Basic functors of differential calculus over
Alexandre Vinogradov, Michael Vinogradov
A program of the course at the 4-th Italian Diffiety School,
(Forino, July 17-29, 2000)
If you have difficulties with HTML,
kindly use TeX, DVI,
PS or PDF-files
- The Diff-prolongations of differntial operators. The
cs,l: Diff+s Diff+l® Diff+s+l.|
- The functor equivalence Diff1 @ DÅ id. The
- The kernel of the universal differential operator
and functors D2 and P(+)2. The Spencer Diff-sequence of the
- The construction and properties of K-modules D(P Ì Q)
and Diff(+)1(P Ì Q).
- Functors Di and P(+)i.
The functor equivalence Pi @ DiÅDi-1. The
- The Spencer Diff-complexes.
- The construction and properties of A-modules
- A-modules Li's as representative objects of
- Algebraic de Rham complexes. The exterior product,
the Lie derivative and their properties.
- The Spencer J-complexes of an algebra A as
as representative objects of the Spencer Diff-complexes.
- The Spencer J-complexes of A-modules and
1. Let F1, F2 be functors on the category of all A-modules,
j1, j2 be corresponding representative objects, that is
for any A-module P, and F : F1®F2 be a natural
transformation. Set P = j2, and define the homomorphism
F* : j2®j1 by the formula
F1(P) = HomA(j1,P), F2(P) = HomA(j2,P)|
id Î HomA(j1,j1) = F1(j1), F(id) Î HomA(j2,j1) = F2(j1).
Prove that for any A-module Q and
any element h Î F1(Q) = HomA(j1,Q)
2. Prove that HomA(P+ÄinS, Q) = HomA·(S,Hom+A(P,Q)).
3. Prove that Diff+sP = Hom+A(Js,P).
4. Prove that
Js+(P) = Js+ÄP, Js(P) = JsÄin 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:
- Giovanni Manno
- Barbara Prinari
Questions and suggestions should go to
Jet NESTRUEV, jet @ diffiety.ac.ru.