Lemma 47.21.6. Let (A, \mathfrak m, \kappa ) be a Noetherian local ring. Let f \in \mathfrak m be a nonzerodivisor. Set B = A/(f). Then A is Gorenstein if and only if B is Gorenstein.
Proof. If A is Gorenstein, then B is Gorenstein by Lemma 47.16.10. Conversely, suppose that B is Gorenstein. Then \mathop{\mathrm{Ext}}\nolimits ^ i_ B(\kappa , B) is zero for i \gg 0 (Lemma 47.21.5). Recall that R\mathop{\mathrm{Hom}}\nolimits (B, -) : D(A) \to D(B) is a right adjoint to restriction (Lemma 47.13.1). Hence
R\mathop{\mathrm{Hom}}\nolimits _ A(\kappa , A) = R\mathop{\mathrm{Hom}}\nolimits _ B(\kappa , R\mathop{\mathrm{Hom}}\nolimits (B, A)) = R\mathop{\mathrm{Hom}}\nolimits _ B(\kappa , B[1])
The final equality by direct computation or by Lemma 47.13.10. Thus we see that \mathop{\mathrm{Ext}}\nolimits ^ i_ A(\kappa , A) is zero for i \gg 0 and A is Gorenstein (Lemma 47.21.5). \square
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