## RESEARCH IN RESIDENCE

#### Galois coverings, finite global dimension and Hochschild homology

*Recouvrements de Galois, dimension globale finie et homologie de Hochschild*

#### 3 – 14 June, 2024

#### Participants

Claude Cibils (Université de Montpellier)

Marcelo Lanzilotta (University of the Republic, Uruguay)

Eduardo do Nascimento Marcos (University of São Paulo)

Andrea Solotar (University of Buenos Aires)

Galois coverings of finite dimensional algebras were first introduced by Bongartz, Gabriel, and Riedtmann in relation to representation theory. We intend to prove that, in case the Galois group is finite, the Hochschild homology of the base algebra is finite if and only if the Hochschild homology of its covering algebra is finite. We also plan to prove that the finiteness of the global dimension is preserved. In doing so, we would be proving that Han’s conjecture is valid for a Galois covering of a finite dimensional algebra A if and only if it is valid for A.

We know, after a classical result by P. Gabriel, that any finite dimensional algebra A over an algebraically closed field k can be presented – up to Morita equivalence – as a quotient of a path algebra kQ of a quiver Q by the ideal I generated by a finite set R of admissible minimal relations. given a simple module S of finite projective dimension, let A’ be the the endomorphism algebra of all the indecomposable projective modules except the projective cover of S. The algebras A and A’ could share several homological properties, we plan to work in this direction.

We believe that the finiteness of the global dimension is also preserved by dropping an inert arrow. After proving this we would be able to reduce Han’s conjecture to quivers with relations with neither inert arrows nor simple modules of finite projective dimension. Due to a result on Morita contexts that we have already proven, we could also reduce Han’s conjecture to quivers without source or sink vertices