In
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, Nagata's compactification theorem, introduced by , implies that every
abstract variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
can be embedded in a
complete variety
In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety , such that for any variety the projection morphism
:X \times Y \to Y
is a closed map (i.e. maps closed sets onto closed sets). This can ...
, and more generally shows that a
separated and
finite type morphism to a
Noetherian scheme
In algebraic geometry, a Noetherian scheme is a scheme that admits a finite covering by open affine subsets \operatorname A_i, where each A_i is a Noetherian ring. More generally, a scheme is locally Noetherian if it is covered by spectra of Noe ...
''S'' can be factored into an
open immersion
This is a glossary of algebraic geometry.
See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometr ...
followed by a
proper morphism In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces.
Some authors call a proper variety over a field k a complete variety. For example, every projective variety over a field k ...
.
Nagata's original proof used the older terminology of
Zariski–Riemann space
In algebraic geometry, a Zariski–Riemann space or Zariski space of a subring ''k'' of a field ''K'' is a locally ringed space whose points are valuation rings containing ''k'' and contained in ''K''. They generalize the Riemann surface of a c ...
s and
valuation theory
In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of the size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of siz ...
, which sometimes made it hard to follow.
Deligne
Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord P ...
showed, in unpublished notes expounded by
Conrad
Conrad may refer to:
People
* Conrad (name)
* Saint Conrad (disambiguation)
Places
United States
* Conrad, Illinois, an unincorporated community
* Conrad, Iowa, a city
* Conrad, Montana, a city
* Conrad Glacier, Washington
Elsewher ...
, that Nagata's proof can be translated into scheme theory and that the condition that ''S'' is Noetherian can be replaced by the much weaker condition that ''S'' is
quasi-compact
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it ...
and quasi-separated. gave another scheme-theoretic proof of Nagata's theorem.
An important application of Nagata's theorem is in defining the analogue in algebraic geometry of
cohomology with compact support
In mathematics, cohomology with compact support refers to certain cohomology theories, usually with some condition requiring that cocycles should have compact support.
Singular cohomology with compact support
Let X be a topological space. Then
:\ ...
, or more generally higher
direct image functors with proper support. The idea is that given a compactifiable morphism
one defines
by choosing a factorization
by an open immersion ''j'' and proper morphism ''p'', and then setting
:
,
where
is the extension by zero functor. One then shows the independence of the definition from the choice of compactification.
In the context of
étale sheaves In mathematics, more specifically in algebra, the adjective étale refers to several closely related concepts:
* Étale morphism
** Formally étale morphism
* Étale cohomology
* Étale topology
* Étale fundamental group
* Étale group scheme
* Ét ...
, this idea was carried out by Deligne in
SGA 4, Exposé XVII. In the context of
coherent sheaves
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
, the statements are more delicate since for an open immersion ''j'', the
inverse image
In mathematics, for a function f: X \to Y, the image of an input value x is the single output value produced by f when passed x. The preimage of an output value y is the set of input values that produce y.
More generally, evaluating f at each ...
functor
does not usually admit a left adjoint. Nonetheless,
exists as a
pro-left adjoint, and Deligne was able to define the functor
as valued in the pro-derived category of coherent sheaves.
[ cf. Appendix by P. Deligne.]
References
*
Stacks Project
The Stacks Project is an open source collaborative mathematics textbook writing project with the aim to cover "algebraic stacks and the algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, ...
Nagata compactification- See Lemma 38.33.8 first, then backtrack
*Stacks Project
Derived lower shriek via compactifications*Stacks Project
Compactly supported cohomology for coherent modules*
*
*
*{{Citation , last1=Nagata , first1=Masayoshi , author1-link=Masayoshi Nagata , title=A generalization of the imbedding problem of an abstract variety in a complete variety , url=http://projecteuclid.org/euclid.kjm/1250524859 , mr=0158892 , year=1963 , journal=Journal of Mathematics of Kyoto University , issn=0023-608X , volume=3 , issue=1 , pages=89–102, doi=10.1215/kjm/1250524859 , doi-access=free
Theorems in algebraic geometry