B-mod category equivalence theorem

Theorem 0.1   B-mod category equivalence theorem.

Let $ \mathcal{A}$ be an abelian category with arbitrary direct sums (or coproducts). Also, let $ P$ in $ \mathcal{A}$ be a compact projective generator and set $ B = (End_{\mathcal{A}} P)^{op}$ . The functor $ hom_\mathcal{A}(P,--)$ yields an equivalence of categories between $ \mathcal{A}$ and the category $ B-mod$ .

Proof. The proof proceeds in two steps. At the first step one shows that the functor

$\displaystyle F(X) = hom_{\mathcal{A}}(P,X)$

is fully faithful, and therefore, at the second step one can apply the Abelian category equivalence lemma to yield the sought for equivalence of categories.



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