The global dimension of an associative algebra A over a a field is a measure of the complexity of its representations. It is 0 if A is a matrix algebra. It is 1 if A is a path algebras of quivers without directed cycles. It is infinite if A is the algebra of dual numbers.
I will give a brief introduction to Hochschild homology (1945), in order to explain Han’s conjecture (2006): for finite-dimensional algebras, the Hochschild homology should control the finiteness of the global dimension.
Next, I will present some progress made in showing Han’s conjecture, using the relative version of Hochschild homology (1956) with respect to a subalgebra B. This theory was little used until recently. Now we have a Jacobi-Zariski long nearly exact sequence which relates the usual and relative versions of Hochschild homology. Its gap to be exact is approximated by a spectral sequence which has Tor functors in its first page, of B-tensor powers of A/B. This tool enables to show, for instance, that the class of algebras verifying Han’s conjecture is closed by bounded extensions of algebras.
These results have been obtained in joint work with M. Lanzilotta, E. N. Marcos and A. Solotar.
This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.
