The Stacks project

Email of Martin Olsson dated Sep 9, 2021.

Lemma 85.24.5. Let $(\mathcal{C}, \mathcal{O}_\mathcal {C})$ be a ringed site. Assume $\mathcal{C}$ has fibre products. Let $\{ U_ i \to X\} _{i \in I}$ be a covering in $\mathcal{C}$. For $i \in I$ let $E_ i$ be an object of $D(\mathcal{O}_{U_ i})$ and for $i, j \in I$ let

\[ \rho _{ij} : E_ i|_{\mathcal{C}/U_{ij}} \longrightarrow E_ j|_{\mathcal{C}/U_{ij}} \]

be an isomorphism in $D(\mathcal{O}_{U_{ij}})$ where $U_{ij} = U_ i \times _ X U_ j$. Assume

  1. the $\rho _{ij}$ satisfy the cocycle condition on $U_ i \times _ X U_ j \times _ X U_ k$ for all $i, j, k \in I$,

  2. $\mathop{\mathcal{E}\! \mathit{xt}}\nolimits ^ p_{\mathcal{O}_{U_ i}}(E_ i, E_ i) = 0$ for all $p < 0$ and $i \in I$, and

  3. there exists a $t \in \mathbf{Z}$ such that $H^ p(E_ i) = 0$ for $p < t$ and all $i \in I$.

Then there exists a unique pair $(E, \rho _ i)$ where $E$ is an object of $D(\mathcal{O}_ X)$ and $\rho _ i : E|_{U_ i} \to E_ i$ are isomorphisms in $D(\mathcal{O}_{U_ i})$ compatible with the $\rho _{ij}$.

First proof. In this proof we deduce the lemma from the very general Lemma 85.24.3. We urge the reader to look at the second proof in stead.

We may replace $\mathcal{C}$ with $\mathcal{C}/X$. Thus we may and do assume $X$ is the final object of $\mathcal{C}$ and that $\mathcal{C}$ has all finite limits.

Let $\mathcal{B}$ be the full subcategory of $\mathcal{C}$ consisting of $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ such that there exists an $i(U) \in I$ and a morphism $a_ U : U \to U_{i(U)}$. We denote $E_ U = a_ U^*E_{i(U)}$ in $D(\mathcal{O}_ U)$ the pullback (restriction) of $E_ i$ via $a_ U$. Given a morphism $a : U \to U'$ of $\mathcal{B}$ we obtain a morphism $(a_{U'} \circ a, a_ U) : U \to U_{i(U')} \times _ X U_{i(U)} = U_{i(U')i(U)}$ and hence an isomorphism

\[ \rho _ a : a^*E_{U'} = a^*a_{U'}^*E_{i(U')} \xrightarrow {(a_{U'} \circ a, a_ U)^*\rho _{i(U')i(U)}} a_{U}^*E_{i(U)} = E_{U} \]

in $D(\mathcal{O}_ U)$. The data $\mathcal{B}, E_ U, \rho _ a$ are as in Situation 85.24.1; the isomorphisms $\rho _ a$ satisfy the cocycle condition exactly because of condition (1) in the statement of the lemma (details omitted).

We are going to apply Lemma 85.24.3 with $\mathcal{B}$, $E_ U$, $\rho _ a$ as above and with $\mathcal{D} = \mathcal{C}$ and $f : \mathcal{C} \to \mathcal{D}$ the identity morphism. Assumptions (1) and (2)(a) of Lemma 85.24.3 we have seen above. Assumption (2)(b) of Lemma 85.24.3 is clear. Assumption (2)(c) of Lemma 85.24.3 holds because $\{ U_ i \to X\} $ is a covering1. Assumption (3) of Lemma 85.24.3 holds because we have assumed the vanishing of all negative Ext sheaves of $E_ i$ which certainly implies that for any object $U$ lying over $U_ i$ the negative self-Exts of $E_ i|_ U$ are zero. Assumption (4) of Lemma 85.24.3 holds because we have assumed the cohomology sheaves of each $E_ i$ are zero to the left of $t$.

We obtain a unique solution $(E, \rho _ U)$. Setting $\rho _ i = \rho _{U_ i}$ the lemma follows. $\square$

Second proof. We sketch a more direct proof. Denote $K$ the Čech hypercovering of $X$ associated to the covering $\{ U_ i \to X\} _{i \in I}$, see Hypercoverings, Example 25.3.4. Thus for example $K_0 = \{ U_ i \to X\} _{i \in I}$ and $K_1 = \{ U_ i \times _ X U_ j \to X\} _{i, j \in I}$ and so on. Let $((\mathcal{C}/K)_{total}, \mathcal{O})$, $a$, $a_ n$ be as in Remark 85.16.6. The objects $E_ i$ determine an object $M_0$ in $D(\mathcal{O}_0) = \prod D(\mathcal{O}_{U_ i})$. Similarly, the isomorphisms $\rho _{ij}$ determine an isomorphism

\[ \alpha : L(f_{\delta _1^1})^*M_0 \longrightarrow L(f_{\delta _0^1})^*M_0 \]

in $D(\mathcal{O}_1)$ satisfying the cocycle condition. By Lemma 85.14.3 we obtain a cartesian simplicial system $(M_ n)$ of the derived category. By the assumed vanishing of the negative Ext sheaves we see that the objects $M_ n$ have vanishing negative self-exts. Thus we find a cartesian object $M$ of $D(\mathcal{O})$ whose associated simplicial system is isomorphic to $(M_ n)$ by Lemma 85.14.7. Since the cohomology sheaves of $M$ are zero in degrees $< t$ we see that by Lemma 85.20.4 we have $M = La^*E$ for some $E$ in $D(\mathcal{O}_ X)$. The isomorphism $La^*E \to M$ restricted to $\mathcal{C}/U_ i$ produces the isomorphisms $\rho _ i$. We omit the verification of the compatibility with the isomorphisms $\rho _{ij}$. $\square$

[1] In fact, it would suffice if the map $\coprod _{i \in I} h_{U_ i} \to h_ X$ becomes surjective on sheafification and the lemma holds in this case with the same proof.

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