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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