Perfect complex
In algebra, a perfect complex of
Other characterizations
Perfect complexes are precisely the compact objects in the unbounded derived category of A-modules.
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
- ^ 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, S2CID 2202294
Bibliography
- MR 0354655.
- Kerz, Moritz; Strunk, Florian; Tamme, Georg (2018). "Algebraic K-theory and descent for blow-ups". Inventiones Mathematicae. 211 (2): 523–577. .
- Lurie, Jacob (2014). "Algebraic K-Theory and Manifold Topology (Math 281), Lecture 19: K-Theory of Ring Spectra" (PDF).
External links
- "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.