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113.8 Cohomology

Remarks on K-flat complexes which are obsoleted by stronger results.

Lemma 113.8.1. 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. $\square$

Remark 113.8.2. This remark used to discuss what we know about pullbacks of K-flat complexes being K-flat or not, but is now obsoleted by Cohomology on Sites, Lemma 21.18.1.

The following lemma computes the cohomology sheaves of the derived limit in a special case.

Lemma 113.8.3. 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

  1. $K_ n$ is an object of $D^+(\mathcal{O})$ for all $n$,

  2. 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 $n \geq n(q)$,

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

  4. 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 15.81.9 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 113.8.4. In Simplicial Spaces, Situation 83.3.3 let $a_0$ be an augmentation towards a site $\mathcal{D}$ as in Simplicial Spaces, Remark 83.4.1. 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

  1. 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

  1. $a^{-1}$ sends $\mathcal{A}$ into $\mathcal{A}_{total}$ and

  2. $a^{-1}$ 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

  1. $R^ qa_*$ sends $\mathcal{A}_{total}$ into $\mathcal{A}$ for all $q$, and

  2. $Ra_*$ 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 83.9.3 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. $\square$

Remark 113.8.5. This tag used to refer to a section of the chapter on cohomology listing topics to be treated.

Remark 113.8.6. This tag used to refer to a section of the chapter on cohomology listing topics to be treated.

Remark 113.8.7. This tag used to refer to the special case of Cohomology on Sites, Lemma 21.29.3 pertaining to the situation described in Cohomology on Sites, Lemma 21.30.9.

Remark 113.8.8. This tag used to refer to the special case of Cohomology on Sites, Lemma 21.29.4 pertaining to the situation described in Cohomology on Sites, Lemma 21.30.9.

Remark 113.8.9. This tag used to refer to the special case of Cohomology on Sites, Lemma 21.29.7 pertaining to the situation described in Cohomology on Sites, Lemma 21.30.9.

Remark 113.8.10. This tag used to refer to the special case of Cohomology on Sites, Lemma 21.29.3 pertaining to the situation described in Étale Cohomology, Lemma 58.94.5.

Remark 113.8.11. This tag used to refer to the special case of Cohomology on Sites, Lemma 21.29.4 pertaining to the situation described in Étale Cohomology, Lemma 58.94.5.

Remark 113.8.12. This tag used to refer to the special case of Cohomology on Sites, Lemma 21.29.7 pertaining to the situation described in Étale Cohomology, Lemma 58.94.5.

Remark 113.8.13. This tag used to refer to the special case of Cohomology on Sites, Lemma 21.29.3 pertaining to the situation described in Étale Cohomology, Lemma 58.96.4.

Remark 113.8.14. This tag used to refer to the special case of Cohomology on Sites, Lemma 21.29.4 pertaining to the situation described in Étale Cohomology, Lemma 58.96.4.

Remark 113.8.15. This tag used to refer to the special case of Cohomology on Sites, Lemma 21.29.5 pertaining to the situation described in Étale Cohomology, Lemma 58.96.4.

Remark 113.8.16. This tag used to refer to the special case of Cohomology on Sites, Lemma 21.29.6 pertaining to the situation described in Étale Cohomology, Lemma 58.96.4.

Remark 113.8.17. This tag used to refer to the special case of Cohomology on Sites, Lemma 21.29.7 pertaining to the situation described in Étale Cohomology, Lemma 58.96.4.

Remark 113.8.18. This tag used to refer to the special case of Cohomology on Sites, Lemma 21.29.3 pertaining to the situation described in Étale Cohomology, Lemma 58.97.4.

Remark 113.8.19. This tag used to refer to the special case of Cohomology on Sites, Lemma 21.29.4 pertaining to the situation described in Étale Cohomology, Lemma 58.97.4.

Remark 113.8.20. This tag used to refer to the special case of Cohomology on Sites, Lemma 21.29.5 pertaining to the situation described in Étale Cohomology, Lemma 58.97.4.

Remark 113.8.21. This tag used to refer to the special case of Cohomology on Sites, Lemma 21.29.7 pertaining to the situation described in Étale Cohomology, Lemma 58.97.4.

Remark 113.8.22. This tag used to be in the chapter on étale cohomology, but is no longer suitable there because of a reorganization. The content of the tag was the following: Étale Cohomology, Lemma 58.75.3 can be used to prove that if $f : X \to Y$ is a separated, finite type morphism of schemes and $Y$ is Noetherian, then $Rf_!$ induces a functor $D_{ctf}(X_{\acute{e}tale}, \Lambda ) \to D_{ctf}(Y_{\acute{e}tale}, \Lambda )$. An example of this argument, when $Y$ is the spectrum of a field and $X$ is a curve is given in The Trace Formula, Proposition 62.13.1.

Lemma 113.8.23. 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

  1. for any open $j : U \to X$ with $f \circ j$ separated there is a canonical isomorphism $j^! \circ f^! = (f \circ j)^!$, and

  2. these isomorphisms for $U \subset U' \subset X$ are compatible with the isomorphisms in More Étale Cohomology, Lemma 61.5.3.

Proof. Immediate consequence of More Étale Cohomology, Lemmas 61.5.1 and 61.5.3. $\square$

Proposition 113.8.24. 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

  1. the functor $f^!$ is the one constructed in More Étale Cohomology, Lemma 61.5.1,

  2. for any open $j : U \to X$ with $f \circ j$ separated there is a canonical isomorphism $f_! \circ j_! = (f \circ j)_!$, and

  3. these isomorphisms for $U \subset U' \subset X$ are compatible with the isomorphisms in More Étale Cohomology, Lemma 61.3.13.

Proof. See More Étale Cohomology, Sections 61.4 and 61.5. $\square$

Lemma 113.8.25. 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 61.5.1. $\square$

Lemma 113.8.26. 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 61.4.12 and 61.5.3. $\square$


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