which functor is the projective line?

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 discussionsin the philosophy of mathematics, and subtle changes in the mathematical community’s opinion onthis issue have had monumental influence 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 thesame as the information needed to encapsulate it. For example, we view a complex vector space VCas the same as a real vector space VR equipped with an automorphism J satisfying J2 = −Id. By“the same,” we mean that it is possible to pass from one description to the other and back withoutany loss of information. Indeed, given a complex vector space VC, we can view it as a real vectorspace VR, and by letting J : VR → VR be the automorphism defined by J(v) = i ·v, we obtain the pair(VR, J), as desired. Conversely, if we start with the information (VR, J), we can define a complexstructure on VR by declaring (a+ bi) · v = a · v+ b · J(v). Thus, these two notions of complex vectorspace 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 methodof writing down a particular set of rules gives the same vector space, whether that means writingdown a complete addition and multiplication table for VC, or first specifying only the structure of areal vector space VR, then declaring (via the automorphism J) how the complex number i will act onVR, and finally letting the vector space axioms along with the fact i generates C over R do the restof the work. Naturally, vector spaces over C are not alone in this respect: almost all mathematicalobjects have the (often extremely useful) feature that they can be described in several different ways.

Consider, for a moment, the two-element group Z/2. There is a seemingly endless list of ways tospecify—and, accordingly, study—this group. We might describe it, as our notation suggests, as thequotient of Z by the ideal of even numbers. We could describe it in terms of generators and relations,〈a|a2 = e〉; or as the unique group of order 2; or as the Galois group Gal(C/R). We could, morewhimsically, describe it as the group whose set of elements is {cabbages, kings} with the followingmultiplication table.

× cabbages kingscabbages 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 nonidentitygroup 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 ϕ : G→ Z/2, the subgroup kerϕof G has index 2, and conversely, given a subgoup H with [G : H] = 2, we get a homomorphismG → G/H ∼= Z/2. The aim of this article is to explain the principle that any mathematical objectwhatsoever can be described in a similar way, and furthermore, that this observation can be usedto great profit in several areas of mathematics.

1

2 DANIEL K. BISS

Notice first of all that, although any object—for now, we consider only groups—can be specifiedin such terms, these descriptions may be far less simple and elegant than the index 2 subgroupclassification just mentioned. Indeed, by merely passing from Z/2 to Z/3, we find a substantialincrease in complexity: there are exactly two nonidentity group homomorphisms from G to Z/3 forevery normal subgroup H of G with [G : H] = 3. The appearance of the normality condition comesfrom the fact that any subgroup of index 2 is automatically normal, whereas this is not the case forindex 3 subgroups. Also, there are two homomorphisms per subgroup rather than one because, oncewe fix the kernel H in G, it still remains to be decided which coset of H gets sent to 1 in Z/3 andwhich coset goes to 2.

Furthermore, if we replace the groups Z/2 and Z/3 with a classical group C such as C = GLnCor C = U(n), the problem of determining the group homomorphisms from arbitrary groups G toC encompasses essentially all of representation theory! Thus, understanding a group explicitly viaan 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 simpleexample, let C1 and C2 be two groups. Then although we may have no way of understandingthe set of group homomorphisms from an arbitrary group G to C1, denoted Hom(G,C1), or theset Hom(G,C2) of homomorphisms from G to C2, it is very easy to see that Hom(G,C1 × C2) =Hom(G,C1)×Hom(G,C2). Accordingly, it is not so hard to imagine that this seemingly intractibleway of specifying groups might be very useful in studying products.

This philosophy represents the radical conclusion of a trajectory in mathematical fashion thathas existed for literally centuries, the move to abstraction and generality. For as long as Westernmathematics has been studied by an organized community, there has been a push to frame discussionsin a more and more conceptual way, to remove specifics and excise unnecessary restrictions infavor of the broad picture. From a semantic point of view, this movement reached its zenith withthe acceptance of category theory in mainstream mathematical parlance—suddenly, mathematicalobjects 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 thehuman mind to cling to; instead, Z/2 is essentially identified with the symbol you see on this pieceof paper! But the phenomenon is more than semantic and psychological: the unexpected payoffof this strand of thought is that the language of category theory can in the end be used to createnew objects that turn out to be instrumental in studying the old ones. Ultimately, this—at firstblush—hopelessly abstract point of view has its say about the most concrete of issues. It is thisinterplay 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 ideasin the standard undergraduate curriculum. A much more thorough discussion of these and relatedmatters is provided in [6]. In Section 3, we introduce the notion of functors and define representableand corepresentable functors, again with many examples, and explain how the Yoneda lemma allowsus to completely understand a category via representable functors. Next, in Section 4, we provide anintroduction to the category of finitely generated commutative C-algebras, and its connection withalgebraic geometry. We explain how to associate a geometric object with a C-algebra and discussbriefly the question of which geometric objects can be realized in this way. Lastly, in Section 5 weindicate how passing from the category of C-algebras to an associated functor category allows us toperform constructions that were impossible in the original category and therefore gives us greatlyincreased flexibility. Our main point is that the complex projective line CP1 can be understood asa functor on this category, but not as an object in it; thus, we give a careful characterization of thefunctor represented by CP1.

WHICH FUNCTOR IS THE PROJECTIVE LINE? 3

In order to reach our main conclusion in Section 5, it was necessary to make some unorthodoxexpository choices. In particular, we begin by defining C-algebras and then quickly move to theconstruction of nonaffine complex varieties. An ordinary discussion would include a more compre-hensive outline of the commutative algebra involved, as well as a precise definition of the category ofvarieties 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 webother working over C at all, for the Nullstellensatz is simply mentioned, without any indications ofa proof. On the other hand, this seemed like the only way to communicate the power of the “objectsare functors” point of view without assuming substantially more background, which was after allour 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 finitely generated abelian groups is more or less a closed subject, because wehave a well-understood list of isomorphism classes, as well as knowledge about how the list behaveswith respect to operations such as products, taking subgroups, forming quotients, and so on. Bycontrast, even though all finite simple groups are known, and there is a theoretical way of buildingup an arbitrary finite group out of simple ones, the theory is not considered finished, because thereis no practical way to construct a list of all isomorphism classes. Similarly, the goal of ring theoryis to classify rings up to isomorphism, the goal of field theory is to classify fields up to isomorphism(and, accordingly, the subject of finite fields is considered to be completely understood), the goal oftopology is to classify topological spaces, or manifolds, or CW complexes, up to homeomorphism orhomotopy equivalence, the goal of algebraic geometry is to classify algebraic varieties or schemes upto isomorphism, and so forth. Throughout these markedly different areas of mathematics, the basictheme is that we have some sort of object, a set with some kind of structure, we have a notion ofwhat it means for a map to preserve that structure, and therefore a notion of isomorphism, and ourgoal is to classify the objects up to isomorphism. This leads us to make the following definition.

Definition 2.1. A category C consists of a collection Ob(C) of objects, and, corresponding to everypair of objects A and B in Ob(C), a set HomC(A,B) of morphisms. A morphism f in HomC(A,B)will often be denoted f : A → B. For each object A, the set HomC(A,A) contains a distinguishedidentity morphism IdA. Furthermore, for every triple of objects A,B, and C from Ob(C), we assumethe existence of a composition map ◦ : HomC(A,B) × HomC(B,C) → HomC(A,C). This data isrequired to satisfy the following axioms:

Identity: For all objects A and B, and all morphisms f : A → B, the equations f ◦ IdA = fand IdB ◦ f = f are satisfied.

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 generalization—most areas of mathematics can be squeezed intothis framework. A category is simply an abstract collection of things that we call objects; we arenot concerned with what the objects are or look like, just that they exist. All that we ask of them isthat we have a list of “morphisms” between all pairs of objects, and that someone give us a rule forcomposing these morphisms. Although, given objects A and B of a category C and a morphism f inHomC(A,B), we use the suggestive notation f : A→ B, this does not mean that there exist elementsof A that are being mapped to elements of B according to some rule called f ; it only means thatwe have objects called A and B, and an “arrow” between them that we call f . (Nevertheless, oldnotational 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 issometimes derisively called, “abstract nonsense.” Indeed, the original intentions of category theorywere to provide a convenient language for expressing certain universal mathematical concepts, ratherthan to furnish us with a tool that could actually be used in proving new theorems. We will nowgive a series of examples that will hopefully demonstrate that the theory easily meets the primarygoal, and then spend the rest of the article sketching some instances in which category-inspiredconstructions manage serendipitously to provide the groundwork for deep new mathematical ideas.

So, let us give some examples of categories. The first example, and the example that will be mostimportant to us later on, is the category Set of sets. The objects of this category comprise all sets; forany two sets X and Y , HomSet(X,Y ) is the set of all functions f : X → Y . Another category thatis 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 define HomGrp(G,H) to be the set of all group homomorphisms fromG to H. Similarly, we can form the categories Rng of all rings with identity and identity-preservingring homomorphisms, Ab of all abelian groups and group homomorphisms, Fld of fields and fieldhomomorphisms, and Top of all topological spaces and continuous maps. Moreover, given a fieldF or a ring R, we can consider the categories VectF of all finite-dimensional F -vector spaces andF -linear maps and R-Mod of all finitely generated R-modules and R-linear maps.

It is worth spending a moment to examine some of the similarities and differences between thevarious categories just described. For example, Fld has the rather unusual property that everymorphism in this category is an injective map of sets; it is easily seen that this fails for the otherexamples mentioned. On the other hand, although any map between fields is one-to-one, it isfairly difficult to write down all the maps between any given pair of fields. By contrast, if wehave two objects from the category VectF (i.e., two F -vector spaces V and W of dimensions mand n respectively), then HomVectF

(V,W ) can be identified with Mn×m(F ), the n × m matricesover F . This brings us to another point: given two morphisms in HomVectF

(V,W ) (that is, giventwo n ×m matrices over F ), we can add them to obtain another linear map. This addition givesHomVectF

(V,W ) a group structure, something that is also satisfied in Ab and R-Mod but in noneof the rest of the categories we have listed.

Let us study another special feature satisfied by some of these categories. Consider first theobject Z in the category Rng. This object has a rather amazing property: for any ring R, the setHomRng(Z, R) consists of just one morphism; namely, the homomorphism f : Z → R defined byf(1) = 1R and hence satisfying f(n) = n·1R, the element obtained by adding the element 1R to itselfn times. An object satisfying this condition is said to be “initial”; that is, an object A in a categoryC is initial if for every object X in C the set HomC(A,X) has exactly one element. Similarly, theobject A is said to be terminal if for any object X the set HomC(X,A) has exactly one element. Thecategories Ab, Grp, VectF , and R-Mod each have an object that is both initial and terminal. Theobjects are, respectively, the trivial group, the trivial group, the trivial (zero-dimensional) vectorspace, and the trivial module. The categories Set and Top have both initial and terminal objects,but the two are distinct: specifically, 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 ithas no terminal object. Indeed, suppose that R is a terminal object. Then for every prime numberp, the set HomRng(Z/p,R) must have precisely one element, say fp : Z/p→ R. But fp(1) = 1R, so

p · 1R = p · fp(1) = fp(p · 1) = fp(0) = 0R,

and therefore p · 1R = 0 for all p. Hence, in particular, 2 · 1R = 0 = 3 · 1R, and subtracting, we findthat 1R = 0R, which contradicts the definition of a ring with identity. (Notice that this argumentactually only required the existence of homomorphisms fp : Z/p → R for two distinct primes toobtain a contradiction; the uniqueness of the fp was not exploited. Thus, Rng does not even havean object that is a “universal head” for morphism arrows.) For similar reasons, the category Fld has


no terminal object, but the situation there is even worse—it has no initial object either. Indeed, everyfield has a characteristic (either a prime p or else 0), and if two fields have different characteristics,then there can be no morphisms between them, so no field can be initial. In the subcategory of fieldsof characteristic p, however, Fp is an initial object, and in the subcategory of fields of characteristic0, the initial object is Q.

Before we go on, there is one more definition we must make. Recall that a category is meant tobe an abstraction of a mathematical theory in which there are objects we would like to classify up tosome notion of isomorphism. Accordingly, we would like to introduce the definition of isomorphismin an arbitrary category.

Definition 2.2. Let C be a category, and let A and B be objects in C. They are said to be isomorphicif there are morphisms f : A → B and g : B → A in HomC(A,B) and HomC(B,A), respectively,such that f ◦ g = IdB and g ◦ f = IdA. In this case, f and g are said to be isomorphisms.

It is useful to try to understand why this definition corresponds to our usual notion of isomorphismin the examples that we have provided. Furthermore, a good exercise to give the reader some practicein navigating these definitions is to show that initial and terminal objects in any category are uniqueup to isomorphism. Later, we will give a more sophisticated explanation for this fact, but it wouldbe 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 groupsor rings. Although categories already represent a valuable conceptual tool, most of the significantapplications of the categorical point of view arise when we attempt to compare one category withanother. Indeed, the underlying principle of category theory is that a mathematical discipline oughtto 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 introducemaps between categories.

Definition 3.1. For categories C and D, a covariant functor F : C → D consists of a mapF : Ob(C)→ Ob(D) and, for all A and B in Ob(C), a map F : HomC(A,B)→ HomD(F (A), F (B)),such that the following axioms hold:

(1) For any A in Ob(C), F (IdA) = IdF (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 different 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 needto consider functor-like rules that assign to each morphism in C a morphism in D heading in the“opposite” direction.

Definition 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 : HomC(A,B)→ HomD(F (B), F (A)),such that the following axioms hold:

(1) For any A in Ob(C), F (IdA) = IdF (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 theseprefixes 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 first several examplesall fall under the heading of “forgetful functors.” That is, if we have two categories, one of whichhas more structure than the other, then we obtain a functor from the first category to the secondsimply by “forgetting” the extra structure. For example, a group G is a set with some extra algebraicproperties (namely, a multiplication law). By ignoring the product and simply considering G to be aset, we obtain a functor U : Grp→ Set. This is actually a functor because any group homomorphismis 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 ofsets.

There are many such forgetful functors among the various categories discussed so far. For example,by forgetting the multiplicative structure of a ring or field, we get functors Rng → Ab and Fld →Ab. Similarly, forgetting the scalar multiplication gives functors R-Mod → Ab and VectF → Ab.Moreover, by ignoring all of the algebraic structures involved, we obtain functors from any of thesecategories 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; forexample, the initial object Z of Rng is sent to a rather unremarkable object in Set when we forgetthe ring structure. Also, objects that are not isomorphic in one category might become isomorphicwhen a functor is applied. For example, the groups S3 and Z/6 have isomorphic images under theforgetful functor U : Grp→ Set, as do the rings Z/2×Z/2 and F4 under the functor U : Rng → Ab.Nonetheless, a systematic study of the functors between various categories will give us substantialpower in understanding the categories themselves.

Before going on to discuss the most crucial instances of functors that we will need, let us returnmomentarily to the example with which we began. Recall that we explained how a complex vectorspace VC could be identified with a real vector space VR equipped with a linear transformationJ : VR → VR satisfying J2 = −Id. Let us see how to cast this discussion in the categorical frameworkwe have created. Viewing a complex vector space VC as a real vector space VR corresponds to applyinga forgetful functor U : VectC → VectR that causes us to “forget” how to multiply by i, since thecategory VectR knows nothing about imaginary numbers. Thus, the automorphism J is a convenientway of reminding us of what we have forgotten in this process.

This story can, of course, be generalized. Indeed, let a field K be a finite separable extension ofa field F . Then once again we have a forgetful functor U : VectK → VectF , because given a rule forscalar multiplication by elements of K, we get scalar multiplication by elements of F for free. Letus try to express what we have forgotten by applying this functor. Naturally, we have forgotten therules for scalar multiplication by elements of K that do not happen to lie in F . Fortunately, it is notdifficult to enumerate the set K\F . Indeed, recall that there exists an element α in K generating Kover F, meaning that the only subfield of K containing both α and all of F is K itself. Equivalently,every element of K can be expressed as a polynomial in α with coefficients in F . Thus, to list theelements of K, it suffices to write down polynomials in α over F.

Let f in F [x] be the minimal polynomial of α; that is, f(x) = xn + bn−1xn−1 + · · · + b0 is an

irreducible polynomial with coefficients bi in F , and f(α) = 0. Then we have a homomorphismρ : F [x]/(f(x)) → K defined by ρ(g(x)) = g(α), and it is an isomorphism. Indeed, it is one-to-onebecause any polynomial with coefficients in F having α as a zero must be a multiple of f and hencelie in the ideal (f(x)), and it is onto because its image contains both α and all of F . Therefore, anyelement of K has a unique representation of the form cn−1α

n−1 + · · ·+ c1α+ c0 with ci in F. As aresult, given aK-vector space VK , if we apply the forgetful functor U and view it as an F -vector spaceVF , then to recover the information we have lost, it suffices to recall the rule for scalar multiplicationby α. Furthermore, the only condition we need to place on this scalar multiplication rule is that it


satisfies the polynomial f , since α itself satisfies the equation f(α) = 0 but no polynomial of lowerorder. Summarizing, we have the following.

Proposition 3.3. Let F ⊂ K be a finite extension of fields such that K is generated over Fby an element α whose minimial polynomial is f(x) = xn + bn−1x

n−1 + · · · + b0. Then a K-vector space VK is equivalent to an F -vector space VF along with a map A : VF → VF satisfyingAn + bn−1A

n−1 + · · ·+ b1A = −b0.Notice that when we replace the data (F,K, α, f,A) by the special case of (R,C, i, x2 + 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 theensuing property: for any group G, the set of nontrivial homomorphisms from G to Z/2 is equal tothe set of index two subgroups H of G. This leads to the idea that we might understand an object ina category by enumerating the morphisms into or out of it. More precisely, let us make the followingdefinition.

Definition 3.4. Let C be a category and A an object of C. Then the functor represented by A isthe covariant functor yA : C→ Set defined as follows:

(1) For all X in Ob(C), yA(X) = HomC(A,X).(2) For all morphisms f : X → Y, yA(f) : HomC(A,X)→ HomC(A, Y ) is post-composition with

f , i.e., (yA(f))(g) = f ◦ g.Similarly, the functor corepresented by A is the contravariant functor yA : C→ Set defined by:

(1) For all X in Ob(C), yA(X) = HomC(X,A).(2) For all morphisms f : X → Y, yA(f) : HomC(Y,A)→ HomC(X,A) is pre-composition with

f , i.e., (yA(f))(g) = g ◦ f .Thus, the assertion that Z/2 is the unique group such that for all groups G the set HomGrp(G,Z/2)

is made up of the trivial homomorphism together with a family of homomorphisms indexed by thesubgroups of G of index 2 is equivalent to the statement that there is no other group H withyH = yZ/2. This fact can be generalized to a famous lemma of Yoneda. To state the lemma, weneed to make one more definition concerning functors.

Definition 3.5. Let F,G : C → D be two covariant (respectively, contravariant) functors. Amorphism η of these functors is a family of morphisms ηA : F (A)→ G(A), one for each object A inC, such that for all morphisms f : A→ B in C it is the case that G(f)◦ηA = ηB◦F (f) : F (A)→ G(B)(respectively, ηA ◦ F (f) = G(f) ◦ ηB : F (B)→ G(A)).

This definition of morphisms actually makes the collection of covariant or contravariant functorsfrom C to D into a category itself. Notice now that, if we have two objects A and B in C and amorphism f : A → B, then we get an induced morphism of functors yf : yB → yA (respectively,yf : yA → yB) by pre-composing (respectively, post-composing) with f . Finally, we are ready topresent the Yoneda lemma [10].

Lemma 3.6 (Yoneda). Suppose A and B are two objects in a category C, and let η : yB → yA beany morphism of functors. Then there is a morphism f : A → B such that η = yf . Similarly, ifζ : yA → yB is a morphism of functors, then there exists a morphism g : A→ B such that ζ = yg.

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 objectcan 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 familiarizeourselves with the notion of representable functors in action. Suppose first that A is an initial

8 DANIEL K. BISS

object in a category C. Then, for any object X, we have yA(X) = HomC(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 toisomorphism, as we indicated earlier. Similarly, if B is a terminal object in C, then for any X wehave yB(X) = HomC(X,B), so the functor corepresented by B is the functor sending each objectto a singleton. By Yoneda’s lemma, terminal objects are also unique up to isomorphism.

Returning to yet another example given in the introduction, we consider groups C1 and C2. Forany group G, we have yC1×C2(G) = HomGrp(G,C1 × C2) = HomGrp(G,C1) × HomGrp(G,C2) =Y C1(G)× yC2(G). Thus, yC1×C2 = yC1 × yC2 . Moreover, a similar statement holds in any categoryin 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 define products!

For a more sophisticated example, the fundamental group π1 is a functor from the category ofpointed topological spaces to Grp that is represented in the homotopy category by the circle S1.This is part of a very broad-reaching theme in algebraic topology in which functors represented andcorepresented by various spaces are used in analyzing geometry and topology. In fact, essentiallyevery 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 forstudying categories. Rather, we will begin with a category that seems interesting but is somehowsmaller 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 andfunctor will furnish us with a beautiful technique for constructing mathematical objects that mightnot 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 froma category C to the category of sets as an expansion of C can be extremely useful. In order to doso, we must introduce a new category, the category C-Alg. An object of this category is a finitelygenerated commutative C-algebra, that is, a commutative ring R that contains the complex numbersC as a distinguished subring. The hypothesis of finite generation simply means that there is a finiteset of elements x1, . . . , xn of R such that no proper subring of R contains C ∪ {x1, . . . , xn}. Fromtime to time, we will have to make reference to standard results from commutative algebra in ourstudy 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 wesketch, the reader is encouraged to consult [4], [5], or [8].

A morphism of C-algebras R1 and R2 is a ring homomorphism f : R1 → R2 that is the identityon the subring C. For example, C is a C-algebra, but the conjugation map a+bi 7→ a−bi, which is aring homomorphism from C to itself, is not a C-algebra homomorphism, because it is not the identityon C. On the other hand, the polynomial ring C[x] in one variable over C is also a C-algebra, and forany polynomial f(x), the map Ff : C[x]→ C[x] sending x to f(x) is a C-algebra map. In particular,for any polynomial ϕ in C[x], we have Ff (ϕ) = ϕ(f(x)). Likewise, for any complex number z, themap ez : C[x]→ C defined by ez(ϕ) = ϕ(z) is a C-algebra map.

Let us now introduce some objects of C-Alg and examine the functors they represent. Considerfirst the object C. For any C-algebra R, all C-algebra maps f : C → R must be the identity onC; therefore there is a unique such map. Hence, C is an initial object in C-Alg. The next easiestexample 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 ϕ be an arbitraryelement of C[x]. By definition, we must have ϕ = anx

n +an−1xn−1 + · · ·+a1x+a0 for some complex


numbers a0, a1, . . . , an and some integer n. Now, we compute

f(ϕ) = f(anxn + · · ·+ a1x+ a0)

= f(an)f(x)n + · · ·+ f(a1)f(x) + f(a0)= anf(x)n + · · ·+ a1f(x) + a0.

Therefore, the value of the map f at ϕ is determined entirely by the element f(x) if R. Hence,for each element of r in R, we get a unique map fr : C[x] → R defined by setting fr(x) = r. Inother words, HomC-Alg(C[x], R) = R, or, to be completely precise, the object C[x] represents theforgetful functor U : C-Alg → Set sending a C-algebra to its underlying set: yC[x] = U .

Similarly, a C-algebra map f : C[x, y] → R is determined by the two elements f(x) and f(y) ofR, via the formula

f

n∑i=1

m∑j=1

aijxiyj

=n∑

i=1

m∑j=1

aijf(x)if(y)j ,

so we have HomC-Alg(C[x, y], R) = R × R. Precisely speaking, the object C[x, y] represents thefunctor sending a C-algebra R to the underlying set of R×R. More generally, for any positive integern, a C-algebra map f : C[x1, . . . , xn] → R is determined by the n elements f(x1), . . . , f(xn) of R,whence HomC-Alg(C[x1, . . . , xn], R) = Rn. Thus, the object C[x1, . . . , xn] represents the functorsending a C-algebra R to the underlying set of Rn. We can express this in symbols by lettingPn : Set→ Set signify the functor taking a set X to Xn. Then yC[x1,...,xn] = Pn ◦ U .

There is more to be said. Let I be an ideal in C[x1, . . . , xn]. Then we may form the C-algebraC[x1, . . . , xn]/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, itis the case that a C-algebra map f : C[x]/I → R is determined by the element f(x) of R. Indeed, ageneral element of C[x]/I is of the form anx

n + · · ·+a1x+a0, and we have f(anxn + · · ·+a1x+a0) =

anf(x)n + · · ·+ a1f(x) + a0. Thus, we might be led to believe that C[x]/I, like C[x], represents theforgetful functor sending a C-algebra R to its underlying set.

However, this is not the case (which, incidentally, is fortunate, since the Yoneda lemma wouldotherwise imply that C[x] ∼= C[x]/I, which is simply false). Indeed, let ψ belong to I, and supposethat f : C[x]/I → R is a map with f(x) = r. Then we have 0 = f(0) = f(ψ) = ψ(r). Thus, for all ψin I, we must have ψ(r) = 0. This means that HomC-Alg(C[x]/I,R) can only include those r fromR with ψ(r) = 0 for all ψ in I. Indeed, it is not hard to check that HomC-Alg(C[x]/I,R) is equal tothe set of all r in R with ψ(r) = 0 for all ψ in I. Moreover, by the Hilbert basis theorem [1, ch. 7], Imust be a finitely generated ideal, say with generating set {ψ1, . . . , ψk}. Then every element ψ of Ican be written as ψ =

∑ki=1 ϕiψi with ϕi in C[x], and so ψ(r) =

∑ki=1 ϕi(r)ψi(r). Hence, if ψi(r) = 0

for all i, then ψ(r) = 0. In other words, HomC-Alg(C[x]/I,R) = {r ∈ R|ψi(r) = 0 for all i}. It isnot difficult to see that this result has the following generalization.

Proposition 4.1. Let I be the ideal in C[x1, . . . , xn] generated by the set {ψ1, . . . , ψk}. Then forevery C-algebra R, we have

HomC-Alg(C[x1, . . . , xn]/I,R) = {(r1, . . . , rn) ∈ Rn|ψ(r1, . . . , rn) = 0 for all ψ ∈ I}= {(r1, . . . , rn) ∈ Rn|ψi(r1, . . . , rn) = 0 for i = 1, . . . , k}.

The goal of this section is to indicate why we might want to have at our disposal functors fromC-Alg to Set that are not representable. To this end, we seek an alternate, more geometric wayof viewing C-algebras. Let us start with the universal example C[x]. This C-algebra, more or lessby definition, 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[x1, . . . , xn] consists of allpolynomial functions on the space Cn. Indeed, given any n-tuple of complex numbers (z1, . . . , zn)and any polynomial ϕ(x1, . . . , xn) in C[x1, . . . , xn], we may substitute the complex numbers zi for theindeterminates xi to obtain a complex number ϕ(z1, . . . , zn). Thus, it is perhaps not too far-fetchedto think of C[x1, . . . , xn] as somehow corresponding to the set Cn.

The next example of a C-algebra we consider is the algebra RI = C[x1, . . . , xn]/I for an idealI in C[x1, . . . , xn]. Now, given an element ϕ1 of C[x1, . . . , xn], we obtain by reducing modulo I anelement ϕ1 of RI . To continue in the same vein as before, we would like to view ϕ1 as a polynomialon Cn, by evaluating ϕ1(z1, . . . , zn) = ϕ1(z1, . . . , zn). Unfortunately, there are many elements ϕ2

in C[z1, . . . , zn] with ϕ1 = ϕ2, and there is no guarantee that ϕ1(z1, . . . , zn) = ϕ2(z1, . . . , zn), sothis function might not be well-defined. However, all is not lost. If ϕ1 = ϕ2, then we know thatϕ1 = ϕ2+ψ for some ψ from I, so ϕ1(z1, . . . , zn) = ϕ2(z1, . . . , zn)+ψ(z1, . . . , zn). Thus, if it happensto be the case that ψ(z1, . . . , zn) = 0 for all ψ in I, then we have ϕ1(z1, . . . , zn) = ϕ2(z1, . . . , zn).With this in mind, we let Z(I) be the subset of Cn defined by

Z(I) = {(z1, . . . , zn) ∈ Cn|ψ(z1, . . . , zn) = 0 for all ψ ∈ I}.

The upshot of the above discussion is that, if I is any ideal in C[x1, . . . , xn], we may view RI =C[x1, . . . , xn]/I as a ring of polynomial functions on the set Z(I) contained in Cn. In the same sensethat the set Cn corresponded to the C-algebra C[x1, . . . , xn], we can think of Z(I) as correspondingto RI .

For example, let I = (xy) in C[x, y]. Then Z(I) = {(z1, z2) ∈ C2|z1z2 = 0} = {(z1, z2) ∈ C2|z1 =0 or z2 = 0}. That is, Z(I) consists of the coordinate axes in C2. Moreover, an element ϕ(x, y) inC[x, y]/I must be of the form

ϕ(x, y) =m∑

i=0

n∑j=0

aijxiyj

for some complex numbers aij . But if i ≥ 1 and j ≥ 1, then xiyj belongs to (xy) = I, implying thataijxiyj = 0. Thus, we can actually express ϕ in the form

ϕ(x, y) =

(m∑

i=1

bixi

)+

n∑j=1

cjyj

+ a0.

In other words, ϕ must be the sum of a polynomial on the x-axis that vanishes at x = 0, a polynomialon the y-axis vanishing at y = 0, and a scalar. Basically, the restriction of ϕ to the x-axis can be anypolynomial ϕx, and the restriction to the y-axis may be any polynomial ϕy, so long as ϕx(0) = ϕy(0),which is of course necessary for the function ϕ to be well-defined at the origin.

In any case, we now have a procedure that associates to an ideal I in C[x1, . . . , xn] the subsetZ(I) of Cn. Recall that I is said to be a radical ideal if an element ψ of C[x1, . . . , xn] lies in Iwhenever ψr belongs to I for some r ≥ 1. It follows from Hilbert’s Nullstellensatz [1, ch. 5] that ifwe restrict attention to radical ideals I, our correspondence

C[x1, . . . , xn]/I ←→ Z(I) ⊂ Cn

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[x1, . . . , xn] and C[x1, . . . , xm] respectively, it is possible

to interpret a map f : RI = C[x1, . . . , xn]/I −→ C[x1, . . . , xm]/J = RJ of C-algebras in terms ofthese subsets. Indeed, RI and RJ are made up of polynomial functions on the two sets Z(I) andZ(J). Suppose that we have a polynomial map ξ : Z(J) → Z(I) and an element ϕ in RI . Then ϕis actually a polynomial function ϕ : Z(I)→ C, and by composing ϕ with ξ we obtain a polynomialϕ ◦ ξ : Z(J)→ C, in other words, an element of RJ . Thus the polynomial map ξ gives rise to a ring


homomorphism fξ : RI → RJ . It can be shown that all homomorphisms from RI to RJ arise in thismanner. That is, one can show that in this way we obtain a one-to-one correspondence betweenHomC-Alg(RI , RJ) and the set of polynomial maps ξ : Z(J)→ Z(I).

Now that we have a description of the category C-Alg in terms of subsets of complex vectorspaces, it is worth examining with a little more care what types of geometric objects can be realizedin this fashion. Consider first the set U = C\{0}; let us try to find a commutative C-algebra R towhich it corresponds. In other words, R should have the property that, for any positive integer nand any ideal I in C[x1, . . . , xn], the set M of polynomial maps from Z(I) to U is in one-to-onecorrespondence with the set HomC-Alg(R,RI). NowM is precisely the set of polynomial maps fromZ(I) to C that avoid the value 0; in other words, it is the set of invertible polynomial maps fromZ(I) to C. Here, we say that a polynomial map ξ is invertible if there is another polynomial mapξ−1 : Z(I)→ C such that ξ(p)·ξ−1(p) = 1 holds for all p in Z(I). We have already seen that the set ofall polynomial functions from Z(I) to C is the set HomC-Alg(C[x], RI) = RI . It is not hard to checkthat multiplication of polynomial maps as just described corresponds to ordinary multiplication onRI . Hence, the set of invertible functions of this type is precisely the set of invertible elements in RI .Equivalently,M is equal to the subset of HomC-Alg(C[x], RI) consisting of all ring homomorphismsf such that f(x) is a unit in RI .

Roughly speaking, the set M behaves as though x were a unit of C[x], since it consists of allmaps out of C[x] sending x to an invertible element. To make this precise, consider the idealJ = (xy − 1) in C[x, y] and the ring RJ = C[x, y]/J. In passing from C[x] to RJ , we have firstadjoined a free variable y, and then decreed that y be the multiplicative inverse of x. In otherwords, RJ is the C-algebra obtained by declaring that x in C[x] must be invertible. Hence, forany C-algebra S, the set HomC-Alg(RJ , S) is equal to the set of units of S. Indeed, consider ahomomorphism f : RJ → 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(xy) = 1. Hence, f(x) must be a unit in S, and f(y) (andhence all of f) is determined by f(x) via the uniqueness of inverses. Thus, we have established thatM = HomC-Alg(RJ , RI), so the functor we are interested in is given by

RI 7−→ HomC-Alg(C[x, y]/(xy − 1), RI).

Restated, this tells us that RJ represents the functor that takes RI to the set of polynomialfunctions from Z(I) to U. What, then, is the connection between Z(J) and U? Well, Z(J) is preciselythe subset of C2 consisting of all pairs (z1, z2) with z1z2 = 1. But the map f : U → Z(J) defined byf(z) = (z, z−1) is a bijective rational map. Thus, U and Z(J) are isomorphic in the category whoseobjects 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 Cbecause it consists of all polynomial maps from C to C. Similarly, the C-algebra Z(J) is supposed toconsist 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 7→ z−1, which is obviouslynot defined on all of C, is well-defined on U. Therefore, Z(J) should be isomorphic to the C-algebraobtained from C[x] by formally adjoining the element x−1. This simply reflects the isomorphismf : C[x, y]/(xy − 1)→ C[x, x−1] defined by f(x) = x and f(y) = x−1.

However, the story grows stickier rather quickly. To illustrate this, consider the the set V =C2\{(0, 0)}. If this space could be realized in the form Z(I) for some ideal I in C[x1, . . . , xn], thenby making slight modifications in the foregoing discussion, we would be able to recover RI by formallyinverting the polynomials in C[x, y] that take nonzero values away from the origin (0, 0) in C2. Butall such polynomials are constant, and hence already invertible. Indeed, consider ϕ ∈ C[x, y] and

12 DANIEL K. BISS

assume that ϕ is not constant. Then

ϕ(x, y) =m∑

i=0

n∑j=0

aijxiyj

for some complex numbers aij , not all 0. Reordering the sum gives us

ϕ(x, y) =m∑

i=0

ψi(y)xi,

where ψi(y) =∑n

j=0 aijyj . If ψi(y) = 0 for all i > 0, then ϕ(x, y) = ψ0(y). By assumption, ψ0(y)

is not constant, so it has some root z, and then (1, z) is a root of ϕ(x, y). On the other hand, ifψi(y) 6= 0 for some i > 0, then for some nonzero z in C the complex number ψi(z) is nonzero. Hence,plugging in the value z for the y variable, we obtain a nonconstant polynomial ϕ(x, z) in x, whichmust have some root w. Then (w, z) is a root of ϕ(x, y). Therefore, any nonconstant polynomial ϕhas a root away from the origin (0, 0), and so RI

∼= C[x, y], which is plainly a contradiction. Thisdashes any hope of realizing V as a space Z(I).

Hence, given a subset of Cn, the question of whether it can be expressed in the form Z(I) is asubtle one. However, we now leave this question and address a more far-reaching issue, namely, theconstruction of a space that is not a subset of a complex vector space. We will manage to producea natural geometric object that can not be embedded in Cn for any n, but does nevertheless have aconcrete description as a (nonrepresentable) functor C-Alg → Set.

5. Complex projective space, the functor

We are finally ready to present the example that is the payoff for all our work, a geometric objectwhose algebraic description is mostly easily conveyed as a functor C-Alg → Set. The object inquestion will be one-dimensional complex projective space CP1. This is defined to be the space ofnonzero vectors (z1, z2) in C2, modulo the equivalence relation (z1, z2) ∼ (λz1, λz2) for all nonzerocomplex numbers λ. Equivalently, CP1 is the set of one-dimensional complex vector subspaces ofC2. Surely it is believable that CP1 is an object in which geometers might have interest; indeed, itand its higher-dimensional analogues play fundamental roles in algebraic and differential geometry,as well as in topology. In order to see how this new creature CP1 fits into the algebraic frameworkthat we have constructed, it will be necessary to understand it more explicitly.

Given a nonzero vector (z1, z2) in C2, we denote its equivalence class in CP1 by [z1, z2]. Notice thatif z2 6= 0, then we have [z1, z2] = [z1/z2, 1] . Thus, denoting by U1 the subset of CP1 that comprisesall elements [z1, z2] with z2 6= 0, we have an isomorphism κ1 : U1−→C given by κ1 ([z1, z2]) = z1/z2.Similarly, we denote by U2 the subset of CP1 consisting of all elements [z1, z2] with z1 6= 0; this setis also isomorphic to C, via the map κ2 ([z1, z2]) = z2/z1. Since U1 ∪ U2 = CP1, we have coveredCP1 by two subsets that we understand very well. Indeed, CP1 = U1 ∪ {[1, 0]} ; for this reason, CP1

is often described as a copy of C (namely, U1) together with a “point at infinity” (namely, [1, 0]).It will also be important for us to understand the intersection U1 ∩ U2. We have κ1(U1 ∩ U2) =

C\{0}, since κ1([z1, z2]) = z1/z2 and z1 6= 0 on U1 ∩ U2. Similarly, κ2(U1 ∩ U2) = C\{0}. Thus, wehave two isomorphisms

κ1, κ2 : U1 ∩ U2−→C\{0}satisfying κ1 = 1/κ2.

We are finally ready to address the functor P1 determined by CP1. This is the functor C-Alg →Set that takes a C-algebra RI = C[x1, . . . , xn]/I to the set P1(RI) of polynomial maps from thecorresponding subset Z(I) of Cn to CP1. It is this set P1(RI) that we now undertake to study. Fix


a polynomial map ξ : Z(I)→ CP1, and set

Z1 = {p ∈ Z(I)|ξ(p) ∈ U1}

andZ2 = {p ∈ Z(I)|ξ(p) ∈ U2}.

Then we have have two polynomial maps ξ1 : Z1 → C and ξ2 : Z2 → C defined by

Ziξi−→ Ui

κi−→ C

for i = 1, 2. Since κ1 · κ2 ≡ 1, these maps are related by the property that ξ1 · ξ2 ≡ 1 on the setZ1 ∩ Z2.

We now need to find an algebraic expression of the maps ξ1 and ξ2. For an element f of RI , letZf denote the subset of Z(I) consisting of all points p with f(p) 6= 0. Then as we saw in the lastsection, the ring of functions on Zf can be described as the ring Rf obtained from R by inverting f.Thus, if Zf is contained in U1, then the map ξ1|Zf

is necessarily of the form g/fn for some elementg of R and some nonnegative integer n. It is a basic result of algebraic geometry that U1 can becovered by a finite collection of subsets Zf1 , . . . , Zfs (see [4]). We therefore have

ξ1|Zfj=

gj

fnj

j

for elements gj of R and nonnegative integers nj . Furthmore, setting

n = max{n1, . . . , ns},

and replacing gj by fn−nj

j gj , we may assume that

ξ1|Zfj=gj

fnj

.

Moreover, by construction, it must be the case that for every pair j and k, the two maps gj/fnj

and gk/fnk agree when restricted to Zfj

∩ Zfk. In other words, the equation

gjfnk − gkf

nj = 0

must hold on Zfj∩Zfk

, or equivalently, in the ring Rfjfk. This is not quite the same as demanding

that this equation be satisfied in R, since the act of inverting the elements fj and fk to createthe ring Rfjfk

can affect the multiplication somewhat. More specifically, since the element fjfk isnow a unit, if x belongs to R and if there exists a nonnegative integer m such that the equation(fjfk)m

x = 0 holds in R, then dividing both sides by (fjfk)m demonstrates that x itself must bezero in Rfjfk

. In fact, one can show that this is the only additional relation obtained by formingRfjfk

. Thus, returning to our situation, the compatibility criterion implies that we must have arelation

(5.1) (fjfk)mjk(gjf

nk − gkf

nj

)= 0

for mjk a sufficiently large nonnegative integer. Once again, if we set

m = max1≤j,k≤s

{mjk},

then equation (5.1) is satisfied with mjk replaced by m.Hence ξ1 : Z1 → C is specified by the data(

g1fn1

, . . . ,gs

fns

,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 ξ1 : Z2 → C is determined by the information(G1

FN1

, . . . ,GS

FNS

,M

)where N is a nonnegative integer, S is a positive integer, Fj and Gj are elements of R for allj = 1, 2, . . . , S, and M is a nonnegative integer such that for all j and k satisfying 1 ≤ j, k ≤ S, therelation

(5.2) (FjFk)M (GjF

Nk −GkF

Nj

)= 0

holds. In order that ξ1 and ξ2 glue together to furnish us with a map ξ : Z(I) → CP1, they mustsatisfy

ξ1 · ξ2|Z1∩Z2 ≡ 1.

That is, for whenever 1 ≤ j ≤ s and 1 ≤ k ≤ S, the element(gj/f

nj

)·(Gk/F

Nk

)must be the constant

function 1 on Zfj∩ ZFk

. Equivalently, there must exist a nonnegative integer pjk such that

(5.3) (fjFk)pjk(gjGk − fn

j FNk

)= 0.

As before, settingp = max

1≤j≤s1≤k≤S

{pjk},

we find that equation (5.3) holds with pjk replaced by p.We have so far seen that a polynomial map ξ : Z(I)→ CP1 presents us with a baroque array of

data in the form of various fs, gs, F s, and Gs as described above, and it is clear from the constructionthat these data determine ξ. We now turn to the question of when a collection of fs, gs, F s, and Gsadmitting positive integers n, N , m, M, and p satisfying equations (5.1), (5.2), and (5.3) gives riseto a map ξ : Z(I)→ CP1. By construction, it is clear that the data provides us with a map

ξ :

s⋃j=1

Zfj

∪ S⋃

j=1

ZFj

−→ CP1.

Thus, we need only check that the Zfjand ZFj

cover Z(I), that is, that the locus of points ofZ(I) on which all of the fj and Fj vanish is empty. By the Nullstellensatz, this is equivalent tothe statement that the set F = {f1, . . . , fs, F1, . . . , FS} generates the unit ideal in R. Hence, everycollection of fs, gs, F s, 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 F is R itself, determines a uniquemap ξ : Z(I)→ CP1.

We are still not quite finished. Although we have found a collection of algebraic data thatcompletely determines any polnomial function ξ : Z(I)→ CP1, we do not yet know when two datasets specify the same map. So, let

f1, g1, . . . , fs, gs, F1, G1, . . . , FS , GS , n, N , m, M , p

be another data set satisfying equations (5.1), (5.2), and (5.3) such that F generates the unit ideal inR. This data then determines polynomial functions ξ1 : Z1 → C and ξ2 : Z2 → C which glue to givea map ξ : Z(I)→ CP1. Then ξ = ξ if and only if we have ξ1|Z1∩Z1

= ξ1|Z1∩Z1, ξ2|Z2∩Z2

= ξ2|Z2∩Z2,

ξ1|Z1∩Z2· ξ2|Z1∩Z2

≡ 1, and ξ2|Z2∩Z1· ξ1|Z2∩Z1

≡ 1. In other words, we must have nonnegativeintegers `1jk, `

2jk, `

3jk, and `4jk for appropriate j and k such that

(5.4)(fj fk

)`1jk(gj f

nk − gkf

nj

)= 0 for 1 ≤ j ≤ s, 1 ≤ k ≤ s,


(5.5)(FjFk

)`2jk(GjF

Nk − GkF

Nj

)= 0 for 1 ≤ j ≤ S, 1 ≤ k ≤ S,

(5.6)(fjFk

)`3jk(gjGk − fn

j FNk

)= 0 for 1 ≤ j ≤ s, 1 ≤ k ≤ S,

and

(5.7)(fjFk

)`3jk(gjGk − f n

j FNk

)= 0 for 1 ≤ j ≤ s, 1 ≤ k ≤ S.

We have now finally accumulated enough facts to obtain a complete understanding of the functorP1.

Theorem 5.1. The functor P1 : C-Alg → Set takes a C-algebra R to the set of all collections ofdata

f1, g1, . . . , fs, gs, F1, G1, . . . , FS , GS , n,N,m,M, p,

where fj , Fj , gj , and Gj are elements of R and n, N, m, M, and p nonnegative integers satisfyingequations (5.1), (5.2), and (5.3) and the set F = {f1, . . . , fs, F1, . . . , Fs} generates the unit ideal ofR, modulo the relation that another data set

f1, g1, . . . , fs, gs, F1, G1, . . . , FS , GS , n, N , m, M , p

is equivalent to the first if and only if there are nonnegative integers `1jk, `2jk, `

3jk, and `4jk, satisfying

equations (5.4), (5.5), (5.6), and (5.7).

This gives us an entirely algebraic description of the object CP1 as a functor P1 : C-Alg → Set. Inother words, we have derived a completely explicit expression of CP1 without ever having to leavethe comparatively tractable category of finitely generated commutative C-algebras. However, as wewill now see, CP1 does not itself correspond to any object of C-Alg.

Theorem 5.2. There is no finitely generated commutative C-algebra RJ such that the functorC-Alg → Set given by S 7→ HomC-Alg(RJ , S) is isomorphic to the functor P1. Equivalently, thereis no ideal J in C[x1, . . . , xn] with CP1 ∼= Z(J).

Proof. Suppose there were such a ring RJ , so that CP1 ∼= Z(J). We have already seen that the setof polynomial maps from Z(J) to C is equal to the set HomC-Alg(C[x], RJ), which is equal to RJ .

Let us now compute this set directly; it is clearly the same as the set of polynomial maps from CP1

to C, for Z(J) ∼= CP1. Fix a polynomial map ξ : CP1 → C. Then restricting to U1 we obtain a map

ξ|U1 : U1 −→ C.

Recall that U1∼= C, so this is equivalent to a map C→ C. By definition, the set of polynomial maps

from C to C is simply C[x], so ξ|U1 = ϕ for some polynomial ϕ in C[x].Next, recall that CP1 = U1∪{[1, 0]}. Denote the complex number ξ([1, 0]) by z. Since ξ, by virtue

of being polynomial, is continuous, we know that for all p in some open neighborhood U of [1, 0],the distance between ξ(p) and z is at most 1. In particular, ξ is uniformly bounded on U. Since U isopen and contains [1, 0], there must be some positive ε such that U contains all points of the form[1, w] for w a nonzero complex number of modulus less than ε. Equivalently, U contains all pointsof the form [1/w, 1] . As w ranges over all nonzero complex numbers with modulus less than ε, thenumber 1/w ranges over all complex numbers with modulus greater than 1/ε. Thus, the polynomialϕ takes bounded values on the set of all w whose modulus exceeds 1/ε. But we know that any suchpolynomial is constant. Therefore, ξ|U1 must be constant—hence, by continuity, so must ξ itself.

We conclude that the set of polynomial maps from CP1 to C is just C itself, whence one concludesRJ = C. But this implies that the functor C-Alg → Set sending S to HomC-Alg(RJ , 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 P1

described in Theorem 5.1, the proof is complete. �

As a result, although CP1 does not arise as a C-algebra, in studying it we are able to takeadvantage of the machinery of commutative algebra via the language of functors. Thus, by shiftingattention from the category C-Alg to the category of functors from C-Alg to Set, we give ourselvesthe tools to study more general geometric objects. Naturally, CP1 is just one among many suchgadgets, beginning with its higher-dimensional analogues CPn with n ≥ 2. This is a tremendouslyvaluable gain; indeed, mathematicians had been aware of the correspondence between C-algebrasand subsets of Cn for years, but until the revolution in algebraic geometry that began in the middleof the twentieth century, there was no systematic approach available for applying this relationshipto more complicated algebro-geometric objects. It was only with the seminal achievements of theFrench school beginning with Weil [9] and culminating in the work of Grothendieck [3] that therepresentation of arbitrary objects of complex algebraic geometry—known as varieties—as functorswas exploited fully. This point of view has become a dominant force in current research (for aparticularly spectacular example, consult [7]), and has opened up a beautiful interplay betweenalgebra and geometry that touches fields from differential geometry to algebraic number theory.

Acknowledgments

I would like to thank Johan de Jong and Max Lieblich for helpful discussions; Ravi Vakil and theanonymous referee for useful comments on the text of the article; and Dan Dugger for expositoryinspiration 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 I–IV, 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, A1-homotopy theory of schemes, Inst. Hautes Etudes Sci. Publ. Math. 90 (1999),

45–143.

[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 mathematicsfrom MIT in 2002. He began a five-year tenure as a Clay Mathematical Institute Long-Term PrizeFellow 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 theteaching and exposition of mathematics. This is his second expository article to be published inthe Monthly; current related projects include a book whose goal is to communicate the spirit ofmathematics to a lay audience in an entirely non-technical manner. Wish him luck—or, better still,offer him a publishing contract.

Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637E-mail address: [email protected]

which functor is the projective line?

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