Representation Theory by Alexander Zimmermann

Author:Alexander Zimmermann
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Proposition 3.5.43

Let be an algebra. Then the functor from Corollary 3.5.41 induces equivalences

and if is a Noetherian -algebra over a commutative ring we get

as triangulated categories.

Proof

We shall construct for each object in an object in with in . We call a projective resolution of .

Without loss of generality, shifting if necessary, we may assume that is isomorphic to a complex whose negative degree components are , and such that . Indeed, if

such that for , then by Lemma 3.5.21, i.e.

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