The Stacks project

Lemma 47.9.1. Let $A$ be a ring and let $I \subset A$ be a finitely generated ideal. There exists a right adjoint $R\Gamma _ Z$ (47.9.0.1) to the inclusion functor $D_{I^\infty \text{-torsion}}(A) \to D(A)$. In fact, if $I$ is generated by $f_1, \ldots , f_ r \in A$, then we have

\[ R\Gamma _ Z(K) = (A \to \prod \nolimits _{i_0} A_{f_{i_0}} \to \prod \nolimits _{i_0 < i_1} A_{f_{i_0}f_{i_1}} \to \ldots \to A_{f_1\ldots f_ r}) \otimes _ A^\mathbf {L} K \]

functorially in $K \in D(A)$.

Proof. Say $I = (f_1, \ldots , f_ r)$ is an ideal. Let $K^\bullet $ be a complex of $A$-modules. There is a canonical map of complexes

\[ (A \to \prod \nolimits _{i_0} A_{f_{i_0}} \to \prod \nolimits _{i_0 < i_1} A_{f_{i_0}f_{i_1}} \to \ldots \to A_{f_1\ldots f_ r}) \longrightarrow A. \]

from the extended Čech complex to $A$. Tensoring with $K^\bullet $, taking associated total complex, we get a map

\[ \text{Tot}\left( K^\bullet \otimes _ A (A \to \prod \nolimits _{i_0} A_{f_{i_0}} \to \prod \nolimits _{i_0 < i_1} A_{f_{i_0}f_{i_1}} \to \ldots \to A_{f_1\ldots f_ r})\right) \longrightarrow K^\bullet \]

in $D(A)$. We claim the cohomology modules of the complex on the left are $I$-power torsion, i.e., the LHS is an object of $D_{I^\infty \text{-torsion}}(A)$. Namely, we have

\[ (A \to \prod \nolimits _{i_0} A_{f_{i_0}} \to \prod \nolimits _{i_0 < i_1} A_{f_{i_0}f_{i_1}} \to \ldots \to A_{f_1\ldots f_ r}) = \mathop{\mathrm{colim}}\nolimits K(A, f_1^ n, \ldots , f_ r^ n) \]

by More on Algebra, Lemma 15.28.13. Moreover, multiplication by $f_ i^ n$ on the complex $K(A, f_1^ n, \ldots , f_ r^ n)$ is homotopic to zero by More on Algebra, Lemma 15.28.6. Since

\[ H^ q\left( LHS \right) = \mathop{\mathrm{colim}}\nolimits H^ q(\text{Tot}(K^\bullet \otimes _ A K(A, f_1^ n, \ldots , f_ r^ n))) \]

we obtain our claim. On the other hand, if $K^\bullet $ is an object of $D_{I^\infty \text{-torsion}}(A)$, then the complexes $K^\bullet \otimes _ A A_{f_{i_0} \ldots f_{i_ p}}$ have vanishing cohomology. Hence in this case the map $LHS \to K^\bullet $ is an isomorphism in $D(A)$. The construction

\[ R\Gamma _ Z(K^\bullet ) = \text{Tot}\left( K^\bullet \otimes _ A (A \to \prod \nolimits _{i_0} A_{f_{i_0}} \to \prod \nolimits _{i_0 < i_1} A_{f_{i_0}f_{i_1}} \to \ldots \to A_{f_1\ldots f_ r})\right) \]

is functorial in $K^\bullet $ and defines an exact functor $D(A) \to D_{I^\infty \text{-torsion}}(A)$ between triangulated categories. It follows formally from the existence of the natural transformation $R\Gamma _ Z \to \text{id}$ given above and the fact that this evaluates to an isomorphism on $K^\bullet $ in the subcategory, that $R\Gamma _ Z$ is the desired right adjoint. $\square$


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