Syzygy modules with semidualizing or G-projective summands (Q2576980)

A module \(C\) over a noetherian local ring \(R\) is semidualizing if \(\text{End}(C)=R\) and \(\text{Ext}^R_i(C,C)=0\) for all \(i>0\). The author shows that \(R\) must be regular if (and only if) the residue field \(k\) has a syzygy module with a semidualizing direct summand. He gives two proofs, one of which uses Dutta's theorem that \(R\) must be regular if a syzygy module of \(k\) has a proper free direct summand. In a similar vein, it is shown that \(R\) must be Gorenstein if the \(n\)-th syzygy module of \(k\) has a direct summand of finite Gorenstein dimension, provided \(n\leq \text{depth} R+2\). It remains an open question whether the restriction on \(n\) can be omitted.