# Algebra by MacLane S., G. Birkhoff PDF By MacLane S., G. Birkhoff

This e-book goals to give sleek algebra from first ideas, which will be
accessible to undergraduates or graduates, and this through combining commonplace
materials and the wanted algebraic manipulations with the overall options
which make clear their which means and significance.

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It is not the case that Ext preserves inverse limits or direct limits (even with the assumption that the preordered set is directed). 3. Theorem. (i) An R-module B is injective if and only if Ext1R (A, B) = 0 for all A. (ii) An R-module A is projective if and only if Ext1R (A, B) = 0 for all B. 4. 32 (Extension) Let A and C be R-modules. An extension of C by A is an exact sequence f g 0 → A → B → C → 0. : Two extensions and of C by A are equivalent if there exists a chain map from one to the other that is the identity on A and on C: : GA 0 :  GB 1  GA 0 GC ϕ  1 GC GB G0 G0 (in this case ϕ is an isomorphism by the five lemma).

The relation ∼ is an equivalence relation on the class of n-extensions of C by A. We denote by [ ] the class of and by YextnA (C, A) the collection of all equivalence classes of n-extensions of C by A. We assume that YextnA (C, A) is a set (which is the case if A is a module category). If n ≥ 2, the Baer sum of [ ] and [ ] is the class of the n-extension + : ¯n → Bn−1 ⊕ B ¯ 0→A→B n−1 → · · · → B2 ⊕ B2 → B1 → C → 0, ¯n is the pushout of A → Bn , A → B and B ¯1 is the pullback of where B n B1 → C, B1 → C.

C → CI op by ( M )i = M for all i and ( , lim) is an adjoint pair. ←− In particular, if C is abelian, then lim is left exact. 4. Let G : D → C be a functor and assume that inverse limits exist in D and C. For any preordered set I, the functor G induces a functor op op G : C I → DI with G(A)i = G(Ai ) and G(τ )i = G(τi ) for each inverse system A and morphism τ : A → B. We say that G preserves inverse op limits if for each preordered set I, lim G ∼ G lim as functors from DI to ←− = ←− C. Theorem.

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