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30.22 Cohomology and base change, III

In this section we prove the simplest case of a very general phenomenon that will be discussed in Derived Categories of Schemes, Section 36.22. Please see Remark 30.22.2 for a translation of the following lemma into algebra.

Lemma 30.22.1. Let $A$ be a Noetherian ring and set $S = \mathop{\mathrm{Spec}}(A)$. Let $f : X \to S$ be a proper morphism of schemes. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module flat over $S$. Then

  1. $R\Gamma (X, \mathcal{F})$ is a perfect object of $D(A)$, and

  2. for any ring map $A \to A'$ the base change map

    \[ R\Gamma (X, \mathcal{F}) \otimes _ A^{\mathbf{L}} A' \longrightarrow R\Gamma (X_{A'}, \mathcal{F}_{A'}) \]

    is an isomorphism.

Proof. Choose a finite affine open covering $X = \bigcup _{i = 1, \ldots , n} U_ i$. By Lemmas 30.7.1 and 30.7.2 the Čech complex $K^\bullet = {\check C}^\bullet (\mathcal{U}, \mathcal{F})$ satisfies

\[ K^\bullet \otimes _ A A' = R\Gamma (X_{A'}, \mathcal{F}_{A'}) \]

for all ring maps $A \to A'$. Let $K_{alt}^\bullet = {\check C}_{alt}^\bullet (\mathcal{U}, \mathcal{F})$ be the alternating Čech complex. By Cohomology, Lemma 20.23.6 there is a homotopy equivalence $K_{alt}^\bullet \to K^\bullet $ of $A$-modules. In particular, we have

\[ K_{alt}^\bullet \otimes _ A A' = R\Gamma (X_{A'}, \mathcal{F}_{A'}) \]

as well. Since $\mathcal{F}$ is flat over $A$ we see that each $K_{alt}^ n$ is flat over $A$ (see Morphisms, Lemma 29.25.2). Since moreover $K_{alt}^\bullet $ is bounded above (this is why we switched to the alternating Čech complex) $K_{alt}^\bullet \otimes _ A A' = K_{alt}^\bullet \otimes _ A^{\mathbf{L}} A'$ by the definition of derived tensor products (see More on Algebra, Section 15.59). By Lemma 30.19.2 the cohomology groups $H^ i(K_{alt}^\bullet )$ are finite $A$-modules. As $K_{alt}^\bullet $ is bounded, we conclude that $K_{alt}^\bullet $ is pseudo-coherent, see More on Algebra, Lemma 15.64.17. Given any $A$-module $M$ set $A' = A \oplus M$ where $M$ is a square zero ideal, i.e., $(a, m) \cdot (a', m') = (aa', am' + a'm)$. By the above we see that $K_{alt}^\bullet \otimes _ A^\mathbf {L} A'$ has cohomology in degrees $0, \ldots , n$. Hence $K_{alt}^\bullet \otimes _ A^\mathbf {L} M$ has cohomology in degrees $0, \ldots , n$. Hence $K_{alt}^\bullet $ has finite Tor dimension, see More on Algebra, Definition 15.66.1. We win by More on Algebra, Lemma 15.74.2. $\square$

Remark 30.22.2. A consequence of Lemma 30.22.1 is that there exists a finite complex of finite projective $A$-modules $M^\bullet $ such that we have

\[ H^ i(X_{A'}, \mathcal{F}_{A'}) = H^ i(M^\bullet \otimes _ A A') \]

functorially in $A'$. The condition that $\mathcal{F}$ is flat over $A$ is essential, see [Hartshorne].

Comments (2)

Comment #3418 by James on

is flatness of F needed for condition (2)? The proof seems to suggest not but I am probably wrong

Comment #3480 by on

Dear James, please look at Section 30.7 to see that the answer to your question is affirmative. It is the raison d'etre of that section. BUT please be very careful in using this property, because it is something about the particular complex and not a property of the total direct image in the derived category.

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