Problem 2 Construct an example of a nonassociative 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 finitedimensional vector spaces L_{1} and L_{2} over a field F prove that Hom_{F}(L_{1},L_{2}) @ L_{1}^{*}ÄL_{2}.
Problem 6 Compute the module Z_{3}Ä_{Z} Z_{6}.
Problem 7 Let {e_{1},...,e_{m}} and {f_{1},...,f_{n}} be free generators of free modules M and N respectively. Prove that { e_{i}Ä 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

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^{¥}(M_{1}) and C^{¥}(M_{2}), where M_{1} and M_{2} are smooth manifolds, the set of morphisms Mor(C^{¥}(M_{1}),C^{¥}(M_{2})) is a nonempty and nonzero subset of Hom(C^{¥}(M_{2}),C^{¥}(M_{1})).
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 finitedimensional 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


Problem 16
Prove that if P is a projective module then the sequence
(see the previous problem)

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
Problem 20 Is the module of derivations D(C^{¥}(K)), where K is the coordinate cross in R^{2}, 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]/ (x^{n}).
Problem 24 Describe D(A) if A = Z_{m}[x]/ (x^{n}).
Problem 25 Describe D(A) if A = Z_{2}.
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]/ (x^{n}).
Problem 28 Describe Diff_{k}(A,A) if A = Z_{m}[x]/ (x^{n}).
Problem 29
Prove the formula

Problem 30
Prove the formula

Problem 31
Construct a natural isomorphism

Problem 32 Prove that the map S : D(Diff_{k1}^{+}(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^{¥}(R^{n}).
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(R^{3})® C^{¥}(R^{3}), (f_{1},f_{2},f_{3})®¶df_{1}x_{1}+¶df_{2}x_{2}+¶df_{3}x_{3}. (You may take the Maxwell equation or, better, abelian 2form gauge theory instead of the divergence.)
Problem 36 Prove that g_{k+1} : D_{k+1}(P)® D_{k}(Diff_{1}^{+}(P)) is a differential operator of order 1.
Problem 37
Prove that the kernel of the composition

Problem 38
Prove that the composition S_{k,l}

Problem 39 Check that S_{k1,l+1}°S_{k,l} = 0.
Problem 40 Prove that J^{1}(M) = M ´ T^{*}(M).
Problem 41 Show that de^{x}e^{x} dx ¹ 0 if the module of 1forms 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 cogluing map: c^{k,l}(aj_{k+l}(p)) = aj_{k}(j_{l}(p)).
Problem 44 Check that the wedge product Ù : L^{k}Ä_{A}L^{1}®L^{k+1} is equal to the composition L^{k}Ä_{A}L^{1}®lJ^{1}(L^{k}) ®y^{d}L^{k+1}, where l(wÄ da) = (1)^{k}(j_{1}(aw)aj_{1}(w)).