Titlebook: Homology; Saunders Mac Lane Book 1995 Springer-Verlag Berlin Heidelberg 1995 Abelian group.Factor.algebra.auditor.cohomology.collaboration

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Classics in Mathematicshttp://image.papertrans.cn/h/image/428145.jpg

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Modules, Diagrams, and Functors,Homology theory deals repeatedly with the formal properties of functions and their composites. The functions concerned are usually homomorphisms of modules or of related algebraic systems. The formal properties are subsumed in the statement that the homomorphisms constitute a category. This chapter will examine the notions of module and category.

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Cohomology of Groups,The cohomology of a group . provides our first example of the functors Ext.(.) — with . the group ring and . = .. These cohomology groups may be defined directly in terms of a standard “bar resolution”. In low dimensions they arise in problems of group extensions by . in all dimensions they have a topological interpretation (§11).

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Tensor and Torsion Products,Let . be a right .-module and . a left .-module — a situation we may indicate as ., .. Their . .?. is the abelian group generated by the symbols .?. for .∈. and .∈. subject to the relations

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Cohomology of Algebraic Systems,The homology of algebraic systems is an instance of relative homological algebra.

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Extensions and Resolutions,sions, suitably classified by a congruence relation, are the elements of a group Ext.(.). To calculate this group, we present . as the quotient .=./. of a free module .; this process can be iterated as .=./., .=./.,… to give an exact sequence?→.→.→?→.→.→.→0called a “free resolution” of .. The complex Horn (., .) has cohomology Ext.(.).

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