Lemma 42.68.30 (02Q0)—The Stacks project

Loading [MathJax]/extensions/tex2jax.js

Table of contents
Part 3: Topics in Scheme Theory
Chapter 42: Chow Homology and Chern Classes
Section 42.68: Appendix A: Alternative approach to key lemma
Lemma 42.68.30 (cite)

previous definition
next definition

numberstags

history
statistics
2 tags refer to this tag

Lemma 42.68.30. Let $A$ be a Noetherian local ring. Let $a, b, c \in A$. Let $M$ be a finite $A$-module with $\dim (\text{Supp}(M)) = 1$. Assume $a, b, c$ are nonzerodivisors on $M$. Then

\[ d_ M(a, bc) = d_ M(a, b) d_ M(a, c) \]

and $d_ M(a, b)d_ M(b, a) = 1$.

Proof. The first statement follows from Lemma 42.68.24 applied to $M/abcM$ and endomorphisms $\alpha , \beta , \gamma $ given by multiplication by $a, b, c$. The second comes from Lemma 42.68.15. $\square$

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