189-570A: Higher Algebra I
Due: Monday, September 18.
Notations: In this assignment,
Sets, Groups, Rings,
VSK and ModR
will denote the categories
of Sets, groups, rings with unit, vector spaces over a field K and
modules over a ring R
1. Explain how the following assignments extend naturally to
a. The assignment F from Rings to Groups which to an
object R of the former associates the group Rx of
invertible elements of R (under multiplication).
b. The assignment GLn which to an object of Rings
associates the group GLn(R) of invertible n x n matrices
with entries in R.
c. The assignmet from the objects of Rings to itself which
to a ring R associates the ring Mn(R) of n x n matrices with
entries in R.
2. Show that the functor F of exercise 1 is naturally
equivalent to GL1 and that
GLn is naturally equivalent to
the composition F Mn of F and Mn.
3. A group G may be realised as a category with a single object *,
in which the arrows are indexed by elements of G and the
composition of arrows corresponds to the multiplication in the group.
Let G denote this category. Show that a functor F from G
to a category C is equivalent to the datum of a homomorphism
from G to the automorphism group of
F(*). (Of particular interest are the cases where C
the category of sets or of vector spaces: in those cases such a functor is
called a permutation representation and a linear representation of
4. Let C be any category, and fix any object * of C.
a. Show that the assignment from C to Sets which to
A in Ob(C) associates the set Hom(*,A) extends naturally to a covariant
functor from C to Sets.
b. Show that the assignment from C to Sets which to
A in Ob(C) associates the set Hom(A,*) extends
naturally to a contravariant
functor from C to Sets.
c. If C is equal to the category VSK,
show that hom(*,A) and hom(A,*) can themselves be viewed as vector
spaces in a natural way, and that hom(*,-) and hom(-,*) can be
viewed as functors into the category VSK.
Note that the same is true if VSK is
replaced by the category of ModR.
5. Show that
a. If 0 --> V' --> V --> V'' --> 0 is an exact sequence of vector
spaces, the resulting sequences
0 --> hom(W,V') --> hom(W,V) --> hom(W,V'') --> 0
0 --> hom(V'',W) --> hom(V,W) --> hom(V',W) --> 0
are also exact. (One says that the functors hom(W,-) and
hom(-,W) are exact.)
b. If VSK is replaced by the category
ModR, show that the sequences
0 --> hom(W,V') --> hom(W,V) --> hom(W,V'')
0 --> hom(V'',W) --> hom(V,W) --> hom(V',W)
continue to be exact, but that
the R-module homomorphisms hom(W,V)--> hom(W,V'') and
hom(V,W) --> hom(V',W) need not be surjective in general.
6. A functor F from a category C to Sets is said to be
representable if there is an object X of C such that
F is naturally equivalent to Hom(X,-). Show that the functor
from Rings to Sets which to every ring R associates
the set of solutions (x,y)
to the equation
with x,y in R
is a representable functor.
7. Let G be a finite group acting on a finite set X.
For any g in G, let FP(g) denote the number of fixed points for g acting on x. Show that
a. The sum of FP(g) as g ranges over G is equal to the cardinality of G,
if and only if the action of G on X is transitive.
b. The sum of FP(g)2 as g ranges over G
is equal to twice the cardinality of G,
if and only if the action of G on X is doubly transitive, i.e., if G acts transitively on the complement of the diagonal in the Cartesian product of
X with itself.
8. Let p be a prime. Show that every group of order pn
has a non-trivial center. Conclude that every such group is solvable.
9. Let G be the group GLn(Fp) of
n x n matrices with entries in the field with p elements.
What is the cardinality of G?
10. Using the result of question 9, compute the
number of k-dimensional subspaces
contained in an n-dimensional vector space over