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- Noncommutative_algebraic_geometry 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, taking noncommutative stack quotients etc.). 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 multiply by points; 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 motivations 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 why not looking at their geometric aspects.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 analogues 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.".
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- Noncommutative_algebraic_geometry wikiPageWikiLink Affine_space.
- Noncommutative_algebraic_geometry wikiPageWikiLink Alexander_Grothendieck.
- Noncommutative_algebraic_geometry wikiPageWikiLink Ample_line_bundle.
- Noncommutative_algebraic_geometry wikiPageWikiLink Brauer_group.
- Noncommutative_algebraic_geometry wikiPageWikiLink Category:Algebraic_geometry.
- Noncommutative_algebraic_geometry wikiPageWikiLink Complex_number.
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- Noncommutative_algebraic_geometry wikiPageWikiLink Cotangent_bundle.
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- Noncommutative_algebraic_geometry wikiPageWikiLink Descent_(mathematics).
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- Noncommutative_algebraic_geometry wikiPageWikiLink Dixmier_conjecture.
- Noncommutative_algebraic_geometry wikiPageWikiLink Dmitri_Kaledin.
- Noncommutative_algebraic_geometry wikiPageWikiLink Fred_Van_Oystaeyen.
- Noncommutative_algebraic_geometry wikiPageWikiLink Gabriel–Rosenberg_reconstruction_theorem.
- Noncommutative_algebraic_geometry wikiPageWikiLink Homogeneous_coordinate_ring.
- Noncommutative_algebraic_geometry wikiPageWikiLink Homological_algebra.
- Noncommutative_algebraic_geometry wikiPageWikiLink Jacobian_conjecture.
- Noncommutative_algebraic_geometry wikiPageWikiLink Jean-Pierre_Serre.
- Noncommutative_algebraic_geometry wikiPageWikiLink K-theory.
- Noncommutative_algebraic_geometry wikiPageWikiLink Local_ring.
- Noncommutative_algebraic_geometry wikiPageWikiLink Ludmil_Katzarkov.
- Noncommutative_algebraic_geometry wikiPageWikiLink Mathematical_Sciences_Research_Institute.
- Noncommutative_algebraic_geometry wikiPageWikiLink Mathematics.
- Noncommutative_algebraic_geometry wikiPageWikiLink Maxim_Kontsevich.
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- Noncommutative_algebraic_geometry wikiPageWikiLink Ring_(mathematics).
- Noncommutative_algebraic_geometry wikiPageWikiLink Scheme_(mathematics).
- Noncommutative_algebraic_geometry wikiPageWikiLink Simple_ring.
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- Noncommutative_algebraic_geometry wikiPageWikiLink Symbol_of_a_differential_operator.
- Noncommutative_algebraic_geometry wikiPageWikiLink Topos.
- Noncommutative_algebraic_geometry wikiPageWikiLink Very_ample_line_bundle.
- Noncommutative_algebraic_geometry wikiPageWikiLink Weyl_algebra.
- Noncommutative_algebraic_geometry wikiPageWikiLink Yuri_I._Manin.
- Noncommutative_algebraic_geometry wikiPageWikiLink Yuri_Manin.
- Noncommutative_algebraic_geometry wikiPageWikiLinkText "Derived noncommutative algebraic geometry".
- Noncommutative_algebraic_geometry wikiPageWikiLinkText "Equivariant noncommutative algebraic geometry".
- Noncommutative_algebraic_geometry wikiPageWikiLinkText "Noncommutative algebraic geometry".
- Noncommutative_algebraic_geometry wikiPageWikiLinkText "noncommutative algebraic geometry".
- Noncommutative_algebraic_geometry hasPhotoCollection Noncommutative_algebraic_geometry.
- Noncommutative_algebraic_geometry id "Kapranov%27s%20noncommutative%20geometry".
- Noncommutative_algebraic_geometry id "equivariant+noncommutative+algebraic+geometry".
- Noncommutative_algebraic_geometry id "noncommutative+algebraic+geometry".
- Noncommutative_algebraic_geometry id "noncommutative+scheme".
- Noncommutative_algebraic_geometry title "Kapranov's noncommutative geometry".
- Noncommutative_algebraic_geometry title "equivariant noncommutative algebraic geometry".
- Noncommutative_algebraic_geometry title "noncommutative algebraic geometry".
- Noncommutative_algebraic_geometry title "noncommutative scheme".
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- Noncommutative_algebraic_geometry subject Category:Algebraic_geometry.
- Noncommutative_algebraic_geometry hypernym Branch.
- Noncommutative_algebraic_geometry type Article.
- Noncommutative_algebraic_geometry type Organisation.
- Noncommutative_algebraic_geometry type Article.
- Noncommutative_algebraic_geometry 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, taking noncommutative stack quotients etc.).".
- Noncommutative_algebraic_geometry label "Noncommutative algebraic geometry".
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