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 ${\displaystyle D(A)}$ 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 ${\displaystyle {\mathcal {O}}_{X}}$ 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 ${\displaystyle (X,{\mathcal {O}}_{X})}$, an ${\displaystyle {\mathcal {O}}_{X}}$-module is called pseudo-coherent if for every integer ${\displaystyle n\geq 0}$, locally, there is a free presentation of finite type of length n; i.e.,

${\displaystyle L_{n}\to L_{n-1}\to \cdots \to L_{0}\to F\to 0}$.

A complex F of ${\displaystyle {\mathcal {O}}_{X}}$-modules is called pseudo-coherent if, for every integer n, there is locally a quasi-isomorphism ${\displaystyle L\to F}$ where L has degree bounded above and consists of finite free modules in degree ${\displaystyle \geq n}$. 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

1. See, e.g., Ben-Zvi, Francis & Nadler (2010)
2. Lemma 2.6. of Kerz, Strunk & Tamme (2018)
3. 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

• 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

• "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.