B2 Course. Differential Calculus over Commutative Algebras
- Generalities on unitary commutative algebras.
- Modules over an algebra, basic constructions.
- Categories and functors.
Differential Calculus over Commutative Algebras
- Differential operators of order k with values in a module P: Diffk(P).
- Bi-module structure on Diffk(P).
- Lie Algebra of derivations.
- The universal operator Д. The gluing homomorphism ck,s.
- The functor Diffk and its rapresentative object, module of l-jets.
- The module of infinite jets.
- Basic constructions with jets.
- The functor D and its representative object: the module of 1-forms.
- The algebra of differential forms.
- Basic operation with differential forms.
- Spencer and de Rham cohomologies.
- Geometric interpretation of jets and forms, the Cartan distribution.
- M. F. Atiyah, I. G. MacDonald, Introduction to Commutative Algebra, Westview Press (1969).
- I. S. Krasil’shchik (in collaboration with B. Prinari), Lectures on Linear Differential Operators over Commutative Algebras (The 1st Italian Diffiety School, July, 1998),
The Diffiety Institute Preprint Series, DIPS 1/99.
- I. S. Krasil’shchik,
Calculus over commutative algebras: a concise user guide,
Acta Appl. Math., Volume 49, Issue 3, December 1997, pp. 235-248.
See also The Diffiety Institute Preprints,
- Jet Nestruev,
Smooth manifolds and Observables,
Springer-Verlag, Graduate Texts in Mathematics, Vol. 220 (2002).
- I. S. Krasil’shchik, V. V. Lychagin,
A. M. Vinogradov,
Geometry of Jet Spaces
and Nonlinear Differential Equations, Advanced Studies in Contemporary
Mathematics, 1 (1986), Gordon and Breach, New York, London. xx+441