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.
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
Comments (2)
Comment #3418 by James on
Comment #3480 by Johan on