Remark 49.13.1. Let $Y \to X$ be a locally quasi-finite syntomic morphism of schemes. What does the pair $(\det (\mathop{N\! L}\nolimits _{Y/X}), \delta (\mathop{N\! L}\nolimits _{Y/X}))$ look like locally? Choose affine opens $V = \mathop{\mathrm{Spec}}(B) \subset Y$, $U = \mathop{\mathrm{Spec}}(A) \subset X$ with $f(V) \subset U$ and an integer $n$ and $f_1, \ldots , f_ n \in A[x_1, \ldots , x_ n]$ such that $B = A[x_1, \ldots , x_ n]/(f_1, \ldots , f_ n)$. Then

and $(f_1, \ldots , f_ n)/(f_1, \ldots , f_ n)^2$ is free with generators the classes $\overline{f}_ i$. See proof of Lemma 49.10.1. Thus $\det (L_{B/A})$ is free on the generator

and the section $\delta (\mathop{N\! L}\nolimits _{B/A})$ is the element

by definition.

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