Gorenstein homological algebra for rngs and Lie superalgebras

TL;DR

This paper extends Gorenstein homological algebra concepts from rings to arbitrary abelian categories, applying the theory to Lie superalgebras and their categories, leading to new formulas and insights.

Contribution

It generalizes Gorenstein homological algebra to abelian categories and applies this to Lie superalgebras, introducing new formulas for Serre functors and Gorenstein extension groups.

Findings

Gorenstein homological algebra is extended to arbitrary abelian categories.

The theory is applied to the BGG category O for Lie superalgebras.

New formulas for Serre functors and Gorenstein extension groups are derived.

Abstract

We generalise notions of Gorenstein homological algebra for rings to the context of arbitrary abelian categories. The results are strongest for module categories of rngs with enough idempotents. We also reformulate the notion of Frobenius extensions of noetherian rings into a setting which allows for direct generalisation to arbitrary abelian categories. The abstract theory is then applied to the BGG category O for Lie superalgebras, which can now be seen as a "Frobenius extension" of the corresponding category for the underlying Lie algebra and is therefore "Gorenstein". In particular we obtain new and more general formulae for the Serre functors and instigate the theory of Gorenstein extension groups.