Loading web-font TeX/Caligraphic/Regular

The Stacks project

Lemma 13.42.3. Let \mathcal{A} be an abelian category. Let A_ n be an inverse system of objects of D(\mathcal{A}). Assume

  1. there exist integers a \leq b such that H^ i(A_ n) = 0 for i \not\in [a, b], and

  2. the inverse systems H^ i(A_ n) of \mathcal{A} are essentially constant for all i \in \mathbf{Z}.

Then A_ n is an essentially constant system of D(\mathcal{A}) whose value A satisfies that H^ i(A) is the value of the constant system H^ i(A_ n) for each i \in \mathbf{Z}.

Proof. By Remark 13.12.4 we obtain an inverse system of distinguished triangles

\tau _{\leq a}A_ n \to A_ n \to \tau _{\geq a + 1}A_ n \to (\tau _{\leq a}A_ n)[1]

Of course we have \tau _{\leq a}A_ n = H^ a(A_ n)[-a] in D(\mathcal{A}). Thus by assumption these form an essentially constant system. By induction on b - a we find that the inverse system \tau _{\geq a + 1}A_ n is essentially constant, say with value A'. By Lemma 13.42.2 we find that A_ n is an essentially constant system. We omit the proof of the statement on cohomologies (hint: use the final part of Lemma 13.42.2). \square


Comments (0)


Your email address will not be published.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.