Lemma 15.91.14. Let $A$ be a ring and let $I \subset A$ be an ideal. Let $(K_ n)$ be an inverse system of objects of $D(A)$ such that for all $f \in I$ and $n$ there exists an $e = e(n, f)$ such that $f^ e$ is zero on $K_ n$. Then for $K \in D(A)$ the object $K' = R\mathop{\mathrm{lim}}\nolimits (K \otimes _ A^\mathbf {L} K_ n)$ is derived complete with respect to $I$.

**Proof.**
Since the category of derived complete objects is preserved under $R\mathop{\mathrm{lim}}\nolimits $ it suffices to show that each $K \otimes _ A^\mathbf {L} K_ n$ is derived complete. By assumption for all $f \in I$ there is an $e$ such that $f^ e$ is zero on $K \otimes _ A^\mathbf {L} K_ n$. Of course this implies that $T(K \otimes _ A^\mathbf {L} K_ n, f) = 0$ and we win.
$\square$

## 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Your email address will not be published.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: