The Stacks project

59.96 Finite cohomological dimension

We continue the discussion started in Section 59.95.

Definition 59.96.1. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of schemes. The cohomological dimension of $f$ is the smallest element

\[ \text{cd}(f) \in \{ 0, 1, 2, \ldots \} \cup \{ \infty \} \]

such that for any abelian torsion sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ we have $R^ if_*\mathcal{F} = 0$ for $i > \text{cd}(f)$.

Lemma 59.96.2. Let $K$ be a field.

  1. If $f : X \to Y$ is a morphism of finite type schemes over $K$, then $\text{cd}(f) < \infty $.

  2. If $\text{cd}(K) < \infty $, then $\text{cd}(X) < \infty $ for any finite type scheme $X$ over $K$.

Proof. Proof of (1). We may assume $Y$ is affine. We will use the induction principle of Cohomology of Schemes, Lemma 30.4.1 to prove this. If $X$ is affine too, then the result holds by Lemma 59.95.8. Thus it suffices to show that if $X = U \cup V$ and the result is true for $U \to Y$, $V \to Y$, and $U \cap V \to Y$, then it is true for $f$. This follows from the relative Mayer-Vietoris sequence, see Lemma 59.50.2.

Proof of (2). We will use the induction principle of Cohomology of Schemes, Lemma 30.4.1 to prove this. If $X$ is affine, then the result holds by Proposition 59.95.6. Thus it suffices to show that if $X = U \cup V$ and the result is true for $U$, $V$, and $U \cap V $, then it is true for $X$. This follows from the Mayer-Vietoris sequence, see Lemma 59.50.1. $\square$

Lemma 59.96.3. Cohomology and direct sums. Let $n \geq 1$ be an integer.

  1. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of schemes with $\text{cd}(f) < \infty $. Then the functor

    \[ Rf_* : D(X_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z}) \longrightarrow D(Y_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z}) \]

    commutes with direct sums.

  2. Let $X$ be a quasi-compact and quasi-separated scheme with $\text{cd}(X) < \infty $. Then the functor

    \[ R\Gamma (X, -) : D(X_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z}) \longrightarrow D(\mathbf{Z}/n\mathbf{Z}) \]

    commutes with direct sums.

Proof. Proof of (1). Since $\text{cd}(f) < \infty $ we see that

\[ f_* : \textit{Mod}(X_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z}) \longrightarrow \textit{Mod}(Y_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z}) \]

has finite cohomological dimension in the sense of Derived Categories, Lemma 13.32.2. Let $I$ be a set and for $i \in I$ let $E_ i$ be an object of $D(X_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z})$. Choose a K-injective complex $\mathcal{I}_ i^\bullet $ of $\mathbf{Z}/n\mathbf{Z}$-modules each of whose terms $\mathcal{I}_ i^ n$ is an injective sheaf of $\mathbf{Z}/n\mathbf{Z}$-modules representing $E_ i$. See Injectives, Theorem 19.12.6. Then $\bigoplus E_ i$ is represented by the complex $\bigoplus \mathcal{I}_ i^\bullet $ (termwise direct sum), see Injectives, Lemma 19.13.4. By Lemma 59.51.7 we have

\[ R^ qf_*(\bigoplus \mathcal{I}_ i^ n) = \bigoplus R^ qf_*(\mathcal{I}_ i^ n) = 0 \]

for $q > 0$ and any $n$. Hence we conclude by Derived Categories, Lemma 13.32.2 that we may compute $Rf_*(\bigoplus E_ i)$ by the complex

\[ f_*(\bigoplus \mathcal{I}_ i^\bullet ) = \bigoplus f_*(\mathcal{I}_ i^\bullet ) \]

(equality again by Lemma 59.51.7) which represents $\bigoplus Rf_*E_ i$ by the already used Injectives, Lemma 19.13.4.

Proof of (2). This is identical to the proof of (1) and we omit it. $\square$

Lemma 59.96.4. Let $f : X \to Y$ be a proper morphism of schemes. Let $n \geq 1$ be an integer. Then the functor

\[ Rf_* : D(X_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z}) \longrightarrow D(Y_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z}) \]

commutes with direct sums.

Proof. It is enough to prove this when $Y$ is quasi-compact. By Morphisms, Lemma 29.28.5 we see that the dimension of the fibres of $f : X \to Y$ is bounded. Thus Lemma 59.92.2 implies that $\text{cd}(f) < \infty $. Hence the result by Lemma 59.96.3. $\square$

Lemma 59.96.5. Let $X$ be a quasi-compact and quasi-separated scheme such that $\text{cd}(X) < \infty $. Let $\Lambda $ be a torsion ring. Let $E \in D(X_{\acute{e}tale}, \Lambda )$ and $K \in D(\Lambda )$. Then

\[ R\Gamma (X, E \otimes _\Lambda ^\mathbf {L} \underline{K}) = R\Gamma (X, E) \otimes _\Lambda ^\mathbf {L} K \]

Proof. There is a canonical map from left to right by Cohomology on Sites, Section 21.48. Let $T(K)$ be the property that the statement of the lemma holds for $K \in D(\Lambda )$. We will check conditions (1), (2), and (3) of More on Algebra, Remark 15.59.11 hold for $T$ to conclude. Property (1) holds because both sides of the equality commute with direct sums, see Lemma 59.96.3. Property (2) holds because we are comparing exact functors between triangulated categories and we can use Derived Categories, Lemma 13.4.3. Property (3) says the lemma holds when $K = \Lambda [k]$ for any shift $k \in \mathbf{Z}$ and this is obvious. $\square$

Lemma 59.96.6. Let $f : X \to Y$ be a proper morphism of schemes. Let $\Lambda $ be a torsion ring. Let $E \in D(X_{\acute{e}tale}, \Lambda )$ and $K \in D(Y_{\acute{e}tale}, \Lambda )$. Then

\[ Rf_*E \otimes _\Lambda ^\mathbf {L} K = Rf_*(E \otimes _\Lambda ^\mathbf {L} f^{-1}K) \]

in $D(Y_{\acute{e}tale}, \Lambda )$.

Proof. There is a canonical map from left to right by Cohomology on Sites, Section 21.48. We will check the equality on stalks at $\overline{y}$. By the proper base change (in the form of Lemma 59.92.3 where $Y' = \overline{y}$) this reduces to the case where $Y$ is the spectrum of an algebraically closed field. This is shown in Lemma 59.96.5 where we use that $\text{cd}(X) < \infty $ by Lemma 59.92.2. $\square$


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