Lemma 15.91.18. In Situation 15.91.15. The functor which sends $K \in D(A)$ to the derived limit $K' = R\mathop{\mathrm{lim}}\nolimits ( K \otimes _ A^\mathbf {L} K_ n^\bullet )$ is the left adjoint to the inclusion functor $D_{comp}(A) \to D(A)$ constructed in Lemma 15.91.10.

**First proof.**
The assignment $K \leadsto K'$ is a functor and $K'$ is derived complete with respect to $I$ by Lemma 15.91.16. By a formal argument (omitted) we see that it suffices to show $K \to K'$ is an isomorphism if $K$ is derived complete with respect to $I$. This is Lemma 15.91.17.
$\square$

**Second proof.**
Denote $K \mapsto K^\wedge $ the adjoint constructed in Lemma 15.91.10. By that lemma we have

In Lemma 15.29.6 we have seen that the extended alternating Čech complex

is a colimit of the Koszul complexes $K^ n = K(A, f_1^ n, \ldots , f_ r^ n)$ sitting in degrees $0, \ldots , r$. Note that $K^ n$ is a finite chain complex of finite free $A$-modules with dual (as in Lemma 15.74.15) $R\mathop{\mathrm{Hom}}\nolimits _ A(K^ n, A) = K_ n$ where $K_ n$ is the Koszul cochain complex sitting in degrees $-r, \ldots , 0$ (as usual). Thus it suffices to show that

This follows from Lemma 15.74.16. $\square$

## 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Your email address will not be published.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.

## Comments (0)

There are also: