Perfect complex explained
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
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; 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.
, an
-module is called pseudo-coherent if for every integer
, locally, there is a
free presentation of finite type of length
n; i.e.,
Ln\toLn-1\to … \toL0\toF\to0
.
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
Bibliography
- Book: Berthelot . Pierre . Pierre Berthelot (mathematician) . Alexandre Grothendieck . Alexandre Grothendieck . Luc Illusie . Luc Illusie . 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 . 1971 . 225 . . Berlin; New York . fr . xii+700 . true . 10.1007/BFb0066283 . 978-3-540-05647-8 . 0354655.
- 10.1007/s00222-017-0752-2 . Algebraic K-theory and descent for blow-ups . 2018 . Kerz . Moritz . Strunk . Florian . Tamme . Georg . Inventiones Mathematicae . 211 . 2 . 523–577 . 1611.08466 . 2018InMat.211..523K .
- Web site: Lurie . Jacob . Algebraic K-Theory and Manifold Topology (Math 281), Lecture 19: K-Theory of Ring Spectra. . 2014.
External links
Notes and References
- See, e.g.,
- Lemma 2.6. of
