Lemma 47.23.1. Properties (A), (B), (C), (D), and (E) of More on Algebra, Section 15.51 hold for P(k \to R) =“R is a Gorenstein ring”.
Proof. Since we already know the result holds for Cohen-Macaulay instead of Gorenstein, we may in each step assume the ring we have is Cohen-Macaulay. This is not particularly helpful for the proof, but psychologically may be useful.
Part (A). Let K/k be a finitely generated field extension. Let R be a Gorenstein k-algebra. We can find a global complete intersection A = k[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c) over k such that K is isomorphic to the fraction field of A, see Algebra, Lemma 10.158.11. Then R \to R \otimes _ k A is a relative global complete intersection. Hence R \otimes _ k A is Gorenstein by Lemma 47.21.7. Thus R \otimes _ k K is too as a localization.
Proof of (B). This is clear because a ring is Gorenstein if and only if all of its local rings are Gorenstein.
Part (C). Let A \to B \to C be flat maps of Noetherian rings. Assume the fibres of A \to B are Gorenstein and B \to C is regular. We have to show the fibres of A \to C are Gorenstein. Clearly, we may assume A = k is a field. Then we may assume that B \to C is a regular local homomorphism of Noetherian local rings. Then B is Gorenstein and C/\mathfrak m_ B C is regular, in particular Gorenstein (Lemma 47.21.3). Then C is Gorenstein by Lemma 47.21.8.
Part (D). This follows from Lemma 47.21.8. Part (E) is immediate as the condition does not refer to the ground field. \square
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