Sylvia Wiegand, University of Nebraska - Lincoln
2022 AWM Research Symposium
Homological and Combinatorial Aspects of Commutative Algebra

Consider the following properties for a commutative Noetherian local ring (R,m,k): (AR) Every finitely generated R-module M such that, for every i>0, Ext_R^i(M,M⊕R)=0 is a free module. (HW) Every torsion-free finitely generated R-module M with rank such that M⊗_R M∗ is MCM is a free module. Two conjectures related to these properties are: the Auslander-Reiten Conjecture (ARC), that every local ring R satisfies (AR); and the Huneke-R Wiegand Conjecture (HWC), that every Gorenstein local ring satisfies (HW). Our focus is on quasi-fiber product rings — rings for which there exists a regular sequence x in m such that m/(x) decomposes into a nontrivial direct sum of ideals of R/(x). We show that quasi-fiber product rings satisfy a sharpened form of (ARC) and we make some observations related to (HWC).

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