# Bounded extension algebras and Han's conjecture

Given a finite dimensional algebra $$A$$ over a field $$k$$, Han’s conjecture relates two homological invariants associated to $$A$$: its global dimension and its Hochschild homology. In the commutative case – non necessarily finite dimensional but finitely generated – the finiteness of the global dimension is equivalent to the fact that $$A$$ is geometrically regular, see for example [3, 9.3.13]. More precisely, Han’s conjecture states that for $$A$$ finite dimensional, $$A$$ is smooth if and only if $$HH_n(A)=0$$ for $$n\gg0$$. The direct implication known to be is true.

We proved in [1] that the class $$\mathcal{H}$$ of finite dimensional algebras which verify Han’s conjecture [2] is closed under split bounded extensions. More precisely if $$A=B\oplus M$$ is such an extension, then $$A\in\mathcal{H}$$ if and only if $$B\in\mathcal{H}$$. The proofs make use of the Jacobi–Zariski long exact sequence, and of the reduced relative bar resolution.

I will explain this result and a generalization for algebras that are non necessarily split. As a consequence of our work, we prove that in some cases, adding or deleting arrows to a quiver – even adding or deleting certain relations – does not change the situation with respect to Han’s conjecture.

This talk contains joint work with Claude Cibils, Marcelo Lanzilotta and Eduardo Marcos.

[1] Claude Cibils, Marcelo Lanzilotta, Eduardo N. Marcos and Andrea Solotar. Split bounded extension algebras and Han’s conjecture. To appear in Pacific Journal of Mathematics (arXiv:1908.11130).

[2] Yang Han. Hochschild (co)homology dimension. J. London Math. Soc. 73 (2006), pp. 657–668.

[3] Charles Weibel. An introduction to homological algebra. Cambridge University Press, 1994.