**DIPS 1/99**[tex source, PostScript, PDF file, dvi]-
**Title:**LECTURES ON LINEAR DIFFERENTIAL OPERATORS

OVER COMMUTATIVE ALGEBRAS

(The 1st Italian Diffiety School - July, 1998)**Author:**Joseph KRASIL'SHCHIK

in collaboration with BARBARA PRINARILectures given in July 1998 at the 1st Italian Diffiety School in Forino, Italy.

These Lecture Notes contain introductory material to algebraic theory of linear differential operators over commutative algebras. The basics of this theory were exposed in a short paper by A. M. Vinogradov in 1972.

The initial course, held in Forino (Summer 1998) at the

*First Italian Diffiety School*, consisted of ten lectures and a number of practical lessons, but in preparation of the printed version it became clear that a more logical structure should contain five parts (which are also called*lectures*below):- general introduction to
*commutative algebra*(rings, algebras, and modules) together with main concepts of the*category theory*- first two lectures, - main functors of
*differential calculus*over commutative algebras - lecture 3, - their
*representative objects*(jets and differential forms) - lecture 4, - relations to geometry serving as a bridge to the next course in
*geometry of differential equations*- the last lecture.

We conclude these notes with a series of exercises which are an essential part of the main text.

51 pages, LaTeX-2e (AmSLaTeX 1.2).

To compile source files (**.tex**) one needs the title page style files titlatex.tex.

- general introduction to