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 has finite projective dimension.

Compact object

A basic example is a perfect complex in the derived category of quasi-coherent sheaves.

A triangulated category is said to be compactly generated if it is generated by compact objects.^[1]

A compact object in the ∞-category of (say right) module spectra over a ring spectrum is often called perfect; ^[2] see also module spectrum.

Pseudo-coherent sheaf

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

L_{n}\to L_{n-1}\to \cdots \to L_{0}\to F\to 0

.

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

References

Ben-Zvi, D., Francis, J., and D. Nadler. Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry.
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) (in French). Berlin; New York: Springer-Verlag. xii+700. doi:10.1007/BFb0066283. ISBN 978-3-540-05647-8. MR 0354655.

External links

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[1] ttps://www.preschema.com/teaching/ktheory-ws17/lect3.pdf

[2] ttp://www.math.harvard.edu/~lurie/281notes/Lecture19-Rings.pdf

[3] Lemma 2.6. of https://arxiv.org/pdf/1611.08466.pdf

Perfect complex

Compact object

Pseudo-coherent sheaf

See also

References

External links