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:
Vector bundles over correspond to projective finitely generated
modules over via the
functor which associates to a vector bundle its module of sections.
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.
I. S. Krasil'shchik, A. M. Verbovetsky, "Homological Methods in Equations of Mathematical Physics", Open Ed. and Sciences, Opava (Czech Rep.), 1998; Eprint
arXiv:math/9808130v2.
G. Sardanashvily, Lectures on Differential Geometry of Modules and Rings, Lambert Academic Publishing, 2012; Eprint
arXiv:0910.1515 [math-ph] 137 pages.
A. M. Vinogradov, "The Logic Algebra for the Theory of Linear Differential Operators", Dokl. Akad. Nauk SSSR, 295(5) (1972) 1025-1028; English transl. in Soviet Math. Dokl.13(4) (1972), 1058-1062.
Vinogradov, A. M. (2001). Cohomological Analysis of Partial Differential Equations and Secondary Calculus. American Mathematical Soc.
ISBN9780821897997.
A. M. Vinogradov, "Some new homological systems associated with differential calculus over commutative algebras" (Russian), Uspechi Mat.Nauk, 1979, 34 (6), 145-150;English transl. in Russian Math. Surveys, 34(6) (1979), 250-255.