If $\pi$ does not preserve multiplication, then $A \cong R \#_\xi H$ where $R$ is a pre-bialgebra in $_H^H\mathcal{YD}$ and $\xi: R \otimes R \rightarrow H$. The question is whether $A$ can be twisted by a cocycle to a Radford biproduct. We show that the correct setting for this problem is that of dual quasi-bialgebras and that $A$ can always be twisted by a gauge transformation to a bosonization $Q \#H$ where $Q$ is a connected dual quasi-bialgebra in $_H^H\mathcal{YD}$. This work is joint with A. Ardizzoni and C. Menini.

We consider the case when $X \to X/\mathrm{Aut}_k(X)$ is not a Katz-Gabber cover for the group $\mathrm{Aut}_k(X)$, so that $\mathrm{Aut}_k(X)$ is strictly larger than $G$. We show that then either $X$ belongs to an exceptional family of curves of genus $0$ or $1$, or the action of $\mathrm{Aut}_k(X)$ on $X$ mixes the tame and totally ramified orbits. The exceptional case in which $X$ has genus 1 is of particular interest because it leads to some new explicit formulas for power series $\sigma(t) = \sum_{i = 1}^\infty a_i t^i$ which define automorphisms of $k[[t]]$ of order $p^2$ when $p=2$.

We consider the ideal $I_M$ of End$_{\cal C} (T[1])$ given by all endomorphisms that factor through $M$, and show that the End$ (T)-$module Hom$(T,M)$ has infinite projective dimension precisely when $I_M$ is non-zero. Examples indicate that the objects $M$ of $\cal C$ such that $I_M\neq 0$ lie on hammocks in $\cal C$.

Given any subcategory ${\cal U}$ of $\cal D$, not all torsion theories $({\cal U}, {\cal U}^{\perp})$ are t-structures. Therefore we press on to give a characterization of t-structures in $\cal D$ on the Auslander-Reiten quiver. The simplicity of the way t-structures are visualized on the Auslander-Reiten quiver gives a subtle hint with regard to the description of the "t-structure triangles" in $\cal D$, which turns out to be faithful to our intuition by a rigorous proof with the involvement of the octahedral axiom.

LRB algebras are directed quasi-hereditary algebras and hence have finite global dimension. In this talk, we discuss how to use combinatorial topology to compute the global dimension of an LRB algebra and in fact arbitrary Ext between simple modules. In particular we show that the Leray number of a flag complex is precisely the global dimension of an associated LRB algebra. Our main technical tools are classifying spaces of small categories and Quillen's Theorem A.

This is joint work with Stuart Margolis and Franco Saliola.

The well known such classes of Frobenius categories are the subcategories of finitely presented modules over preprojective algebras associated to the elements in Coxeter groups as done in [GLS] and [BIRSc]. The stable categories of these categories have cluster tilting objects and cluster structure. It was shown in [AIRT] that certain Frobenius categories with cluster tilting objects are triangle equivalent to generalized cluster categories associated to algebras of global dimension 2. In particular it is true for the above categories associated to the elements of Coxeter groups. Another interesting Frobenius category is the category of representations of circle, with the stable category being the continuous cluster category [IT].