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.