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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