Lemma 21.46.6 (08G2)—The Stacks project

Processing math: 100%

Table of contents
Part 1: Preliminaries
Chapter 21: Cohomology on Sites
Section 21.46: Tor dimension
Lemma 21.46.6 (cite)

numberstags

Lemma 21.46.6. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(K, L, M, f, g, h)$ be a distinguished triangle in $D(\mathcal{O})$ . Let $a, b \in \mathbf{Z}$ .

If $K$ has tor-amplitude in $[a + 1, b + 1]$ and $L$ has tor-amplitude in $[a, b]$ then $M$ has tor-amplitude in $[a, b]$ .
If $K$ and $M$ have tor-amplitude in $[a, b]$ , then $L$ has tor-amplitude in $[a, b]$ .
If $L$ has tor-amplitude in $[a + 1, b + 1]$ and $M$ has tor-amplitude in $[a, b]$ , then $K$ has tor-amplitude in $[a + 1, b + 1]$ .

Proof. Omitted. Hint: This just follows from the long exact cohomology sequence associated to a distinguished triangle and the fact that $- \otimes _\mathcal {O}^{\mathbf{L}} \mathcal{F}$ preserves distinguished triangles. The easiest one to prove is (2) and the others follow from it by translation. $\square$

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