The Stacks project

When $A$ is an excellent ring this is [Proposition 1, Kato-Milnor-K].

Lemma 42.68.46 (Key Lemma). Let $A$ be a $2$-dimensional Noetherian local domain with fraction field $K$. Let $f, g \in K^*$. Let $\mathfrak q_1, \ldots , \mathfrak q_ t$ be the height $1$ primes $\mathfrak q$ of $A$ such that either $f$ or $g$ is not an element of $A^*_{\mathfrak q}$. Then we have

\[ \sum \nolimits _{i = 1, \ldots , t} \text{ord}_{A/\mathfrak q_ i}(d_{A_{\mathfrak q_ i}}(f, g)) = 0 \]

We can also write this as

\[ \sum \nolimits _{\text{height}(\mathfrak q) = 1} \text{ord}_{A/\mathfrak q}(d_{A_{\mathfrak q}}(f, g)) = 0 \]

since at any height one prime $\mathfrak q$ of $A$ where $f, g \in A^*_{\mathfrak q}$ we have $d_{A_{\mathfrak q}}(f, g) = 1$ by Lemma 42.68.33.

Proof. Since the tame symbols $d_{A_{\mathfrak q}}(f, g)$ are additive (Lemma 42.68.30) and the order functions $\text{ord}_{A/\mathfrak q}$ are additive (Algebra, Lemma 10.121.1) it suffices to prove the formula when $f = a \in A$ and $g = b \in A$. In this case we see that we have to show

\[ \sum \nolimits _{\text{height}(\mathfrak q) = 1} \text{ord}_{A/\mathfrak q}(\det \nolimits _\kappa (A_{\mathfrak q}/(ab), a, b)) = 0 \]

By Proposition 42.68.43 this is equivalent to showing that

\[ e_ A(A/(ab), a, b) = 0. \]

Since the complex $A/(ab) \xrightarrow {a} A/(ab) \xrightarrow {b} A/(ab) \xrightarrow {a} A/(ab)$ is exact we win. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 02QJ. Beware of the difference between the letter 'O' and the digit '0'.