Totally reflexive modules over rings that are close to Gorenstein (Q2219010)

Over a commutative Noetherian ring, \textit{M. Auslander} and \textit{M. Bridger} [Stable module theory. Providence, RI: American Mathematical Society (AMS) (1969; Zbl 0204.36402)] extended the notion of finitely generated projective modules to totally reflexive modules, i.e. modules occurring as the cokernels of differentials in exact chain complexes of finitely generated free modules whose dual complexes are also exact. There has been a growing interest in identifying the local rings over which every totally reflexive module is projective or equivalently free, i.e. the so-called ``G-regular rings'' coined by Takahashi. Regular local rings and Golod local rings that are not hypersurfaces are among the examples of G-regular rings. However, a singular Gorenstein local ring is never G-regular. In the paper under review, the authors prove that any non-Gorenstein quotient of small colength of a deeply embedded equicharacteristic Artinian Gorenstein local ring is G-regular. Their investigation relies on the fact that an Artinian local ring is G-regular if there exists a projective-test module that happens to be a direct summand of a syzygy of the canonical module of the ring. Overall, this is an interesting and readable paper.