Lemma 47.9.2 (0BJB)—The Stacks project

Lemma 47.9.2. Let $A \to B$ be a ring homomorphism and let $I \subset A$ be a finitely generated ideal. Set $J = IB$. Set $Z = V(I)$ and $Y = V(J)$. Then

\[ R\Gamma _ Z(M_ A) = R\Gamma _ Y(M)_ A \]

functorially in $M \in D(B)$. Here $(-)_ A$ denotes the restriction functors $D(B) \to D(A)$ and $D_{J^\infty \text{-torsion}}(B) \to D_{I^\infty \text{-torsion}}(A)$.

Proof. This follows from uniqueness of adjoint functors as both $R\Gamma _ Z((-)_ A)$ and $R\Gamma _ Y(-)_ A$ are right adjoint to the functor $D_{I^\infty \text{-torsion}}(A) \to D(B)$, $K \mapsto K \otimes _ A^\mathbf {L} B$. Alternatively, one can use the description of $R\Gamma _ Z$ and $R\Gamma _ Y$ in terms of alternating Čech complexes (Lemma 47.9.1). Namely, if $I = (f_1, \ldots , f_ r)$ then $J$ is generated by the images $g_1, \ldots , g_ r \in B$ of $f_1, \ldots , f_ r$. Then the statement of the lemma follows from the existence of a canonical isomorphism

\begin{align*} & M_ A \otimes _ A (A \to \prod \nolimits _{i_0} A_{f_{i_0}} \to \prod \nolimits _{i_0 < i_1} A_{f_{i_0}f_{i_1}} \to \ldots \to A_{f_1\ldots f_ r}) \\ & = M \otimes _ B (B \to \prod \nolimits _{i_0} B_{g_{i_0}} \to \prod \nolimits _{i_0 < i_1} B_{g_{i_0}g_{i_1}} \to \ldots \to B_{g_1\ldots g_ r}) \end{align*}

for any $B$-module $M$. $\square$

The Stacks project

Comments (0)

Post a comment