The Stacks project

Lemma 47.21.7. If $A \to B$ is a local complete intersection homomorphism of rings and $A$ is a Noetherian Gorenstein ring, then $B$ is a Gorenstein ring.

Proof. By More on Algebra, Definition 15.32.2 we can write $B = A[x_1, \ldots , x_ n]/I$ where $I$ is a Koszul-regular ideal. Observe that a polynomial ring over a Gorenstein ring $A$ is Gorenstein: reduce to $A$ local and then use Lemmas 47.15.10 and 47.21.4. A Koszul-regular ideal is by definition locally generated by a Koszul-regular sequence, see More on Algebra, Section 15.31. Looking at local rings of $A[x_1, \ldots , x_ n]$ we see it suffices to show: if $R$ is a Noetherian local Gorenstein ring and $f_1, \ldots , f_ c \in \mathfrak m_ R$ is a Koszul regular sequence, then $R/(f_1, \ldots , f_ c)$ is Gorenstein. This follows from Lemma 47.21.6 and the fact that a Koszul regular sequence in $R$ is just a regular sequence (More on Algebra, Lemma 15.29.7). $\square$


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