Mathematisches Institut WS 2001/02 der Universität München Set 4 Prof. Dr. B. Pareigis

Problem set for Advanced Algebra

(13) Let X be a set and V := 𝕂X be the free 𝕂-module over X. Show that X ⟶ V ⟶ T (V) defines a free algebra over X, i.e. for every 𝕂-algebra A and every map f : X ⟶ A there is a unique homomorphism of 𝕂-algebras g : T (V) ⟶ A such that the diagram

[figure: X; T (V); f; A; g]

commutes.

We write 𝕂⟨X⟩ := T (𝕂X) and call it the polynomial ring over 𝕂 in the non-commuting variables X.

(14) Let X be a set and V := 𝕂X be the free 𝕂-module over X. Show that X ⟶ V ⟶ S(V) defines a free commutative algebra over X, i.e. for every commutative 𝕂-algebra A and every map f : X ⟶ A there is a unique homomorphism of 𝕂-algebras g : S(V) ⟶ A such that the diagram

[figure: X; S(V); f; A; g]

commutes.

(15) (a) Let S(V) and ι : V ⟶ S(V) be a symmetric algebra. Show that there is a unique homomorphism of algebras Δ : S(V) ⟶ S(V) ⊗ S(V) with Δ(v) = v ⊗ 1 + 1 ⊗ v for all v ∈ V.

(b) Show that (Δ ⊗ 1)Δ = (1 ⊗ Δ)Δ : S(V) ⟶ S(V) ⊗ S(V) ⊗ S(V).

(16) Let V be a 𝕂-module and A be a 𝕂-algebra.

(a) Let f : V ⟶ A be a homomorphism of 𝕂-modules satisfying f(v)^{2} = 0 for all v ∈ V. Then f(v)f(v′) = −f(v′)f(v) for all v, v′ ∈ V.

(b) Let 2 be invertible in 𝕂 (e.g. 𝕂 a field of characteristic ≠ 2). Let f : V ⟶ A be a homomorphism of 𝕂-modules satisfying f(v)f(v′) = −f(v′)f(v) for all v, v′ ∈ V. Then f(v)^{2} = 0 for all v ∈ V.

Due date: Tuesday, 13.11.2001, 16:15 in Lecture Hall 138