Mathematisches Institut WS 2001/02 der Universität München Set 8 Prof. Dr. B. Pareigis

Problem set for Advanced Algebra

(29) Let F : C ⟶ Set be a covariant representable functor and let f : M ⟶ N be a monomorphism in C. Show that F(f) : F(M) ⟶ F(N) is a monomorphism, too.

(30) (a) Let U : Ab ⟶ Set be the underlying functor. Show that

U is a representable functor.

(b) Show that F : ℚ -Vec ∋ V ↦ U(ℚ^{n} ⊗_{ℚ} V) ∈ Set is a representable functor.

(31) Show that the functor F : Ab ∋ M ↦ U(ℤ/(n) ⊗_{ℤ} M) ∈ Set is not representable (for n ≥ 2).

(32) For a 𝕂-algebra A let A^{∗} denote the multiplicative group of elements that are invertible under the multiplication of A. Let U : Gr ⟶ Set be the underlying functor. Show that the functor F : 𝕂 -Alg ∋ A ↦ U(A^{∗}) ∈ Set is a representable functor.

Due date: Tuesday, 11.12.2001, 16:15 in Lecture Hall 138