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

   cs,l:    Diff+s Diff+l® Diff+s+l.

2. The functor equivalence  Diff₁ @ DÅ id. The functor transformations

   k1:    Diff(+)1® D.

3. The kernel of the universal differential operator

   Ä+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 Ì Q) and  Diff⁽⁺⁾₁(P Ì Q).
5. Functors D_i and P⁽⁺⁾_i. The functor equivalence P_i @ D_iÅD_i-1. The functor transformations

   ki:   P(+)i® Di.

6. The Spencer Diff-complexes.
7. The construction and properties of A-modules PÄ_in Q and  Hom^·_A(P,Q).
8. A-modules Lⁱ'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.

1. Let F₁, F₂ be functors on the category of all A-modules, j₁, j₂ be corresponding representative objects, that is

F1(P) = HomA(j1,P),    F2(P) = HomA(j2,P)

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

F* = F(id),

id Î Hom_A(j₁,j₁) = F₁(j₁),   F(id) Î Hom_A(j₂,j₁) = F₂(j₁).

Prove that for any A-module Q and any element h Î F₁(Q) = Hom_A(j₁,Q)

F(h) = h°F*

2. Prove that Hom_A(P⁺Ä_inS, Q) = Hom_A^·(S,Hom⁺_A(P,Q)).

3. Prove that Diff⁺_sP = Hom⁺_A(J^s,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.

1. Giovanni Manno
2. Barbara Prinari