Lemma 20.37.1 (0D60)—The Stacks project

Processing math: 100%

Table of contents
Part 1: Preliminaries
Chapter 20: Cohomology of Sheaves
Section 20.37: Derived limits
Lemma 20.37.1 (cite)

next lemma

numberstags

Lemma 20.37.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. For $U \subset X$ open the functor $R\Gamma (U, -)$ commutes with $R\mathop{\mathrm{lim}}\nolimits$ . Moreover, there are short exact sequences

$0 \to R^1\mathop{\mathrm{lim}}\nolimits H^{m - 1}(U, K_ n) \to H^ m(U, R\mathop{\mathrm{lim}}\nolimits K_ n) \to \mathop{\mathrm{lim}}\nolimits H^ m(U, K_ n) \to 0$

for any inverse system $(K_ n)$ in $D(\mathcal{O}_ X)$ and any $m \in \mathbf{Z}$ .

Proof. The first statement follows from Injectives, Lemma 19.13.6. Then we may apply More on Algebra, Remark 15.86.10 to $R\mathop{\mathrm{lim}}\nolimits R\Gamma (U, K_ n) = R\Gamma (U, R\mathop{\mathrm{lim}}\nolimits K_ n)$ to get the short exact sequences. $\square$

Comments (0)

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).

Comments (0)

Post a comment