**189-570A:** Higher Algebra I

## Assignment 1

## Due: Monday, September 18.

Notations: In this assignment,
**Sets**, **Groups**, **Rings**,
**VS**_{K} and **Mod**_{R}
will denote the categories
of Sets, groups, rings with unit, vector spaces over a field K and
modules over a ring R
respectively.

1. Explain how the following assignments extend naturally to
covariant functors.

a. The assignment F from **Rings** to **Groups** which to an
object R of the former associates the group R^{x} of
invertible elements of R (under multiplication).

b. The assignment GL_{n} which to an object of **Rings**
associates the group GL_{n}(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 M_{n}(R) of n x n matrices with
entries in R.

2. Show that the functor F of exercise 1 is naturally
equivalent to GL_{1} and that
GL_{n} is naturally equivalent to
the composition F M_{n} of F and M_{n}.

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**
is
the category of sets or of vector spaces: in those cases such a functor is
called a *permutation representation* and a *linear representation* of
G, respectively.)

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 **VS**_{K},
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 **VS**_{K}.
Note that the same is true if **VS**_{K} is
replaced by the category of **Mod**_{R}.

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

and

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 **VS**_{K} is replaced by the category
**Mod**_{R}, show that the sequences

0 --> hom(W,V') --> hom(W,V) --> hom(W,V'')

and

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

x^{2}-3 y^{5}=17

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 p^{n}
has a non-trivial center. Conclude that every such group is solvable.

9. Let G be the group GL_{n}(F_{p}) 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
F_{p}.