Canonical Resolutions over Koszul Algebras

TL;DR

This paper extends classical resolutions for polynomial rings to graded Koszul algebras, providing more explicit and minimal resolutions for powers of the homogeneous maximal ideal, improving upon previous methods.

Contribution

It introduces a generalized approach to resolutions over Koszul algebras that are more explicit and minimal than existing techniques.

Findings

Constructed explicit minimal resolutions for powers of the homogeneous maximal ideal.

Generalized classical resolutions from polynomial rings to Koszul algebras.

Enhanced resolution methods compared to prior work by Green and Martínez-Villa.

Abstract

We generalize Buchsbaum and Eisenbud's resolutions for the powers of the maximal ideal of a polynomial ring to resolve powers of the homogeneous maximal ideal over graded Koszul algebras. Our approach has the advantage of producing resolutions that are both more explicit and minimal compared to those previously discovered by Green and Mart\'{\i}nez-Villa \cite{GreenMartinezVilla} or Mart\'{\i}nez-Villa and Zacharia \cite{MartinezVillaZacharia}.