The Stacks project

83.24 Glueing complexes

This section is the continuation of Cohomology, Section 20.41. 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 83.13 and 83.14. We will also reprove the unbounded version as it is proved in [six-I].

Here is the situation we are interested in.

Situation 83.24.1. 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 83.24.1 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 83.24.2. In Situation 83.24.1. 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 83.16.5. 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 83.15. 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 83.24.1 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 83.14.1. 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 83.14.6 we obtain $E$. Uniqueness up to unique isomorphism follows from Lemmas 83.14.4 and 83.14.5. $\square$

Lemma 83.24.3 (BBD glueing lemma). In Situation 83.24.1. 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 83.24.2 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 83.18.4 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 83.24.1 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 83.24.2 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 83.24.4 (Unbounded BBD glueing lemma). In Situation 83.24.1. 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 83.24.3. The only change is that $E$ is an object of $D_{\mathcal{A}_{total}}(\mathcal{O})$ and hence we use Lemma 83.23.3 to obtain $F$ with $E = a^*F$ instead of Lemma 83.18.4. $\square$


Comments (1)

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.


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