The Stacks project

Lemma 15.91.17. In Situation 15.91.15. Let $K \in D(A)$. The following are equivalent

  1. $K$ is derived complete with respect to $I$, and

  2. the canonical map $K \to R\mathop{\mathrm{lim}}\nolimits (K \otimes _ A^\mathbf {L} K_ n^\bullet )$ is an isomorphism of $D(A)$.

Proof. If (2) holds, then $K$ is derived complete with respect to $I$ by Lemma 15.91.16. Conversely, assume that $K$ is derived complete with respect to $I$. Consider the filtrations

\[ K_ n^\bullet \supset \sigma _{\geq -r + 1}K_ n^\bullet \supset \sigma _{\geq -r + 2}K_ n^\bullet \supset \ldots \supset \sigma _{\geq -1}K_ n^\bullet \supset \sigma _{\geq 0}K_ n^\bullet = A \]

by stupid truncations (Homology, Section 12.15). Because the construction $R\mathop{\mathrm{lim}}\nolimits (K \otimes E)$ is exact in the second variable (Lemma 15.87.11) we see that it suffices to show

\[ R\mathop{\mathrm{lim}}\nolimits \left( K \otimes _ A^\mathbf {L} (\sigma _{\geq p}K_ n^\bullet / \sigma _{\geq p + 1}K_ n^\bullet ) \right) = 0 \]

for $p < 0$. The explicit description of the Koszul complexes above shows that

\[ R\mathop{\mathrm{lim}}\nolimits \left( K \otimes _ A^\mathbf {L} (\sigma _{\geq p}K_ n^\bullet / \sigma _{\geq p + 1}K_ n^\bullet ) \right) = \bigoplus \nolimits _{i_1, \ldots , i_{-p}} T(K, f_{i_1}\ldots f_{i_{-p}}) \]

which is zero for $p < 0$ by assumption on $K$. $\square$

Comments (0)

There are also:

  • 14 comment(s) on Section 15.91: Derived Completion

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Your email address will not be published.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 091Z. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 15.91.17 as pdf