Koszul pairs and applications

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.

💡 Research Summary

The paper develops a unified framework for Koszul theory over a semisimple base ring R by introducing the notion of an “almost‑Koszul pair” (A, C). Here A is a connected graded R‑algebra (i.e. A⁰ = R and A¹ is a free R‑module) and C is a compatible connected graded R‑coring (C⁰ = R, C¹ free). Compatibility means that there exists an R‑linear isomorphism θ₁ : A¹ → C¹ which identifies the multiplication μ : A¹⊗A¹ → A² with the comultiplication Δ : C² → C¹⊗C¹ after applying θ₁ on each factor.

From such a pair the authors construct three chain complexes K·(A, C), B·(A, C), L·(A, C) and three cochain complexes K·(A, C), B·(A, C), L·*(A, C). Each complex is built from alternating tensor products of A and C, mimicking the classical bar and cobar constructions but now intertwining the algebra and coring structures. A key “equivalence theorem” shows that exactness of any one of these six complexes forces exactness of all the others. Consequently, a pair (A, C) is called a Koszul pair when these complexes are exact and, moreover, when A (resp. C) satisfies the usual Koszul condition as an algebra (resp. coring).

The central structural result is a bi‑directional characterization: a connected graded R‑algebra A is Koszul if and only if there exists a connected graded R‑coring C such that (A, C) forms a Koszul pair. This removes the need to construct a Koszul dual object “by hand” in the classical sense; the existence of a suitable coring is guaranteed precisely when A is Koszul.

With the Koszul pair in hand, the authors turn to Hochschild (co)homology. The chain complex K·(A, C) provides a resolution of A as an A‑bimodule that is naturally isomorphic to the standard Hochschild complex HH·(A). Dually, the cochain complex K·*(A, C) computes Hochschild cohomology. Because K·(A, C) is built from the coring C, the Hochschild (co)homology of a Koszul algebra can be expressed in terms of the cobar complex of C, dramatically simplifying calculations, especially for algebras with non‑quadratic relations.

A major application concerns twisted tensor products. Let A₁ and A₂ be Koszul algebras and τ : A₂¹⊗A₁¹ → A₁¹⊗A₂¹ a twisting map satisfying the usual compatibility conditions. The paper proves that the twisted tensor product A₁ ⊗_τ A₂ is again Koszul. The proof proceeds by taking the Koszul corings C₁ and C₂ of A₁ and A₂, forming the twisted coring C = C₁ ⊗_τ C₂, and showing that (A₁ ⊗_τ A₂, C) is a Koszul pair. This result extends the well‑known fact that ordinary tensor products of Koszul algebras are Koszul to the more subtle twisted setting, opening the door to constructing new Koszul algebras from known ones via deformation‑type procedures.

The authors also revisit Froberg’s theorem, which classically states that a quadratic monomial algebra is Koszul. By exploiting the flexibility of Koszul pairs, they obtain a generalized criterion that applies to algebras whose defining relations may have mixed degrees. The key is to construct a coring C from the relation module in such a way that (A, C) satisfies the Koszul pair conditions; the existence of such a C provides a sufficient condition for A to be Koszul, thereby extending Froberg’s theorem beyond the strictly quadratic case.

The final sections present a variety of examples—path algebras of quivers, quantum matrix algebras, and non‑standard graded polynomial algebras—illustrating how to build the appropriate corings and verify the Koszul pair conditions. In each case the authors compute Hochschild (co)homology via the coring perspective and confirm that the results agree with known computations when available.

In summary, the paper introduces a robust, pair‑centric approach to Koszul theory that works over any semisimple base ring, unifies algebraic and coalgebraic viewpoints, simplifies Hochschild (co)homology calculations, guarantees the Koszul property for twisted tensor products, and yields a broad generalization of Froberg’s theorem. The methodology promises further applications in non‑commutative geometry, quantum groups, and homological aspects of deformation theory.