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

Perfect complexes are precisely the compact objects in the unbounded derived category [math]\displaystyle{ D(A) }[/math] of A-modules. They are also precisely the dualizable objects in this category.

A compact object in the ∞-category of (say right) module spectra over a ring spectrum is often called perfect; see also module spectrum.

Pseudo-coherent sheaf

When the structure sheaf [math]\displaystyle{ \mathcal{O}_X }[/math] 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 [math]\displaystyle{ (X, \mathcal{O}_X) }[/math], an [math]\displaystyle{ \mathcal{O} X }[/math]-module is called pseudo-coherent if for every integer [math]\displaystyle{ n \ge 0 }[/math], locally, there is a free presentation of finite type of length n; i.e.,

[math]\displaystyle{ L_n \to L_{n-1} \to \cdots \to L_0 \to F \to 0 }[/math].

A complex F of [math]\displaystyle{ \mathcal{O}_X }[/math]-modules is called pseudo-coherent if, for every integer n, there is locally a quasi-isomorphism [math]\displaystyle{ L \to F }[/math] where L has degree bounded above and consists of finite free modules in degree [math]\displaystyle{ \ge n }[/math]. 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, doi:10.1090/S0894-0347-10-00669-7 

Bibliography

• Berthelot, Pierre, ed (1971) (in fr). 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. 225. Berlin; New York: Springer-Verlag. pp. xii+700. doi:10.1007/BFb0066283. ISBN 978-3-540-05647-8. 
• Kerz, Moritz; Strunk, Florian; Tamme, Georg (2018). "Algebraic K-theory and descent for blow-ups". Inventiones Mathematicae 211 (2): 523–577. doi:10.1007/s00222-017-0752-2. Bibcode: 2018InMat.211..523K. 
• Lurie, Jacob (2014). "Algebraic K-Theory and Manifold Topology (Math 281), Lecture 19: K-Theory of Ring Spectra.". https://www.math.ias.edu/~lurie/281notes/Lecture19-Rings.pdf. 

External links

• "Determinantal identities for perfect complexes". https://mathoverflow.net/questions/354214/determinantal-identities-for-perfect-complexes. 
• "An alternative definition of pseudo-coherent complex". https://mathoverflow.net/questions/200540/an-alternative-definition-of-pseudo-coherent-complex. 
• "15.74 Perfect complexes". http://stacks.math.columbia.edu/tag/0656. 
• "perfect module". https://ncatlab.org/nlab/show/perfect+module.