Bilinear forms on Grothendieck groups of triangulated categories

We extend the theory of bilinear forms on the Green ring of a finite group developed by Benson and Parker to the context of the Grothendieck group of a triangulated category with Auslander-Reiten triangles, taking only relations given by direct sum decompositions. We examine the non-degeneracy of the bilinear form given by dimensions of homomorphisms, and show that the form may be modified to give a Hermitian form for which the standard basis given by indecomposable objects has a dual basis given by Auslander-Reiten triangles. An application is given to the homotopy category of perfect complexes over a symmetric algebra, with a consequence analogous to a result of Erdmann and Kerner.Comment: arXiv admin note: substantial text overlap with arXiv:1301.470

Auslander-Reiten conjecture and Auslander-Reiten duality

Motivated by a result of Araya, we extend the Auslander-Reiten duality theorem to Cohen-Macaulay local rings. We also study the Auslander-Reiten conjecture, which is rooted in Nakayama's work on finite dimensional algebras. One of our results detects a certain condition that forces the conjecture to hold over local rings of positive depth.Comment: 16 pages, to appear in Journal of Algebr

The Eventual Vanishing of Self-Extensions

In this thesis we examine modules that have eventually vanishing self-extensions with a view towards better understanding the Auslander-Reiten Condition. In the first part, we focus on modules over symmetric algebras using techniques from the representation theory of Artin algebras. More specifically, we examine how the shape of a component in the Auslander-Reiten quiver of a symmetric algebra is related to the vanishing of self-extensions of the modules it contains. We obtain several restrictions on the possible shape of such a component containing a module with eventually vanishing self-extensions. For many algebras we are able to describe completely which components may contain a module with eventually vanishing self-extensions. We determine the degree in which these extensions must begin to vanish for every module in such a component. In particular, we identify new conditions which guarantee that a symmetric algebra satisfies a generalized version of the Auslander-Reiten Condition. In the second part, we consider all Noetherian rings. Motivated by the results in the first part of the thesis, we introduce the finitistic extension degree of a ring. We show that rings for which this invariant is finite satisfy the generalized version of the Auslander-Reiten Condition. These results extend recent results for rings satisfying another cohomological condition known as Auslander\u27s Condition. We also investigate the relationship between the finiteness of the finitistic extension degree and Auslander\u27s Condition

Correlation of Warmth and Loneliness

My Research Methods course spent the semester studying the association between loneliness and physical warmth. We decided to replicate two studies by Bargh and Shalev (2011) in which they found that people feel less lonely when they are physically warm. In one study, we measured physical warmth and loneliness and did not replicate Bargh and Shalev\u27s finding that loneliness is associated with warmth. In a second study, we manipulated warmth. One hundred and forty participants were randomly assigned to one of two groups. One group held a warm pack for one minute and the other group held a cold pack for one minute. All participants completed a product evaluation of the hot/cold pack. Then, all participants completed a measure of loneliness. Previous research found that participants who held a warm pack felt less lonely than those who held a cold pack. Our results did not support this finding