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 10.136.13. 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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