| dbo:abstract |
Noncommutative algebraic geometry is a branch of mathematics, and more specifically a direction in noncommutative geometry, that studies the geometric properties of formal duals of non-commutative algebraic objects such as rings as well as geometric objects derived from them (e.g. by gluing along localizations or taking noncommutative stack quotients). For example, noncommutative algebraic geometry is supposed to extend a notion of an algebraic scheme by suitable gluing of spectra of noncommutative rings; depending on how literally and how generally this aim (and a notion of spectrum) is understood in noncommutative setting, this has been achieved in various level of success. The noncommutative ring generalizes here a commutative ring of regular functions on a commutative scheme. Functions on usual spaces in the traditional (commutative) algebraic geometry have a product defined by pointwise multiplication; as the values of these functions commute, the functions also commute: a times b equals b times a. It is remarkable that viewing noncommutative associative algebras as algebras of functions on "noncommutative" would-be space is a far-reaching geometric intuition, though it formally looks like a fallacy. Much of the motivation for noncommutative geometry, and in particular for the noncommutative algebraic geometry, is from physics; especially from quantum physics, where the algebras of observables are indeed viewed as noncommutative analogues of functions, hence having the ability to observe their geometric aspects is desirable. One of the values of the field is that it also provides new techniques to study objects in commutative algebraic geometry such as Brauer groups. The methods of noncommutative algebraic geometry are analogs of the methods of commutative algebraic geometry, but frequently the foundations are different. Local behavior in commutative algebraic geometry is captured by commutative algebra and especially the study of local rings. These do not have a ring-theoretic analogue in the noncommutative setting; though in a categorical setup one can talk about stacks of local categories of quasicoherent sheaves over noncommutative spectra. Global properties such as those arising from homological algebra and K-theory more frequently carry over to the noncommutative setting. (en) |
| dbo:wikiPageExternalLink |
http://www.math.northwestern.edu/~jnkf/writ/thezrev.pdf https://arxiv.org/abs/alg-geom/9506012 http://www.mpim-bonn.mpg.de/preprints/send%3Fbid=1947 http://www.msri.org/publications/ln/msri/2000/interact/rosenberg/1/index.html http://www.mpim-bonn.mpg.de/preprints/send%3Fbid=1948 https://arxiv.org/abs/math/9812158 https://dx.doi.org/10.1023/A:1000479824211 http://projecteuclid.org/euclid.rmi/1063050166 http://www.mpim-bonn.mpg.de/preprints/send%3Fbid=56 http://www.mpim-bonn.mpg.de/preprints/send%3Fbid=57 http://imperium.lenin.ru/~kaledin/seoul http://imperium.lenin.ru/~kaledin/tokyo/final.pdf http://imperium.lenin.ru/~kaledin/tokyo/final.tex https://arxiv.org/abs/0806.0107 https://arxiv.org/abs/0811.4770 https://arxiv.org/abs/math/0501166 https://arxiv.org/abs/math/0611806 https://arxiv.org/abs/math/9802041 https://dx.doi.org/10.1006/aima.1994.1087 https://dx.doi.org/10.1023/A:1002470302976 http://www.numdam.org/item%3Fid=BSMF_1962__90__323_0 https://mathoverflow.net/q/10512 |
| dbo:wikiPageID |
25493347 (xsd:integer) |
| dbo:wikiPageLength |
13721 (xsd:nonNegativeInteger) |
| dbo:wikiPageRevisionID |
1055869174 (xsd:integer) |
| dbo:wikiPageWikiLink |
dbr:Quasicoherent_sheaf dbr:En-ring dbr:Representation_theory dbr:Primitive_ideal dbr:Non-commutative_localization dbr:Non-commutative_projective_geometry dbr:Topos_theory dbr:Deformation_theory dbr:Derived_algebraic_geometry dbr:Derived_noncommutative_algebraic_geometry dbr:Jacobian_conjecture dbr:Commutative_property dbr:Complex_number dbr:Mathematical_Sciences_Research_Institute dbr:Mathematics dbr:Maxim_Kontsevich dbr:Alexander_L._Rosenberg dbr:Graded_commutative_ring dbr:Operad dbr:Commutative_algebra dbr:Compositio_Mathematica dbr:Yuri_Manin dbr:Spectrum_of_a_ring dbr:Stack_(mathematics) dbr:Weyl_algebra dbr:Dixmier_conjecture dbr:Gabriel–Rosenberg_reconstruction_theorem dbr:K-theory dbr:Local_ring dbr:Affine_space dbr:Alexander_Grothendieck dbr:Algebra dbr:Algebraic_geometry dbc:Noncommutative_geometry dbr:Brauer_group dbr:Non-commutative dbr:Ring_(mathematics) dbr:Jacob_Lurie dbr:Jean-Pierre_Serre dbr:Cotangent_bundle dbr:Abelian_category dbc:Algebraic_geometry dbr:Advances_in_Mathematics dbr:Homogeneous_coordinate_ring dbr:Homological_algebra dbr:Pierre_Gabriel dbr:Fred_Van_Oystaeyen dbr:Michael_Artin dbr:Observables dbr:Scheme_(mathematics) dbr:Symbol_of_a_differential_operator dbr:Simple_ring dbr:Irreducible_representations dbr:Noncommutative_geometry dbr:Proj_construction dbr:Ring_of_regular_functions dbr:Dixmier dbr:Pointwise_multiplication dbr:Journal_für_die_reine_und_angewandte_Mathematik dbr:Descent_theory dbr:Primitive_spectrum dbr:Projective_algebraic_variety dbr:Very_ample_line_bundle dbr:Stack_quotient dbr:Sheaf_theory dbr:Dmitri_Kaledin dbr:Ludmil_Katzarkov |
| dbp:id |
Kapranov%27s%20noncommutative%20geometry (en) equivariant+noncommutative+algebraic+geometry (en) noncommutative+algebraic+geometry (en) noncommutative+scheme (en) |
| dbp:title |
Kapranov's noncommutative geometry (en) equivariant noncommutative algebraic geometry (en) noncommutative algebraic geometry (en) noncommutative scheme (en) |
| dbp:wikiPageUsesTemplate |
dbt:= dbt:Citation_needed dbt:Harv dbt:Main dbt:Nlab dbt:No_footnotes dbt:Reflist dbt:Ring_theory_sidebar dbt:Rquote dbt:Isbn |
| dct:subject |
dbc:Noncommutative_geometry dbc:Algebraic_geometry |
| gold:hypernym |
dbr:Branch |
| rdf:type |
dbo:Organisation |
| rdfs:comment |
Noncommutative algebraic geometry is a branch of mathematics, and more specifically a direction in noncommutative geometry, that studies the geometric properties of formal duals of non-commutative algebraic objects such as rings as well as geometric objects derived from them (e.g. by gluing along localizations or taking noncommutative stack quotients). One of the values of the field is that it also provides new techniques to study objects in commutative algebraic geometry such as Brauer groups. (en) |
| rdfs:label |
Noncommutative algebraic geometry (en) |
| owl:sameAs |
freebase:Noncommutative algebraic geometry wikidata:Noncommutative algebraic geometry https://global.dbpedia.org/id/4t451 |
| prov:wasDerivedFrom |
wikipedia-en:Noncommutative_algebraic_geometry?oldid=1055869174&ns=0 |
| foaf:isPrimaryTopicOf |
wikipedia-en:Noncommutative_algebraic_geometry |
| is dbo:academicDiscipline of |
dbr:Michael_Artin |
| is dbo:knownFor of |
dbr:Alexander_L._Rosenberg__Alexander_L._Rosenberg__1 |
| is dbo:wikiPageRedirects of |
dbr:Noncommutative_scheme |
| is dbo:wikiPageWikiLink of |
dbr:Noncommutative_scheme dbr:Bertrand_Toën dbr:Derived_algebraic_geometry dbr:Derived_noncommutative_algebraic_geometry dbr:List_of_geometry_topics dbr:Timeline_of_category_theory_and_related_mathematics dbr:Generic_matrix_ring dbr:Geometric_group_theory dbr:Alexander_L._Rosenberg dbr:Glossary_of_algebraic_geometry dbr:Glossary_of_areas_of_mathematics dbr:Calabi–Yau_algebra dbr:Gabriel–Rosenberg_reconstruction_theorem dbr:Algebraic_geometry dbr:Noncommutative_ring dbr:Chelsea_Walton dbr:Kameshwar_C._Wali dbr:Associative_algebra dbr:Fred_Van_Oystaeyen dbr:Grothendieck_category dbr:Michael_Artin dbr:Noncommutative_geometry dbr:Noncommutative_projective_geometry dbr:Noncommutative_standard_model |
| is dbp:fields of |
dbr:Michael_Artin |
| is dbp:knownFor of |
dbr:Alexander_L._Rosenberg |
| is foaf:primaryTopic of |
wikipedia-en:Noncommutative_algebraic_geometry |