189-571B: Higher Algebra II

Practice Final

To be discussed: Tuesday, April 17.

This practice final is designed to help you in your revisions in preparation for the actual final exam that will be held on Wednesday, April 18.

You are strongly advised to work through each problem carefully. If you have any questions, you can e-mail them to me (although I will be out of town on the week of April 8-15, I will try to read my e-mail and to respond to questions as they arrive).

I will also be available to answer your questions during office hours on Monday, April 16 from 10-12 and from 2-4.

There will be a review session on the material on Tuesday April 17 at 3:30. Come with questions (either about the practice final, or anything that has been covered in class).

1. Give an example of two finite-dimensional aglebras over R which are not isomorphic, yet become isomorphic as C-algebras after tensoring with C.

2. Which of the rings below are primitive? Semi-primitive? Explain.

a) The ring of residue classes modulo N, where N is square-free.

b) The ring F[e]/(e^2), where F is a field.

c) The ring M_n(Z) of integral nxn matrices.

d) The group ring C[G], where C is the field of complex numbers and G is a finite group.

e) The ring B of upper-triangular matrices with entries in a field F.

3. Let G be a finite group and let p be a prime which does not divide the order of G. Let F be the finite field with p elements, and let F^~ denote its algebraic closure. Finally, let

r: G --> GL_n(F^~)

be a representation of G. State whether the following are true or false. If the statement is true, give a proof, if not, give a counterexample.

a) If the character of r takes values in F, then r is isomorphic to a representation with values in GL_n(F).

b) Same statement as in part a), but with F replaced by the field R of real numbers.

4. Let R=Z[s]/(s²+5) be the ring of integers of the field obtained by adjoining to Q a square root of -5. Let M be the ideal in R generated by (2,1+s). Show that the direct sum M + M of M with itself is isomorphic to R+R, so that M+M is free. Conclude that M is projective. Show that M is not a free R-module.

5. Describe the Lie Algebras which are the tangent spaces at the identity of the following Lie groups:

a) The group GL_n(R) of real invertible n x n matrices.

b) The group SL_n(R) of matrices of determinant 1.

c) The group B of invertible upper-triangular matrices.

d) The group of SO₃(R) of rotations in three-dimensional euclidean space.

6. Let G be a finite group and let A be a module over the integral group ring Z[G]. Show that the group

H¹(G,A) = Ext¹_Z[G](Z,A)

can be identified with the quotient of Z¹(G,A) by B¹(G,A), where Z¹(G,A) is the group of crossed homomorphisms of G into A, as described in class - i.e., the set of functions f:G-->A satisfying

f(rs) = f(r) + rf(s)

and B¹(G,A) is the group of 1-coboundaries, i.e., functions f:G-->A of the form (for some a in A)

f(s) = sa -a

7. For any Z-module M, let M[n] denote the n-torsion submodule of M, and let nM denote the elements in M which are divisible by n. Contruct exact sequences

0 --> Ext(nM,Z) --> Ext(M,Z) --> Ext(M[n],Z) --> 0,

0 --> Hom(M,Z) --> Hom(nM,Z) --> Ext(M/nM,Z) --> Ext(M,Z) --> Ext(nM,Z) -->0,

and show that Hom(M,Z) is isomorphic to Hom(nM,Z).

Deduce that

(a) M[n]=0 if and only if Ext(M,Z)/nExt(M,Z) = 0.

(b) If M/nM=0, then Ext(M,Z)[n]=0. Give a counterexample to show that the converse is not true.

(c) If Ext(M,Z)[n]=Hom(M,Z)=0, then M/nM=0.

8. Let G be a finite group, let R=Z[G] be the integral group ring, and let I be the augmentation ideal of R, i.e., the kernel of the augmentation map R-->Z.

a) Show that the exact sequence

0 --> I --> R --> Z --> 0

of R-modules is non-split (i.e., defines a non-trivial extension) if G is non-trivial.

b) Compute Ext¹_R(Z,Z).

c) Suppose that G is a group of p-power order, and that R=F_p[G] is the group ring of G with coefficients in the field with p elements. Let M=Z/pZ be the R-module defined by making G act trivially on M. Compute Ext¹_R(M,M).