2.11. Tensor products 33

called the nth symmetric power, respectively exterior power of

V . If {vi} is a basis of V , can you construct a basis of SnV, ∧nV ?

If dimV = m, what are their dimensions?

(e) If k has characteristic zero, find a natural identification of

SnV

with the space of T ∈ V

n

such that T = sT for all transpositions s,

and find a natural identification of

∧nV

with the space of T ∈ V

n

such that T = −sT for all transpositions s.

(f) Let A : V → W be a linear operator. Then we have an op-

erator A

n

: V

n

→ W

n

and its symmetric and exterior powers

SnA

:

SnV

→

SnW

,

∧nA

:

∧nV

→

∧nW

which are defined in an

obvious way. Suppose that V = W and that dim V = N, and that

the eigenvalues of A are λ1,...,λN . Find

Tr(SnA)

and

Tr(∧nA).

(g) Show that

∧N

A = det(A)Id, and use this equality to give a

one-line proof of the fact that det(AB) = det(A) det(B).

Remark 2.11.4. Note that a similar definition to the above can be

used to define the tensor product V ⊗A W , where A is any ring, V

is a right A-module, and W is a left A-module. Namely, V ⊗A W

is the abelian group which is the quotient of the group V • W freely

generated by formal symbols v ⊗ w, v ∈ V , w ∈ W , modulo the

relations

(v1 + v2) ⊗ w − v1 ⊗ w − v2 ⊗ w,

v ⊗ (w1 + w2) − v ⊗ w1 − v ⊗ w2,

va ⊗ w − v ⊗ aw, a ∈ A.

Exercise 2.11.5. Let K be a field, and let L be an extension of K.

If A is an algebra over K, show that A ⊗K L is naturally an algebra

over L. Show that if V is an A-module, then V ⊗K L has a natural

structure of a module over the algebra A ⊗K L.

Problem 2.11.6. Throughout this problem, we let k be an arbi-

trary field (not necessarily of characteristic zero and not necessarily

algebraically closed).

If A and B are two k-algebras, then an (A, B)-bimodule will

mean a k-vector space V with both a left A-module structure and a

right B-module structure which satisfy (av) b = a (vb) for any v ∈ V ,

a ∈ A, and b ∈ B. Note that both the notions of “left A-module”