Lemma 42.68.39 (02Q9)—The Stacks project

Processing math: 100%

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 42.68.39 (cite)

previous lemma
next subsection

numberstags

history
statistics

Lemma 42.68.39. Let $A$ be a Noetherian local domain of dimension $1$ with fraction field $K$ . For $x \in K \setminus \{ 0, 1\}$ we have

$d_ A(x, 1 -x) = 1$

Proof. Write $x = a/b$ with $a, b \in A$ . The hypothesis implies, since $1 - x = (b - a)/b$ , that also $b - a \not= 0$ . Hence we compute

$d_ A(x, 1 - x) = d_ A(a, b - a)d_ A(a, b)^{-1}d_ A(b, b - a)^{-1}d_ A(b, b)$

Thus we have to show that $d_ A(a, b - a) d_ A(b, b) = d_ A(b, b - a) d_ A(a, b)$ . This is Lemma 42.68.38. $\square$

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