Homework 4, February 26

Math 402 – Spring Semester, 1990 Second Course in Abstract Algebra, II Prof. Daniel Grayson

2:00 PM MWF, Section F1, 243 Altgeld Hall

These problems are due Monday, March 5.

Do p. 377, problems 1, 2, (3), (5), 6, 7, and p. 111, problems 1, 6, 7.

The exercises in parentheses are to be turned in only if you are uncertain whether you got the correct answer.

The following exam question is to be done independently.

Exam question 1: Suppose F is a field, and V is a finite dimensional F -vector space. Suppose U_{1} ⊆ U_{2} ⊆ ⋯ ⊆ U_{n} are F -subspaces of V. Let U_{1} ∧ U_{2} ∧ ⋯ ∧ U_{n} denote the image of U_{1} ⊗_{F} U_{2} ⊗_{F} ⋯ ⊗_{F} U_{n} → ⋀^{n} V. If X is a subspace satisfying U_{p} ⊆ X ⊆ U_{p+1}, then show that there is a natural isomorphism of

\frac{U_{1} ∧ ⋯ ∧ U_{p} ∧ U_{p+1} ∧ U_{p+2} ∧ ⋯ ∧ U_{n}}{U_{1} ∧ ⋯ ∧ U_{p} ∧ X ∧ U_{p+2} ∧ ⋯ ∧ U_{n}}

with

(U_{1} ∧ ⋯ ∧ U_{p}) ⊗_{F} (\frac{U_{p+1}}{X} ∧ \frac{U_{p+2}}{X} ∧ ⋯ ∧ \frac{U_{n}}{X} )

You might begin by trying to find a convenient set of generators and relations for U_{1} ∧ ⋯ ∧ U_{n}. Notice that since you are asked to find a natural isomorphism, it would be a bad idea for the isomorphism’s definition to depend on choices of basis.