Homological Properties of Structures in Commutative Algebra and Algebraic Combinatorics

Abstract

The purpose of this work is to understand homological properties of structures appearing in commutative algebra and algebraic combinatorics, objects such as commutative rings and associated structures, such as ideals and modules, or simplicial complexes. In particular, we study vanishing conditions for Ext and Tor in connection with homological dimensions of the modules involved, the representation theory of maximal Cohen-Macaulay modules, and various homological properties of simplicial complexes though the lens of combinatorial commutative algebra. Specifically, we study when a Cohen-Macaulay local ring has only trivial vanishings of Ext or Tor, and provide sufficient numerical criterion under which these condition are satisfied. We apply these results to establish new cases of the famous Auslander-Reiten conjecture; other conditions on Ext and Tor are also explored in connection with this conjecture. We also study the connection between classifically studied representation types of the category of maximal Cohen-Macaulay modules of a Cohen-Macaulay local ring and newly introduced representation types which study those maximal Cohen-Macaulay modules that are not locally free on the punctured spectrum. We provide a classification theorem in dimension 1, and discuss partial results and obstacles in higher dimension. We also explore combinatorial constructions such as the nerve complex of a simplicial complex, and introduce the new notion higher nerve complexes. We explore their connection with order complexes of posets, in particular the face poset of a simplicial complex, and we prove that the depth and h-vector of the Stanley-Reisner ring of a simplicial complex can be computed in a nice way from the reduced homologies of these higher nerve complexes. We expand upon our study of these notions by studying balanced simplicial complexes, and using this abstraction we prove that, while one cannot characterize which of Serre's conditions are satisfied by a simplicial complex via the reduced homologies of higher nerve complexes, one can pin it down to one of two possible values. We also provide a depth formula for arbitrary balanced simplicial complexes and consider total Euler characteristics of links; using the latter, we provide some applications to the study of Gorenstein* complexes. Finally, we introduce the notion of minimal Cohen-Macaulay simplicial complexes and provide some necessary and sufficient conditions for this property. We conclude by showing that many recently introduced counterexamples to longstanding conjectures in the literature are minimal Cohen-Macaulay.