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.

**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.
$\square$

## Comments (2)

Comment #3418 by James on

Comment #3480 by Johan on