Splitting Algebras: Koszul, Cohen-Macaulay and Numerically Koszul

TL;DR

This paper investigates the algebraic properties of splitting algebras derived from finite posets, establishing conditions under which they are Koszul or Cohen-Macaulay, and clarifying the relationship between numerical Koszulity and Koszulity.

Contribution

It proves that the algebra R is Koszul if and only if the poset is Cohen-Macaulay and shows that numerically Koszul does not necessarily imply Koszul for these algebras.

Findings

R is Koszul if and only if the poset is Cohen-Macaulay.

Koszulity of R implies Koszulity of the algebra A.

Numerically Koszul does not imply Koszul for these algebras.

Abstract

We study a finite dimensional quadratic graded algebra R defined from a finite ranked poset. This algebra has been central to the study of the splitting algebra of the poset, A, as introduced by Gelfand, Retakh, Serconek and Wilson . The algebra A is known to be quadratic when the poset satisfies a combinatorial condition known as uniform, and R is the quadratic dual of an associated graded algebra of A. We prove that R is Koszul and the poset is uniform if and only if the poset is Cohen-Macaulay. Koszulity of R implies Koszulity of A. We also show that when R is Koszul, the cohomology of the order complex of the poset can be identified with certain cohomology groups defined internally to the ring R. Finally, we settle in the negative the long-standing question: Does numerically Koszul imply Koszul for algebras of the form R?