The Stacks project

Lemma 49.10.3. Let $k$ be a field. Let $B = k[x_1, \ldots , x_ n]/(f_1, \ldots , f_ n)$ be a global complete intersection over $k$ of dimension $0$. Then $\omega _{B/k}$ is invertible.

Proof. By Noether normalization, see Algebra, Lemma 10.115.4 we see that there exists a finite injection $k \to B$, i.e., $\dim _ k(B) < \infty $. Hence $\omega _{B/k} = \mathop{\mathrm{Hom}}\nolimits _ k(B, k)$ as a $B$-module. By Dualizing Complexes, Lemma 47.15.8 we see that $R\mathop{\mathrm{Hom}}\nolimits (B, k)$ is a dualizing complex for $B$ and by Dualizing Complexes, Lemma 47.13.3 we see that $R\mathop{\mathrm{Hom}}\nolimits (B, k)$ is equal to $\omega _{B/k}$ placed in degree $0$. Thus it suffices to show that $B$ is Gorenstein (Dualizing Complexes, Lemma 47.21.4). This is true by Dualizing Complexes, Lemma 47.21.7. $\square$

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