The Stacks project

Remark 47.16.6. Let $(A, \mathfrak m)$ and $\omega _ A^\bullet$ be as in Lemma 47.16.5. By More on Algebra, Lemma 15.69.2 we see that $\omega _ A^\bullet$ has injective-amplitude in $[-d, 0]$ because part (3) of that lemma applies. In particular, for any $A$-module $M$ (not necessarily finite) we have $\mathop{\mathrm{Ext}}\nolimits ^ i_ A(M, \omega _ A^\bullet ) = 0$ for $i \not\in \{ -d, \ldots , 0\}$.