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>D(A)</math> of ''A''-modules.<ref>See, e.g., {{harvtxt|Ben-Zvi|Francis|Nadler|2010}}</ref> They are also precisely the dualizable objects in this category.<ref>Lemma 2.6. of {{harvtxt|Kerz|Strunk|Tamme|2018}}</ref>

A compact object in the ∞-category of (say right) module spectra over a ring spectrum is often called perfect;<ref>{{harvtxt|Lurie|2014}}</ref> see also module spectrum.

== Pseudo-coherent sheaf == When the structure sheaf <math>\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>(X, \mathcal{O}_X)</math>, an <math>\mathcal{O}_X</math>-module is called pseudo-coherent if for every integer <math>n \ge 0</math>, locally, there is a free presentation of finite type of length ''n''; i.e., :<math>L_n \to L_{n-1} \to \cdots \to L_0 \to F \to 0</math>.

A complex ''F'' of <math>\mathcal{O}_X</math>-modules is called pseudo-coherent if, for every integer ''n'', there is locally a quasi-isomorphism <math>L \to F</math> where ''L'' has degree bounded above and consists of finite free modules in degree <math>\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 == {{Reflist}} * {{Citation|last1=Ben-Zvi|first1=David|last2=Francis|first2=John|last3=Nadler|first3=David|title=Integral transforms and Drinfeld centers in derived algebraic geometry|journal=Journal of the American Mathematical Society| volume=23|year=2010|issue=4|pages=909&ndash;966|mr=2669705|doi=10.1090/S0894-0347-10-00669-7|arxiv=0805.0157|s2cid=2202294}} ==Bibliography== *{{cite book | editor-last = Berthelot | editor-first = Pierre | editor-link = Pierre Berthelot (mathematician) | editor2=Alexandre Grothendieck | editor2-link=Alexandre Grothendieck | editor3=Luc Illusie | editor3-link=Luc Illusie | title = Séminaire de Géométrie Algébrique du Bois Marie - 1966-67 - Théorie des intersections et théorème de Riemann-Roch - (SGA 6) (Lecture notes in mathematics '''225''') | series = Lecture Notes in Mathematics | year = 1971 | volume = 225 | publisher = Springer-Verlag | location = Berlin; New York | language = fr | pages = xii+700 | no-pp = true |doi=10.1007/BFb0066283 |isbn= 978-3-540-05647-8 | mr = 0354655 }} *{{cite journal |doi=10.1007/s00222-017-0752-2 |title=Algebraic K-theory and descent for blow-ups |date=2018 |last1=Kerz |first1=Moritz |last2=Strunk |first2=Florian |last3=Tamme |first3=Georg |journal=Inventiones Mathematicae |volume=211 |issue=2 |pages=523–577 |arxiv=1611.08466 |bibcode=2018InMat.211..523K }} *{{cite web |last1=Lurie |first1=Jacob |author-link=Jacob Lurie |title=Algebraic K-Theory and Manifold Topology (Math 281), Lecture 19: K-Theory of Ring Spectra. |url=https://www.math.ias.edu/~lurie/281notes/Lecture19-Rings.pdf |date=2014}}

== External links == *{{cite web |title=Determinantal identities for perfect complexes |url=https://mathoverflow.net/questions/354214/determinantal-identities-for-perfect-complexes |website=MathOverflow}} *{{cite web |title=An alternative definition of pseudo-coherent complex |url=https://mathoverflow.net/questions/200540/an-alternative-definition-of-pseudo-coherent-complex |website=MathOverflow}} *{{Cite web |title=15.74 Perfect complexes |url=http://stacks.math.columbia.edu/tag/0656 |website=The Stacks project}} *{{Cite web |title=perfect module |url=https://ncatlab.org/nlab/show/perfect+module |website=ncatlab.org}} Category:Abstract algebra

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