This case was already treated in the proof of the lemma above but for clarity we repeat the argument.) Recall that $W_\alpha = W_\beta \cup U_\alpha $ for some $U_\alpha \in \mathcal{B}$ in this case. By induction hypothesis we have a solution $(K_{W_\beta }, \{ \rho ^{W_\beta }_ U\} _{U \in \mathcal{B}_\beta })$ for the system $S_\beta $. Then we can consider the collection $\mathcal{B}_\alpha ^* = \mathcal{B}_\alpha \cup \{ W_\beta \} $ of opens of $W_\alpha $ and we see that we obtain a system $(\{ K_ U\} _{U \in \mathcal{B}_\alpha ^*}, \{ \rho _ V^ U\} _{V \subset U\text{ with }U, V \in \mathcal{B}_\alpha ^*})$. Note that this new system also satisfies condition (3) by Lemma 20.39.4 applied to the solution $K_{W_\beta }$. For this system we have $W_\alpha = W_\beta \cup U_\alpha $. This reduces us to the case handled in Lemma 20.39.6.

The case where $\alpha $ is a limit ordinal. Recall that $W_\alpha = \bigcup _{\beta < \alpha } W_\beta $ in this case. For $\beta < \alpha $ let $(K_{W_\beta }, \{ \rho ^{W_\beta }_ U\} _{U \in \mathcal{B}_\beta })$ be the solution for $S_\beta $. For $\gamma < \beta < \alpha $ the restriction $K_{W_\beta }|_{W_\gamma }$ endowed with the maps $\rho ^{W_\beta }_ U$, $U \in \mathcal{B}_\gamma $ is a solution for $S_\gamma $. By uniqueness we get unique isomorphisms $\rho _{W_\gamma }^{W_\beta } : K_{W_\beta }|_{W_\gamma } \to K_{W_\gamma }$ compatible with the maps $\rho ^{W_\beta }_ U$ and $\rho ^{W_\gamma }_ U$ for $U \in \mathcal{B}_\gamma $. These maps compose in the correct manner, i.e., $\rho _{W_\delta }^{W_\gamma } \circ \rho _{W_\gamma }^{W_\beta }|_{W_\delta } = \rho ^{W_\delta }_{W_\beta }$ for $\delta < \gamma < \beta < \alpha $. Thus we may apply Lemma 20.39.7 (note that the vanishing of negative exts is true for $K_{W_\beta }$ by Lemma 20.39.4 applied to the solution $K_{W_\beta }$) to obtain $K_{W_\alpha }$ and isomorphisms

\[ \rho _{W_\beta }^{W_\alpha } : K_{W_\alpha }|_{W_\beta } \longrightarrow K_{W_\beta } \]

compatible with the maps $\rho _{W_\gamma }^{W_\beta }$ for $\gamma < \beta < \alpha $.

To show that $K_{W_\alpha }$ is a solution we still need to construct the isomorphisms $\rho _ U^{W_\alpha } : K_{W_\alpha }|_ U \to K_ U$ for $U \in \mathcal{B}_\alpha $ satisfying certain compatibilities. We choose $\rho _ U^{W_\alpha }$ to be the unique map such that for any $\beta < \alpha $ and any $V \in \mathcal{B}_\beta $ with $V \subset U$ the diagram

\[ \xymatrix{ K_{W_\alpha }|_ V \ar[r]_{\rho _ U^{W_\alpha }|_ V} \ar[d]_{\rho _{W_\beta }^{W_\alpha }|_ V} & K_ U|_ V \ar[d]^{\rho _ U^ V} \\ K_{W_\beta } \ar[r]^{\rho _ V^{W_\beta }} & K_ V } \]

commutes. This makes sense because

\[ (\{ K_ V\} _{V \subset U, V \in \mathcal{B}_\beta \text{ for some }\beta < \alpha }, \{ \rho _ V^{V'}\} _{V \subset V'\text{ with }V, V' \subset U \text{ and }V, V' \in \mathcal{B}_\beta \text{ for some }\beta < \alpha }) \]

is a system as in Lemma 20.39.4 on the ringed space $U$ and because $(K_ U, \rho ^ U_ V)$ and $(K_{W_\alpha }|_ U, \rho _ V^{W_\beta }\circ \rho _{W_\beta }^{W_\alpha }|_ V)$ are both solutions for this system. This gives existence and uniqueness. We omit the proof that these maps satisfy the desired compatibilities (it is just bookkeeping). $\square$


Comments (2)

Comment #2796 by Tanya Kaushal Srivastava on

Minor typo: In the second last paragraph of the proof of BBD gluing lemma, I guess we would like the diagram to commute. (Word "commute" is missing).


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