Basic functors of differential calculus over commutative algebras
Alexandre Vinogradov, Michael Vinogradov
A program of the course at the 4-th Italian Diffiety School,
(Forino, July 17-29, 2000)

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  1. The Diff-prolongations of differntial operators. The functor transformations
    cs,l:    Diff+s Diff+l Diff+s+l.
  2. The functor equivalence  Diff1 @ D id. The functor transformations
    k1:    Diff(+)1 D.

  3. The kernel of the universal differential operator
     +2:    Diff+2P P,
    and functors D2 and P(+)2. The Spencer Diff-sequence of the second order.
  4. The construction and properties of K-modules D(P Q) and  Diff(+)1(P Q).
  5. Functors Di and P(+)i. The functor equivalence Pi @ DiDi-1. The functor transformations
    ki:   P(+)i Di.
  6. The Spencer Diff-complexes.
  7. The construction and properties of A-modules Pin Q and  HomA(P,Q).
  8. A-modules Li's as representative objects of functors Di.
  9. Algebraic de Rham complexes. The exterior product, the Lie derivative and their properties.
  10. The Spencer J-complexes of an algebra A as as representative objects of the Spencer Diff-complexes.
  11. The Spencer J-complexes of A-modules and their properties.

Examination problems

1. Let F1F2 be functors on the category of all A-modules, j1j2 be corresponding representative objects, that is
F1(P) = HomA(j1,P),    F2(P) = HomA(j2,P)
for any A-module P, and F :  F1F2 be a natural transformation. Set P = j2, and define the homomorphism F* :  j2j1 by the formula
F* = F(id),
id HomA(j1,j1) = F1(j1),   F(id) HomA(j2,j1) = F2(j1).

Prove that for any A-module Q and any element h F1(Q) = HomA(j1,Q)
F(h) = hF*

2. Prove that HomA(P+inS, Q) = HomA(S,Hom+A(P,Q)).

3. Prove that Diff+sP = Hom+A(Js,P).

4. Prove that
Js+(P) = Js+P,    Js(P) = Jsin P.

5. Using the algebraic definition of Lie derivation deduce all usual formulas for it.

To pass the exam by email one should solve 5 problems.

The exam has been passed by the following students:
  1. Giovanni Manno
  2. Barbara Prinari

Questions and suggestions should go to Jet NESTRUEV, jet @