Invertible unital bimodules over rings with local units, and related exact sequences of groups II

Let $R$ be a ring with a set of local units, and a homomorphism of groups $\underline{\Theta} : \G \to \Picar{R}$ to the Picard group of $R$. We study under which conditions $\underline{\Theta}$ is determined by a factor map, and, henceforth, it defines a generalized crossed product with a same set of local units. Given a ring extension $R \subseteq S$ with the same set of local units and assuming that $\underline{\Theta}$ is induced by a homomorphism of groups $\G \to \Inv{R}{S}$ to the group of all invertible $R$-sub-bimodules of $S$, then we construct an analogue of the Chase-Harrison-Rosenberg seven terms exact sequence of groups attached to the triple $(R \subseteq S, \underline{\Theta})$, which involves the first, the second and the third cohomology groups of $\G$ with coefficients in the group of all $R$-bilinear automorphisms of $R$. Our approach generalizes the works by Kanzaki and Miyashita in the unital case.

💡 Research Summary

The paper develops a comprehensive theory of invertible unital bimodules over rings that possess a set of local units, extending classical results that were previously confined to unital (i.e., rings with a global identity) settings. A “set of local units” E is a collection of idempotents such that for any finite subset of the ring there exists an e in E that simultaneously acts as a left and right unit for all elements of that subset. This notion allows the authors to treat rings that are not necessarily unital but still enjoy enough “local” identity elements to develop module theory.

The first part of the paper establishes the basic properties of unital modules, similar modules, and invertible bimodules in this context. For a unital bimodule M, the right dual M_R = Hom_R‑R(R,M) is shown to be a finitely generated projective module over the commutative ring Z = End_R‑R(R). Moreover, there is a natural isomorphism M ≅ R ⊗Z M_R, which plays a central role in handling tensor products of invertible bimodules. The authors also prove a natural “twist” isomorphism T{M,N}: M ⊗_R N → N ⊗_R M for any two invertible unital bimodules M, N, and verify its weak associativity. These results are the technical backbone for the later construction of generalized crossed products.

Next, the authors consider a group G and a group homomorphism Θ: G → Pic(R), where Pic(R) is the Picard group of invertible unital R‑bimodules. The homomorphism Θ is equivalent to a family of invertible bimodules {Θ_x}{x∈G} together with isomorphisms F{x,y}: Θ_x ⊗R Θ_y → Θ{xy}. The crucial question is whether these data can be assembled into an associative multiplication on the direct sum Δ = ⊕{x∈G} Θ_x. The authors show that associativity holds precisely when the diagram involving the threefold tensor product commutes; this condition can be expressed as the vanishing of a normalized 3‑cocycle α{x,y,z} with values in the unit group U(Z) of Z. When such a factor map exists, Δ becomes a generalized crossed product that shares the same set of local units E′ = ι(E), where ι: R → Θ_1 is the canonical identification.

The paper then focuses on a ring extension R ⊆ S that shares the same set of local units. Assuming that the homomorphism Θ is induced by a map G → Inv_R(S) into the group of all invertible R‑sub‑bimodules of S, the authors construct a generalized crossed product inside S. They develop normalized 2‑ and 3‑cocycles with coefficients in U(Z) and define an abelian group C(R,G,Θ) consisting of isomorphism classes of such crossed products. Proposition 3.2 shows that C(R,G,Θ) is naturally isomorphic to the second cohomology group H^2(G,U(Z)).

The main achievement is the derivation of a seven‑term exact sequence that generalizes the classical Chase‑Harrison‑Rosenberg sequence. The sequence reads

1 → H^1(G,U(Z)) → Pic(R) → Pic(S)^G → H^2(G,U(Z)) → B(R/S) → H^1(G,Pic(S)) → H^3(G,U(Z)),

where B(R/S) denotes the Brauer group of Azumaya algebras over S that split over R. The first, fourth, and seventh terms are the usual G‑cohomology groups with coefficients in U(Z); the middle terms involve the Picard groups of R and S and the Brauer group, mirroring the structure of the original seven‑term sequence for unital Galois extensions of commutative rings. The authors’ construction recovers the results of Kanzaki (for unital rings) and Miyashita (for non‑commutative unital extensions) as special cases, while extending them to the broader setting of rings with local units.

Throughout the paper, the authors discuss the technical obstacles posed by non‑commuting idempotents and the lack of a universally accepted notion of morphisms between rings with local units. By restricting to extensions that preserve the same set of local units, they circumvent these difficulties and obtain a coherent theory that integrates Picard groups, invertible bimodules, and group cohomology. The work opens new avenues for applying cohomological methods to non‑unital, non‑commutative algebraic structures, including non‑commutative Galois theory, categorical localization, and the classification of Azumaya algebras in settings where a global identity element is absent.