By Andreas Gathmann
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Extra info for Commutative Algebra
E. if and only if ϕ is injective. ϕ A sequence M −→ N −→ 0 is exact if and only if im ϕ = N, i. e. if and only if ϕ is surjective. (b) The sequence 0 −→ M −→ 0 is exact if and only if M = 0. ϕ By (a), the sequence 0 −→ M −→ N −→ 0 is exact if and only if ϕ is injective and surjective, i. e. if and only if ϕ is an isomorphism. 3 (Short exact sequences). 2, the first interesting case of an exact sequence occurs if it has at least three non-zero terms. Therefore, an exact sequence of the form ϕ ψ 0 −→ M1 −→ M2 −→ M3 −→ 0 (∗) is called a short exact sequence.
A) ⇒ (b) and (c): Under the assumption (a) we can take α to be the projection from M ⊕ P onto M, and for β the inclusion of P in M ⊕ P. (b) ⇒ (a): If α is a left-sided inverse of ϕ, the diagram 0 ϕ M (α, ψ) id 0 ψ N M⊕P M P 0 id P 0 is commutative with exact rows. 12. (c) ⇒ (a): If β is a right-sided inverse of ψ, the diagram 0 M⊕P M ϕ +β id 0 P ϕ M id ψ N 0 P 0 is commutative with exact rows, where (ϕ + β )(m, p) := ϕ(m) + β (p). 12. 15. Every short exact sequence ϕ ψ 0 −→ U −→ V −→ W −→ 0 of vector spaces over a field K is split exact: if (bi )i∈I is a basis of W we can pick inverse images ci ∈ ψ −1 (bi ) by the surjectivity of ψ.
In fact, up to isomorphisms every short exact sequence is of these forms: in a short exact sequence as in (∗) the second map ψ is surjective, and thus im ψ = M3 . Moreover, ker ψ = im ϕ = ϕ(M1 ) ∼ = M1 , where the last isomorphism follows from the injectivity of ϕ. So the given sequence is of the type as in (a). 10 (a), we can also regard it as a sequence as in (b). A nice feature of exact sequences is that there are simple rules to create new sequences from old ones. The simplest way to do this is to split and glue exact sequences as follows.