projective object

Let us consider the category of Abelian groups $ {\bf Ab}_G$ .

Definition 0.1   An object $ P$ of an abelian category $ \mathcal{A}$ is called projective if the functor $ Hom_A (P,−) : \mathcal{A} \to {\bf Ab}_G$ is exact.

Remark.

This is equivalent to the following statement: An object $ P$ is projective if given a short exact sequence $ 0 \to M′ \to M \to M′′ \to 0$ in an Abelian category $ \mathcal{A}$ , one has that:

$\displaystyle 0 \to Hom_{\mathcal{A}}(M′, P) \to Hom_{\mathcal{A}}(M, P) \to Hom_{\mathcal{A}}(M′′, P) \to 0$

is exact in $ {\bf Ab}_G$ .



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