B2 Course. Differential Calculus over Commutative Algebras

Preliminaries

• 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: Diff_k(P).
• Bi-module structure on Diff_k(P).
• Lie Algebra of derivations.
• The universal operator Д. The gluing homomorphism c_k,s.
• The functor Diff_k 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.

References

• 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, DIPS 1/96.
• 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 pp.