On the Ext Ring of Koszul Rings

TL;DR

This paper investigates the Ext ring of Koszul rings, establishing isomorphisms with dual and shriek rings, and applies these results to incidence rings of Koszul posets and monoid rings, providing new computational methods.

Contribution

It introduces new isomorphisms between Ext rings and dual/shriek rings for Koszul rings, with applications to incidence rings and monoid rings.

Findings

Isomorphism between Ext ring and dual/shriek rings for Koszul rings

Method for computing shriek rings of incidence $R$-coring of Koszul posets

Applications to monoid rings of submonoids of $ abla^n$

Abstract

The aim of this article is to study the Ext ring associated to a Koszul $R$-ring and to use it to provide further characterisations of the former. As such, for $R$ being a semisimple ring and $A$ a graded Koszul $R$-ring, we will prove that there is an isomorphism of DG rings between $\mathcal{E}(A):=\mathrm{Ext}^\bullet_A(R,R)$ and $^{\ast-\text{gr}}\mathrm{T}(A) \simeq \mathrm{E}(^{\ast-\text{gr}\!} A)$. Also, the Ext $R$-ring will prove to be isomorphic to the shriek ring of the left graded dual of $A$, namely $\mathcal{E}(A) \simeq (^{\ast-\text{gr}\!} A)^!$. As an application, these isomorphisms will be studied in the context of incidence $R$-(co)rings for Koszul posets. Thus, we will obtain a description and method of computing the shriek ring for $\Bbbk^c[\mathcal{P}]$, the incidence $R$-coring of a Koszul poset. Another application is provided for monoid rings associated to submonoids of $\mathbb{Z}^n$.