# Perfect complex

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. A **perfect module** is a module that is perfect when it is viewed as a complex concentrated at degree zero. For example, if *A* is Noetherian, a module over *A* is perfect if and only if it has finite projective dimension.

## Other characterizations

Perfect complexes are precisely the compact objects in the unbounded derived category of *A*-modules.[1] They are also precisely the dualizable objects in this 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.

## Pseudo-coherent sheaf

When the structure sheaf is not coherent, working with coherent sheaves has awkwardness (namely the kernel of a finite presentation can fail to be coherent). Because of this, SGA 6 Expo I introduces the notion of a **pseudo-coherent sheaf**.

By definition, given a ringed space , an -module is called pseudo-coherent if for every integer , 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 where *L* has degree bounded above and consists of finite free modules in degree . 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.

Roughly speaking, a pseudo-coherent complex may be thought of as a limit of perfect complexes.

## See also

- Hilbert–Burch theorem
- elliptic complex (related notion; discussed at SGA 6 Exposé II, Appendix II.)

## References

- Ben-Zvi, David; Francis, John; Nadler, David (2010), "Integral transforms and Drinfeld centers in derived algebraic geometry",
*Journal of the American Mathematical Society*,**23**(4): 909–966, arXiv:0805.0157, doi:10.1090/S0894-0347-10-00669-7, MR 2669705

- Berthelot, Pierre; Alexandre Grothendieck; Luc Illusie, eds. (1971).
*Séminaire de Géométrie Algébrique du Bois Marie - 1966-67 - Théorie des intersections et théorème de Riemann-Roch - (SGA 6) (Lecture notes in mathematics*(in French). Berlin; New York: Springer-Verlag. xii+700. doi:10.1007/BFb0066283. ISBN 978-3-540-05647-8. MR 0354655.**225**)

## External links

- http://stacks.math.columbia.edu/tag/0656
- http://ncatlab.org/nlab/show/perfect+module
- An alternative definition of pseudo-coherent complex