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 15.29.6. 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. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 47.9: Local cohomology

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0A6R. Beware of the difference between the letter 'O' and the digit '0'.