arXiv subjects

Adrian Manea

Publications and source records attributed to Adrian Manea.

3 recordsLinked to original sources

arXiv2016

Further Properties and Applications of Koszul Pairs

Koszul pairs were introduced in [arXiv:1011.4243] as an instrument for the study of Koszul rings. In this paper, we continue the enquiry of such pairs, focusing on the description of the second component, as a follow-up of the study in [arXiv:1605.05458]. As such, we introduce Koszul corings and prove several equivalent characterizations for them. As applications, in the case of locally finite $R$-rings, we show that a graded $R$-ring is Koszul if and only if its left (or right) graded dual coring is Koszul. Finally, for finite graded posets, we obtain that the respective incidence ring is Koszul if and only if the incidence coring is so.

math.KT↗

arXiv2016

On Koszulity of Finite Posets

We prove in a unifying way several equivalent descriptions of Koszul rings, some of which being well known in the literature. Most of them are stated in terms of coring theoretical properties of $\Tor_n^A(R,R)$. As an application of these characterizations we investigate the Koszulity of the incidence rings for finite graded posets. Based on these results, we describe an algorithm to produce new classes of Koszul posets (i.e. graded posets whose incidence rings are Koszul). Specific examples of Koszul posets are included.

math.KT↗

arXiv2015

On the Ext Ring of Koszul Rings

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 $\mathbb{Z}^n$.

math.KT↗