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189-570A: Higher Algebra I

Assignment 1

Due: Friday, September 17.




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 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 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 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 is the category of sets or of vector spaces.)

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
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 VSK is replaced by the category ModR, 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
x2-3 y5=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.

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 8, compute the number of k-dimensional subspaces contained in an n-dimensional vector space over Fp.