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• Q7049219 subject Q8234760.
• Q7049219 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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• Q7049219 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.).".
• Q7049219 label "Noncommutative algebraic geometry".