Perfect complex

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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[edit]

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[edit]

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[edit]

References[edit]

  • 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

External links[edit]