Hochschild homology and global dimension
Petter Andreas Bergh, Dag Madsen
TL;DR
This paper establishes a link between infinite global dimension and non-vanishing Hochschild homology in high degrees for specific classes of graded algebras over fields of characteristic zero, using Igusa's formula.
Contribution
It demonstrates a new relationship between global dimension and Hochschild homology for Koszul, local, and cellular graded algebras, expanding understanding of their homological properties.
Findings
Infinite global dimension implies Hochschild homology does not vanish in high degrees.
The result applies to Koszul, local, and cellular graded algebras.
The proof utilizes Igusa's formula relating cyclic homology and the graded Cartan determinant.
Abstract
We prove that for certain classes of graded algebras (Koszul, local, cellular), infinite global dimension implies that Hochschild homology does not vanish in high degrees, provided the characteristic of the ground field is zero. Our proof uses Igusa's formula relating the Euler characteristic of relative cyclic homology to the graded Cartan determinant.
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