| dbo:abstract |
In mathematics the differential calculus over commutative algebras is a part of commutative algebra based on the observation that most concepts known from classical differential calculus can be formulated in purely algebraic terms. Instances of this are: 1. * The whole topological information of a smooth manifold is encoded in the algebraic properties of its -algebra of smooth functions as in the Banach–Stone theorem. 2. * Vector bundles over correspond to projective finitely generated modules over via the functor which associates to a vector bundle its module of sections. 3. * Vector fields on are naturally identified with derivations of the algebra . 4. * More generally, a linear differential operator of order k, sending sections of a vector bundle to sections of another bundle is seen to be an -linear map between the associated modules, such that for any elements : where the bracket is defined as the commutator Denoting the set of th order linear differential operators from an -module to an -module with we obtain a bi-functor with values in the category of -modules. Other natural concepts of calculus such as jet spaces, differential forms are then obtained as representing objects of the functors and related functors. Seen from this point of view calculus may in fact be understood as the theory of these functors and their representing objects. Replacing the real numbers with any commutative ring, and the algebra with any commutative algebra the above said remains meaningful, hence differential calculus can be developed for arbitrary commutative algebras. Many of these concepts are widely used in algebraic geometry, differential geometry and secondary calculus. Moreover, the theory generalizes naturally to the setting of graded commutative algebra, allowing for a natural foundation of calculus on supermanifolds, graded manifolds and associated concepts like the Berezin integral. (en) 数学における可換環上の微分法（かかんかんじょうのびぶんほう、英: differential calculus over commutative algebras）は、古典的な微分法における既知の概念の大半を純代数学的な言葉で定式化する研究観察に基づく可換代数学の一分野である。 (ja) Дифференциальное исчисление над коммутативными алгебрами — раздел коммутативной алгебры, возникший в семидесятых годах прошлого века. (ru) |
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http://diffiety.ac.ru/preprint/96/01_96abs.htm http://arxiv.org/abs/math/9808130v2 http://diffiety.ac.ru/preprint/96/02_96abs.htm http://diffiety.ac.ru/preprint/96/03_96abs.htm http://diffiety.ac.ru/preprint/96/04_96abs.htm http://diffiety.ac.ru/preprint/96/05_96abs.htm http://diffiety.ac.ru/preprint/96/06_96abs.htm http://diffiety.ac.ru/preprint/96/07_96abs.htm http://diffiety.ac.ru/preprint/96/08_96abs.htm http://diffiety.ac.ru/preprint/99/01_99abs.htm https://arxiv.org/abs/0910.1515 |
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| rdfs:comment |
数学における可換環上の微分法（かかんかんじょうのびぶんほう、英: differential calculus over commutative algebras）は、古典的な微分法における既知の概念の大半を純代数学的な言葉で定式化する研究観察に基づく可換代数学の一分野である。 (ja) Дифференциальное исчисление над коммутативными алгебрами — раздел коммутативной алгебры, возникший в семидесятых годах прошлого века. (ru) In mathematics the differential calculus over commutative algebras is a part of commutative algebra based on the observation that most concepts known from classical differential calculus can be formulated in purely algebraic terms. Instances of this are: where the bracket is defined as the commutator Seen from this point of view calculus may in fact be understood as the theory of these functors and their representing objects. (en) |
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Differential calculus over commutative algebras (en) 可換環上の微分法 (ja) Дифференциальное исчисление над коммутативными алгебрами (ru) |
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