115.8 Cohomology
Obsolete lemmas about cohomology.
Lemma 115.8.1. Let $I$ be an ideal of a ring $A$. Let $X$ be a scheme over $\mathop{\mathrm{Spec}}(A)$. Let
\[ \ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1 \]
be an inverse system of $\mathcal{O}_ X$-modules such that $\mathcal{F}_ n = \mathcal{F}_{n + 1}/I^ n\mathcal{F}_{n + 1}$. Assume
\[ \bigoplus \nolimits _{n \geq 0} H^1(X, I^ n\mathcal{F}_{n + 1}) \]
satisfies the ascending chain condition as a graded $\bigoplus _{n \geq 0} I^ n/I^{n + 1}$-module. Then the inverse system $M_ n = \Gamma (X, \mathcal{F}_ n)$ satisfies the Mittag-Leffler condition.
Proof.
This is a special case of the more general Cohomology, Lemma 20.35.1.
$\square$
Lemma 115.8.2. Let $I$ be an ideal of a ring $A$. Let $X$ be a scheme over $\mathop{\mathrm{Spec}}(A)$. Let
\[ \ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1 \]
be an inverse system of $\mathcal{O}_ X$-modules such that $\mathcal{F}_ n = \mathcal{F}_{n + 1}/I^ n\mathcal{F}_{n + 1}$. Given $n$ define
\[ H^1_ n = \bigcap \nolimits _{m \geq n} \mathop{\mathrm{Im}}\left( H^1(X, I^ n\mathcal{F}_{m + 1}) \to H^1(X, I^ n\mathcal{F}_{n + 1}) \right) \]
If $\bigoplus H^1_ n$ satisfies the ascending chain condition as a graded $\bigoplus _{n \geq 0} I^ n/I^{n + 1}$-module, then the inverse system $M_ n = \Gamma (X, \mathcal{F}_ n)$ satisfies the Mittag-Leffler condition.
Proof.
This is a special case of the more general Cohomology, Lemma 20.35.2.
$\square$
Lemma 115.8.3. Let $I$ be a finitely generated ideal of a ring $A$. Let $X$ be a scheme over $\mathop{\mathrm{Spec}}(A)$. Let
\[ \ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1 \]
be an inverse system of $\mathcal{O}_ X$-modules such that $\mathcal{F}_ n = \mathcal{F}_{n + 1}/I^ n\mathcal{F}_{n + 1}$. Assume
\[ \bigoplus \nolimits _{n \geq 0} H^0(X, I^ n\mathcal{F}_{n + 1}) \]
satisfies the ascending chain condition as a graded $\bigoplus _{n \geq 0} I^ n/I^{n + 1}$-module. Then the limit topology on $M = \mathop{\mathrm{lim}}\nolimits \Gamma (X, \mathcal{F}_ n)$ is the $I$-adic topology.
Proof.
This is a special case of the more general Cohomology, Lemma 20.35.3.
$\square$
Lemma 115.8.4. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C})$ be a ringed topos. For any complex of $\mathcal{O}_\mathcal {C}$-modules $\mathcal{G}^\bullet $ there exists a quasi-isomorphism $\mathcal{K}^\bullet \to \mathcal{G}^\bullet $ such that $f^*\mathcal{K}^\bullet $ is a K-flat complex of $\mathcal{O}_\mathcal {D}$-modules for any morphism $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C})$ of ringed topoi.
Proof.
This follows from Cohomology on Sites, Lemmas 21.17.11 and 21.18.1.
$\square$
The following lemma computes the cohomology sheaves of the derived limit in a special case.
Lemma 115.8.6. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(K_ n)$ be an inverse system of objects of $D(\mathcal{O})$. Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be a subset. Let $d \in \mathbf{N}$. Assume
$K_ n$ is an object of $D^+(\mathcal{O})$ for all $n$,
for $q \in \mathbf{Z}$ there exists $n(q)$ such that $H^ q(K_{n + 1}) \to H^ q(K_ n)$ is an isomorphism for $n \geq n(q)$,
every object of $\mathcal{C}$ has a covering whose members are elements of $\mathcal{B}$,
for every $U \in \mathcal{B}$ we have $H^ p(U, H^ q(K_ n)) = 0$ for $p > d$ and all $q$.
Then we have $H^ m(R\mathop{\mathrm{lim}}\nolimits K_ n) = \mathop{\mathrm{lim}}\nolimits H^ m(K_ n)$ for all $m \in \mathbf{Z}$.
Proof.
Set $K = R\mathop{\mathrm{lim}}\nolimits K_ n$. Let $U \in \mathcal{B}$. For each $n$ there is a spectral sequence
\[ H^ p(U, H^ q(K_ n)) \Rightarrow H^{p + q}(U, K_ n) \]
which converges as $K_ n$ is bounded below, see Derived Categories, Lemma 13.21.3. If we fix $m \in \mathbf{Z}$, then we see from our assumption (4) that only $H^ p(U, H^ q(K_ n))$ contribute to $H^ m(U, K_ n)$ for $0 \leq p \leq d$ and $m - d \leq q \leq m$. By assumption (2) this implies that $H^ m(U, K_{n + 1}) \to H^ m(U, K_ n)$ is an isomorphism as soon as $n \geq \max {n(m), \ldots , n(m - d)}$. The functor $R\Gamma (U, -)$ commutes with derived limits by Injectives, Lemma 19.13.6. Thus we have
\[ H^ m(U, K) = H^ m(R\mathop{\mathrm{lim}}\nolimits R\Gamma (U, K_ n)) \]
On the other hand we have just seen that the complexes $R\Gamma (U, K_ n)$ have eventually constant cohomology groups. Thus by More on Algebra, Remark 15.86.10 we find that $H^ m(U, K)$ is equal to $H^ m(U, K_ n)$ for all $n \gg 0$ for some bound independent of $U \in \mathcal{B}$. Pick such an $n$. Finally, recall that $H^ m(K)$ is the sheafification of the presheaf $U \mapsto H^ m(U, K)$ and $H^ m(K_ n)$ is the sheafification of the presheaf $U \mapsto H^ m(U, K_ n)$. On the elements of $\mathcal{B}$ these presheaves have the same values. Therefore assumption (3) guarantees that the sheafifications are the same too. The lemma follows.
$\square$
Lemma 115.8.7. In Simplicial Spaces, Situation 85.3.3 let $a_0$ be an augmentation towards a site $\mathcal{D}$ as in Simplicial Spaces, Remark 85.4.1. Suppose given strictly full weak Serre subcategories
\[ \mathcal{A} \subset \textit{Ab}(\mathcal{D}),\quad \mathcal{A}_ n \subset \textit{Ab}(\mathcal{C}_ n) \]
Then
the collection of abelian sheaves $\mathcal{F}$ on $\mathcal{C}_{total}$ whose restriction to $\mathcal{C}_ n$ is in $\mathcal{A}_ n$ for all $n$ is a strictly full weak Serre subcategory $\mathcal{A}_{total} \subset \textit{Ab}(\mathcal{C}_{total})$.
If $a_ n^{-1}$ sends $\mathcal{A}$ into $\mathcal{A}_ n$ for all $n$, then
$a^{-1}$ sends $\mathcal{A}$ into $\mathcal{A}_{total}$ and
$a^{-1}$ sends $D_\mathcal {A}(\mathcal{D})$ into $D_{\mathcal{A}_{total}}(\mathcal{C}_{total})$.
If $R^ qa_{n, *}$ sends $\mathcal{A}_ n$ into $\mathcal{A}$ for all $n, q$, then
$R^ qa_*$ sends $\mathcal{A}_{total}$ into $\mathcal{A}$ for all $q$, and
$Ra_*$ sends $D_{\mathcal{A}_{total}}^+(\mathcal{C}_{total})$ into $D_\mathcal {A}^+(\mathcal{D})$.
Proof.
The only interesting assertions are (4) and (5). Part (4) follows from the spectral sequence in Simplicial Spaces, Lemma 85.9.3 and Homology, Lemma 12.24.11. Then part (5) follows by considering the spectral sequence associated to the canonical filtration on an object $K$ of $D_{\mathcal{A}_{total}}^+(\mathcal{C}_{total})$ given by truncations. We omit the details.
$\square$
Lemma 115.8.26. Let $f : X \to Y$ be a locally quasi-finite morphism of schemes. There exists a unique functor $f^! : \textit{Ab}(Y_{\acute{e}tale}) \to \textit{Ab}(X_{\acute{e}tale})$ such that
for any open $j : U \to X$ with $f \circ j$ separated there is a canonical isomorphism $j^! \circ f^! = (f \circ j)^!$, and
these isomorphisms for $U \subset U' \subset X$ are compatible with the isomorphisms in More Étale Cohomology, Lemma 63.6.3.
Proof.
Immediate consequence of More Étale Cohomology, Lemmas 63.6.1 and 63.6.3.
$\square$
Proposition 115.8.27. Let $f : X \to Y$ be a locally quasi-finite morphism. There exist adjoint functors $f_! : \textit{Ab}(X_{\acute{e}tale}) \to \textit{Ab}(Y_{\acute{e}tale})$ and $f^! : \textit{Ab}(Y_{\acute{e}tale}) \to \textit{Ab}(X_{\acute{e}tale})$ with the following properties
the functor $f^!$ is the one constructed in More Étale Cohomology, Lemma 63.6.1,
for any open $j : U \to X$ with $f \circ j$ separated there is a canonical isomorphism $f_! \circ j_! = (f \circ j)_!$, and
these isomorphisms for $U \subset U' \subset X$ are compatible with the isomorphisms in More Étale Cohomology, Lemma 63.3.13.
Proof.
See More Étale Cohomology, Sections 63.4 and 63.6.
$\square$
Lemma 115.8.28. Let $f : X \to Y$ be a morphism of schemes which is locally quasi-finite. For an abelian group $A$ and a geometric point $\overline{y} : \mathop{\mathrm{Spec}}(k) \to Y$ we have $f^!(\overline{y}_*A) = \prod \nolimits _{f(\overline{x}) = \overline{y}} \overline{x}_*A$.
Proof.
Follows from the corresponding statement in More Étale Cohomology, Lemma 63.6.1.
$\square$
Lemma 115.8.29. Let $f : X \to Y$ and $g : Y \to Z$ be composable locally quasi-finite morphisms of schemes. Then $g_! \circ f_! = (g \circ f)_!$ and $f^! \circ g^! = (g \circ f)^!$.
Proof.
Combination of More Étale Cohomology, Lemmas 63.4.12 and 63.6.3.
$\square$
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