WHICH FUNCTOR IS THE PROJECTIVE LINE?

DANIEL K. BISS
1. Introduction
The question of what a mathematical object really is underlies most of the fundamental discussions
in the philosophy of mathematics, and subtle changes in the mathematical communitys opinion on
this issue have had monumental inuence over the direction that mathematical research has taken.
One basic tenet of the school of the past century is that a mathematical object is in some sense the
same as the information needed to encapsulate it. For example, we view a complex vector space V
C
as the same as a real vector space V
R
equipped with an automorphism J satisfying J
2
= Id. By
the same, we mean that it is possible to pass from one description to the other and back without
any loss of information. Indeed, given a complex vector space V
C
, we can view it as a real vector
space V
R
, and by letting J : V
R
V
R
be the automorphism dened by J(v) = i v, we obtain the pair
(V
R
, J), as desired. Conversely, if we start with the information (V
R
, J), we can dene a complex
structure on V
R
by declaring (a +bi) v = a v +b J(v). Thus, these two notions of complex vector
space convey precisely the same information.
More pedantically, a complex vector space is a collection of elements and a collection of rules
rules that dictate how to add vectors, and how to multiply a vector by a complex scalar. Any method
of writing down a particular set of rules gives the same vector space, whether that means writing
down a complete addition and multiplication table for V
C
, or rst specifying only the structure of a
real vector space V
R
, then declaring (via the automorphism J) how the complex number i will act on
V
R
, and nally letting the vector space axioms along with the fact i generates C over R do the rest
of the work. Naturally, vector spaces over C are not alone in this respect: almost all mathematical
objects have the (often extremely useful) feature that they can be described in several dierent ways.
Consider, for a moment, the two-element group Z/2. There is a seemingly endless list of ways to
specifyand, accordingly, studythis group. We might describe it, as our notation suggests, as the
quotient of Z by the ideal of even numbers. We could describe it in terms of generators and relations,
a[a
2
= e); or as the unique group of order 2; or as the Galois group Gal(C/R). We could, more
whimsically, describe it as the group whose set of elements is cabbages, kings with the following
multiplication table.
cabbages kings
cabbages cabbages kings
kings kings cabbages
However, in a move that might seem pig-headedly convoluted, let us instead consider the charac-
terization of Z/2 as the unique group having the property that for any group G the set of nonidentity
group homomorphisms from G to Z/2 is in one-to-one correspondence with the set of index 2 sub-
groups of G. Indeed, for a given nonidentity group homomorphism : G Z/2, the subgroup ker
of G has index 2, and conversely, given a subgoup H with [G : H] = 2, we get a homomorphism
G G/H

= Z/2. The aim of this article is to explain the principle that any mathematical object
whatsoever can be described in a similar way, and furthermore, that this observation can be used
to great prot in several areas of mathematics.
1
2 DANIEL K. BISS
Notice rst of all that, although any objectfor now, we consider only groupscan be specied
in such terms, these descriptions may be far less simple and elegant than the index 2 subgroup
classication just mentioned. Indeed, by merely passing from Z/2 to Z/3, we nd a substantial
increase in complexity: there are exactly two nonidentity group homomorphisms from G to Z/3 for
every normal subgroup H of G with [G : H] = 3. The appearance of the normality condition comes
from the fact that any subgroup of index 2 is automatically normal, whereas this is not the case for
index 3 subgroups. Also, there are two homomorphisms per subgroup rather than one because, once
we x the kernel H in G, it still remains to be decided which coset of H gets sent to 1 in Z/3 and
which coset goes to 2.
Furthermore, if we replace the groups Z/2 and Z/3 with a classical group C such as C = GL
n
C
or C = U(n), the problem of determining the group homomorphisms from arbitrary groups G to
C encompasses essentially all of representation theory! Thus, understanding a group explicitly via
an analysis of group homomorphisms into it can be exceedingly complicated. On the other hand,
this model can be quite convenient for studying relationships between groups. To give one simple
example, let C
1
and C
2
be two groups. Then although we may have no way of understanding
the set of group homomorphisms from an arbitrary group G to C
1
, denoted Hom(G, C
1
), or the
set Hom(G, C
2
) of homomorphisms from G to C
2
, it is very easy to see that Hom(G, C
1
C
2
) =
Hom(G, C
1
) Hom(G, C
2
). Accordingly, it is not so hard to imagine that this seemingly intractible
way of specifying groups might be very useful in studying products.
This philosophy represents the radical conclusion of a trajectory in mathematical fashion that
has existed for literally centuries, the move to abstraction and generality. For as long as Western
mathematics has been studied by an organized community, there has been a push to frame discussions
in a more and more conceptual way, to remove specics and excise unnecessary restrictions in
favor of the broad picture. From a semantic point of view, this movement reached its zenith with
the acceptance of category theory in mainstream mathematical parlancesuddenly, mathematical
objects were no longer objects at all in any concrete sense of the word, but rather elements of a
category. No longer are the descriptions of Z/2 given earlier anything more than crutches for the
human mind to cling to; instead, Z/2 is essentially identied with the symbol you see on this piece
of paper! But the phenomenon is more than semantic and psychological: the unexpected payo
of this strand of thought is that the language of category theory can in the end be used to create
new objects that turn out to be instrumental in studying the old ones. Ultimately, thisat rst
blushhopelessly abstract point of view has its say about the most concrete of issues. It is this
interplay between the abstract and the concrete that makes up the content of this article.
We begin, in the next section, by introducing the rudiments of the language of category theory.
We provide many fundamental examples, shying away from the more abstruse points of the theory;
the primary goal is to familiarize the reader with the notions of categories and to situate these ideas
in the standard undergraduate curriculum. A much more thorough discussion of these and related
matters is provided in [6]. In Section 3, we introduce the notion of functors and dene representable
and corepresentable functors, again with many examples, and explain how the Yoneda lemma allows
us to completely understand a category via representable functors. Next, in Section 4, we provide an
introduction to the category of nitely generated commutative C-algebras, and its connection with
algebraic geometry. We explain how to associate a geometric object with a C-algebra and discuss
briey the question of which geometric objects can be realized in this way. Lastly, in Section 5 we
indicate how passing from the category of C-algebras to an associated functor category allows us to
perform constructions that were impossible in the original category and therefore gives us greatly
increased exibility. Our main point is that the complex projective line CP
1
can be understood as
a functor on this category, but not as an object in it; thus, we give a careful characterization of the
functor represented by CP
1
.
WHICH FUNCTOR IS THE PROJECTIVE LINE? 3
In order to reach our main conclusion in Section 5, it was necessary to make some unorthodox
expository choices. In particular, we begin by dening C-algebras and then quickly move to the
construction of nonane complex varieties. An ordinary discussion would include a more compre-
hensive outline of the commutative algebra involved, as well as a precise denition of the category of
varieties over C, for which we have no room. Our strategy might seem to have some disadvantages:
this is quite a lot of material for the uninitiated to swallow, and it may indeed be unclear why we
bother working over C at all, for the Nullstellensatz is simply mentioned, without any indications of
a proof. On the other hand, this seemed like the only way to communicate the power of the objects
are functors point of view without assuming substantially more background, which was after all
our goal in writing this article.
2. Categories
What is the goal of group theory? Roughly speaking, it is to classify groups up to isomorphism.
That is, the theory of nitely generated abelian groups is more or less a closed subject, because we
have a well-understood list of isomorphism classes, as well as knowledge about how the list behaves
with respect to operations such as products, taking subgroups, forming quotients, and so on. By
contrast, even though all nite simple groups are known, and there is a theoretical way of building
up an arbitrary nite group out of simple ones, the theory is not considered nished, because there
is no practical way to construct a list of all isomorphism classes. Similarly, the goal of ring theory
is to classify rings up to isomorphism, the goal of eld theory is to classify elds up to isomorphism
(and, accordingly, the subject of nite elds is considered to be completely understood), the goal of
topology is to classify topological spaces, or manifolds, or CW complexes, up to homeomorphism or
homotopy equivalence, the goal of algebraic geometry is to classify algebraic varieties or schemes up
to isomorphism, and so forth. Throughout these markedly dierent areas of mathematics, the basic
theme is that we have some sort of object, a set with some kind of structure, we have a notion of
what it means for a map to preserve that structure, and therefore a notion of isomorphism, and our
goal is to classify the objects up to isomorphism. This leads us to make the following denition.
Denition 2.1. A category C consists of a collection Ob(C) of objects, and, corresponding to every
pair of objects A and B in Ob(C), a set Hom
C
(A, B) of morphisms. A morphism f in Hom
C
(A, B)
will often be denoted f : A B. For each object A, the set Hom
C
(A, A) contains a distinguished
identity morphism Id
A
. Furthermore, for every triple of objects A, B, and C from Ob(C), we assume
the existence of a composition map : Hom
C
(A, B) Hom
C
(B, C) Hom
C
(A, C). This data is
required to satisfy the following axioms:
Identity: For all objects A and B, and all morphisms f : A B, the equations f Id
A
= f
and Id
B
f = f are satised.
Associativity: For all objects A, B, C, and D, and all morphisms f : A B, g : B C,
and h : C D, the equality (h g) f = h (g f) holds.
This is in a sense the ultimate generalizationmost areas of mathematics can be squeezed into
this framework. A category is simply an abstract collection of things that we call objects; we are
not concerned with what the objects are or look like, just that they exist. All that we ask of them is
that we have a list of morphisms between all pairs of objects, and that someone give us a rule for
composing these morphisms. Although, given objects A and B of a category C and a morphism f in
Hom
C
(A, B), we use the suggestive notation f : A B, this does not mean that there exist elements
of A that are being mapped to elements of B according to some rule called f; it only means that
we have objects called A and B, and an arrow between them that we call f. (Nevertheless, old
notational habits do not die easily. In an abuse of language we will occasionally use the term map
as a synonym for morphism, especially in categories where the morphisms really are functions
4 DANIEL K. BISS
of one type or another.) In a sense, this is mathematics with the substance removed, or, as it is
sometimes derisively called, abstract nonsense. Indeed, the original intentions of category theory
were to provide a convenient language for expressing certain universal mathematical concepts, rather
than to furnish us with a tool that could actually be used in proving new theorems. We will now
give a series of examples that will hopefully demonstrate that the theory easily meets the primary
goal, and then spend the rest of the article sketching some instances in which category-inspired
constructions manage serendipitously to provide the groundwork for deep new mathematical ideas.
So, let us give some examples of categories. The rst example, and the example that will be most
important to us later on, is the category Set of sets. The objects of this category comprise all sets; for
any two sets X and Y , Hom
Set
(X, Y ) is the set of all functions f : X Y . Another category that
is important to us is the category Grp of groups. The objects of this category consist of all groups,
and for groups G and H, we dene Hom
Grp
(G, H) to be the set of all group homomorphisms from
G to H. Similarly, we can form the categories Rng of all rings with identity and identity-preserving
ring homomorphisms, Ab of all abelian groups and group homomorphisms, Fld of elds and eld
homomorphisms, and Top of all topological spaces and continuous maps. Moreover, given a eld
F or a ring R, we can consider the categories Vect
F
of all nite-dimensional F-vector spaces and
F-linear maps and R-Mod of all nitely generated R-modules and R-linear maps.
It is worth spending a moment to examine some of the similarities and dierences between the
various categories just described. For example, Fld has the rather unusual property that every
morphism in this category is an injective map of sets; it is easily seen that this fails for the other
examples mentioned. On the other hand, although any map between elds is one-to-one, it is
fairly dicult to write down all the maps between any given pair of elds. By contrast, if we
have two objects from the category Vect
F
(i.e., two F-vector spaces V and W of dimensions m
and n respectively), then Hom
Vect
F
(V, W) can be identied with M
nm
(F), the n m matrices
over F. This brings us to another point: given two morphisms in Hom
Vect
F
(V, W) (that is, given
two n m matrices over F), we can add them to obtain another linear map. This addition gives
Hom
Vect
F
(V, W) a group structure, something that is also satised in Ab and R-Mod but in none
of the rest of the categories we have listed.
Let us study another special feature satised by some of these categories. Consider rst the
object Z in the category Rng. This object has a rather amazing property: for any ring R, the set
Hom
Rng
(Z, R) consists of just one morphism; namely, the homomorphism f : Z R dened by
f(1) = 1
R
and hence satisfying f(n) = n1
R
, the element obtained by adding the element 1
R
to itself
n times. An object satisfying this condition is said to be initial; that is, an object A in a category
C is initial if for every object X in C the set Hom
C
(A, X) has exactly one element. Similarly, the
object A is said to be terminal if for any object X the set Hom
C
(X, A) has exactly one element. The
categories Ab, Grp, Vect
F
, and R-Mod each have an object that is both initial and terminal. The
objects are, respectively, the trivial group, the trivial group, the trivial (zero-dimensional) vector
space, and the trivial module. The categories Set and Top have both initial and terminal objects,
but the two are distinct: specically, the empty set (or space) is the initial object in both categories,
and the one-point set (or space) is the terminal object. As noted, Rng has an initial object Z, but it
has no terminal object. Indeed, suppose that R is a terminal object. Then for every prime number
p, the set Hom
Rng
(Z/p, R) must have precisely one element, say f
p
: Z/p R. But f
p
(1) = 1
R
, so
p 1
R
= p f
p
(1) = f
p
(p 1) = f
p
(0) = 0
R
,
and therefore p 1
R
= 0 for all p. Hence, in particular, 2 1
R
= 0 = 3 1
R
, and subtracting, we nd
that 1
R
= 0
R
, which contradicts the denition of a ring with identity. (Notice that this argument
actually only required the existence of homomorphisms f
p
: Z/p R for two distinct primes to
obtain a contradiction; the uniqueness of the f
p
was not exploited. Thus, Rng does not even have
an object that is a universal head for morphism arrows.) For similar reasons, the category Fld has
WHICH FUNCTOR IS THE PROJECTIVE LINE? 5
no terminal object, but the situation there is even worseit has no initial object either. Indeed, every
eld has a characteristic (either a prime p or else 0), and if two elds have dierent characteristics,
then there can be no morphisms between them, so no eld can be initial. In the subcategory of elds
of characteristic p, however, F
p
is an initial object, and in the subcategory of elds of characteristic
0, the initial object is Q.
Before we go on, there is one more denition we must make. Recall that a category is meant to
be an abstraction of a mathematical theory in which there are objects we would like to classify up to
some notion of isomorphism. Accordingly, we would like to introduce the denition of isomorphism
in an arbitrary category.
Denition 2.2. Let C be a category, and let A and B be objects in C. They are said to be isomorphic
if there are morphisms f : A B and g : B A in Hom
C
(A, B) and Hom
C
(B, A), respectively,
such that f g = Id
B
and g f = Id
A
. In this case, f and g are said to be isomorphisms.
It is useful to try to understand why this denition corresponds to our usual notion of isomorphism
in the examples that we have provided. Furthermore, a good exercise to give the reader some practice
in navigating these denitions is to show that initial and terminal objects in any category are unique
up to isomorphism. Later, we will give a more sophisticated explanation for this fact, but it would
be instructive for those unfamiliar with category theory to attempt to demonstrate it directly.
3. Functors and representability
A category is supposed to be a model for a mathematical theory, such as the theory of groups
or rings. Although categories already represent a valuable conceptual tool, most of the signicant
applications of the categorical point of view arise when we attempt to compare one category with
another. Indeed, the underlying principle of category theory is that a mathematical discipline ought
to be made up of some type of object to be studied and a family of morphisms between objects,
preserving some sort of structure. It is only in keeping with this basic philosophy that we introduce
maps between categories.
Denition 3.1. For categories C and D, a covariant functor F : C D consists of a map
F : Ob(C) Ob(D) and, for all A and B in Ob(C), a map F : Hom
C
(A, B) Hom
D
(F(A), F(B)),
such that the following axioms hold:
(1) For any A in Ob(C), F(Id
A
) = Id
F(A)
.
(2) For any three members A, B, and C of Ob(C) and any two morphisms f : A B and
g : B C, F(g) F(f) = F(g f) : F(A) F(C).
Here, we use the symbol F to mean several dierent things; hopefully no confusion will ensue.
Roughly speaking, a covariant functor from C to D is a rule that assigns to each object of C
an object of D and to each morphism in C a morphism in D. Occasionally, however, we will need
to consider functor-like rules that assign to each morphism in C a morphism in D heading in the
opposite direction.
Denition 3.2. For categories C and D, a contravariant functor F : C D consists of a map F :
Ob(C) Ob(D) and, for all pairs A and B in Ob(C), a map F : Hom
C
(A, B) Hom
D
(F(B), F(A)),
such that the following axioms hold:
(1) For any A in Ob(C), F(Id
A
) = Id
F(A)
.
(2) For any three members A, B, and C of Ob(C) and any two morphisms f : A B and
g : B C, F(f) F(g) = F(g f) : F(C) F(A).
Because of the unwieldy nature of the words covariant and contravariant, we often omit these
prexes from the term functor. Once again, this should not be the source of much confusion.
6 DANIEL K. BISS
Let us now give some examples of functors to illustrate how they work. The rst several examples
all fall under the heading of forgetful functors. That is, if we have two categories, one of which
has more structure than the other, then we obtain a functor from the rst category to the second
simply by forgetting the extra structure. For example, a group G is a set with some extra algebraic
properties (namely, a multiplication law). By ignoring the product and simply considering G to be a
set, we obtain a functor U : Grp Set. This is actually a functor because any group homomorphism
is also a morphism of sets, the identity map of a group is also the identity map of the underlying set,
and the composition law for group homomorphisms is the same as the composition law for maps of
sets.
There are many such forgetful functors among the various categories discussed so far. For example,
by forgetting the multiplicative structure of a ring or eld, we get functors Rng Ab and Fld
Ab. Similarly, forgetting the scalar multiplication gives functors R-Mod Ab and Vect
F
Ab.
Moreover, by ignoring all of the algebraic structures involved, we obtain functors from any of these
categories to Set; by the same token, forgetting the topology on a set induces a functor Top Set.
Notice that functors need not behave well with respect to most properties of these categories; for
example, the initial object Z of Rng is sent to a rather unremarkable object in Set when we forget
the ring structure. Also, objects that are not isomorphic in one category might become isomorphic
when a functor is applied. For example, the groups S
3
and Z/6 have isomorphic images under the
forgetful functor U : Grp Set, as do the rings Z/2 Z/2 and F
4
under the functor U : Rng Ab.
Nonetheless, a systematic study of the functors between various categories will give us substantial
power in understanding the categories themselves.
Before going on to discuss the most crucial instances of functors that we will need, let us return
momentarily to the example with which we began. Recall that we explained how a complex vector
space V
C
could be identied with a real vector space V
R
equipped with a linear transformation
J : V
R
V
R
satisfying J
2
= Id. Let us see how to cast this discussion in the categorical framework
we have created. Viewing a complex vector space V
C
as a real vector space V
R
corresponds to applying
a forgetful functor U : Vect
C
Vect
R
that causes us to forget how to multiply by i, since the
category Vect
R
knows nothing about imaginary numbers. Thus, the automorphism J is a convenient
way of reminding us of what we have forgotten in this process.
This story can, of course, be generalized. Indeed, let a eld K be a nite separable extension of
a eld F. Then once again we have a forgetful functor U : Vect
K
Vect
F
, because given a rule for
scalar multiplication by elements of K, we get scalar multiplication by elements of F for free. Let
us try to express what we have forgotten by applying this functor. Naturally, we have forgotten the
rules for scalar multiplication by elements of K that do not happen to lie in F. Fortunately, it is not
dicult to enumerate the set KF. Indeed, recall that there exists an element in K generating K
over F, meaning that the only subeld of K containing both and all of F is K itself. Equivalently,
every element of K can be expressed as a polynomial in with coecients in F. Thus, to list the
elements of K, it suces to write down polynomials in over F.
Let f in F[x] be the minimal polynomial of ; that is, f(x) = x
n
+ b
n1
x
n1
+ + b
0
is an
irreducible polynomial with coecients b
i
in F, and f() = 0. Then we have a homomorphism
: F[x]/(f(x)) K dened by (g(x)) = g(), and it is an isomorphism. Indeed, it is one-to-one
because any polynomial with coecients in F having as a zero must be a multiple of f and hence
lie in the ideal (f(x)), and it is onto because its image contains both and all of F. Therefore, any
element of K has a unique representation of the form c
n1

n1
+ +c
1
+c
0
with c
i
in F. As a
result, given a K-vector space V
K
, if we apply the forgetful functor U and view it as an F-vector space
V
F
, then to recover the information we have lost, it suces to recall the rule for scalar multiplication
by . Furthermore, the only condition we need to place on this scalar multiplication rule is that it
WHICH FUNCTOR IS THE PROJECTIVE LINE? 7
satises the polynomial f, since itself satises the equation f() = 0 but no polynomial of lower
order. Summarizing, we have the following.
Proposition 3.3. Let F K be a nite extension of elds such that K is generated over F
by an element whose minimial polynomial is f(x) = x
n
+ b
n1
x
n1
+ + b
0
. Then a K-
vector space V
K
is equivalent to an F-vector space V
F
along with a map A : V
F
V
F
satisfying
A
n
+b
n1
A
n1
+ +b
1
A = b
0
.
Notice that when we replace the data (F, K, , f, A) by the special case of (R, C, i, x
2
+ 1, J), we
recover precisely the statement alluded to in the introduction.
We now proceed to the primary construction that will occupy us for the rest of this article. Recall
one last example given in the introduction, in which we described Z/2 as the unique group with the
ensuing property: for any group G, the set of nontrivial homomorphisms from G to Z/2 is equal to
the set of index two subgroups H of G. This leads to the idea that we might understand an object in
a category by enumerating the morphisms into or out of it. More precisely, let us make the following
denition.
Denition 3.4. Let C be a category and A an object of C. Then the functor represented by A is
the covariant functor y
A
: C Set dened as follows:
(1) For all X in Ob(C), y
A
(X) = Hom
C
(A, X).
(2) For all morphisms f : X Y, y
A
(f) : Hom
C
(A, X) Hom
C
(A, Y ) is post-composition with
f, i.e., (y
A
(f))(g) = f g.
Similarly, the functor corepresented by A is the contravariant functor y
A
: C Set dened by:
(1) For all X in Ob(C), y
A
(X) = Hom
C
(X, A).
(2) For all morphisms f : X Y, y
A
(f) : Hom
C
(Y, A) Hom
C
(X, A) is pre-composition with
f, i.e., (y
A
(f))(g) = g f.
Thus, the assertion that Z/2 is the unique group such that for all groups G the set Hom
Grp
(G, Z/2)
is made up of the trivial homomorphism together with a family of homomorphisms indexed by the
subgroups of G of index 2 is equivalent to the statement that there is no other group H with
y
H
= y
Z/2
. This fact can be generalized to a famous lemma of Yoneda. To state the lemma, we
need to make one more denition concerning functors.
Denition 3.5. Let F, G : C D be two covariant (respectively, contravariant) functors. A
morphism of these functors is a family of morphisms
A
: F(A) G(A), one for each object A in
C, such that for all morphisms f : A B in C it is the case that G(f)
A
=
B
F(f) : F(A) G(B)
(respectively,
A
F(f) = G(f)
B
: F(B) G(A)).
This denition of morphisms actually makes the collection of covariant or contravariant functors
from C to D into a category itself. Notice now that, if we have two objects A and B in C and a
morphism f : A B, then we get an induced morphism of functors y
f
: y
B
y
A
(respectively,
y
f
: y
A
y
B
) by pre-composing (respectively, post-composing) with f. Finally, we are ready to
present the Yoneda lemma [10].
Lemma 3.6 (Yoneda). Suppose A and B are two objects in a category C, and let : y
B
y
A
be
any morphism of functors. Then there is a morphism f : A B such that = y
f
. Similarly, if
: y
A
y
B
is a morphism of functors, then there exists a morphism g : A B such that = y
g
.
This implies, in particular, that if two objects represent (or corepresent) isomorphic functors,
then the objects themselves are isomorphic. This has the immensely powerful upshot that an object
can be studied simply by analyzing the functor that it represents (or corepresents).
Before going on to use these ideas to construct new objects, let us take some time to familiarize
ourselves with the notion of representable functors in action. Suppose rst that A is an initial
8 DANIEL K. BISS
object in a category C. Then, for any object X, we have y
A
(X) = Hom
C
(A, X), which is a one-
point set. Thus, an initial object represents the functor that sends each object to a singleton;
in particular, the Yoneda lemma then immediately tells us that initial objects are unique up to
isomorphism, as we indicated earlier. Similarly, if B is a terminal object in C, then for any X we
have y
B
(X) = Hom
C
(X, B), so the functor corepresented by B is the functor sending each object
to a singleton. By Yonedas lemma, terminal objects are also unique up to isomorphism.
Returning to yet another example given in the introduction, we consider groups C
1
and C
2
. For
any group G, we have y
C1C2
(G) = Hom
Grp
(G, C
1
C
2
) = Hom
Grp
(G, C
1
) Hom
Grp
(G, C
2
) =
Y
C1
(G) y
C2
(G). Thus, y
C1C2
= y
C1
y
C2
. Moreover, a similar statement holds in any category
in which there is a notion of products, such as the category of sets, rings, or modules over some ring;
in fact, in an arbitrary category, this is the way to dene products!
For a more sophisticated example, the fundamental group
1
is a functor from the category of
pointed topological spaces to Grp that is represented in the homotopy category by the circle S
1
.
This is part of a very broad-reaching theme in algebraic topology in which functors represented and
corepresented by various spaces are used in analyzing geometry and topology. In fact, essentially
every invariant studied by topologists can be made into a representable or corepresentable functor.
The goal of the rest of this article, however, is not to use representable functors to provide a tool for
studying categories. Rather, we will begin with a category that seems interesting but is somehow
smaller than one would like it to be. We will construct functors that are almost representable,
which we will view as enlargements of the category itself. In this way, the notions of category and
functor will furnish us with a beautiful technique for constructing mathematical objects that might
not be otherwise attainable.
4. C-algebras, geometry, and functors
In the rest of this article, we present an example in which viewing the collection of functors from
a category C to the category of sets as an expansion of C can be extremely useful. In order to do
so, we must introduce a new category, the category C-Alg. An object of this category is a nitely
generated commutative C-algebra, that is, a commutative ring R that contains the complex numbers
C as a distinguished subring. The hypothesis of nite generation simply means that there is a nite
set of elements x
1
, . . . , x
n
of R such that no proper subring of R contains C x
1
, . . . , x
n
. From
time to time, we will have to make reference to standard results from commutative algebra in our
study of the category C-Alg; such theorems can be found in any standard commutative algebra text,
such as [1] or [2]. For a more leisurely discussion of the basic notions of algebraic geometry that we
sketch, the reader is encouraged to consult [4], [5], or [8].
A morphism of C-algebras R
1
and R
2
is a ring homomorphism f : R
1
R
2
that is the identity
on the subring C. For example, C is a C-algebra, but the conjugation map a+bi abi, which is a
ring homomorphism from C to itself, is not a C-algebra homomorphism, because it is not the identity
on C. On the other hand, the polynomial ring C[x] in one variable over C is also a C-algebra, and for
any polynomial f(x), the map F
f
: C[x] C[x] sending x to f(x) is a C-algebra map. In particular,
for any polynomial in C[x], we have F
f
() = (f(x)). Likewise, for any complex number z, the
map e
z
: C[x] C dened by e
z
() = (z) is a C-algebra map.
Let us now introduce some objects of C-Alg and examine the functors they represent. Consider
rst the object C. For any C-algebra R, all C-algebra maps f : C R must be the identity on
C; therefore there is a unique such map. Hence, C is an initial object in C-Alg. The next easiest
example to study is C[x]. Again, let R be any C-algebra, and let f : C[x] R be a C-algebra map.
Then f is the identity on C, and f(x) is some element of R. Furthermore, let be an arbitrary
element of C[x]. By denition, we must have = a
n
x
n
+a
n1
x
n1
+ +a
1
x+a
0
for some complex
WHICH FUNCTOR IS THE PROJECTIVE LINE? 9
numbers a
0
, a
1
, . . . , a
n
and some integer n. Now, we compute
f() = f(a
n
x
n
+ +a
1
x +a
0
)
= f(a
n
)f(x)
n
+ +f(a
1
)f(x) +f(a
0
)
= a
n
f(x)
n
+ +a
1
f(x) +a
0
.
Therefore, the value of the map f at is determined entirely by the element f(x) if R. Hence,
for each element of r in R, we get a unique map f
r
: C[x] R dened by setting f
r
(x) = r. In
other words, Hom
C-Alg
(C[x], R) = R, or, to be completely precise, the object C[x] represents the
forgetful functor U : C-Alg Set sending a C-algebra to its underlying set: y
C[x]
= U.
Similarly, a C-algebra map f : C[x, y] R is determined by the two elements f(x) and f(y) of
R, via the formula
f
_
_
n

i=1
m

j=1
a
ij
x
i
y
j
_
_
=
n

i=1
m

j=1
a
ij
f(x)
i
f(y)
j
,
so we have Hom
C-Alg
(C[x, y], R) = R R. Precisely speaking, the object C[x, y] represents the
functor sending a C-algebra R to the underlying set of RR. More generally, for any positive integer
n, a C-algebra map f : C[x
1
, . . . , x
n
] R is determined by the n elements f(x
1
), . . . , f(x
n
) of R,
whence Hom
C-Alg
(C[x
1
, . . . , x
n
], R) = R
n
. Thus, the object C[x
1
, . . . , x
n
] represents the functor
sending a C-algebra R to the underlying set of R
n
. We can express this in symbols by letting
P
n
: Set Set signify the functor taking a set X to X
n
. Then y
C[x1,...,xn]
= P
n
U.
There is more to be said. Let I be an ideal in C[x
1
, . . . , x
n
]. Then we may form the C-algebra
C[x
1
, . . . , x
n
]/I. First, suppose for simplicity that n = 1, so we are considering the algebra C[x]/I.
Let x in C[x]/I denote the element represented by x. Then for any C-algebra R, much like before, it
is the case that a C-algebra map f : C[x]/I R is determined by the element f( x) of R. Indeed, a
general element of C[x]/I is of the form a
n
x
n
+ +a
1
x+a
0
, and we have f(a
n
x
n
+ +a
1
x+a
0
) =
a
n
f( x)
n
+ +a
1
f( x) +a
0
. Thus, we might be led to believe that C[x]/I, like C[x], represents the
forgetful functor sending a C-algebra R to its underlying set.
However, this is not the case (which, incidentally, is fortunate, since the Yoneda lemma would
otherwise imply that C[x]

= C[x]/I, which is simply false). Indeed, let belong to I, and suppose
that f : C[x]/I R is a map with f(x) = r. Then we have 0 = f(0) = f(

) = (r). Thus, for all

k
i=1

i
with
i
in C[x], and so (r) =

k
i=1

i
(r)
i
(r). Hence, if
i
(r) = 0
for all i, then (r) = 0. In other words, Hom
C-Alg
(C[x]/I, R) = r R[
i
(r) = 0 for all i. It is
not dicult to see that this result has the following generalization.
Proposition 4.1. Let I be the ideal in C[x
1
, . . . , x
n
] generated by the set
1
, . . . ,
k
. Then for
every C-algebra R, we have
Hom
C-Alg
(C[x
1
, . . . , x
n
]/I, R) = (r
1
, . . . , r
n
) R
n
[(r
1
, . . . , r
n
) = 0 for all I
= (r
1
, . . . , r
n
) R
n
[
i
(r
1
, . . . , r
n
) = 0 for i = 1, . . . , k.
The goal of this section is to indicate why we might want to have at our disposal functors from
C-Alg to Set that are not representable. To this end, we seek an alternate, more geometric way
of viewing C-algebras. Let us start with the universal example C[x]. This C-algebra, more or less
by denition, can be thought of as the algebra of polynomial functions on the space C of complex
10 DANIEL K. BISS
numbers. By the same token, for any positive integer n, the ring C[x
1
, . . . , x
n
] consists of all
polynomial functions on the space C
n
. Indeed, given any n-tuple of complex numbers (z
1
, . . . , z
n
)
and any polynomial (x
1
, . . . , x
n
) in C[x
1
, . . . , x
n
], we may substitute the complex numbers z
i
for the
indeterminates x
i
to obtain a complex number (z
1
, . . . , z
n
). Thus, it is perhaps not too far-fetched
to think of C[x
1
, . . . , x
n
] as somehow corresponding to the set C
n
.
The next example of a C-algebra we consider is the algebra R
I
= C[x
1
, . . . , x
n
]/I for an ideal
I in C[x
1
, . . . , x
n
]. Now, given an element
1
of C[x
1
, . . . , x
n
], we obtain by reducing modulo I an
element
1
of R
I
. To continue in the same vein as before, we would like to view
1
as a polynomial
on C
n
, by evaluating
1
(z
1
, . . . , z
n
) =
1
(z
1
, . . . , z
n
). Unfortunately, there are many elements
2
in C[z
1
, . . . , z
n
] with
1
=
2
, and there is no guarantee that
1
(z
1
, . . . , z
n
) =
2
(z
1
, . . . , z
n
), so
this function might not be well-dened. However, all is not lost. If
1
=
2
, then we know that

1
=
2
+ for some from I, so
1
(z
1
, . . . , z
n
) =
2
(z
1
, . . . , z
n
)+(z
1
, . . . , z
n
). Thus, if it happens
to be the case that (z
1
, . . . , z
n
) = 0 for all in I, then we have
1
(z
1
, . . . , z
n
) =
2
(z
1
, . . . , z
n
).
With this in mind, we let Z(I) be the subset of C
n
dened by
Z(I) = (z
1
, . . . , z
n
) C
n
[(z
1
, . . . , z
n
) = 0 for all I.
The upshot of the above discussion is that, if I is any ideal in C[x
1
, . . . , x
n
], we may view R
I
=
C[x
1
, . . . , x
n
]/I as a ring of polynomial functions on the set Z(I) contained in C
n
. In the same sense
that the set C
n
corresponded to the C-algebra C[x
1
, . . . , x
n
], we can think of Z(I) as corresponding
to R
I
.
For example, let I = (xy) in C[x, y]. Then Z(I) = (z
1
, z
2
) C
2
[z
1
z
2
= 0 = (z
1
, z
2
) C
2
[z
1
=
0 or z
2
= 0. That is, Z(I) consists of the coordinate axes in C
2
. Moreover, an element (x, y) in
C[x, y]/I must be of the form
(x, y) =
m

i=0
n

j=0
a
ij
x
i
y
j
for some complex numbers a
ij
. But if i 1 and j 1, then x
i
y
j
belongs to (xy) = I, implying that
a
ij
x
i
y
j
= 0. Thus, we can actually express in the form
(x, y) =
_
m

i=1
b
i
x
i
_
+
_
_
n

j=1
c
j
y
j
_
_
+ a
0
.
In other words, must be the sum of a polynomial on the x-axis that vanishes at x = 0, a polynomial
on the y-axis vanishing at y = 0, and a scalar. Basically, the restriction of to the x-axis can be any
polynomial
x
, and the restriction to the y-axis may be any polynomial
y
, so long as
x
(0) =
y
(0),
which is of course necessary for the function to be well-dened at the origin.
In any case, we now have a procedure that associates to an ideal I in C[x
1
, . . . , x
n
] the subset
Z(I) of C
n
. Recall that I is said to be a radical ideal if an element of C[x
1
, . . . , x
n
] lies in I
whenever
r
belongs to I for some r 1. It follows from Hilberts Nullstellensatz [1, ch. 5] that if
we restrict attention to radical ideals I, our correspondence
C[x
1
, . . . , x
n
]/I Z(I) C
n
is one-to-one. Here it is essential that we work over C; otherwise the Nullstellensatz need not hold.
Moreover, given radical ideals I and J in C[x
1
, . . . , x
n
] and C[x
1
, . . . , x
m
] respectively, it is possible
to interpret a map f : R
I
= C[x
1
, . . . , x
n
]/I C[x
1
, . . . , x
m
]/J = R
J
of C-algebras in terms of
these subsets. Indeed, R
I
and R
J
are made up of polynomial functions on the two sets Z(I) and
Z(J). Suppose that we have a polynomial map : Z(J) Z(I) and an element in R
I
. Then
is actually a polynomial function : Z(I) C, and by composing with we obtain a polynomial
: Z(J) C, in other words, an element of R
J
. Thus the polynomial map gives rise to a ring
WHICH FUNCTOR IS THE PROJECTIVE LINE? 11
homomorphism f

: R
I
R
J
. It can be shown that all homomorphisms from R
I
to R
J
arise in this
manner. That is, one can show that in this way we obtain a one-to-one correspondence between
Hom
C-Alg
(R
I
, R
J
) and the set of polynomial maps : Z(J) Z(I).
Now that we have a description of the category C-Alg in terms of subsets of complex vector
spaces, it is worth examining with a little more care what types of geometric objects can be realized
in this fashion. Consider rst the set U = C0; let us try to nd a commutative C-algebra R to
which it corresponds. In other words, R should have the property that, for any positive integer n
and any ideal I in C[x
1
, . . . , x
n
], the set / of polynomial maps from Z(I) to U is in one-to-one
correspondence with the set Hom
C-Alg
(R, R
I
). Now /is precisely the set of polynomial maps from
Z(I) to C that avoid the value 0; in other words, it is the set of invertible polynomial maps from
Z(I) to C. Here, we say that a polynomial map is invertible if there is another polynomial map

1
: Z(I) C such that (p)
1
(p) = 1 holds for all p in Z(I). We have already seen that the set of
all polynomial functions from Z(I) to C is the set Hom
C-Alg
(C[x], R
I
) = R
I
. It is not hard to check
that multiplication of polynomial maps as just described corresponds to ordinary multiplication on
R
I
. Hence, the set of invertible functions of this type is precisely the set of invertible elements in R
I
.
Equivalently, / is equal to the subset of Hom
C-Alg
(C[x], R
I
) consisting of all ring homomorphisms
f such that f(x) is a unit in R
I
.
Roughly speaking, the set / behaves as though x were a unit of C[x], since it consists of all
maps out of C[x] sending x to an invertible element. To make this precise, consider the ideal
J = (xy 1) in C[x, y] and the ring R
J
= C[x, y]/J. In passing from C[x] to R
J
, we have rst
adjoined a free variable y, and then decreed that y be the multiplicative inverse of x. In other
words, R
J
is the C-algebra obtained by declaring that x in C[x] must be invertible. Hence, for
any C-algebra S, the set Hom
C-Alg
(R
J
, S) is equal to the set of units of S. Indeed, consider a
homomorphism f : R
J
S. Then f is determined by the elements f( x) and f( y) of S. Moreover,
we have the equation f( x)f( y) = f( x y) = 1. Hence, f( x) must be a unit in S, and f( y) (and
hence all of f) is determined by f( x) via the uniqueness of inverses. Thus, we have established that
/ = Hom
C-Alg
(R
J
, R
I
), so the functor we are interested in is given by
R
I
Hom
C-Alg
(C[x, y]/(xy 1), R
I
).
Restated, this tells us that R
J
represents the functor that takes R
I
to the set of polynomial
functions from Z(I) to U. What, then, is the connection between Z(J) and U? Well, Z(J) is precisely
the subset of C
2
consisting of all pairs (z
1
, z
2
) with z
1
z
2
= 1. But the map f : U Z(J) dened by
f(z) = (z, z
1
) is a bijective rational map. Thus, U and Z(J) are isomorphic in the category whose
objects are subsets of complex vector spaces and whose morphisms are rational functions.
There is another way to interpret this situation. The C-algebra C[x] corresponds to the space C
because it consists of all polynomial maps from C to C. Similarly, the C-algebra Z(J) is supposed to
consist of all polynomial functions from U to C. Of course, any element of C[x] gives such a function.
However, there are some additional functions; for example, the map z z
1
, which is obviously
not dened on all of C, is well-dened on U. Therefore, Z(J) should be isomorphic to the C-algebra
obtained from C[x] by formally adjoining the element x
1
. This simply reects the isomorphism
f : C[x, y]/(xy 1) C[x, x
1
] dened by f(x) = x and f(y) = x
1
.
However, the story grows stickier rather quickly. To illustrate this, consider the the set V =
C
2
(0, 0). If this space could be realized in the form Z(I) for some ideal I in C[x
1
, . . . , x
n
], then
by making slight modications in the foregoing discussion, we would be able to recover R
I
by formally
inverting the polynomials in C[x, y] that take nonzero values away from the origin (0, 0) in C
2
. But
all such polynomials are constant, and hence already invertible. Indeed, consider C[x, y] and
12 DANIEL K. BISS
assume that is not constant. Then
(x, y) =
m

i=0
n

j=0
a
ij
x
i
y
j
for some complex numbers a
ij
, not all 0. Reordering the sum gives us
(x, y) =
m

i=0

i
(y)x
i
,
where
i
(y) =

n
j=0
a
ij
y
j
. If
i
(y) = 0 for all i > 0, then (x, y) =
0
(y). By assumption,
0
(y)
is not constant, so it has some root z, and then (1, z) is a root of (x, y). On the other hand, if

i
(y) ,= 0 for some i > 0, then for some nonzero z in C the complex number
i
(z) is nonzero. Hence,
plugging in the value z for the y variable, we obtain a nonconstant polynomial (x, z) in x, which
must have some root w. Then (w, z) is a root of (x, y). Therefore, any nonconstant polynomial
has a root away from the origin (0, 0), and so R
I

= C[x, y], which is plainly a contradiction. This
dashes any hope of realizing V as a space Z(I).
Hence, given a subset of C
n
, the question of whether it can be expressed in the form Z(I) is a
subtle one. However, we now leave this question and address a more far-reaching issue, namely, the
construction of a space that is not a subset of a complex vector space. We will manage to produce
a natural geometric object that can not be embedded in C
n
for any n, but does nevertheless have a
concrete description as a (nonrepresentable) functor C-Alg Set.
5. Complex projective space, the functor
We are nally ready to present the example that is the payo for all our work, a geometric object
whose algebraic description is mostly easily conveyed as a functor C-Alg Set. The object in
question will be one-dimensional complex projective space CP
1
. This is dened to be the space of
nonzero vectors (z
1
, z
2
) in C
2
, modulo the equivalence relation (z
1
, z
2
) (z
1
, z
2
) for all nonzero
complex numbers . Equivalently, CP
1
is the set of one-dimensional complex vector subspaces of
C
2
. Surely it is believable that CP
1
is an object in which geometers might have interest; indeed, it
and its higher-dimensional analogues play fundamental roles in algebraic and dierential geometry,
as well as in topology. In order to see how this new creature CP
1
ts into the algebraic framework
that we have constructed, it will be necessary to understand it more explicitly.
Given a nonzero vector (z
1
, z
2
) in C
2
, we denote its equivalence class in CP
1
by [z
1
, z
2
]. Notice that
if z
2
,= 0, then we have [z
1
, z
2
] = [z
1
/z
2
, 1] . Thus, denoting by U
1
the subset of CP
1
that comprises
all elements [z
1
, z
2
] with z
2
,= 0, we have an isomorphism
1
: U
1
C given by
1
([z
1
, z
2
]) = z
1
/z
2
.
Similarly, we denote by U
2
the subset of CP
1
consisting of all elements [z
1
, z
2
] with z
1
,= 0; this set
is also isomorphic to C, via the map
2
([z
1
, z
2
]) = z
2
/z1. Since U
1
U
2
= CP
1
, we have covered
CP
1
by two subsets that we understand very well. Indeed, CP
1
= U
1
[1, 0] ; for this reason, CP
1
is often described as a copy of C (namely, U
1
) together with a point at innity (namely, [1, 0]).
It will also be important for us to understand the intersection U
1
U
2
. We have
1
(U
1
U
2
) =
C0, since
1
([z
1
, z
2
]) = z
1
/z
2
and z
1
,= 0 on U
1
U
2
. Similarly,
2
(U
1
U
2
) = C0. Thus, we
have two isomorphisms

1
,
2
: U
1
U
2
C0
satisfying
1
= 1/
2
.
We are nally ready to address the functor P
1
determined by CP
1
. This is the functor C-Alg
Set that takes a C-algebra R
I
= C[x
1
, . . . , x
n
]/I to the set P
1
(R
I
) of polynomial maps from the
corresponding subset Z(I) of C
n
to CP
1
. It is this set P
1
(R
I
) that we now undertake to study. Fix
WHICH FUNCTOR IS THE PROJECTIVE LINE? 13
a polynomial map : Z(I) CP
1
, and set
Z
1
= p Z(I)[(p) U
1

and
Z
2
= p Z(I)[(p) U
2
.
Then we have have two polynomial maps
1
: Z
1
C and
2
: Z
2
C dened by
Z
i
i
U
i
i
C
for i = 1, 2. Since
1

2
1, these maps are related by the property that
1

2
1 on the set
Z
1
Z
2
.
We now need to nd an algebraic expression of the maps
1
and
2
. For an element f of R
I
, let
Z
f
denote the subset of Z(I) consisting of all points p with f(p) ,= 0. Then as we saw in the last
section, the ring of functions on Z
f
can be described as the ring R
f
obtained from R by inverting f.
Thus, if Z
f
is contained in U
1
, then the map
1
[
Z
f
is necessarily of the form g/f
n
for some element
g of R and some nonnegative integer n. It is a basic result of algebraic geometry that U
1
can be
covered by a nite collection of subsets Z
f1
, . . . , Z
fs
(see [4]). We therefore have

1
[
Z
f
j
=
g
j
f
nj
j
for elements g
j
of R and nonnegative integers n
j
. Furthmore, setting
n = maxn
1
, . . . , n
s
,
and replacing g
j
by f
nnj
j
g
j
, we may assume that

1
[
Z
f
j
=
g
j
f
n
j
.
Moreover, by construction, it must be the case that for every pair j and k, the two maps g
j
/f
n
j
and g
k
/f
n
k
agree when restricted to Z
fj
Z
f
k
. In other words, the equation
g
j
f
n
k
g
k
f
n
j
= 0
must hold on Z
fj
Z
f
k
, or equivalently, in the ring R
fjf
k
. This is not quite the same as demanding
that this equation be satised in R, since the act of inverting the elements f
j
and f
k
to create
the ring R
fjf
k
can aect the multiplication somewhat. More specically, since the element f
j
f
k
is
now a unit, if x belongs to R and if there exists a nonnegative integer m such that the equation
(f
j
f
k
)
m
x = 0 holds in R, then dividing both sides by (f
j
f
k
)
m
demonstrates that x itself must be
zero in R
fjf
k
. In fact, one can show that this is the only additional relation obtained by forming
R
fjf
k
. Thus, returning to our situation, the compatibility criterion implies that we must have a
relation
(5.1) (f
j
f
k
)
m
jk
_
g
j
f
n
k
g
k
f
n
j
_
= 0
for m
jk
a suciently large nonnegative integer. Once again, if we set
m = max
1j,ks
m
jk
,
then equation (5.1) is satised with m
jk
replaced by m.
Hence
1
: Z
1
C is specied by the data
_
g
1
f
n
1
, . . . ,
g
s
f
n
s
, m
_
where m is a nonnegative integer making equation (5.1) hold for all j and k with 1 j, k s.
14 DANIEL K. BISS
Similarly, the map
1
: Z
2
C is determined by the information
_
G
1
F
N
1
, . . . ,
G
S
F
N
S
, M
_
where N is a nonnegative integer, S is a positive integer, F
j
and G
j
are elements of R for all
j = 1, 2, . . . , S, and M is a nonnegative integer such that for all j and k satisfying 1 j, k S, the
relation
(5.2) (F
j
F
k
)
M
_
G
j
F
N
k
G
k
F
N
j
_
= 0
holds. In order that
1
and
2
glue together to furnish us with a map : Z(I) CP
1
, they must
satisfy

1

2
[
Z1Z2
1.
That is, for whenever 1 j s and 1 k S, the element
_
g
j
/f
n
j
_

_
G
k
/F
N
k
_
must be the constant
function 1 on Z
fj
Z
F
k
. Equivalently, there must exist a nonnegative integer p
jk
such that
(5.3) (f
j
F
k
)
p
jk
_
g
j
G
k
f
n
j
F
N
k
_
= 0.
As before, setting
p = max
1js
1kS
p
jk
,
we nd that equation (5.3) holds with p
jk
replaced by p.
We have so far seen that a polynomial map : Z(I) CP
1
presents us with a baroque array of
data in the form of various fs, gs, Fs, and Gs as described above, and it is clear from the construction
that these data determine . We now turn to the question of when a collection of fs, gs, Fs, and Gs
admitting positive integers n, N, m, M, and p satisfying equations (5.1), (5.2), and (5.3) gives rise
to a map : Z(I) CP
1
. By construction, it is clear that the data provides us with a map
:
_
_
s
_
j=1
Z
fj
_
_

_
_
S
_
j=1
Z
Fj
_
_
CP
1
.
Thus, we need only check that the Z
fj
and Z
Fj
cover Z(I), that is, that the locus of points of
Z(I) on which all of the f
j
and F
j
vanish is empty. By the Nullstellensatz, this is equivalent to
the statement that the set T = f
1
, . . . , f
s
, F
1
, . . . , F
S
generates the unit ideal in R. Hence, every
collection of fs, gs, Fs, and Gs satisfying equations (5.1), (5.2), and (5.3) for some n, N, m, M,
and p and enjoying the property that the ideal of R generated by T is R itself, determines a unique
map : Z(I) CP
1
.
We are still not quite nished. Although we have found a collection of algebraic data that
completely determines any polnomial function : Z(I) CP
1
, we do not yet know when two data
sets specify the same map. So, let

f
1
, g
1
, . . . ,

f
s
, g
s
,

F
1
,

G
1
, . . . ,

F

S
,

G

S
, n,

N, m,

M, p
be another data set satisfying equations (5.1), (5.2), and (5.3) such that

T generates the unit ideal in
R. This data then determines polynomial functions

1
:

Z
1
C and

2
:

Z
2
C which glue to give
a map

: Z(I) CP
1
. Then =

if and only if we have
1
[
Z1

Z1
=

1
[
Z1

Z1
,
2
[
Z2

Z2
=

2
[
Z2

Z2
,

1
[
Z1

Z2

2
[
Z1

Z2
1, and
2
[
Z2

Z1

1
[
Z2

Z1
1. In other words, we must have nonnegative
integers
1
jk
,
2
jk
,
3
jk
, and
4
jk
for appropriate j and k such that
(5.4)
_
f
j

f
k
_

1
jk
_
g
j

f
n
k
g
k
f
n
j
_
= 0 for 1 j s, 1 k s,
WHICH FUNCTOR IS THE PROJECTIVE LINE? 15
(5.5)
_
F
j

F
k
_

2
jk
_
G
j

F

N
k


G
k
F
N
j
_
= 0 for 1 j S, 1 k

S,
(5.6)
_
f
j

F
k
_

3
jk
_
g
j

G
k
f
n
j

F

N
k
_
= 0 for 1 j s, 1 k

S,
and
(5.7)
_

f
j
F
k
_

3
jk
_
g
j
G
k


f
n
j
F
N
k
_
= 0 for 1 j s, 1 k S.
We have now nally accumulated enough facts to obtain a complete understanding of the functor
P
1
.
Theorem 5.1. The functor P
1
: C-Alg Set takes a C-algebra R to the set of all collections of
data
f
1
, g
1
, . . . , f
s
, g
s
, F
1
, G
1
, . . . , F
S
, G
S
, n, N, m, M, p,
where f
j
, F
j
, g
j
, and G
j
are elements of R and n, N, m, M, and p nonnegative integers satisfying
equations (5.1), (5.2), and (5.3) and the set T = f
1
, . . . , f
s
, F
1
, . . . , F
s
generates the unit ideal of
R, modulo the relation that another data set

f
1
, g
1
, . . . ,

f
s
, g
s
,

F
1
,

G
1
, . . . ,

F
S
,

G
S
, n,

N, m,

M, p
is equivalent to the rst if and only if there are nonnegative integers
1
jk
,
2
jk
,
3
jk
, and
4
jk
, satisfying
equations (5.4), (5.5), (5.6), and (5.7).
This gives us an entirely algebraic description of the object CP
1
as a functor P
1
: C-Alg Set. In
other words, we have derived a completely explicit expression of CP
1
without ever having to leave
the comparatively tractable category of nitely generated commutative C-algebras. However, as we
will now see, CP
1
does not itself correspond to any object of C-Alg.
Theorem 5.2. There is no nitely generated commutative C-algebra R
J
such that the functor
C-Alg Set given by S Hom
C-Alg
(R
J
, S) is isomorphic to the functor P
1
. Equivalently, there
is no ideal J in C[x
1
, . . . , x
n
] with CP
1
= Z(J).
Proof. Suppose there were such a ring R
J
, so that CP
1
= Z(J). We have already seen that the set
of polynomial maps from Z(J) to C is equal to the set Hom
C-Alg
(C[x], R
J
), which is equal to R
J
.
Let us now compute this set directly; it is clearly the same as the set of polynomial maps from CP
1
to C, for Z(J)

= CP
1
. Fix a polynomial map : CP
1
C. Then restricting to U
1
we obtain a map
[
U1
: U
1
C.
Recall that U
1

= C, so this is equivalent to a map C C. By denition, the set of polynomial maps
from C to C is simply C[x], so [
U1
= for some polynomial in C[x].
Next, recall that CP
1
= U
1
[1, 0]. Denote the complex number ([1, 0]) by z. Since , by virtue
of being polynomial, is continuous, we know that for all p in some open neighborhood U of [1, 0],
the distance between (p) and z is at most 1. In particular, is uniformly bounded on U. Since U is
open and contains [1, 0], there must be some positive such that U contains all points of the form
[1, w] for w a nonzero complex number of modulus less than . Equivalently, U contains all points
of the form [1/w, 1] . As w ranges over all nonzero complex numbers with modulus less than , the
number 1/w ranges over all complex numbers with modulus greater than 1/. Thus, the polynomial
takes bounded values on the set of all w whose modulus exceeds 1/. But we know that any such
polynomial is constant. Therefore, [
U1
must be constanthence, by continuity, so must itself.
We conclude that the set of polynomial maps from CP
1
to C is just C itself, whence one concludes
R
J
= C. But this implies that the functor C-Alg Set sending S to Hom
C-Alg
(R
J
, S) actually
16 DANIEL K. BISS
sends any ring S to a singleton set. Since that functor is manifestly not the same as the functor P
1
described in Theorem 5.1, the proof is complete.
As a result, although CP
1
does not arise as a C-algebra, in studying it we are able to take
advantage of the machinery of commutative algebra via the language of functors. Thus, by shifting
attention from the category C-Alg to the category of functors from C-Alg to Set, we give ourselves
the tools to study more general geometric objects. Naturally, CP
1
is just one among many such
gadgets, beginning with its higher-dimensional analogues CP
n
with n 2. This is a tremendously
valuable gain; indeed, mathematicians had been aware of the correspondence between C-algebras
and subsets of C
n
for years, but until the revolution in algebraic geometry that began in the middle
of the twentieth century, there was no systematic approach available for applying this relationship
to more complicated algebro-geometric objects. It was only with the seminal achievements of the
French school beginning with Weil [9] and culminating in the work of Grothendieck [3] that the
representation of arbitrary objects of complex algebraic geometryknown as varietiesas functors
was exploited fully. This point of view has become a dominant force in current research (for a
particularly spectacular example, consult [7]), and has opened up a beautiful interplay between
algebra and geometry that touches elds from dierential geometry to algebraic number theory.
Acknowledgments
I would like to thank Johan de Jong and Max Lieblich for helpful discussions; Ravi Vakil and the
anonymous referee for useful comments on the text of the article; and Dan Dugger for expository
inspiration on this and other subjects.
References
[1] M.F. Atiyah and I.G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley, Reading, MA, 1969.
[2] D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Springer-Verlag, New York, 1995.
[3] A. Grothendieck, Elements de geometrie algebrique IIV, Inst. Hautes

Etudes Sci. Publ. Math. 4 (1960), 8 (1961),
11 (1961), 17 (1963), 20 (1964), 24 (1965), and 32 (1967).
[4] J. Harris, Algebraic Geometry, a First Course, Springer-Verlag, New York, 1992.
[5] F.C. Kirwan, Complex Algebraic Curves, Cambridge University Press, Cambridge, 1992.
[6] S. Mac Lane, Categories for the Working Mathematician, 2nd ed., Springer-Verlag, New York, 1998.
[7] F. Morel and V. Voevodsky, A
1
-homotopy theory of schemes, Inst. Hautes

Etudes Sci. Publ. Math. 90 (1999),
45143.
[8] I.R. Shafarevich, Basic Algebraic Geometry I, 2nd ed., Springer-Verlag, Berlin, 1994.
[9] A. Weil, Foundations of Algebraic Geometry, American Mathematical Society, New York, 1946.
[10] N. Yoneda, Letter to S. Mac Lane.
Daniel Biss was born in 1977 in Akron, OH, and grew up in Bloomington, IN. He received an A.B.
summa cum laude in mathematics from Harvard University in 1998, and a Ph.D. in mathematics
from MIT in 2002. He began a ve-year tenure as a Clay Mathematical Institute Long-Term Prize
Fellow in the fall of 2002 at the University of Chicago. His research interests include topology,
algebraic geometry, and Lie theory; Daniel has also for several years taken a great interest in the
teaching and exposition of mathematics. This is his second expository article to be published in
the Monthly; current related projects include a book whose goal is to communicate the spirit of
mathematics to a lay audience in an entirely non-technical manner. Wish him luckor, better still,
oer him a publishing contract.
Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637
E-mail address: daniel@math.uchicago.edu