On the vanishing of $\textrm {Ext}$

Abstract:

In this paper we exhibit certain modules $A$ over a commutative noetherian local ring $(R,\mathfrak {M})$ which test projective dimension of finitely generated modules in the following sense: if ${\operatorname {Ext} ^j}(M,A) = 0$ for all $j \geqq i$, then pd $M < i$. We also show that the module $\mathfrak {M}$ tests in a stronger way: if ${\operatorname {Ext} ^i}(M,\mathfrak {M}) = 0$, then pd $M < i$. In conclusion we show that if $(R,\mathfrak {M})$ is artin, then $R$ is self-injective if and only if ${\operatorname {Ext} ^1}(R/{\mathfrak {M}^n},R) = 0$, where the index of nilpotence of $\mathfrak {M}$ is $n + 1$.