The Stacks project

Lemma 49.12.4. Let $A$ be a ring. Let $n \geq 1$ and $h, f_1, \ldots , f_ n \in A[x_1, \ldots , x_ n]$. Set $B = A[x_1, \ldots , x_ n, 1/h]/(f_1, \ldots , f_ n)$. Assume that $B$ is quasi-finite over $A$. Then there is an isomorphism $B \to \omega _{B/A}$ mapping $\det (\partial f_ i/\partial x_ j)$ to $\tau _{B/A}$.

Proof. Let $J$ be the annihilator of $\mathop{\mathrm{Ker}}(B \otimes _ A B \to B)$. By Lemma 49.12.2 the map $A \to B$ is flat and $J$ is a free $B$-module with generator $\xi $ mapping to $\det (\partial f_ i/\partial x_ j)$ in $B$. Thus the lemma follows from Lemma 49.6.7 and the fact (Lemma 49.10.4) that $\omega _{B/A}$ is an invertible $B$-module. (Warning: it is necessary to prove $\omega _{B/A}$ is invertible because a finite $B$-module $M$ such that $\mathop{\mathrm{Hom}}\nolimits _ B(M, B) \cong B$ need not be free.) $\square$

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