Definition 42.68.29. Let $A$ be a Noetherian local ring with residue field $\kappa $. Let $a, b \in A$. Let $M$ be a finite $A$-module of dimension $1$ such that $a, b$ are nonzerodivisors on $M$. We define the symbol associated to $M, a, b$ to be the element
\[ d_ M(a, b) = \det \nolimits _\kappa (M/abM, a, b) \in \kappa ^* \]
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