Lemma 42.68.37 (02Q6)—The Stacks project

Loading web-font TeX/Math/Italic

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 42.68.37 (cite)

numberstags

Lemma 42.68.37. Let $A$ be a discrete valuation ring with fraction field $K$ . For nonzero $x, y \in K$ we have

$d_ A(x, y) = (-1)^{\text{ord}_ A(x)\text{ord}_ A(y)} \frac{x^{\text{ord}_ A(y)}}{y^{\text{ord}_ A(x)}} \bmod \mathfrak m_ A,$

in other words the symbol is equal to the usual tame symbol.

Proof. By multiplicativity it suffices to prove this when $x, y \in A$ . Let $t \in A$ be a uniformizer. Write $x = t^ bu$ and $y = t^ bv$ for some $a, b \geq 0$ and $u, v \in A^*$ . Set $l = a + b$ . Then $t^{l - 1}, \ldots , t^ b$ is an admissible sequence in $(x)/(xy)$ and $t^{l - 1}, \ldots , t^ a$ is an admissible sequence in $(y)/(xy)$ . Hence by Remark 42.68.14 we see that $d_ A(x, y)$ is characterized by the equation

$[t^{l - 1}, \ldots , t^ b, v^{-1}t^{b - 1}, \ldots , v^{-1}] = (-1)^{ab} d_ A(x, y) [t^{l - 1}, \ldots , t^ a, u^{-1}t^{a - 1}, \ldots , u^{-1}].$

Hence by the admissible relations for the symbols $[x_1, \ldots , x_ l]$ we see that

$d_ A(x, y) = (-1)^{ab} u^ a/v^ b \bmod \mathfrak m_ A$

as desired. $\square$

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