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

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

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. $\square$

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