On local rings of finite syzygy representation type

Let R be a commutative Noetherian local ring. We give a characterization of when the completion of R has an isolated singularity. This result simultaneously improves a theorem of Dao and Takahashi and a theorem of Bahlekeh, Hakimian, Salarian, and Takahashi. As an application, we strengthen the Auslander-Huneke-Leuschke-Wiegand theorem in the form refined by Dao and Takahashi. We further investigate the ascent and descent of finite and countable syzygy representation type along the canonical map from R to its completion. As a consequence, we obtain a complete affirmative answer to Schreyer's conjecture. We also explore analogues of Chen's questions in the context of finite Cohen-Macaulay representation type over Cohen-Macaulay rings. The main result in this direction shows that if R is Cohen-Macaulay and there are only finitely many non-isomorphic indecomposable maximal Cohen-Macaulay modules that are locally free on the punctured spectrum, then either R is a hypersurface or every Gorenstein projective module is projective, and every Gorenstein projective module over the completion of R is a direct sum of finitely generated ones. Finally, we study dominant local rings, introduced by Takahashi, under certain finite representation type conditions, and identify a new class of virtually Gorenstein rings.