Study Guide for Midterm Exam

The midterm exam problems will be very similar to some of the problems listed below. The exam will consist of approximately five such problems.

Module Theory

Problem 1

Let $R$ be a commutative ring (with unity). We have shown that for every $R$ -module $M$ there is an $R$ -module isomorphism ${Hom}_{R} (R, M) ≃ M$ . Now show more generally that for each fixed positive integer $n$ there is also an $R$ -module isomorphism
${Hom}_{R} (⨁_{i = 1}^{n} R, M) ≃ ⨁_{i = 1}^{n} M .$

Problem 2

Let $R$ be a commutative ring (with unity), $X$ a finite set, and $F (X)$ the free $R$ -module on $X$ . Prove there is an $R$ -module isomorphism ${Hom}_{R} (F (X), R) ≃ F (X)$ .

Problem 3

Suppose $f : N \to P$ is a $(S, T)$ -bimodule morphism.

Show that for every $(R, S)$ -bimodule $M$ there is an $(R, T)$ -bimodule morphism
$1_{M} \otimes f : M \otimes_{S} N \to M \otimes_{S} P,$
defined on simple tensors by $m \otimes n \mapsto m \otimes f (n)$ .
Prove that if $f$ is surjective, then so is $1_{M} \otimes f$ .
Show there are isomorphisms of abelian groups $Z_{2} \otimes_{Z} Z ≃ Z_{2}$ and $Z_{2} \otimes_{Z} Q ≃ 0$ , and then explain why this proves tensor product does not in general preserve injections.

Problem 4

Let $M_{1}$ and $M_{2}$ be $R$ -modules and $A_{1} \subseteq M_{1}$ and $A_{2} \subseteq M_{2}$ be submodules. Prove that $A_{1} \times A_{2}$ is (isomorphic to) a submodule of $M_{1} \times M_{2}$ and that there is an $R$ -module isomorphism
$(M_{1} \times M_{2}) / (A_{1} \times A_{2}) ≃ (M_{1} / A_{1}) \times (M_{2} / A_{2}) .$

Problem 5

Let $V$ be a finite-dimensional vector space over a field $F$ and let $v, w \in V$ be nonzero vectors. Prove that $v \otimes w = w \otimes v$ in $V \otimes_{F} V$ if and only if $w = a v$ for some $a \in F$ .

Hints

First prove the following lemma: If ${e_{1}, \dots, e_{n}}$ is a basis for $V$ as an $F$ -vector space, then every element of $V \otimes_{F} V$ can be written uniquely in the form $\sum_{i = 1}^{n} v_{i} \otimes e_{i}$ where $v_{i} \in V$ .

Then suppose $v = c_{1} e_{1} + \dots + c_{n} e_{n}$ and $w = d_{1} e_{1} + \dots + d_{n} e_{n}$ . Under the assumption $v \otimes w = w \otimes v$ , use the above lemma to prove that $d_{i} v = c_{i} w$ for every $i$ .

Note: If necessary, you may use the fact that ${e_{i} \otimes e_{j} ∣ 1 \leq i, j \leq n}$ is an $F$ -vector space basis for $V \otimes_{F} V$ .

Category Theory Problems

Problem 6

Let $F, G : C \to Set$ be functors and $τ : F \Rightarrow G$ be a natural transformation between those functors. We say $τ$ is:

a natural bijection if for every $c \in C$ the set map $τ_{c} : F (c) \to G (c)$ is a bijection;
a natural isomorphism if there is a natural transformation $η : G \Rightarrow F$ such that $η \circ τ = 1_{F}$ and $τ \circ η = 1_{G}$ .

Prove that $τ$ is a natural bijection if and only if it is a natural isomorphism.

Problem 7

Suppose $C$ is a category that has pullbacks and a terminal object.

Prove that $C$ has all (binary) products.
Prove that $C$ has all equalizers.

Hints

If $t$ is the terminal object of $C$ , show that for every pair of objects $a, b$ in $C$ the pullback $a \times_{t} b$ satisfies the universal property of the product $a \times b$ .
Show that the equalizer of a pair of arrows $f, g : a \to b$ may be constructed as the pullback of $a \overset{(1_{a}, f)}{\to} a \times b \overset{(1_{a}, g)}{\leftarrow} a$ .

Problem 8

Suppose $C$ is a category that has all equalizers.

The map that assigns to each pair of parallel arrows $f, g : a \to b$ the equalizer object $Eq (f, g)$ is the object function of a functor to $C$ . What is the arrow function of that functor?
The equalizer functor above is right adjoint to a certain diagonal (or constant) functor from $C$ . Describe:
- a) the other functor;
- b) the natural bijection of the adjunction; and
- c) the unit and counit of the adjunction.

Problem 9

To each category $C$ we can associate an opposite category $C^{op}$ . The objects of $C^{op}$ are the objects of $C$ , and the arrows of $C^{op}$ are arrows $f^{op}$ in one-to-one correspondence $f \mapsto f^{op}$ with the arrows $f$ of $C$ . For each arrow $f : a \to b$ in $C$ , the corresponding arrow is $f^{op} : b \to a$ in $C^{op}$ .

Suppose $F : C \to D$ is a functor. Show that the maps $c \mapsto F (c)$ and $f^{op} \mapsto (F (f))^{op}$ together define a functor $F^{op} : C^{op} \to D^{op}$ .
Consider a functor $F : C^{op} \to D$ . If we write $\overset{―}{F} (f)$ for $F (f^{op})$ , show how the functor $F$ can be expressed directly in terms of the original category $C$ as maps (both denoted $\overset{―}{F}$ ) on objects and arrows in $C$ . How does $\overset{―}{F}$ interact with composition of arrows?

Such maps are sometimes called contravariant functors from $C$ to $D$ , in which case our usual functors are called covariant functors.
For each fixed object $b$ in a category $C$ , show how to define a functor

${Hom}_{C} (-, b) : C^{op} \to S e t .$

This functor is sometimes called the contravariant hom-functor (associated to the object $b$ ).

Problem 10

Suppose $C$ is a category, $F : C \to Set$ is a functor and $r \in C$ is an object of $C$ , and let $H_{r} = {Hom}_{C} (r, -)$ be the hom-out functor associated to the object $r$ . Each natural isomorphism $α : H_{r} \tilde{\to} F$ is called a representation of the functor $F$ and a universal property for the object $r$ .

Suppose $α : H_{r} \tilde{\to} F$ and $β : H_{s} \tilde{\to} F$ are two natural isomorphisms.

Show there is a unique arrow $f : r \to s$ in $C$ such that $β = α \circ H (f)$ .^[1]
Show that the unique arrow $f$ in part (a) is an isomorphism; i.e., there is an arrow $g : s \to r$ in $C$ such that $g \circ f = 1_{r}$ and $f \circ g = 1_{s}$ .

Because of this result, we say that "representations of functors are unique up to unique isomorphism," and also "objects that satisfy a universal property are unique up to unique isomorphism."

Hints

For the first part, to show the existence first consider the natural isomorphism $α^{- 1} \circ β : H_{s} \tilde{\to} H_{r} .$ By Yoneda's Lemma, this corresponds to a unique element of the set $H_{s} (r) = {Hom}_{C} (r, s)$ ; i.e., a unique arrow $f : r \to s .$ Under this correspondence, the natural transformation $α^{- 1} \circ β$ is $H (f)$ , the "precompose with f" natural transformation. Uniqueness follows by a similar argument, since $H (f)$ is uniquely determined by the factorization $β = α \circ H (f)$ .

For the second part of the problem, switch the roles of $r$ and $s$ to show there is a unique arrow $g : s \to r$ such that $α = β \circ H (g)$ . But then $H (g) \circ H (f)$ is the identity natural transformation from $H_{s}$ to itself, and $H (f) \circ H (g)$ is the identity natural transformation from $H_{r}$ to itself. Yoneda's Lemma then guarantees that these most correspond to the identity arrows on $s$ and $r$ , respectively ...

This is if we're considering $H$ as a contravariant functor from $C$ to ${Set}^{C}$ . If we're considering $H$ as a functor from $C^{op}$ to $Set$ , then this should be $H (f^{op})$ . ↩︎