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 Z_m.

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

Problem 6 Compute the module Z₃Ä_Z Z₆.

Problem 7 Let {e₁,...,e_m} and {f₁,...,f_n} be free generators of free modules M and N respectively. Prove that { e_iÄ f_j | 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 m_a is projective iff dimM = 1.

Problem 19 Describe the bundle that corresponds to m_a 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 = C^m(R).

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

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

Problem 25 Describe D(A) if A = Z₂.

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

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

Problem 28 Describe Diff_k(A,A) if A = Z_m[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 Hom_K(P,Q) and the sum is over all indices such that
i₁ < i₂ < ... < i_k,   j₁ < j₂ < ... < j_l-k,   (È_ri_r)È(È_sj_s) = {1,2,...,l}.

Problem 30 Prove the formula

D(a1... alp) = å k > 0  (-1)k+1ai1... aikD(aj1... ajl-kp),

where D Î Diff_s(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(Diff_k-1⁺(P))®Diff_k⁺(P), S(Ñ)(a) = Ñ(a)(1), is a differential operator of order 1.

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

Problem 34 Prove that the gluing map has the form (c_k,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 g_k+1 :   D_k+1(P)® D_k(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 D_k+1(P).

Problem 38 Prove that the composition S_k,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 S_k-1,l+1°S_k,l = 0.

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

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

Problem 42 Let J_alg^k(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

Problem 43 Prove the following formula for the co-gluing map: c^k,l(aj_k+l(p)) = aj_k(j_l(p)).

Problem 44 Check that the wedge product Ù :  L^kÄ_AL¹®L^k+1 is equal to the composition L^kÄ_AL¹®lJ¹(L^k) ®y^dL^k+1, where l(wÄ da) = (-1)^k(j₁(aw)-aj₁(w)).