The Stacks project

83.11 Cohomology and augmentations of ringed simplicial sites

This section is the analogue of Section 83.9 for sheaves of modules.

Consider a simplicial site $\mathcal{C}$ as in Situation 83.3.3. Let $a_0$ be an augmentation towards a site $\mathcal{D}$ as in Remark 83.4.1. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. Let $\mathcal{O}_\mathcal {D}$ be a sheaf of rings on $\mathcal{D}$. Suppose we are given a morphism

\[ a^\sharp : \mathcal{O}_\mathcal {D} \longrightarrow a_*\mathcal{O} \]

where $a$ is as in Lemma 83.4.2. Consequently, we obtain a morphism of ringed topoi

\[ a : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total}), \mathcal{O}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D}) \]

We will think of $g_ n : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n), \mathcal{O}_ n) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total}), \mathcal{O})$ as a morphism of ringed topoi as in Lemma 83.6.1, then taking the composition $a_ n = a \circ g_ n$ (Lemma 83.4.2) as morphisms of ringed topoi we obtain

\[ a_ n : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n), \mathcal{O}_ n) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D}) \]

Using the transition maps $f_\varphi ^{-1}\mathcal{O}_ m \to \mathcal{O}_ n$ we obtain morphisms of ringed topoi

\[ f_\varphi : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n), \mathcal{O}_ n) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ m), \mathcal{O}_ m) \]

such that $a_ n \circ f_\varphi = a_ m$ as morphisms of ringed topoi for all $\varphi : [m] \to [n]$.

Lemma 83.11.1. With notation as above. The morphism $a : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ is flat if and only if $a_ n : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n), \mathcal{O}_ n) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ is flat for $n \geq 0$.

Proof. Since $g_ n : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n), \mathcal{O}_ n) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total}), \mathcal{O})$ is flat, we see that if $a$ is flat, then $a_ n = a \circ g_ n$ is flat as a composition. Conversely, suppose that $a_ n$ is flat for all $n$. We have to check that $\mathcal{O}$ is flat as a sheaf of $a^{-1}\mathcal{O}_\mathcal {D}$-modules. Let $\mathcal{F} \to \mathcal{G}$ be an injective map of $a^{-1}\mathcal{O}_\mathcal {D}$-modules. We have to show that

\[ \mathcal{F} \otimes _{a^{-1}\mathcal{O}_\mathcal {D}} \mathcal{O} \to \mathcal{G} \otimes _{a^{-1}\mathcal{O}_\mathcal {D}} \mathcal{O} \]

is injective. We can check this on $\mathcal{C}_ n$, i.e., after applying $g_ n^{-1}$. Since $g_ n^* = g_ n^{-1}$ because $g_ n^{-1}\mathcal{O} = \mathcal{O}_ n$ we obtain

\[ g_ n^{-1}\mathcal{F} \otimes _{g_ n^{-1}a^{-1}\mathcal{O}_\mathcal {D}} \mathcal{O}_ n \to g_ n^{-1}\mathcal{G} \otimes _{g_ n^{-1}a^{-1}\mathcal{O}_\mathcal {D}} \mathcal{O}_ n \]

which is injective because $g_ n^{-1}a^{-1}\mathcal{O}_\mathcal {D} = a_ n^{-1}\mathcal{O}_\mathcal {D}$ and we assume $a_ n$ was flat. $\square$

Lemma 83.11.2. With notation as above. For a $\mathcal{O}_\mathcal {D}$-module $\mathcal{G}$ there is an exact complex

\[ \ldots \to g_{2!}(a_2^*\mathcal{G}) \to g_{1!}(a_1^*\mathcal{G}) \to g_{0!}(a_0^*\mathcal{G}) \to a^*\mathcal{G} \to 0 \]

of sheaves of $\mathcal{O}$-modules on $\mathcal{C}_{total}$. Here $g_{n!}$ is as in Lemma 83.6.1.

Proof. Observe that $a^*\mathcal{G}$ is the $\mathcal{O}$-module on $\mathcal{C}_{total}$ whose restriction to $\mathcal{C}_ m$ is the $\mathcal{O}_ m$-module $a_ m^*\mathcal{G}$. The description of the functors $g_{n!}$ on modules in Lemma 83.6.1 shows that $g_{n!}(a_ n^*\mathcal{G})$ is the $\mathcal{O}$-module on $\mathcal{C}_{total}$ whose restriction to $\mathcal{C}_ m$ is the $\mathcal{O}_ m$-module

\[ \bigoplus \nolimits _{\varphi : [n] \to [m]} f_\varphi ^*a_ n^*\mathcal{G} = \bigoplus \nolimits _{\varphi : [n] \to [m]} a_ m^*\mathcal{G} \]

The rest of the proof is exactly the same as the proof of Lemma 83.9.1, replacing $a_ m^{-1}\mathcal{G}$ by $a_ m^*\mathcal{G}$. $\square$

Lemma 83.11.3. With notation as above. For an $\mathcal{O}$-module $\mathcal{F}$ on $\mathcal{C}_{total}$ there is a canonical complex

\[ 0 \to a_*\mathcal{F} \to a_{0, *}\mathcal{F}_0 \to a_{1, *}\mathcal{F}_1 \to a_{2, *}\mathcal{F}_2 \to \ldots \]

of $\mathcal{O}_\mathcal {D}$-modules which is exact in degrees $-1, 0$. If $\mathcal{F}$ is an injective $\mathcal{O}$-module, then the complex is exact in all degrees and remains exact on applying the functor $\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {D}}(\mathcal{G}, -)$ for any $\mathcal{O}_\mathcal {D}$-module $\mathcal{G}$.

Proof. To construct the complex, by the Yoneda lemma, it suffices for any $\mathcal{O}_\mathcal {D}$-modules $\mathcal{G}$ on $\mathcal{D}$ to construct a complex

\[ 0 \to \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {D}}(\mathcal{G}, a_*\mathcal{F}) \to \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {D}}(\mathcal{G}, a_{0, *}\mathcal{F}_0) \to \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {D}}(\mathcal{G}, a_{1, *}\mathcal{F}_1) \to \ldots \]

functorially in $\mathcal{G}$. To do this apply $\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(-, \mathcal{F})$ to the exact complex of Lemma 83.11.2 and use adjointness of pullback and pushforward. The exactness properties in degrees $-1, 0$ follow from the construction as $\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(-, \mathcal{F})$ is left exact. If $\mathcal{F}$ is an injective $\mathcal{O}$-module, then the complex is exact because $\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(-, \mathcal{F})$ is exact. $\square$

Lemma 83.11.4. With notation as above for any $K$ in $D^+(\mathcal{O})$ there is a spectral sequence $(E_ r, d_ r)_{r \geq 0}$ in $\textit{Mod}(\mathcal{O}_\mathcal {D})$ with

\[ E_1^{p, q} = R^ qa_{p, *} K_ p \]

converging to $R^{p + q}a_*K$. This spectral sequence is functorial in $K$.

Proof. Let $\mathcal{I}^\bullet $ be a bounded below complex of injective $\mathcal{O}$-modules representing $K$. Consider the double complex with terms

\[ A^{p, q} = a_{p, *}\mathcal{I}^ q_ p \]

where the horizontal arrows come from Lemma 83.11.3 and the vertical arrows from the differentials of the complex $\mathcal{I}^\bullet $. The lemma says rows of the double complex are exact in positive degrees and evaluate to $a_*\mathcal{I}^ q$ in degree $0$. Thus the total complex associated to the double complex computes $Ra_*K$ by Homology, Lemma 12.25.4. On the other hand, since restriction to $\mathcal{C}_ p$ is exact (Lemma 83.3.5) the complex $\mathcal{I}_ p^\bullet $ represents $K_ p$ in $D(\mathcal{C}_ p)$. The sheaves $\mathcal{I}_ p^ q$ are totally acyclic on $\mathcal{C}_ p$ (Lemma 83.6.2). Hence the cohomology of the columns are the sheaves $R^ qa_{p, *}K_ p$ by Leray's acyclicity lemma (Derived Categories, Lemma 13.16.7) and Cohomology on Sites, Lemma 21.14.3. We conclude by applying Homology, Lemma 12.25.3. $\square$


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