Mathematisches Institut WS 2001/02 der Universität München Set 2 Prof. Dr. B. Pareigis

Problem set for Advanced Algebra

(5) Let V be a finite dimensional vector space. Let B = (v_{i}|i = 1, …, n) be a basis of V and (v_{i}^{∗}|i = 1, …, n) be the dual dual basis of the dual space V^{∗}. Show that ∑_{i=1}^{n} v_{i} ⊗ v_{i}^{∗} ∈ V ⊗ V^{∗} does not depend on the choice of the basis B and that

∀v ∈ V : ∑_{i} v_{i}^{∗}(v)v_{i} = v

holds.

(Hint: Find an isomorphism End(V) ≅ V ⊗ V^{∗} and show that id_{V} is mapped to ∑_{i=1}^{n} v_{i} ⊗ v_{i}^{∗} under this isomorphism.)

(6) (a) Let M_{R}, _{R}N, M′_{R}, and _{R}N′ be R-modules. Show that the following is a homomorphism of abelian groups:

μ : Hom_{R}(M, M′)⊗_{ℤ}Hom_{R}(N, N′) ∋ f⊗g ↦ f⊗_{R}g ∈ Hom(M⊗_{R}N, M′⊗_{R}N′).

(b) Find examples where μ is not injective and where μ is not surjective.

(c) Explain why f ⊗g is a decomposable tensor whereas f ⊗_{R} g is not a tensor.

(7) Give a complete proof of Theorem 1.22. In (5) show how Hom_{T} (M., N.) becomes an S-R-bimodule.

(8) Find an example of M, N ∈ 𝕂- Mod -𝕂 such that M ⊗_{𝕂} N ≅ N ⊗_{𝕂} M. (Hint: You may use 𝕂 := L × L, _{𝕂}M := _{𝕂}𝕂, and N_{𝕂} := 𝕂_{𝕂}. Define a right 𝕂-structure on M by (m, n)(a, b) := (ma, na) and a left 𝕂-structure on N by (a, b)(r, s) := (br, bs).)

Due date: Tuesday, 30.10.2001, 16:15 in Lecture Hall 138