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Tag 0D65

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 [BBD, Theorem 3.2.4] 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.

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). $\square$

    The code snippet corresponding to this tag is a part of the file cohomology.tex and is located in lines 8359–8891 (see updates for more information).

    \section{Glueing complexes}
    \label{section-glueing-complexes}
    
    \noindent
    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
    \cite[Theorem 3.2.4]{BBD}
    in the setting of topological spaces and open coverings.
    
    \begin{lemma}
    \label{lemma-glue}
    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
    \begin{enumerate}
    \item an object $A$ of $D(\mathcal{O}_U)$,
    \item an object $B$ of $D(\mathcal{O}_V)$, and
    \item an isomorphism $c : A|_{U \cap V} \to B|_{U \cap V}$.
    \end{enumerate}
    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
    \begin{enumerate}
    \item an object $E$ of $D(\mathcal{O}_X)$,
    \item a morphism $a : A \to E|_U$ of $D(\mathcal{O}_U)$,
    \item a morphism $b : B \to E|_V$ of $D(\mathcal{O}_V)$, 
    \end{enumerate}
    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 $b \circ g$.
    \end{lemma}
    
    \begin{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 $\Hom$, see
    Lemma \ref{lemma-mayer-vietoris-hom}.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-vanishing-and-glueing}
    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 $Y$.
    \begin{enumerate}
    \item 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 $Y$.
    \item Assume $K, L$ are in $D(\mathcal{O}_X)$ such that
    for $V \in \mathcal{B}$ we have
    $\Ext^i(K|_{f^{-1}V}, L|_{f^{-1}V}) = 0$ for $i < 0$.
    Then $\Ext^i(K|_{f^{-1}V}, L|_{f^{-1}V}) = 0$ for $i < 0$
    for all opens $V \subset Y$ and
    the rule $V \mapsto \Hom(K|_{f^{-1}V}, L|_{f^{-1}V})$ is a sheaf on $Y$.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Lemma \ref{lemma-unbounded-describe-higher-direct-images} 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).
    
    \medskip\noindent
    To prove (2) apply (1) to the complex $R\SheafHom(K, L)$ using
    Lemma \ref{lemma-section-RHom-over-U} to do the translation.
    \end{proof}
    
    \begin{situation}
    \label{situation-locally-given}
    Let $(X, \mathcal{O}_X)$ be a ringed space. We are given
    \begin{enumerate}
    \item a collection of opens $\mathcal{B}$ of $X$,
    \item for $U \in \mathcal{B}$ an object $K_U$ in $D(\mathcal{O}_U)$,
    \item for $V \subset U$ with $V, U \in \mathcal{B}$ an isomorphism
    $\rho^U_V : K_U|_V \to K_V$ in $D(\mathcal{O}_V)$,
    \end{enumerate}
    such that whenever we have $W \subset V \subset U$ with $U, V, W$ in
    $\mathcal{B}$, then $\rho^U_W = \rho^V_W \circ \rho ^U_V|_W$.
    \end{situation}
    
    \noindent
    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 $Y$.
    
    \medskip\noindent
    In Situation \ref{situation-locally-given} a {\it 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.
    
    \begin{lemma}
    \label{lemma-uniqueness}
    In Situation \ref{situation-locally-given} assume
    \begin{enumerate}
    \item $X = \bigcup_{U \in \mathcal{B}} U$ and
    for $U, V \in \mathcal{B}$ we have
    $U \cap V = \bigcup_{W \in \mathcal{B}, W \subset U \cap V} W$,
    \item for any $U \in \mathcal{B}$ we have $\Ext^i(K_U, K_U) = 0$
    for $i < 0$.
    \end{enumerate}
    If a solution $(K, \rho_U)$ exists, then it is unique up to unique isomorphism
    and moreover $\Ext^i(K, K) = 0$ for $i < 0$.
    \end{lemma}
    
    \begin{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 \ref{topology-lemma-create-map-from-subcollection}.
    Set $\mathcal{O}_Y = f_*\mathcal{O}_X$.
    Then $K, K'$ and $\mathcal{B}$ are as in
    Lemma \ref{lemma-vanishing-and-glueing} part (2).
    Hence we obtain the vanishing of negative exts for $K$ and we see that
    the rule
    $$
    V \longmapsto \Hom(K|_{f^{-1}V}, K'|_{f^{-1}V})
    $$
    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.
    \end{proof}
    
    \begin{remark}
    \label{remark-uniqueness}
    With notation and assumptions as in Lemma \ref{lemma-uniqueness}.
    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 \ref{lemma-uniqueness}.
    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.
    \end{remark}
    
    \begin{lemma}
    \label{lemma-solution-in-finite-case}
    In Situation \ref{situation-locally-given} assume
    \begin{enumerate}
    \item $X = U_1 \cup \ldots \cup U_n$ with $U_i \in \mathcal{B}$,
    \item for $U, V \in \mathcal{B}$ we have
    $U \cap V = \bigcup_{W \in \mathcal{B}, W \subset U \cap V} W$,
    \item for any $U \in \mathcal{B}$ we have $\Ext^i(K_U, K_U) = 0$
    for $i < 0$.
    \end{enumerate}
    Then a solution exists and is unique up to unique isomorphism.
    \end{lemma}
    
    \begin{proof}
    Uniqueness was seen in Lemma \ref{lemma-uniqueness}. We may prove the lemma
    by induction on $n$. The case $n = 1$ is immediate.
    
    \medskip\noindent
    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 \ref{remark-uniqueness}.
    By Lemma \ref{lemma-glue} 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 \ref{remark-uniqueness} and the compatibility of
    $\rho_{U_1}$ and $\rho_{U_2}$ with $\rho_{U_1, U_2}$. Having said this
    we apply Lemma \ref{lemma-uniqueness} to see that we obtain a unique
    isomorphism $K|_{U'} \to K_{U'}$ compatible with the maps $\tau_{U'}$ and
    $\rho^U_{U'}$ for $U' \in \mathcal{B}'$.
    
    \medskip\noindent
    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 \ref{lemma-uniqueness} 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.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-glueing-increasing-union}
    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 $\Ext^i(K_\alpha, K_\alpha) = 0$ 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 $\rho_\beta^\alpha$.
    \end{lemma}
    
    \begin{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
    $$
    C^\bullet = \colim_{\beta < \alpha} (j_{\beta, \alpha})_!I_\beta^\bullet
    $$
    (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
    $$
    (C^\bullet)_x =
    \colim_{\beta' < \alpha}
    \left((j_{\beta', \alpha})_!I_{\beta'}^\bullet\right)_x =
    \colim_{\beta \leq \beta' < \alpha} (I_{\beta'}^\bullet)_x
    \longleftarrow
    (I_\beta^\bullet)_x
    $$
    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 \ref{lemma-solution-in-finite-case}
    (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
    $$
    K^\bullet = \colim_{\alpha \in E} (W_\alpha \to X)_!I_\alpha^\bullet
    $$
    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 $\rho^\alpha_\beta$.
    \end{proof}
    
    \noindent
    Using transfinite induction we can prove the result in the general case.
    
    \begin{theorem}[BBD gluing lemma]
    \label{theorem-glueing-bbd-general}
    \begin{reference}
    Special case of \cite[Theorem 3.2.4]{BBD}
    without boundedness assumption.
    \end{reference}
    In Situation \ref{situation-locally-given} assume
    \begin{enumerate}
    \item $X = \bigcup_{U \in \mathcal{B}} U$,
    \item for $U, V \in \mathcal{B}$ we have
    $U \cap V = \bigcup_{W \in \mathcal{B}, W \subset U \cap V} W$,
    \item for any $U \in \mathcal{B}$ we have $\Ext^i(K_U, K_U) = 0$
    for $i < 0$.
    \end{enumerate}
    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.
    \end{theorem}
    
    \begin{proof}
    A pair $(K, \rho_U)$ is called a solution in the text above.
    The uniqueness follows from Lemma \ref{lemma-uniqueness}.
    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 \ref{lemma-solution-in-finite-case}.
    In the general case we argue in exactly the same manner,
    using transfinite induction and
    Lemma \ref{lemma-glueing-increasing-union}.
    
    \medskip\noindent
    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 \ref{lemma-uniqueness} on the ringed space $W_\alpha$.
    
    \medskip\noindent
    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 $\alpha$.
    
    \medskip\noindent
    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 \ref{lemma-uniqueness} 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 \ref{lemma-solution-in-finite-case}.
    
    \medskip\noindent
    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 \ref{lemma-glueing-increasing-union}
    (note that the vanishing of negative exts is true for
    $K_{W_\beta}$ by Lemma \ref{lemma-uniqueness} 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
    $\gamma < \beta < \alpha$.
    
    \medskip\noindent
    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 \ref{lemma-uniqueness} 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).
    \end{proof}

    Comments (2)

    Comment #2796 by Tanya Kaushal Srivastava on September 4, 2017 a 1:25 pm UTC

    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).

    Comment #2899 by Johan (site) on October 7, 2017 a 3:23 pm UTC

    THanks, fixed here.

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