Basic functors of differential calculus over commutative algebras

Alexandre Vinogradov, Michael Vinogradov

A program of the course at the 4st Italian Diffiety School, Forino, July 17 — 29, 2000

1. The Diff-prolongations of differntial operators. The functor trans-

formations

c_{s,l}: Diff_{s}^{+} Diff_{l}^{+} → Diff_{s+l}^{+} .

2. The functor equivalence Diff_{1} ≃ D ⊕ id. The functor transforma-

tions

κ_{1} : Diff_{1}^{(+)} → D.

3. The kernel of the universal differential operator

D_{2}^{+} : Diff_{2}^{+} P → P,

and functors D_{2} 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 D_{i} and P_{i}^{(+)}. The functor equivalence P_{i} ≃ D_{i} ⊕ D_{i−1}. The functor transformations

κ_{i}: P_{i}^{(+)} → D_{i}.

6. The Spencer Diff-complexes.

7. The construction and properties of A-modules P ⊗_{in}Q and Hom_{A}^{•}(P, Q).

8. A-modules Λ^{i}’s as representative objects of functors D_{i}.

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

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

1. Let Φ_{1}, Φ_{2} be functors on the category of all A-modules, φ_{1}, φ_{2} be corresponding representative objects, that is

Φ_{1}(P) = Hom_{A}(φ_{1}, P), Φ_{2}(P) = Hom_{A}(φ_{2}, P)

for any A-module P, and F: Φ_{1} → Φ_{2} be a natural transformation. Set P = φ_{2}, and define the homomorphism F^{∗}: φ_{2} → φ_{1} by the formula

F^{∗} = F (id),

id ∈ Hom_{A}(φ_{1}, φ_{1}) = Φ_{1}(φ_{1}), F (id) ∈ Hom_{A}(φ_{2}, φ_{1}) = Φ_{2}(φ_{1}).

Prove that for any A-module Q and any element h ∈ Φ_{1}(Q) = Hom_{A}(φ_{1}, Q)

F (h) = h ∘ F^{∗}

2. Prove that Hom_{A}(P^{+} ⊗_{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} ⊗ P, J^{s}(P) = J^{s} ⊗_{in} P.

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