Introduction to Differential Calculus over Commutative Algebras
Sergey Igonin, Alexander Verbovetsky, Raffaele Vitolo
A program of the course at the 4-th Italian Diffiety School,
(Forino, July 17-29, 2000)

1. Rings
2. Modules, tensor product
3. Free and projective modules, vector bundles
4. Categories and functors
5. Differential operators from algebraic viewpoint
6. Multiderivations
7. Functors of differential calculus
8. Ghosts
9. Jets
10. Compatibility complex
11. Differential forms, de Rham complex
12. Inner product, Lie derivative
13. Symbols of differential operators, Poisson bracket

Examination problems
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Problem 1 When Zm = Z/mZ is a field?

Problem 2 Construct an example of a non-associative ring.

Problem 3 Construct an example of an infinite field other than \mathbbQ, R, and \mathbbC.

Problem 4 Describe ideals of Zm.

Problem 5 For finite-dimensional vector spaces L1 and L2 over a field F prove that HomF(L1,L2) @ L1*ÄL2.

Problem 6 Compute the module Z3ÄZ Z6.

Problem 7 Let {e1,...,em} and {f1,...,fn} be free generators of free modules M and N respectively. Prove that { eiÄ fj | i = 1,...,m j = 1,...,n } are free generators of the module MÄN.

Problem 8 For free modules L and M of finite rank over a ring A prove that the map
 F :   L*ÄM®HomA(L,M),    F(xÄm)(l) = x(l)·m,       x Î L*,    m Î M,    l Î L
is an isomorphism of modules over A.

Problem 9 Construct a category C such that the objects of C are algebras of smooth functions on smooth manifolds and for any two objects C¥(M1) and C¥(M2), where M1 and M2 are smooth manifolds, the set of morphisms Mor(C¥(M1),C¥(M2)) is a non-empty and non-zero subset of Hom(C¥(M2),C¥(M1)).

Problem 10 Construct a functor F from the category of finite sets to the category of vector spaces over R such that for any finite set S the dimension of F(S) equals the number of elements of S.

Problem 11 Prove that in the category of finite-dimensional vector spaces the functors V®(V*)* and V® V are equivalent.

Problem 12 Prove that in the category of all vector spaces the functors from the previous problem are not equivalent.

Problem 13 Prove that the composition of two covariant functors or two contravariant functors is a covariant functor. What is the composition of covariant and contravariant functors?

Problem 14 Prove that if every module over a ring A is free then A is a field.

Problem 15 For any short exact sequence
 0® L® M® N® 0
of modules over a ring A and any module P over A construct the sequence
 0® HomA(P,L)® HomA(P,M)® HomA(P,N)
and prove that it is exact.

Problem 16 Prove that if P is a projective module then the sequence (see the previous problem)
 0®HomA(P,L)®HomA(P,M)®HomA(P,N)®0
is exact.

Problem 17 Prove that the Möbius bundle is nontrivial.

Problem 18 Let a Î M be a point of a smooth manifold M. Prove that the ideal ma is projective iff dimM = 1.

Problem 19 Describe the bundle that corresponds to ma if

1. M = R;
2. M = S1.

Problem 20 Is the module of derivations D(C¥(K)), where K is the coordinate cross in R2, projective?

Problem 21 Describe D(A) if A = C(R) the algebra of continuous functions.

Problem 22 Describe D(A) if A = Cm(R).

Problem 23 Describe D(A) if A = R[x]/ (xn).

Problem 24 Describe D(A) if A = Zm[x]/ (xn).

Problem 25 Describe D(A) if A = Z2.

Problem 26 Describe Diffk(A,A) if A = C¥(K), where K is the cross.

Problem 27 Describe Diffk(A,A) if A = R[x]/ (xn).

Problem 28 Describe Diffk(A,A) if A = Zm[x]/ (xn).

Problem 29 Prove the formula
 [...[[D°Ñ,a1],a2]...,al] = å [...[[D,ai1],ai2]...,aik]° [...[[Ñ,aj1],aj2]...,ajl-k],
where D, Ñ Î HomK(P,Q) and the sum is over all indices such that
i1 < i2 < ... < ik,   j1 < j2 < ... < jl-k,   (Èrir)È(Èsjs) = {1,2,...,l}.

Problem 30 Prove the formula
 D(a1... alp) = å k > 0 (-1)k+1ai1... aikD(aj1... ajl-kp),
where D Î Diffs(P,Q), s < l, and the sum is as in the previous problem.

Problem 31 Construct a natural isomorphism
 Diff+(P,Diff+(Q,R)) = Diff+(P,Diff(Q,R)).

Problem 32 Prove that the map S :   D(Diffk-1+(P))®Diffk+(P), S(Ñ)(a) = Ñ(a)(1), is a differential operator of order 1.

Problem 33 Check in coordinate form that the mapping Dk :  Diffk+(A)® A is a differential operator of order k for A = C¥(Rn).

Problem 34 Prove that the gluing map has the form (ck,l(D))(a) = D(a)(1).

Problem 35 Describe the ghost complex for the operator div :   D(R3)® C¥(R3), (f1,f2,f3)®df1x1+df2x2+df3x3. (You may take the Maxwell equation or, better, abelian 2-form gauge theory instead of the divergence.)

Problem 36 Prove that gk+1 :   Dk+1(P)® Dk(Diff1+(P)) is a differential operator of order 1.

Problem 37 Prove that the kernel of the composition
 Dk(Diff1+(P))®Dk-1(Diff1+(Diff1+(P)))® Dk-1(Diff2+(P))
is isomorphic to Dk+1(P).

Problem 38 Prove that the composition Sk,l
 Dk(Diffl+(P))®Dk-1(Diff1+(Diffl+(P)))® Dk-1(Diffl+1+(P))
is a differential operator of order 1.

Problem 39 Check that Sk-1,l+1°Sk,l = 0.

Problem 40 Prove that J1(M) = M ´ T*(M).

Problem 41 Show that dex-ex dx ¹ 0 if the module of 1-forms was defined in the category of all C¥(R)-modules.

Problem 42 Let Jalgk(P) be the module of jets of a module P in the category of all C¥(M)-modules. Assume that P is a projective finitely generated module. Prove that the jet module in the category of geometric modules has the form

Jgeomk(P) = Jalgk(P) / Ça Î M maJalgk(P).

Problem 43 Prove the following formula for the co-gluing map: ck,l(ajk+l(p)) = ajk(jl(p)).

Problem 44 Check that the wedge product Ù :  LkÄAL1®Lk+1 is equal to the composition LkÄAL1®lJ1(Lk) ®ydLk+1, where l(wÄ da) = (-1)k(j1(aw)-aj1(w)).

Passing the exam during the school required 12 solved problems. To pass the exam by email one should solve 20 problems.

The exam has been passed by the following students:
1. Emanuele Castagna (14 problems: 1- 7,9 - 11,13,15 - 17)
2. Stefania Donadio (12 problems: 2,3,6,9 - 12,15,16,21,23,25)
3. Giovanna Ilardi (20 problems: 1 - 8,10 - 17,21 - 23,25)
4. Massimiliano Malgieri (12 problems: 2,3,6,9 - 12,15,16,21,23,25)

Questions and suggestions should go to Jet NESTRUEV, jet @ diffiety.ac.ru.