The Stacks project

Lemma 15.95.5. Let $A$ be a ring and $I \subset A$ an ideal. Suppose given $K_ n \in D(A/I^ n)$ and maps $K_{n + 1} \to K_ n$ in $D(A/I^{n + 1})$. If

  1. $A$ is Noetherian,

  2. $K_1$ is bounded above, and

  3. the maps induce isomorphisms $K_{n + 1} \otimes _{A/I^{n + 1}}^\mathbf {L} A/I^ n \to K_ n$,

then $K = R\mathop{\mathrm{lim}}\nolimits K_ n$ is a derived complete object of $D^-(A)$ and $K \otimes _ A^\mathbf {L} A/I^ n \to K_ n$ is an isomorphism for all $n$.

Proof. Suppose that $H^ i(K_1) = 0$ for $i > b$. Then we can find a complex of free $A/I$-modules $P_1^\bullet $ representing $K_1$ with $P_1^ i = 0$ for $i > b$. By Lemma 15.74.3 we can, by induction on $n > 1$, find complexes $P_ n^\bullet $ of free $A/I^ n$-modules representing $K_ n$ and maps $P_ n^\bullet \to P_{n - 1}^\bullet $ representing the maps $K_ n \to K_{n - 1}$ inducing isomorphisms (!) of complexes $P_ n^\bullet /I^{n - 1}P_ n^\bullet \to P_{n - 1}^\bullet $.

Thus we have arrived at the situation where $R\mathop{\mathrm{lim}}\nolimits K_ n$ is represented by $P^\bullet = \mathop{\mathrm{lim}}\nolimits P_ n^\bullet $, see Lemma 15.86.1 and Remark 15.86.6. The complexes $P_ n^\bullet $ are uniformly bounded above complexes of flat $A/I^ n$-modules and the transition maps are termwise surjective. Then $P^\bullet $ is a bounded above complex of flat $A$-modules by Lemma 15.27.4. It follows that $K \otimes _ A^\mathbf {L} A/I^ t$ is represented by $P^\bullet \otimes _ A A/I^ t$. We have $P^\bullet \otimes _ A A/I^ t = \mathop{\mathrm{lim}}\nolimits P_ n^\bullet \otimes _ A A/I^ t$ termwise by Lemma 15.27.4. The transition maps $P_{n + 1}^\bullet \otimes _ A A/I^ t \to P_ n^\bullet \otimes _ A A/I^ t$ are isomorphisms for $n \geq t$. Hence we have $\mathop{\mathrm{lim}}\nolimits P_ n^\bullet \otimes _ A A/I^ t = R\mathop{\mathrm{lim}}\nolimits P_ n^\bullet \otimes _ A A/I^ t$. By assumption and our choice of $P_ n^\bullet $ the complex $P_ n^\bullet \otimes _ A A/I^ t = P_ n^\bullet \otimes _{A/I^ n} A/I^ t$ represents $K_ n \otimes _{A/I^ n}^\mathbf {L} A/I^ t = K_ t$ for all $n \geq t$. We conclude

\[ P^\bullet \otimes _ A A/I^ t = R\mathop{\mathrm{lim}}\nolimits P_ n^\bullet \otimes _ A A/I^ t = R\mathop{\mathrm{lim}}\nolimits K_ t = K_ t \]

In other words, we have $K \otimes _ A^\mathbf {L} A/I^ t = K_ t$. This proves the lemma as it follows that $K$ is derived complete by Proposition 15.92.2. $\square$

Comments (0)

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 09AU. Beware of the difference between the letter 'O' and the digit '0'.