7.3 Tensor Product as an Algebra

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Throughout this section R denotes a commutative ring. Thus we need not

distinguish between left or right R-modules. The R-module A is called an

R-algebra if it has a ring structure that satisfies (ra1)a2 = a1(ra2), r 2

R, a1, a2 2 A. If A,B are R-algebras, we shall give a natural R-algebra

structure on the tensor product AR B. Recall from Section 2 that AR B

is already an R-module with scalar multiplication satisfying

r(a  b) = ra  b = a  rb,

a 2 A, b 2 B, r 2 R.

To obtain an R-algebra structure on A R B, we shall apply Exercise 9

of the previous section. Indeed, we map

f : A × B × A × B −! A R B,

by setting f(a1, b1, a2, b2) = a1a2  b1b2, a1, a2 2 A, b1, b2 2 B. Then f is

clearly multilinear; this gives a mapping

_ : (A R B) R (A R B) −! A R B.

Thus we define the multiplication on ARB by setting (a1b1) · (a2b2) =

a1a2  b1b2, a1, a2 2 A, b1, b2 2 B. One now has the targeted result:

Proposition 7.3.1 Let A,B be R-algebras. Then there is an R-algebra

structure on A R B such that (a1  b1) · (a2  b2) = a1a2  b1b2.

Exercises 7.3

1. Let F be a field, and let A be a finite-dimensional F-algebra that is

also an integral domain. Prove that A is a field, algebraic over F. (Of

course, this is simply a restatement of Exercise 9 of Section 2.1.)

2. Let A1,A2 be commutative R-algebras. Prove that A1 R A2 satisfies

a universal condition reminiscient of that for direct sums of Rmodules.

Namely, there exist R-algebra homomorphisms μi : Ai !

A1RA2, i = 1, 2 satisfying the following. If B is any commutative Ralgebra

such that there exist R-algebra homomorphisms _i : Ai ! B,

164 CHAPTER 7. TENSOR PRODUCTS

there there exists a unique R-algebra homomorphism _ : A1RA2 ! B

such that each diagram

Ai B

A1 R A2

-

􀀀

􀀀

􀀀

􀀀􀀀_ @

@

@

@ _i @R

μi _

commutes.

3. Prove that R[x] R R[y] _= R[x, y] as R-algebras.

4. Let G1,G2 be finite groups with F-group algebras as in Section 5.10.

Prove that F[G1 × G2] _= FG1 F FG2.

5. Let A be an algebra over the commutative ring R. We say that A

iLs a graded R-algebra if A admits a direct sum decomposition A = 1

r=0 Ar, where Ar · As _ Ar+s for all r, s _ 0. (We shall discuss

graded algebras in somewhat more detail in the next section.) We say

that the graded R-algebra A is graded-commutative (or just commutative

!) if whenever ar 2 Ar, as 2 As we have aras = (−1)rsasar.

Now let A =

L1

r=0 Ar, B =

L1

s=0 Bs be graded-commutative Ralgebras.

Prove that there is a graded-commutative algebra structure

on A R B satisfying

(ar  bs) · (ap  bq) = (−1)sp(arap  bsbq),

ar 2 Ar, ap 2 Ap, bs 2 Bs, bq 2 Bq. This is usually the intended

meaning of “tensor product” in the category of graded-commutative

R-algebras.

7.4. TENSOR, SYMMETRIC AND EXTERIOR ALGEBRA 165

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