The Stacks project

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

\[ \mathop{N\! L}\nolimits _{B/A} = \left( (f_1, \ldots , f_ n)/(f_1, \ldots , f_ n)^2 \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} B \text{d} x_ i\right) \]

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

\[ \text{d}x_1 \wedge \ldots \wedge \text{d}x_ n \otimes (\overline{f}_1 \wedge \ldots \wedge \overline{f}_ n)^{\otimes -1} \]

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

\[ \delta (\mathop{N\! L}\nolimits _{B/A}) = \det (\partial f_ j/ \partial x_ i) \cdot \text{d}x_1 \wedge \ldots \wedge \text{d}x_ n \otimes (\overline{f}_1 \wedge \ldots \wedge \overline{f}_ n)^{\otimes -1} \]

by definition.

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