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The Stacks project

Lemma 47.21.9. Let (A, \mathfrak m, \kappa ) be a Noetherian local Gorenstein ring of dimension d. Let E be the injective hull of \kappa . Then \text{Tor}_ i^ A(E, \kappa ) is zero for i \not= d and \text{Tor}_ d^ A(E, \kappa ) = \kappa .

Proof. Since A is Gorenstein \omega _ A^\bullet = A[d] is a normalized dualizing complex for A. Also E is the only nonzero cohomology module of R\Gamma _\mathfrak m(\omega _ A^\bullet ) sitting in degree 0, see Lemma 47.18.1. By Lemma 47.9.5 we have

E \otimes _ A^\mathbf {L} \kappa = R\Gamma _\mathfrak m(\omega _ A^\bullet ) \otimes _ A^\mathbf {L} \kappa = R\Gamma _\mathfrak m(\omega _ A^\bullet \otimes _ A^\mathbf {L} \kappa ) = R\Gamma _\mathfrak m(\kappa [d]) = \kappa [d]

and the lemma follows. \square


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