Mathematisches Institut WS 2001/02 der Universität München Set 9 Prof. Dr. B. Pareigis

Problem set for Advanced Algebra

(33) Show that there is a natural isomorphism Map(A × B, C) ≅ Map(B, Map(A, C)). Determine the associated unit and counit.

(34) Show that the underlying functor U : Ab ⟶ Gr, that forgets the commutativity of Abelian groups, has a left adjoint functor F : Gr ⟶ Ab. Determine the associated unit and counit. (Hint: Consider the construction of the commutator factor group.)

(35) Show that the underlying functor U : Ab ⟶ comMon from the category of Abelian groups to the category of commutative monoids, that forgets the existence of inverses in groups, has a left adjoint functor F : comMon ⟶ Ab. Determine the associated unit and counit. (Hint: Consider the construction of fractions for rational numbers.)

(36) Let f : R ⟶ S be a ring homomorphism. Show that the underlying functor U : S - Mod ⟶ R - Mod from the category of S-modules to the category of R-modules, that forgets the Smodule structure on modules and replaces it by the R-module structure r · m := f(r)m, has a left adjoint functor F : R - Mod ⟶ S - Mod and a right adjoint functor G : R - Mod ⟶ S - Mod. Determine the associated units and counits.

Due date: Tuesday, 18.12.2001, 16:15 in Lecture Hall 138