The Stacks project

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. $\square$