In algebra, a perfect complex of
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
over a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
''A'' is an object in the derived category of ''A''-modules that is quasi-isomorphic to a
bounded complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of t ...
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 In mathematics, the adjective Noetherian is used to describe Category_theory#Categories.2C_objects.2C_and_morphisms, objects that satisfy an ascending chain condition, ascending or descending chain condition on certain kinds of subobjects, meaning t ...
, a module over ''A'' is perfect if and only if it is finitely generated and of finite
projective dimension
In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characterizati ...
.
Other characterizations
Perfect complexes are precisely the
compact object
In astronomy, the term compact star (or compact object) refers collectively to white dwarfs, neutron stars, and black holes. It would grow to include exotic stars if such hypothetical, dense bodies are confirmed to exist. All compact objects ha ...
s in the unbounded derived category
of ''A''-modules. They are also precisely the
dualizable object In category theory, a branch of mathematics, a dual object is an analogue of a dual vector space from linear algebra for objects in arbitrary monoidal categories. It is only a partial generalization, based upon the categorical properties of duali ...
s in this category.
A compact object in the ∞-category of (say right)
module spectra over a
ring spectrum
In stable homotopy theory, a ring spectrum is a spectrum ''E'' together with a multiplication map
:''μ'': ''E'' ∧ ''E'' → ''E''
and a unit map
: ''η'': ''S'' → ''E'',
where ''S'' is the sphere spectrum. These maps have to satisfy a ...
is often called perfect;
[http://www.math.harvard.edu/~lurie/281notes/Lecture19-Rings.pdf ] see also
module spectrum In algebra, a module spectrum is a spectrum (topology), spectrum with an action of a ring spectrum; it generalizes a module (mathematics), module in abstract algebra.
The ∞-category of (say right) module spectra is stable ∞-category, stable; he ...
.
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.
By definition, given a
ringed space
In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf of ...
, an
-module is called pseudo-coherent if for every integer
, locally, there is a
free presentation
In algebra, a free presentation of a module ''M'' over a commutative ring ''R'' is an exact sequence of ''R''-modules:
:\bigoplus_ R \ \overset \to\ \bigoplus_ R \ \overset\to\ M \to 0.
Note the image under ''g'' of the standard basis generate ...
of finite type of length ''n''; i.e.,
:
.
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
*
Hilbert–Burch theorem
In mathematics, the Hilbert–Burch theorem describes the structure of some free resolutions of a quotient of a local or graded ring in the case that the quotient has projective dimension 2. proved a version of this theorem for polynomial ...
*
elliptic complex In mathematics, in particular in partial differential equations and differential geometry, an elliptic complex generalizes the notion of an elliptic operator to sequences. Elliptic complexes isolate those features common to the de Rham cohomology, ...
(related notion; discussed at SGA 6 Exposé II, Appendix II.)
References
*
*
Further reading
* https://mathoverflow.net/questions/354214/determinantal-identities-for-perfect-complexes
An alternative definition of pseudo-coherent complex
External links
*http://stacks.math.columbia.edu/tag/0656
*http://ncatlab.org/nlab/show/perfect+module
Abstract algebra
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