The Stacks project

84.24 Glueing complexes

This section is the continuation of Cohomology, Section 20.43. The goal is to prove a slight generalization of [Theorem 3.2.4, BBD]. Our method will be a tiny bit different in that we use the material from Sections 84.13 and 84.14. We will also reprove the unbounded version as it is proved in [six-I].

Advice to the reader: We suggest the reader first look at the statement of Lemma 84.24.5 as well as the second proof of this lemma.

Here is the situation we are interested in.

Situation 84.24.1. Let $(\mathcal{C}, \mathcal{O}_\mathcal {C})$ be a ringed site. We are given

  1. a category $\mathcal{B}$ and a functor $u : \mathcal{B} \to \mathcal{C}$,

  2. an object $E_ U$ in $D(\mathcal{O}_{u(U)})$ for $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$,

  3. an isomorphism $\rho _ a : E_ U|_{\mathcal{C}/u(V)} \to E_ V$ in $D(\mathcal{O}_{u(V)})$ for $a : V \to U$ in $\mathcal{B}$

such that whenever we have composable arrows $b : W \to V$ and $a : V \to U$ of $\mathcal{B}$, then $\rho _{a \circ b} = \rho _ b \circ \rho _ a|_{\mathcal{C}/u(W)}$.

We won't be able to prove anything about this without making more assumptions. An interesting case is where $\mathcal{B}$ is a full subcategory such that every object of $\mathcal{C}$ has a covering whose members are objects of $\mathcal{B}$ (this is the case considered in [BBD]). For us it is important to allow cases where this is not the case; the main alternative case is where we have a morphism of sites $f : \mathcal{C} \to \mathcal{D}$ and $\mathcal{B}$ is a full subcategory of $\mathcal{D}$ such that every object of $\mathcal{D}$ has a covering whose members are objects of $\mathcal{B}$.

In Situation 84.24.1 a solution will be a pair $(E, \rho _ U)$ where $E$ is an object of $D(\mathcal{O}_\mathcal {C})$ and $\rho _ U : E|_{\mathcal{C}/u(U)} \to E_ U$ for $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$ are isomorphisms such that we have $\rho _ a \circ \rho _ U|_{\mathcal{C}/u(V)} = \rho _ V$ for $a : V \to U$ in $\mathcal{B}$.

Lemma 84.24.2. In Situation 84.24.1. Assume negative self-exts of $E_ U$ in $D(\mathcal{O}_{u(U)})$ are zero. Let $L$ be a simplicial object of $\text{SR}(\mathcal{B})$. Consider the simplicial object $K = u(L)$ of $\text{SR}(\mathcal{C})$ and let $((\mathcal{C}/K)_{total}, \mathcal{O})$ be as in Remark 84.16.5. There exists a cartesian object $E$ of $D(\mathcal{O})$ such that writing $L_ n = \{ U_{n, i}\} _{i \in I_ n}$ the restriction of $E$ to $D(\mathcal{O}_{\mathcal{C}/u(U_{n, i})})$ is $E_{U_{n, i}}$ compatibly (see proof for details). Moreover, $E$ is unique up to unique isomorphism.

Proof. Recall that $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K_ n) = \prod _{i \in I_ n} \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/u(U_{n, i}))$ and similarly for the categories of modules. This product decomposition is also inherited by the derived categories of sheaves of modules. Moreover, this product decomposition is compatible with the morphisms in the simplicial semi-representable object $K$. See Section 84.15. Hence we can set $E_ n = \prod _{i \in I_ n} E_{U_{n, i}}$ (“formal” product) in $D(\mathcal{O}_ n)$. Taking (formal) products of the maps $\rho _ a$ of Situation 84.24.1 we obtain isomorphisms $E_\varphi : f_\varphi ^*E_ n \to E_ m$. The assumption about compostions of the maps $\rho _ a$ immediately implies that $(E_ n, E_\varphi )$ defines a simplicial system of the derived category of modules as in Definition 84.14.1. The vanishing of negative exts assumed in the lemma implies that $\mathop{\mathrm{Hom}}\nolimits (E_ n[t], E_ n) = 0$ for $n \geq 0$ and $t > 0$. Thus by Lemma 84.14.7 we obtain $E$. Uniqueness up to unique isomorphism follows from Lemmas 84.14.5 and 84.14.6. $\square$

Lemma 84.24.3 (BBD glueing lemma). In Situation 84.24.1. Assume

  1. $\mathcal{C}$ has equalizers and fibre products,

  2. there is a morphism of sites $f : \mathcal{C} \to \mathcal{D}$ given by a continuous functor $u : \mathcal{D} \to \mathcal{C}$ such that

    1. $\mathcal{D}$ has equalizers and fibre products and $u$ commutes with them,

    2. $\mathcal{B}$ is a full subcategory of $\mathcal{D}$ and $u : \mathcal{B} \to \mathcal{C}$ is the restriction of $u$,

    3. every object of $\mathcal{D}$ has a covering whose members are objects of $\mathcal{B}$,

  3. all negative self-exts of $E_ U$ in $D(\mathcal{O}_{u(U)})$ are zero, and

  4. there exists a $t \in \mathbf{Z}$ such that $H^ i(E_ U) = 0$ for $i < t$ and $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$.

Then there exists a solution unique up to unique isomorphism.

Proof. By Hypercoverings, Lemma 25.12.3 there exists a hypercovering $L$ for the site $\mathcal{D}$ such that $L_ n = \{ U_{n, i}\} _{i \in I_ n}$ with $U_{i, n} \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$. Set $K = u(L)$. Apply Lemma 84.24.2 to get a cartesian object $E$ of $D(\mathcal{O})$ on the site $(\mathcal{C}/K)_{total}$ restricting to $E_{U_{n, i}}$ on $\mathcal{C}/u(U_{n, i})$ compatibly. The assumption on $t$ implies that $E \in D^+(\mathcal{O})$. By Hypercoverings, Lemma 25.12.4 we see that $K$ is a hypercovering too. By Lemma 84.18.4 we find that $E = a^*F$ for some $F$ in $D^+(\mathcal{O}_\mathcal {C})$.

To prove that $F$ is a solution we will use the construction of $L_0$ and $L_1$ given in the proof of Hypercoverings, Lemma 25.12.3. (This is a bit inelegant but there does not seem to be a completely straightforward way around it.)

Namely, we have $I_0 = \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$ and so $L_0 = \{ U\} _{U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})}$. Hence the isomorphism $a^*F \to E$ restricted to the components $\mathcal{C}/u(U)$ of $\mathcal{C}/K_0$ defines isomorphisms $\rho _ U : F|_{\mathcal{C}/u(U)} \to E_ U$ for $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$ by our choice of $E$.

To prove that $\rho _ U$ satisfy the requirement of compatibility with the maps $\rho _ a$ of Situation 84.24.1 we use that $I_1$ contains the set

\[ \Omega = \{ (U, V, W, a, b) \mid U, V, W \in \mathcal{B}, a : U \to V, b : U \to W\} \]

and that for $i = (U, V, W, a, b)$ in $\Omega $ we have $U_{1, i} = U$. Moreover, the component maps $f_{\delta ^1_0, i}$ and $f_{\delta ^1_1, i}$ of the two morphisms $K_1 \to K_0$ are the morphisms

\[ a : U \to V \quad \text{and}\quad b : U \to V \]

Hence the compatibility mentioned in Lemma 84.24.2 gives that

\[ \rho _ a \circ \rho _ V|_{\mathcal{C}/u(U)} = \rho _ U \quad \text{and}\quad \rho _ b \circ \rho _ W|_{\mathcal{C}/u(U)} = \rho _ U \]

Taking $i = (U, V, U, a, \text{id}_ U) \in \Omega $ for example, we find that we have the desired compatibility. The uniqueness of $F$ follows from the uniqueness of $E$ in the previous lemma (small detail omitted). $\square$

Lemma 84.24.4 (Unbounded BBD glueing lemma). In Situation 84.24.1. Assume

  1. $\mathcal{C}$ has equalizers and fibre products,

  2. there is a morphism of sites $f : \mathcal{C} \to \mathcal{D}$ given by a continuous functor $u : \mathcal{D} \to \mathcal{C}$ such that

    1. $\mathcal{D}$ has equalizers and fibre products and $u$ commutes with them,

    2. $\mathcal{B}$ is a full subcategory of $\mathcal{D}$ and $u : \mathcal{B} \to \mathcal{C}$ is the restriction of $u$,

    3. every object of $\mathcal{D}$ has a covering whose members are objects of $\mathcal{B}$,

  3. all negative self-exts of $E_ U$ in $D(\mathcal{O}_{u(U)})$ are zero, and

  4. there exist weak Serre subcategories $\mathcal{A}_ U \subset \textit{Mod}(\mathcal{O}_ U)$ for all $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ satisfying conditions (1), (2), and (3),

  5. $E_ U \in D_{\mathcal{A}_ U}(\mathcal{O}_ U)$.

Then there exists a solution unique up to unique isomorphism.

Proof. The proof is exactly the same as the proof of Lemma 84.24.3. The only change is that $E$ is an object of $D_{\mathcal{A}_{total}}(\mathcal{O})$ and hence we use Lemma 84.23.3 to obtain $F$ with $E = a^*F$ instead of Lemma 84.18.4. $\square$

Here is an example application of the general theory above.

reference

Lemma 84.24.5. 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 84.24.3. 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 84.24.1; 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 84.24.3 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 84.24.3 we have seen above. Assumption (2)(b) of Lemma 84.24.3 is clear. Assumption (2)(c) of Lemma 84.24.3 holds because $\{ U_ i \to X\} $ is a covering1. Assumption (3) of Lemma 84.24.3 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 84.24.3 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 84.16.6. 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 84.14.3 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 84.14.7. Since the cohomology sheaves of $M$ are zero in degrees $< t$ we see that by Lemma 84.20.4 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.

Comments (2)

Comment #5440 by David Hansen on

The formatting of condition (4) in Lemma 0DCC is rather strange - you have to click on the conditions (1), (2), or (3) to read them, and you can't easily read all of them at once.

Comment #5664 by on

@#5440. Yes this is true. In fact, a terrible amount of notation is required for even just stating this. This lemma is kind of just awful. The idea of having the is to sort of say: we have the complexes already over pullbacks of objects from . The idea of conditions (1), (2), and (3) is that we are given some collection of sheaves whose cohomology is sort of locally uniformly bounded (and the last condition stipulates E_U have cohomology sheaves of this type). Note that the collection of objects in condition (2) is different from the category in Situation 84.24.1 which is the same as the collection of objects in mentioned in part (2) of the statement of the lemma. Sigh!

Request: if somebody ever wants to use this lemma in a paper, then please make some precise suggestions as to how to elucidate all of these conditions a bit better in the text. Thanks!


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