[Paper Review] Crossed Bimodules over Rings and Shukla Cohomology

Nguyễn Tiến Quang, Pham Thi Cuc|arXiv (Cornell University)|Jan 4, 2013

Algebraic structures and combinatorial models11 references3 citations

TL;DR

This paper establishes a classification of ring extensions of the type of a crossed bimodule over rings using Shukla cohomology, leveraging Ann-category theory to link E-systems and crossed bimodules. It proves a bijection between equivalence classes of such extensions and the second Shukla cohomology group $ H^2_{ ext{Shu}}(Q, ext{Ker} hinspace d) $, generalizing prior results on Hochschild and group cohomology in this context.

ABSTRACT

In this paper we present some applications of Ann-category theory to classification of crossed bimodules over rings, classification of ring extensions of the type of a crossed bimodule.

Motivation & Objective

• To classify ring extensions of the type of a crossed bimodule over rings using cohomological methods.
• To establish a categorical equivalence between E-systems and crossed bimodules over rings via Ann-category theory.
• To generalize previous results on group and Hochschild cohomology to the setting of ring extensions.
• To show that equivalence classes of such extensions are in bijection with elements of the second Shukla cohomology group $ H^2_{ ext{Shu}}(Q, ext{Ker} hinspace d) $.

Proposed method

• Introduces the concept of an E-system and proves its isomorphism with the category of crossed bimodules over rings.
• Uses strict Ann-categories to model E-systems and establishes a categorical equivalence between E-systems and a subcategory of strict Ann-categories.
• Applies Ann-functor theory to construct a correspondence between ring extensions and Ann-functors from the discrete category $ \text{Dis}Q $ to an Ann-category $ \mathcal{A} $.
• Constructs a crossed product extension $ E_0 = [B, \varphi, g', h', Q] $ associated to an Ann-functor $ F $, using the data of the E-system.
• Defines a morphism $ \eta: E_0 \to E $ via representatives $ e_u \in E $, showing it is a ring isomorphism under the given constraints.
• Proves that the extension $ \mathcal{E} $ is equivalent to $ \mathcal{E}_F $ via the map $ \eta $, with $ \varepsilon \circ \eta = \varepsilon_0 $, establishing the classification via cohomology.

Experimental results

Research questions

• RQ1 How can ring extensions of the type of a crossed bimodule over rings be classified using cohomological invariants?
• RQ2 What is the relationship between E-systems, crossed bimodules over rings, and Ann-categories?
• RQ3 How does Shukla cohomology classify such ring extensions, and how does it generalize Hochschild cohomology in this context?
• RQ4 What is the role of Ann-functors in constructing and classifying these extensions?
• RQ5 Is there a canonical bijection between equivalence classes of such extensions and elements of a cohomology group?

Key findings

• There is a natural bijection between equivalence classes of ring extensions of type $ (B, D, d, \theta) $ and the second Shukla cohomology group $ H^2_{\text{Shu}}(Q, \text{Ker} hinspace d) $, where $ \psi: Q \to \text{Coker} hinspace d $ is induced by the extension.
• The category of regular E-systems is isomorphic to the category of crossed bimodules over rings, establishing a categorical equivalence.
• The construction of an extension $ \mathcal{E}_F $ from an Ann-functor $ F: \text{Dis}Q \to \mathcal{A} $ is shown to be equivalent to the original extension $ \mathcal{E} $, via a ring isomorphism $ \eta $.
• The proof shows that the map $ \eta $ preserves both addition and multiplication, using the relations $ e_u + e_v = g'(u,v) + e_{u+v} $ and $ e_u e_v = h'(u,v) + e_{uv} $, which are derived from the E-system structure.
• The equivalence of extensions is verified by showing $ \varepsilon \circ \eta = \varepsilon_0 $, confirming that the cohomological invariant classifies the extension up to equivalence.
• The result generalizes earlier work by Baues and Pirashvili on $ \partial $-extensions and Hochschild cohomology, now extended to the case of rings via Shukla cohomology.