In algebra, a perfect complex of modules over a commutative ring A is an object in the derived category of A-modules that is quasi-isomorphic to a bounded complex of finite projective A-modules.
Perfect complexes are precisely the compact objects in the unbounded derived category
[2] A compact object in the ∞-category of (say right) module spectra over a ring spectrum is often called perfect;[3] see also module spectrum.
Because of this, SGA 6 Expo I introduces the notion of a pseudo-coherent sheaf.
, locally, there is a free presentation of finite type of length n; i.e., A complex F of
-modules is called pseudo-coherent if, for every integer n, there is locally a quasi-isomorphism
If the complex consists only of the zero-th degree term, then it is pseudo-coherent if and only if it is so as a module.