Mes: maig de 2023

Roozbeh Hazrat (University of Western Sydney)

Sandpile Graphs and Graph Algebras

Abstract: We give a down to earth introduction to seemingly two very different topics, one about sandpile models (a model about spreading objects along networks) and the other is how to associate interesting algebras to graphs. We then relate these two topics, via the concept of monoids.

Manuel Saorín (Universidad de Murcia)

On an overlooked conjecture

Abstract: The concept of flat object can be defined in any Grothendieck category. In 2007 Juan Cuadra and Daniel Simson conjectured that any locally finitely presented Grothendieck with enough flats has enough projectives. Since by (an extended version of) Gabriel-Popescu’s theorem, any Grothendieck category is equivalent to the quotient $(\mathrm{Mod}-\mathcal{A})/\mathcal{T}$, where is $\mathcal A$ a preadditive category and $\mathcal T$ is a hereditary torsion class of $\mathrm{Mod}-\mathcal{A})$, in order to tackle the conjecture one needs to ask first what are the conditions on $\mathcal T$ for the mentioned quotient to: 1) be locally finitely presented; 2) have enough flats; 3) have enough projectives. In this talk we will identify those conditions in the particular case when $\mathcal T$ is also a torsion free class (i.e. it is a TTF class) in which case, due to a generalization of Jan’s bijection, there is a uniquely determined idempotent ideal $\mathcal I$ of $\mathcal A$ such that consists of the right $\mathcal A$-modules (=additive functors $\mathcal{A}^{op}\to\mathrm{Ab}$) that vanish on $\mathcal I$. It turns out that $(\mathrm{Mod}-\mathcal{A})/\mathcal{T}$ has enough projectives in this case if, and only, if $\mathcal I$ is the trace of a projective right $\mathcal A$-module. From this point of view, as the much more popular telescope conjecture of Ravenel, this restricted version of Cuadra-Simson’s conjecture is a particular case of the general question of when a given idempotent ideal $\mathcal I$ of a (pre)additive category, e.g. of a ring, is the trace of a (special type of) projective $\mathcal A$-module. We will give some partial positive answers to Cuadra-Simson’s conjecture when $\mathcal T$ is a TTF class.

Raimund Preusser (Nanjing University of Information Science and Technology)

Graded Bergman algebras

Abstract: This talk is about an ongoing research project with Roozbeh Hazrat and Huanhuan Li. Recall that for a (unital and associative) ring R, the V-monoid of R is the set of isomorphism classes of finitely generated projective left R-modules. It becomes an abelian monoid with direct sum. George Bergman has shown that any conical finitely generated abelian monoid with an order unit can be realised as the V-monoid of a hereditary algebra. We want to obtain a graded version of this result as follows. For an abelian group G, a G-monoid is an abelian monoid M together with an action of G on M via monoid homomorphisms. Our goal is to show that any conical finitely presented G-monoid with an order unit can be realised as the graded V-monoid of a G-graded algebra which is hereditary.