### 42.68.45 Application to the key lemma

In this section we apply the results above to show the analogue of the key lemma (Lemma 42.6.3) with the tame symbol $d_ A$ constructed above. Please see Remark 42.6.4 for the relationship with Milnor $K$-theory.

reference
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.
$\square$

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