The Stacks project

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

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

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 20.33.3. $\square$

Proof. Lemma 20.32.6 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 20.42.1 to do the translation. $\square$

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 20.45.2 part (2). Hence we obtain the vanishing of negative exts for $K$ and we see that the rule

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

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. $\square$

Proof. Uniqueness was seen in Lemma 20.45.4. 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 20.45.5. By Lemma 20.45.1 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

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

To verify this is a solution use the property of the map $\rho _{U_1, U_2}$ described in Remark 20.45.5 and the compatibility of $\rho _{U_1}$ and $\rho _{U_2}$ with $\rho _{U_1, U_2}$. Having said this we apply Lemma 20.45.4 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 20.45.4 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. $\square$

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. Using transfinite recursion we will construct for all $\beta < \alpha$ a map of complexes

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

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

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

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

(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

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

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 20.45.6 (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

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

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

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

Proof. A pair $(K, \rho _ U)$ is called a solution in the text above. The uniqueness follows from Lemma 20.45.4. 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 20.45.6. In the general case we argue in exactly the same manner, using transfinite induction and Lemma 20.45.7.

First we use transfinite recursion to choose opens $W_\alpha \subset X$ for any ordinal $\alpha$. Namely, we set $W_0 = \emptyset$. If $\alpha = \beta + 1$ is a successor, 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

is a system as in Lemma 20.45.4 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 20.45.4 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 20.45.6.

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 20.45.7 (note that the vanishing of negative exts is true for $K_{W_\beta }$ by Lemma 20.45.4 applied to the solution $K_{W_\beta }$) to obtain $K_{W_\alpha }$ and isomorphisms

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

commutes. This makes sense because

is a system as in Lemma 20.45.4 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). $\square$