The Stacks Project


Tag 09AW

Chapter 15: More on Algebra > Section 15.83: Taking limits of complexes

Lemma 15.83.4. 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 $D(A/I^{n + 1})$. Assume

  1. $A$ is $I$-adically complete,
  2. $K_1$ is a perfect object, and
  3. the maps induce isomorphisms $K_{n + 1} \otimes_{A/I^{n + 1}}^\mathbf{L} A/I^n \to K_n$.

Then $K = R\mathop{\rm lim}\nolimits K_n$ is a perfect, 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. Combine Lemmas 15.83.3 and 15.83.2 (to get derived completeness). $\square$

    The code snippet corresponding to this tag is a part of the file more-algebra.tex and is located in lines 22640–22653 (see updates for more information).

    \begin{lemma}
    \label{lemma-Rlim-perfect-gives-complete}
    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 $D(A/I^{n + 1})$. Assume
    \begin{enumerate}
    \item $A$ is $I$-adically complete,
    \item $K_1$ is a perfect object, and
    \item the maps induce isomorphisms
    $K_{n + 1} \otimes_{A/I^{n + 1}}^\mathbf{L} A/I^n \to K_n$.
    \end{enumerate}
    Then $K = R\lim K_n$ is a perfect, derived complete object of $D(A)$
    and $K \otimes_A^\mathbf{L} A/I^n \to K_n$ is an isomorphism for all $n$.
    \end{lemma}
    
    \begin{proof}
    Combine Lemmas \ref{lemma-Rlim-perfect-gives-perfect} and
    \ref{lemma-Rlim-pseudo-coherent-gives-complete-pseudo-coherent}
    (to get derived completeness).
    \end{proof}

    Comments (0)

    There are no comments yet for this tag.

    Add a comment on tag 09AW

    Your email address will not be published.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 lower-right corner).

    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 box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

    This captcha seems more appropriate than the usual illegible gibberish, right?