Homological Algebra

Introduction:

Homology theory was originally a part of algebraic topology, concerned with finding numerical invariants of spaces. Its methods grew out of an analysis of the behaviour of path integrals in complex analysis, but in the 1920s and 30s it became clear that the invariants involved were better viewed as numbers associated to certain abelian groups. In the 1940s the theory of the "homology groups" of a space was developed further and Eilenberg and MacLane applied these tools to certain spaces associated to groups; they linked up the groups that resulted with various other invariants that had been developed earlier by group theorists for tackling purely algebraic problems. Combinig the two outlooks, it became clear that these methods, suitably adapted, gave significant tools in many other areas of algebra. The connections between algebraic topology, homological algebra and areas of "pure" algebra, number theory and algebraic geometry have remained very strong. In fact the spin-off from homological algebra in these other areas has been central in their development over the last 30 years.

A. Homotopical algebra:

The classical areas of homological algebra developed from homology theory in this way gave "Abelian" information only. The new methods recently developed at Bangor, Strasbourg and other centers have their origins in homotopy theory and hence are better suited to giving "non-abelian" information. The term HOMOTOPICAL ALGEBRA has been used for this general area, although this term is also used for describing a larger related area of ideas. The key ideas are those of a crossed module, and more generally of a cat^n - group. The first of these is more fully discussed in the sheet on algebra, the second is a higher dimensional analogue of it.

C. Non-Abelian cohomology:

Given groups G and H, can one find all groups E that contain a copy of H as a normal subgroup in such a way that the quotient group E/H is isomorphic to G? This "extension problem" is analysed by the cohomology groups of G if H is Abelian. If H is not Abelian the problem is very much more tricky. One needs non-Abelian cohomology! The situation can be studied from a topological point of view which, with the new methods that recently have become available, suggests various ideas for arriving at an adequate analysis of the problem. These ideas also allow an attack to be made on related problems in commutative algebra, and other areas.

D. Other related problems:

The type of problems being studied at Bangor has been illustrated above in the context of groups, but these problems have important analogues in other contexts, and the development of these analogues is a fruitful line of enquiry. The applications of crossed modules, cat^n-algebras etc. are providing interesting and challenging problems at various levels, including that of the problems is such that there is material to keep the research team occupied for many years to come.

Crossed Modules of Algebras

A crossed module of algebras has a structure in between that of a module and an ideal. Vert little is known about this structure even in the commutative case. Brief history about that is as follows: Crossed modules of groups were introduced by Whitehead in his investigations into the structure of the second relative homotopy and also by Peiffer. Since that time crossed modules have become an important tool in others contexts, for instant, Loday and Guin-Walery-Loday, etc..The commutative algebra version of crossed modules has been used, in essence rather than in name, by Lichtenbaum-Schlessinger also the work of Gerstenhaber essentially involves the notion of crossed modules in associative and commutative algebras, cf. Lue , and has been shown to be closely related to Koszul complexes, T. Porter.Nizar in his Ph.D. thesis, he investigated the algebraic and categorical structure of the categories of crossed modules of algebras and developed certain aspects of the resulting theory of crossed modules. His work is motivated by the following projects.

1) Categorical tools to enable the manipulation of crossed modules.

2) Crossed modules are generalisation of both modules and ideals and any algebra is a crossed module, so it is of interest to see generalisation of algebra theoretic concepts and structures to crossed modules.

3) A theory of localisaion

4) An abstract axiomatisation of categories of crossed modules analogous to the Abelian Categories of modules categories.

In my Ph.D. thesis I stutied the theory of crossed modules of commutative algebras in terms of simplicial context. For more information about this, see Simplicial Commutative Algebras section of this home page.