I looked at that section of *Local Fields*, and I believe this is about the trace pairing. For a finite, flat morphism of Dedekind schemes, $f:X\to Y$, the pushforward sheaf $f_*\mathcal{O}_X$ is a locally free $\mathcal{O}_Y$-module, let's say of (constant) rank $n$. It is also an algebra, i.e., multiplication is a $\mathcal{O}_Y$-module homomorphism, $$ \theta_f : f_*\mathcal{O}_X\otimes_{\mathcal{O}_Y} f_*\mathcal{O}_X \to f_*\mathcal{O}_X.$$ By adjunction of tensor and hom, this is equivalent to an $\mathcal{O}_Y$-module homomorphism, $$ \widetilde{\theta}_f:f_*\mathcal{O}_X \to \textit{Hom}_{\mathcal{O}_Y}(f_*\mathcal{O}_X,f_*\mathcal{O}_X).$$ But the target module, being End of a locally free sheaf, has a trace map in the usual sense of trace of matrices, $$\text{Tr}_{f_*\mathcal{O}_Y}:\textit{Hom}_{\mathcal{O}_Y}(f_*\mathcal{O}_X,f_*\mathcal{O}_X) \to \mathcal{O}_Y.$$ The composition with $\widetilde{\theta}_f$ is a trace map for $f$, $$ \text{Tr}_f:f_*\mathcal{O}_X \to \mathcal{O}_Y.$$ The composition of this trace map with $\theta_f$ is a $\mathcal{O}_Y$-bilinear pairing on the locally free $\mathcal{O}_Y$-module $f_*\mathcal{O}_X$, $$ T_f: f_*\mathcal{O}_X\otimes_{\mathcal{O}_Y} f_*\mathcal{O}_X \to \mathcal{O}_Y.$$ This pairing is often called the "trace pairing". Again, by adjunction of tensor and hom, it is equivalent to an $\mathcal{O}_Y$-module homomorphism, $$\widetilde{T}_f:f_*\mathcal{O}_X \to \textit{Hom}_{\mathcal{O}_Y}(f_*\mathcal{O}_X,\mathcal{O}_Y).$$
In particular, taking the associated $n^{\text{th}}$ exterior power of both sides, i.e., taking the determinant, gives a morphism of invertible $\mathcal{O}_Y$-modules, $$\text{det}(\widetilde{T}_f): \text{det}(f_*\mathcal{O}_X) \to \textit{Hom}_{\mathcal{O}_Y}(\text{det}(f_*\mathcal{O}_X),\mathcal{O}_Y).$$ If $f$ is generically étale, then $\text{det}(\widetilde{T}_f)$ is injective. In this case, by definition, the **determinant** of $f$ is defined to be the unique invertible ideal sheaf $\mathfrak{d}_f$ of $\mathcal{O}_Y$ such that
$$ \text{Image}(\text{det}(\widetilde{T}_f)) = \textit{Hom}_{\mathcal{O}_Y}(\text{det}(f_*\mathcal{O}_X),\mathcal{O}_Y)\otimes_{\mathcal{O}_Y}\mathfrak{d}_f,$$ as subsheaves of $\textit{Hom}_{\mathcal{O}_Y}(\text{det}(f_*\mathcal{O}_X),\mathcal{O}_Y)$, i.e., $\text{det}(\widetilde{T}_f)$ is an isomorphism,
$$ \text{det}(\widetilde{T}_f): \text{det}(f_*\mathcal{O}_X) \xrightarrow{\cong} \textit{Hom}_{\mathcal{O}_Y}(\text{det}(f_*\mathcal{O}_X),\mathcal{O}_Y)\otimes_{\mathcal{O}_Y}\mathfrak{d}_f.$$

On the other hand, the **different** of $f$, $\mathfrak{D}_f$, is defined to be the ~~maximal~~ minimal invertible ideal sheaf on $X$, $$\iota:\mathfrak{D}_f \hookrightarrow \mathcal{O}_X,$$ such that for the associated transpose, $$\iota^\dagger: \mathcal{O}_X \to \textit{Hom}_{\mathcal{O}_X}(\mathfrak{D}_f,\mathcal{O}_X), $$ with its induced pushforward map under $f_*$, $$f_*\iota^\dagger: f_*\mathcal{O}_X \to f_*\textit{Hom}_{\mathcal{O}_X}(\mathfrak{D}_f,\mathcal{O}_X), $$ the $\mathcal{O}_Y$-module homomorphism $\text{Tr}_f$ factors through $f_*\iota^\dagger$, i.e., there exists a (unique) $\mathcal{O}_Y$-module homomorphism, $$\tau_f:f_*\textit{Hom}_{\mathcal{O}_X}(\mathfrak{D}_f,\mathcal{O}_X) \to \mathcal{O}_Y,$$ such that $\text{Tr}_f$ equals $\tau_f\circ f_*\iota^\dagger$. The **ramification divisor** of $f$, $R_f$, is the unique effective Cartier divisor of $X$ whose associated invertible sheaf equals $\mathfrak{D}_f$. One of the basic computations, Chapter III.3, Proposition 6 on p. 50 of *Local Fields*, states that the pushforward Cartier divisor $f_*R_f$ equals the effective Cartier divisor on $Y$ associated to the ideal sheaf $\mathfrak{d}_f$, i.e., $$\mathfrak{d}_f = \text{Nm}_f(\mathfrak{D}_f),$$ where $\text{Nm}_f$ is the **norm** associated to $f$. In particular, the $\mathcal{O}_Y$-module homomorphism $\text{det}(\widetilde{T}_f)$ above now becomes an isomorphism, $$ \text{det}(\widetilde{T}_f):\text{det}(f_*\mathcal{O}_X) \xrightarrow{\cong} \textit{Hom}_{\mathcal{O}_Y}(\text{det}(f_*\mathcal{O}_X),\mathcal{O}_Y)(-f_*R_f).$$ Using adjunction of tensor and hom one last time, this is an isomorphism of invertible $\mathcal{O}_Y$-modules, $$\text{det}(f_*\mathcal{O}_X) \otimes_{\mathcal{O}_Y} \text{det}(f_*\mathcal{O}_X) \xrightarrow{\cong} \mathcal{O}_Y(-f_*R_f).$$ This is the identity that you asked about.