\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

The Stacks project

29.22 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 35.21. Please see Remark 29.22.2 for a translation of the following lemma into algebra.

Lemma 29.22.1. 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 29.7.1 and 29.7.2 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 20.24.6 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 28.24.2). 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 15.57). By Lemma 29.19.2 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 15.62.18. 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 15.63.1. We win by More on Algebra, Lemma 15.69.2. $\square$

Remark 29.22.2. A consequence of Lemma 29.22.1 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 (1)

Comment #3418 by James on

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

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