# Seminars in Algebra

## Fall 2012

## Spring 2012

## April 11th 2012, 13.00-14.00 in 656

**Title:** TBA

**Speaker:** Osamu Iyama

## March 26th 2012, 13.00-14.00 in 656

**Title:** TBA

**Speaker:** Hugh Thomas

## March 23rd 2012, 13.15-14.15 in room 656

**Title:** Companion bases associated to quivers of cluster-tilted algebras

**Speaker:** Mark Parsons

## March 21st 2012, 13.00-14.00 in Room 656

**Title:** Recollements of Abelian Categories and Derived Dimensions

**Speaker:** Chrysostomos Psaroudakis

**Abstract:** Abstract: We study several homological properties concerning abelian categories A, B, and C involved in a recollement situation (A,B,C). In particular we give bounds for the global dimension, the finitistic, and the representation dimension of B in terms of the corresponding dimensions of A and/or C. We also study when a recollement (A,B,C) lifts to a recollement of triangulated categories (D^b(A),D^b(B),D^b(C)) between the bounded derived categories of A and/or C, and we apply these results in order to provide bounds for the dimension of D^b(B) in terms of the dimensions of D^b(A) and D^b(C).

## March 14th 2012, 13.00-14.00 in 656

**Title:** Realizing stable categories as derived categories

**Speaker:** Kota Yamaura

**Abstract:** In my talk, we discuss a relationship between the stable categories of modules over self-injective algebras and the bounded derived categories of the categories of modules over algebras. In 1990's, D. Happel showed the following result: Let \(\Lambda\) be an algebra and \(A\) the trivial extension of \(\Lambda\). He introduced a natural positively grading on \(A\), and proved that \(\Lambda\) has finite global dimension if and only if there exists a triangle-equivalence \(\mathsf{D}^{\mathrm{b}}(\mathrm{mod}\Lambda) \simeq \underline{\mathrm{mod}}^{\mathbb{Z}}A\). From his result, it is natural to ask that for a positively graded self-injective algebra \(A\), when is \(\underline{\mathrm{mod}}^{\mathbb{Z}}A\) triangle-equivalent to \(\mathsf{D}^{\mathrm{b}}(\mathrm{mod}\Lambda)\) for some algebra \(\Lambda\). The aim of this talk is to give a complete answer of this question.

## March 7th 2012, 13.00-14.00 in 656

**Title:** Cluster tilting and quadratic forms

**Speaker:** Yuya Mizuno

**Abstract:** We will discuss the connection between cluster tilting theory and quadratic forms. In particular, we give the higher analog of Gabriel's theorem. We introduce the notion of cluster-roots and consider the criterion of them.

## February 22nd 2012, 13.00-14.00 in 656

**Title:** Pseudo - tilting theory

**Speaker:** Idun Reiten

**Abstract:** The talk is based upon joint work with Osamu iyama. We investigate a generalization of tilting modules of projective dimension at most one, which we call pseudo - tilting modules.The work was inspired by investigating connections between cluster categories and the associated cluster tilted algebras

## February 15th 2012, 13.00-14.00 in 656

**Title:** Degeneration and composition series

**Speaker:** Nils M. Nornes

**Abstract:** Let k be a field, A a finite-dimensional k-algebra and d a natural number. Degeneration is a partial order on the set of isomorphism classes of d-dimensional A-modules. It can be described as follows: M degenerates to N if there exists a short exact sequence 0 → X → X+M → N → 0. We will show that when M degenerates to N there exist composition series M_1,…,M_d=M and N_1,…,N_d=N such that M_i degenerates to N_i for all i.

## February 6th 2012, 13.00-14.00 in F6 i Gamle fyssik

**Title:** The axioms for n-angulated categories

**Speaker:** Marius Thaule

**Abstract:** We will discuss the axioms for an n-angulated category, recently introduced by Geiss, Keller and Oppermann. In particular, we introduce a higher octahedral axiom, and show that it is equivalent to the mapping cone axiom for an n-angulated category. For a triangulated category, the mapping cone axiom, our octahedral axiom and the classical octahedral axiom are all equivalent. This is joint work with Petter Andreas Bergh.

## February 1st 2012, 13:00-14:00 in 656

**Title:** Detecting Gorenstein algebras

**Speaker:** Petter Andreas Bergh

**Abstract:** This is a report on joint work with Steffen and Dave Jorgensen. Given an additive category, we define a triangle functor from the homotopy category of acyclic complexes to a certain Verdier quotient. This functor is full and faithful. When the additive category is the category of projective modules over an Artin ring, then the functor is dense if and only if the ring is Gorenstein.