On a question of Takahashi concerning direct summands of syzygies

Abstract

For a finitely generated module $M$ of finite Gorenstein dimension over a commutative noetherian local ring $(R,\mathfrak{m},\Bbbk)$, we prove that the natural map $\mathrm{Ext}_R(\Bbbk,M)\longrightarrow\mathrm{Ext}_R(\Bbbk,M/\mathfrak{m} M)$ is zero unless $R$ is Gorenstein. More generally, the same conclusion holds whenever $\mathrm{Ext}_R^i(M,R)=0$ for $i\gg0$. This gives a Gorenstein analogue of a regularity criterion of Avramov and Iyengar. The proof uses the right $\mathrm{Ext}_R(\Bbbk,\Bbbk)$-module structure of bounded cohomology studied by the author in a previous work. As an application, we show that $R$ is Gorenstein if a nonzero quotient of a finite direct sum of syzygies of $\Bbbk$ has finite Gorenstein dimension. In particular, this answers affirmatively a question of Takahashi concerning direct summands of syzygies of the residue field.