Endomorphism algebras over commutative rings and torsion in self tensor products

Let $R$ be a commutative Noetherian local ring. We study tensor products involving a finitely generated $R$-module $M$ through the natural action of its endomorphism ring. In particular, we study torsion properties of self tensor products in the case where $\operatorname{End}_R(M)$ has an $R^*$-algebra structure, and prove that if $M$ is indecomposable, then $M \otimes_{\operatorname{End}_R(M)} M$ must always have torsion in this case under mild hypotheses.Comment: 8 page

Higher Nerves of Simplicial Complexes

We investigate generalized notions of the nerve complex for the facets of a simplicial complex. We show that the homologies of these higher nerve complexes determine the depth of the Stanley-Reisner ring $k[\Delta]$ as well as the $f$-vector and $h$-vector of $\Delta$. We present, as an application, a formula for computing regularity of monomial ideals.Comment: We rewrite Section 4 to fix some errors and clarify the proof

VIBRATIONAL MODE-SPECIFIC AUTODETACHMENT AND COUPLING OF CH2CN-

The Cyanomethyl Anion, CH$_{2}$CN-, and neutral radical have been studied extensively, with several findings of autodetachment about the totally symmetric transition, as well as high resolution experiments revealing symmetrically forbidden and weak vibrational features. We report photoelectron spectra using the Velocity-Mapped Imaging Technique in 1-2 wn increments over a range of 13460 to 15384 wn that has not been previously examined. These spectra include excitation of the ground state cyanomethyl anion into the direct detachment thresholds of previously reported vibrational modes for the neutral radical. Significant variations from Franck-Condon behavior were observed in the branching ratios for resolved vibrational features for excitation in the vicinity of the thresholds involving the nub{3} and nub{5} modes. These are consistent with autodetachment from rovibrational levels of a dipole bound state acting as a resonance in the detachment continuum. The autodetachment channels involve single changes in vibrational quantum number, consistent with the vibrational propensity rule but in some cases reveal relaxation to a different vibrational mode indicating coupling between the modes and/or a breakdown of the normal mode approximation

Homological Properties of Structures in Commutative Algebra and Algebraic Combinatorics

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

On the Independent Domination Number of Regular Graphs

A set S of vertices in a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. In this paper, we consider questions about independent domination in regular graphs

On a generalization of Ulrich modules and its applications

We study a modified version of the classical Ulrich modules, which we call $c$-Ulrich. Unlike the traditional setting, $c$-Ulrich modules always exist. We prove that these modules retain many of the essential properties and applications observed in the literature. Additionally, we reveal their significance as obstructions to Cohen-Macaulay properties of tensor products. Leveraging this insight, we show the utility of these modules in testing the finiteness of homological dimensions across various scenarios.Comment: 22 page