Math 593: Homework 7

October October 31, 2014

1. Basic properties of tensor product. Let M,N and L be modules over a commutative ring. Show that

i. M ⊗ N ∼= N ⊗ M;

ii. (M ⊗ N) ⊗ L ∼= M ⊗ (N ⊗ L);

iii. (M ⊕ N) ⊗ L ∼= (M ⊗ L) ⊕ (N ⊗ L).

iv. If M ∼= M 0 and N ∼= N 0, then M ⊗ N ∼= M 0 ⊗ N 0.

2. Change of Rings. Let A → B → C be maps of commutative rings. Show that for an A-module ∼ M, we have an isomorphism C ⊗B (B ⊗A M) = C ⊗A M. 3. Tensor Products in the category of A-algebras. Let A be a commutative ring. Let R and S be two commutative A-algebras. a). Prove that there is a natural A-algebra structure on R ⊗A S. b). Show that there are natural A-algebra maps R → R ⊗A S and S → R ⊗A S which make the tensor product R ⊗A S a coproduct in the category of commutative A-algebras. [A coproduct is an object satisfying the following universal property: given any commutative A-algebra T with A-algebra maps R → T and S → T , there is a unique A-algebra homomorphism R⊗S → T making the appropriate diagrams commute.] c). Show that coproducts exist in the category of commutative rings. [Hint: over what ring will you tensor?] 4. Find (with proof) explicit isomorphisms between the following algebras (all rings commutative):

∼ a. An isomorphism of B-algebras B ⊗A A[x, y] = B[x, y] where B is an A-algebra. ∼ b. An isomorphism of R/I-algebras, R/I⊗RR[x1, . . . , xd]/(f1, . . . , ft) = R/I[x1, . . . , xd]/(f 1,..., f t) P i P i where rix = rix means the coefficients are reduced modulo I.

5 5 ∼ 5 c. F5 ⊗Z Z[x, y]/(x + 5xy + y ) = F5[x, y]/(x + y) .

5. Prove or Disprove (if you prove it, be clear about what category your isomorphism is in):

∼ • R ⊗Z Q = R.

1 ∼ • C ⊗R C = C. ∼ • C ⊗R R = R. ∼ • C ⊗C C = C. ∼ • k[x] ⊗k k[y] = k[x, y]

3 2 6. Find all prime ideals in Z10 ⊗Z (Z[x, y]/(x ) ⊗Z[x,y] Z[x, y]/(y )).

7. If M and N are finitely generated modules over a PID of ranks r1, r2 respectively, and list of invariant factors f1, . . . , ft and g1, . . . , gs respectively, find a decomposition of M ⊗R N into cyclic modules; describe the rank, and the last invariant factor.

8. A QR problem. Let Mat22(C) be the space of 2 × 2 matrices with complex entries. Fix an element A ∈ Mat22(C). Show that self-map L of Mat22(C) sending X to AX − XA is a C-vector  1 0  space endomorphism. When A = , find its Jordan form (your answer should be a 4 × 4 0 0 matrix).

9. Simple tensors. An element (tensor) of M ⊗R N is a simple tensor if it can be written m ⊗ n for some m ∈ M and n ∈ N. a). Let V and W be vector spaces of dimensions d and e, respectively, over the field k. Show that there is an isomorphism between V ⊗k W with the vector space of d × e matrices which identifies the (non-zero) simple tensors in V ⊗k W with the rank one matrices. b). Let F be a field of cardinality q. Find the fraction of simple tensors in Fd ⊗ Fe. As q gets large, what can be said about the likelihood that a given tensor is simple. If F = R, what can be said d e ∼ de about the measure of the set of simple tensors in R ⊗ R = R ?

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