Basic functors of differential calculus over
commutative algebras
Alexandre Vinogradov, Michael Vinogradov
A program of the course at the 4th Italian Diffiety School,
(Forino, July 1729, 2000)
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 The Diffprolongations of differntial operators. The
functor transformations
c_{s,l}: Diff^{+}_{s} Diff^{+}_{l}® Diff^{+}_{s+l}. 

 The functor equivalence Diff_{1} @ DÅ id. The
functor transformations
k_{1}: Diff^{(+)}_{1}® D. 

 The kernel of the universal differential operator
Ä^{+}_{2}: Diff^{+}_{2}P® P, 

and functors D_{2} and P^{(+)}_{2}. The Spencer Diffsequence of the
second order.
 The construction and properties of Kmodules D(P Ì Q)
and Diff^{(+)}_{1}(P Ì Q).
 Functors D_{i} and P^{(+)}_{i}.
The functor equivalence P_{i} @ D_{i}ÅD_{i1}. The
functor transformations
k_{i}: P^{(+)}_{i}® D_{i}. 

 The Spencer Diffcomplexes.
 The construction and properties of Amodules
PÄ_{in} Q
and Hom^{·}_{A}(P,Q).
 Amodules L^{i}'s as representative objects of
functors D_{i}.
 Algebraic de Rham complexes. The exterior product,
the Lie derivative and their properties.
 The Spencer Jcomplexes of an algebra A as
as representative objects of the Spencer Diffcomplexes.
 The Spencer Jcomplexes of Amodules and
their properties.
Examination problems
1. Let F_{1}, F_{2} be functors on the category of all Amodules,
j_{1}, j_{2} be corresponding representative objects, that is
F_{1}(P) = Hom_{A}(j_{1},P), F_{2}(P) = Hom_{A}(j_{2},P) 

for any Amodule P, and F : F_{1}®F_{2} be a natural
transformation. Set P = j_{2}, and define the homomorphism
F^{*} : j_{2}®j_{1} by the formula
id Î Hom_{A}(j_{1},j_{1}) = F_{1}(j_{1}), F(id) Î Hom_{A}(j_{2},j_{1}) = F_{2}(j_{1}).
Prove that for any Amodule Q and
any element h Î F_{1}(Q) = Hom_{A}(j_{1},Q)
2. Prove that Hom_{A}(P^{+}Ä_{in}S, Q) = Hom_{A}^{·}(S,Hom^{+}_{A}(P,Q)).
3. Prove that Diff^{+}_{s}P = Hom^{+}_{A}(J^{s},P).
4. Prove that
J^{s}_{+}(P) = J^{s}_{+}ÄP, J^{s}(P) = J^{s}Ä_{in} 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.