[Paper Review] Koszul pairs. Applications

Pascual Jara, Javier López Peña+1 more|arXiv (Cornell University)|Nov 18, 2010

Algebraic structures and combinatorial models20 references3 citations

TL;DR

This paper introduces Koszul pairs—compatible connected graded rings and corings—showing that a ring is Koszul if and only if it forms a Koszul pair with some coring. By associating exact chain and cochain complexes to such pairs, the authors establish a homological criterion for Koszulity and apply it to prove that twisted tensor products of Koszul rings remain Koszul, generalizing Fröberg's theorem.

ABSTRACT

Let $R$ be a semisimple ring. A pair $(A,C)$ is called almost-Koszul if $A$ is a connected graded $R$-ring and $C$ is a compatible connected graded $R$-coring. To an almost-Koszul pair one associates three chain complexes and three cochain complexes such that one of them is exact if and only if the others are so. In this situation $(A,C)$ is said to be Koszul. One proves that a connected $R$-ring $A$ is Koszul if and only if there is a connected $R$-coring $C$ such that $(A,C)$ is Koszul. This result allows us to investigate the Hochschild (co)homology of Koszul rings. We apply our method to show that the twisted tensor product of two Koszul rings is Koszul. More examples and applications of Koszul pairs, including a generalization of Fr\oberg Theorem, are discussed in the last part of the paper.

Motivation & Objective

• To define and characterize Koszul pairs as compatible graded rings and corings over a semisimple base ring.
• To establish a homological criterion for Koszulity using exactness of associated chain and cochain complexes.
• To prove that a connected ring is Koszul if and only if it forms a Koszul pair with some connected coring.
• To apply the theory to compute or analyze Hochschild (co)homology of Koszul rings.
• To demonstrate that the twisted tensor product of two Koszul rings is itself Koszul, generalizing Fröberg’s theorem.

Proposed method

• Define an almost-Koszul pair as a compatible pair (A,C) where A is a connected graded R-ring and C is a compatible connected graded R-coring over a semisimple ring R.
• Associate three chain complexes and three cochain complexes to each almost-Koszul pair (A,C).
• Establish that (A,C) is Koszul if and only if one of these complexes is exact, which implies all others are exact.
• Use the exactness criterion to prove that a connected R-ring A is Koszul if and only if there exists a connected R-coring C such that (A,C) is Koszul.
• Apply the theory to compute Hochschild (co)homology of Koszul rings via the Koszul pair framework.
• Use the framework to show that the twisted tensor product of two Koszul rings is Koszul, extending Fröberg’s theorem.

Experimental results

Research questions

• RQ1 When is a connected graded R-ring A Koszul, and how can this property be characterized via a compatible coring?
• RQ2 What is the relationship between the exactness of associated chain and cochain complexes and the Koszul property of a pair (A,C)?
• RQ3 Can the Koszul property of a ring be equivalently defined through the existence of a compatible coring making (A,C) a Koszul pair?
• RQ4 How does the Koszul pair framework facilitate the study of Hochschild (co)homology of Koszul rings?
• RQ5 Is the twisted tensor product of two Koszul rings itself Koszul, and can this be proven using the Koszul pair theory?

Key findings

• A connected R-ring A is Koszul if and only if there exists a connected R-coring C such that the pair (A,C) is Koszul.
• The Koszul property of a pair (A,C) is equivalent to the exactness of one (and hence all) of the associated chain or cochain complexes.
• The theory enables a systematic study of Hochschild (co)homology for Koszul rings through the lens of Koszul pairs.
• The twisted tensor product of two Koszul rings is proven to be Koszul, extending Fröberg’s classical result to a broader class of algebras.
• The framework generalizes Fröberg’s theorem by providing a homological criterion for Koszulity in the context of twisted tensor products.