The Stacks project

Lemma 47.21.3. A regular local ring is Gorenstein. A regular ring is Gorenstein.

Proof. Let $A$ be a regular ring of finite dimension $d$. Then $A$ has finite global dimension $d$, see Algebra, Lemma 10.110.8. Hence $\mathop{\mathrm{Ext}}\nolimits ^{d + 1}_ A(M, A) = 0$ for all $A$-modules $M$, see Algebra, Lemma 10.109.8. Thus $A$ has finite injective dimension as an $A$-module by More on Algebra, Lemma 15.69.2. It follows that $A[0]$ is a dualizing complex, hence $A$ is Gorenstein by the remark following the definition. $\square$

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