# Compact and perfect objects

Let R be a ring. Let D(R) be the derived category of R-modules. An object K of D(R) is perfect if it is quasi-isomorphic to a finite complex of finite projective R-modules. An object K of D(R) is called compact if and only of the functor Hom_{D(R)}(K, – ) commutes with arbitrary direct sums. In the previous post I mentioned two results on perfect complexes which I added to the stacks project today. Both are currently in the second chapter on algebra of the stacks project. Here are the statements with corresponding tags:

1. An object K of D(R) is perfect if and only if it is compact. This is Proposition Tag 07LT.
2. If I ⊂ R is an ideal of square zero and K ⊗^L R/I is a perfect object of D(R/I), then K is a perfect object of D(R). This is Lemma Tag 07LU.

Enjoy! If anybody knows a reference for the first result which predates the paper “Morita theory for derived categories” by Rickard I’d love to hear about it. Thanks.
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