A Course in Homological Algebra by P. J. Hilton, U. Stammbach (auth.)

By P. J. Hilton, U. Stammbach (auth.)

Show description

Read or Download A Course in Homological Algebra PDF

Best linear books

Partial Differential Equations II: Qualitative Studies of Linear Equations

This moment within the sequence of 3 volumes builds upon the elemental idea of linear PDE given in quantity 1, and pursues extra complicated issues. Analytical instruments brought right here comprise pseudodifferential operators, the practical research of self-adjoint operators, and Wiener degree. The booklet additionally develops simple differential geometrical techniques, situated approximately curvature.

Extra info for A Course in Homological Algebra

Example text

O is the category of spaces-with-base-point (see [21]). :~ -+ (fj, where the subscript h indicates that the morphisms are to be regarded as (based) homotopy classes of (based) continuous functions. :~ and then 7t factors as 7t = 1fQ. :h-+Illb). (g) We saw in Chapter I how the set 9Jl~(A, B) = HomA(A,B) may be given the structure of an abelian group. If we hold A fixed and define II. 21b by 9Jl~(A, - )(B) = 9Jl~(A, B), then 9Jl~(A, -) is a functor. More generally, for any category (£: and object A of (£:, (£:(A, -) is a functor from (£: to 6.

21 b). (c) Let A be an object of9Jl~ and let G be an abelian group. We saw in Section I. a left A-module. 47 2. Functors Hom z ( -, G) thus appears as a contravariant functor from 9Jl~ to 9Jl~. Further examples will appear as exercises. Finally we make the following definitions. Recall from Section 1 the notion of a full subcategory. Consistent with that definition, we now define a functor F : (t-:D as full if F maps (t(A, B) onto :D(F A, F B) for all objects A, B in (t, and as faithful if F maps (t(A, B) injectively to :D(F A, F B).

2. Let G be a divisible abelian group. Then X is an injective A-module. = Homz(A, G) Proof. Let J1: A-+B be a monomorphism of A-modules, and let a: A-+X a homomorphism of A-modules. We have to show that there exists P: B-+ X such that PJ1 = a. 1 to a homomorphism of abelian groups a' : A -+ G. Since G is injecti ve, there exists P' : B -+ G such that P' J1 = a'. 1 we obtain a homomorphism of A-modules p: B-+A. Finally by the naturality of 1], the diagram is commutative. 3. 3. Every A-module A is a sub module of a cofree, hence injective, A-module.

Download PDF sample

Rated 4.50 of 5 – based on 17 votes