189-571B: Higher Algebra II
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
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 Mn(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
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.
r: G --> GLn(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 GLn(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 GLn(R) of real invertible n x n matrices.
b) The group SLn(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
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) = Ext1Z[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).
(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 Ext1R(Z,Z).
c) Suppose that G is a group of p-power order, and that
R=Fp[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.