About: Nagata's compactification theorem

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In algebraic geometry, Nagata's compactification theorem, introduced by Nagata , implies that every abstract variety can be embedded in a complete variety, and more generally shows that a separated and finite type morphism to a Noetherian scheme S can be factored into an open immersion followed by a proper morphism. , where is the extension by zero functor. One then shows the independence of the definition from the choice of compactification.

Attributes	Values
rdfs:label	Nagata's compactification theorem (en)
rdfs:comment	In algebraic geometry, Nagata's compactification theorem, introduced by Nagata , implies that every abstract variety can be embedded in a complete variety, and more generally shows that a separated and finite type morphism to a Noetherian scheme S can be factored into an open immersion followed by a proper morphism. , where is the extension by zero functor. One then shows the independence of the definition from the choice of compactification. (en)
dct:subject	Theorems in algebraic geometry
Wikipage page ID	37613671 (xsd:integer)
Wikipage revision ID	1112351355 (xsd:integer)
Link from a Wikipage to another Wikipage	Brian Conrad Valuation (algebra) Inverse image functor Compact space Complete variety Stacks Project Coherent sheaves Algebraic geometry Abstract variety Direct image with compact support Proper morphism Zariski–Riemann space Theorems in algebraic geometry Cohomology with compact support Pierre Deligne Glossary of scheme theory Finite morphism Noetherian scheme SGA 4 Étale sheaf dbr:Nlab:pro-left+adjoint
Link from a Wikipage to an external page	https://stacks.math.columbia.edu/tag/0F3T https://stacks.math.columbia.edu/tag/0F7H https://stacks.math.columbia.edu/tag/0G4Z http://projecteuclid.org/euclid.kjm/1250524859 http://projecteuclid.org/euclid.kjm/1250524969 http://math.stanford.edu/~conrad/papers/nagatafinal.pdf%7Cfirst=B%7Clast=Conrad%7Ctitle=Deligne's
sameAs	Nagata's compactification theorem Nagata's compactification theorem Nagata's compactification theorem Nagata's compactification theorem Nagata's compactification theorem
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Harvtxt dbt:Reflist dbt:Harvs
authorlink	Masayoshi Nagata (en)
last	Nagata (en)
year	1962 (xsd:integer) 1963 (xsd:integer)
has abstract	In algebraic geometry, Nagata's compactification theorem, introduced by Nagata , implies that every abstract variety can be embedded in a complete variety, and more generally shows that a separated and finite type morphism to a Noetherian scheme S can be factored into an open immersion followed by a proper morphism. Nagata's original proof used the older terminology of Zariski–Riemann spaces and valuation theory, which sometimes made it hard to follow.Deligne showed, in unpublished notes expounded by Conrad, 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 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, 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, this idea was carried out by Deligne in SGA 4, Exposé XVII. In the context of coherent sheaves, the statements are more delicate since for an open immersion j, the inverse image functor does not usually admit a left adjoint. Nonetheless, exists as a , and Deligne was able to define the functor as valued in the pro-derived category of coherent sheaves. (en)
prov:wasDerivedFrom	wikipedia-en:Nagata's_compactification_theorem?oldid=1112351355&ns=0
page length (characters) of wiki page	4054 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Nagata's_compactification_theorem
is Link from a Wikipage to another Wikipage of	Masayoshi Nagata Glossary of algebraic geometry Smooth completion Proper morphism List of theorems Noetherian scheme Nagata compactification
is Wikipage redirect of	Nagata compactification
is known for of	Masayoshi Nagata
is known for of	Masayoshi Nagata
is foaf:primaryTopic of	wikipedia-en:Nagata's_compactification_theorem

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