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 is finitely generated and of 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]- Hilbert–Burch theorem
- elliptic complex (related notion; discussed at SGA 6 Exposé II, Appendix II.)
References
[edit]- ^ See, e.g., Ben-Zvi, Francis & Nadler (2010)
- ^ Lemma 2.6. of Kerz, Strunk & Tamme (2018)
- ^ Lurie (2014)
- 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, S2CID 2202294
Bibliography
[edit]- 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 225). Lecture Notes in Mathematics (in French). Vol. 225. Berlin; New York: Springer-Verlag. xii+700. doi:10.1007/BFb0066283. ISBN 978-3-540-05647-8. MR 0354655.
- Kerz, Moritz; Strunk, Florian; Tamme, Georg (2018). "Algebraic K-theory and descent for blow-ups". Inventiones Mathematicae. 211 (2): 523–577. arXiv:1611.08466. Bibcode:2018InMat.211..523K. doi:10.1007/s00222-017-0752-2.
- Lurie, Jacob (2014). "Algebraic K-Theory and Manifold Topology (Math 281), Lecture 19: K-Theory of Ring Spectra" (PDF).
External links
[edit]- "Determinantal identities for perfect complexes". MathOverflow.
- "An alternative definition of pseudo-coherent complex". MathOverflow.
- "15.74 Perfect complexes". The Stacks project.
- "perfect module". ncatlab.org.