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The Stacks project

Remark 50.18.2. Let A be a ring. Let P = A[x_1, \ldots , x_ n]. Let f_1, \ldots , f_ n \in P and set B = P/(f_1, \ldots , f_ n). Assume A \to B is quasi-finite. Then B is a relative global complete intersection over A (Algebra, Definition 10.136.5) and (f_1, \ldots , f_ n)/(f_1, \ldots , f_ n)^2 is free with generators the classes \overline{f}_ i by Algebra, Lemma 10.136.12. Consider the following diagram

\xymatrix{ \Omega _{A/\mathbf{Z}} \otimes _ A B \ar[r] & \Omega _{P/\mathbf{Z}} \otimes _ P B \ar[r] & \Omega _{P/A} \otimes _ P B \\ & (f_1, \ldots , f_ n)/(f_1, \ldots , f_ n)^2 \ar[u] \ar@{=}[r] & (f_1, \ldots , f_ n)/(f_1, \ldots , f_ n)^2 \ar[u] }

The right column represents \mathop{N\! L}\nolimits _{B/A} in D(B) hence has cohomology \Omega _{B/A} in degree 0. The top row is the split short exact sequence 0 \to \Omega _{A/\mathbf{Z}} \otimes _ A B \to \Omega _{P/\mathbf{Z}} \otimes _ P B \to \Omega _{P/A} \otimes _ P B \to 0. The middle column has cohomology \Omega _{B/\mathbf{Z}} in degree 0 by Algebra, Lemma 10.131.9. Thus by Lemma 50.18.1 we obtain canonical B-module maps

\Omega ^ p_{B/\mathbf{Z}} \longrightarrow \Omega ^ p_{A/\mathbf{Z}} \otimes _ A \det (\mathop{N\! L}\nolimits _{B/A})

whose composition with \Omega ^ p_{A/\mathbf{Z}} \to \Omega ^ p_{B/\mathbf{Z}} is multiplication by \delta (\mathop{N\! L}\nolimits _{B/A}).


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