### 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.