## Regular implies locally factorial

I’ve decided to end the series of posts on geometric class field theory (not that I’d been posting much anyway). This is partially because I’m in the process of writing up my own interpretation and proofs of these results, which will hopefully appear eventually. The other reason is that Deligne’s proof, which is the one I was going to present, is explained well in several places, e.g. Edward Frenkel’s article “Lectures on the Langlands program and conformal field theory” and Peter Toth’s master’s thesis.

Instead, I’ll try to post here when I encounter ideas which I think aren’t well enough documented and deserve to be written up. Today I want to prove the often stated but rarely proved fact that a regular local ring is a unique factorization domain. According to Wikipedia, in 1958 Nagata reduced the theorem to the case of a three-dimensional regular local ring, and in 1959 Auslander and Buchsbaum proved the result in dimension three. Their proof has been simplified over the years, although it still involves a fairly complicated induction on dimension, and in my mind doesn’t give much insight into why the theorem is true. Below I’ll present a beautiful geometric argument which was explained to me by Dennis Gaitsgory. Much of what follows is based on unpublished notes from a class he taught on the theory of schemes.

Although we are trying to prove a theorem in commutative algebra, we will use non-affine schemes in the proof. Let’s tautologically reformulate the statement in scheme theory.

Recall that a scheme is called regular provided that all its local rings are regular. Similarly, a scheme is called locally factorial provided that all its local rings are unique factorization domains, hereafter abbreviated to UFDs. So we are trying to prove the following.

**Theorem** A regular scheme is locally factorial.

There is some potential for confusion in the term “locally factorial.” If a scheme has an open covering by affine schemes with each a UFD, then is factorial, but the converse is false. In fact, it may happen that a factorial scheme contains no affine open subschemes with a UFD, e.g. any smooth curve of positive genus over a field.

We’ll use the following geometric criterion for local factoriality. Recall that is normal provided that the local ring at every point is an integrally closed domain. With everything in sight being a property of local rings, it suffices to formulate the criterion for affine schemes. Actually, we will assume throughout that is noetherian and integral as well, i.e. that is a noetherian domain. The noetherian hypothesis is essential for the kind of results we’re considering, but integrality just makes for cleaner statements: if the local rings of are domains then is the disjoint union of its irreducible components (because is noetherian!), which are themselves integral. So we are merely avoiding connectedness hypotheses and related complications.

**Proposition 1** A noetherian affine integral scheme is locally factorial if and only if

- is normal, and
- for any open whose complement has codimension at least two, every line bundle on extends to a line bundle on .

Thus an arbitrary locally noetherian scheme is locally factorial if and only if it can be covered by open affines satisfying the criterion above.

If this is what local factoriality really means, it begs the question: what does normality really mean? We’ll need to answer this question before we prove Proposition 1. Recall that a scheme is regular in codimension one provided that all of its one-dimensional local rings are regular. This hypothesis helps us to pass from Cartier divisors to Weil divisors, since it ensures that the local ring at a codimension one point is a discrete valuation ring.

**Proposition 2** A noetherian integral affine scheme is normal if and only if is regular in codimension one and the following equivalent conditions hold:

- no effective Cartier divisor on has an embedded point,
- for some (equivalently, any) line bundle and the inclusion of an open set whose complement has codimension at least two, the canonical morphism is an isomorphism, and
- Serre’s condition holds, i.e. for any coherent sheaf on whose support has codimension at least two, we have .

Condition 1 is pleasantly geometrical while condition 3 is just technical, but condition 2 deserves further discussion. It might be called the “Hartogs condition,” after the Hartogs theorem in complex analysis which says that a holomorphic function extends over loci of codimension at least two. We might also compare it to condition 2 in Proposition 1, which says that line bundles extend over loci of codimension at least two, whereas condition 2 in Proposition 2 says that such an extension, if it exists, is unique.

We will prove the statements above in the order opposite to the one in which they were stated, but first we need a lemma.

**Lemma 1** Let be a noetherian integral affine scheme. Then a rational function on is regular if and only if it is regular along every associated point of every effective Cartier divisor.

*Proof.* For the non-tautological direction, suppose we are given a rational function on which is not regular, so . Then : geometrically this is the ideal of functions on which vanish on , which is nonzero because does not contain . Let be an associated point of the -module .

Since is an ideal in , we see that is an associated point of the Cartier divisor . It remains to show that . Suppose in , so that there exists with . If , then the equation implies that vanishes in , so that . But is an associated point of , which means it is an irreducible component of the support of . This is a contradiction, so in fact .

*Proof of Proposition 2*. First we prove that normality is equivalent to condition 1 under the given assumptions.

For the “only if” direction, observe that since all the one-dimensional local rings of a noetherian normal scheme are integrally closed noetherian domains, they are discrete valuation rings (see e.g. Proposition 9.2 in Atiyah-Macdonald), and in particular regular. Let be an effective Cartier divisor and an associated point of . We need to show that is an irreducible component of , or equivalently that has codimension one in . Shrinking if necessary, we can take the Cartier divisor to be principal, cut out by some nonzero function . Now corresponds to a prime which is the annihilator of a nonzero function on , so there is some such that

.

We need to prove is one-dimensional, for which it suffices to show that is principal. To this end we may localize further and replace by , so that in particular is an integrally closed noetherian local domain with maximal ideal . Consider the fractional ideal

,

where is the fraction field of , and form the product . The reader may object that we do not know that is invertible, but that can be seen as follows: since and is maximal, we must have or . In the former case the elements of are integral over because they stabilize the finitely generated -submodule , so by our normality assumption. But now by the definition of we have , i.e. , a contradiction. Thus is indeed invertible, so that there exist some and such that . But then for some , so that has the property that , i.e. .

The “if” direction is easier. Lemma 1 says that is the intersection in its fraction field of the local rings where is an associated point of an effective Cartier divisor on , so it suffices to show that is normal under this condition on . Condition 1 tells us that has codimension one, and is regular in codimension one.

Now let us prove that condition 1 implies condition 2 for regular in codimension one. The assertion is local, so it suffices to consider . Thus we are trying to prove that is an isomorphism. The map is injective because is integral, and for surjectivity suppose we are given . Then is regular along every subscheme of codimension one, which includes all associated points of effective Cartier divisors on by condition 1. Thus by Lemma 1.

Next we show that condition 2 implies condition 1 under our hypotheses. If is a rational function on which is regular along every subscheme of codimension one, then is regular, since condition 2 says that can be extended over loci of codimension at least two. Thus where runs over prime ideals of height one. Since the are normal so is , and we proved that normality implies condition 1.

Finally, we check that condition 3 is equivalent to condition 2. For now is any coherent sheaf. Given an open subscheme , write for the inclusion, and consider the short exact sequence

.

This induces the long exact sequence

.

Now let be the complement of the support of , so we obtain an isomorphism

.

If condition 2 holds and the support of has codimension at least 2, then the left hand side vanishes, so that condition 3 holds as well. On the other hand, if we assume condition 3 and has codimension at least 2, we can take . The left side is identified with , and since is affine the vanishing of the global sections implies as desired.

.

*Proof of Proposition 1.* First assume that is locally factorial. We omit the standard argument that a unique factorization domain is normal. To prove condition 2, note that is reduced, so we may apply Corollary 7.1.19 of Qing Liu’s Algebraic Geometry and Arithmetic Curves to identify isomorphism classes of line bundles on or with the class group of Cartier divisors. Proposition 7.2.16 of loc. cit. says that the Cartier class group is identified with the Weil class group (actually, the hypotheses of that result are that is noetherian, regular, and integral; but the proof invokes our main theorem and then actually uses local factoriality). Now recall that the Weil class group is insensitive to the removal of loci of codimension at least two.

Now suppose that conditions 1 and 2 hold. Recall that a noetherian domain is a unique factorization domain if and only if every height one prime ideal is principal, so it suffices to show that the local rings of have the latter property. In fact, we will prove that every prime Weil divisor is a Cartier divisor, which means that the ideal sheaf is a line bundle. Since the local ring of at is a discrete valuation ring, the restriction of to the generic point of is a trivial line bundle. Thus there is an open set with dense in such that is a line bundle over . We have , so that is a line bundle over . Observe that the complement of has codimension at least two in , so there is a line bundle with . The map induces , where and the latter isomorphism uses condition 2 of Proposition 2. Since is integral must be injective, and we claim that maps isomorphically onto . To see that lands in , notice that factors through , which vanishes because it is a generically zero map of line bundles on . The image where is a Cartier divisor, and we have just shown that . Now , and since has codimension at least two this implies that is the maximal reduced closed subscheme of , and that any component of is an embedded point of . Thus by condition 1 of Proposition 2.

Finally we come to the proof of the theorem, which ingeniously uses the notion of determinant of a perfect complex.

If is a vector bundle on a scheme , recall that the determinant is the line bundle on obtained by taking the top exterior power of locally (the rank of may vary but is at least locally constant). More generally, if is a bounded complex of vector bundles on , we put

.

This construction is compatible with localization and functorial in the sense that a quasi-isomorphism of perfect complexes induces an isomorphism between their determinant lines.

We can even apply to a coherent sheaf if it has a finite resolution by vector bundles. The result will not depend on the choice of resolution by the functoriality mentioned above. Any coherent sheaf on a regular quasi-projective variety has such a resolution, although we will not use this fact. We only need the following lemma.

For the rest of this post, will be a regular local ring of dimension with maximal ideal and residue field .

**Lemma 2** Any finitely generated -module has a free resolution of length at most .

*Proof**.* We can construct a free resolution of , possibly of infinite length, in the following manner. Instead of picking an arbitrary set of generators of , we lift a basis of : by Nakayama’s lemma this is the same as a minimal set of generators for . This minimal set of generators yields an epimorphism , where is a free module of the appropriate rank. Similarly, one can choose a minimal set of generators for the kernel of , thereby obtaining , and so on. It is not hard to see that this minimality condition is equivalent to the vanishing of all differentials in the complex . But then we have

so it suffices to show that for , since this implies for (no need to apply Nakayama’s lemma: this is obvious because $F^{-i}$ is free).

For this, recall the existence of the Koszul resolution , which is a free resolution of the -module of length . Thus for we have

.

*Proof of the theorem.* We reduce immediately to the local case, so is still a regular local ring and as usual. First we must prove that is normal, for which it suffices to show that condition 3 of Proposition 2 holds (note that is a domain because its associated graded algebra is isomorphic to a polynomial ring, and in particular is a domain). So fix an -module whose support has codimension at least two. This implies that we can find satisfying which are a regular sequence in the sense that if , then the Koszul complex

is exact, where the first nonzero arrow sends and the second sends , . Consider the following segment of the long exact sequence gotten by applying :

.

Now is an isomorphism because , and vanishes for the same reason. It follows that as desired.

Finally, we must prove condition 2 of Proposition 1 holds, and it is now that our investment in geometry pays off. We reduce to the following situation: is the closed point, , and is a line bundle on which we want to extend to . First, we claim we can extend to a coherent sheaf on . Indeed, if is the inclusion, then is a quasi-coherent extension of , and since is noetherian is the filtered colimit of its coherent subsheaves. Thus there exists a coherent such that . Now apply Lemma 2 to , so that we obtain a bounded free resolution of and hence can make sense of . This is a line bundle on with the property that

.

Sorry for the obscenely long post, but after all, it has been almost a year.

## Grothendieck’s functions-sheaves correspondence

Today begins a shift in perspective for this series of posts. Whereas before we took as our basic objects of study the (abelian) étale covers of a smooth projective curve , from now on we will study (rank ) -adic local systems on , which can be considered as a “linearized” version of the problem. More precisely, since understanding the structure of the fundamental group (here is the geometric generic point, not that it really matters) is tantamount to understanding all étale covers of , the Tannakian philosophy says that we should consider the monoidal category of finite-dimensional representations of , from which the group can be recovered. Just as in topology, representations of the fundamental group are equivalent to local systems.

In fact, the other side of the Artin reciprocity map also has a useful interpretation in terms of -adic local systems: characters of the Picard group (more precisely, its profinite completion) correspond to “multiplicative” rank local systems on the Picard scheme . This situation is very special to finite fields! A multiplicative local system on can be pulled back along the Abel-Jacobi map , given by on points, and global unramified class field theory says that this pullback establishes a bijection between multiplicative local systems on and rank local systems on . This reformulation of class field theory is true over any field whatsoever, and has a beautiful geometric proof due to Deligne which we will hopefully get to next time.

Fix a prime (traditionally one assumes is not equal to , the characteristic of our finite ground field , but for us is fine). Let be an algebraic closure of the -adic numbers . The totally disconnected topology on makes it better suited to our purposes than the complex numbers, although it is worth mentioning that they are isomorphic as discrete fields. For the most part we will only care that is an algebraically closed field of characteristic zero.

Let be a connected scheme and a geometric point.

**Definition** An *-adic local system* on is a finite-dimensional continuous -representation of . The dimension of the representation is called the *rank* of .

As our terminology and notation suggest, -adic local systems can be thought of as locally constant sheaves of -vector spaces. This is not literally true, however: in order to get nontrivial local systems one must first consider locally constant étale sheaves over finite coefficient rings, pass to pro-systems of these sheaves, then localize (or kill torsion) to obtain a -linear category, which one can prove is monoidally equivalent to representations of the fundamental group. It seems clear that there is no such procedure for complex coefficients. Although these beasts are not literally sheaves, most of our sheafy intuition and technique applies, whence the power of this approach.

We will choose our notation accordingly: for instance, if and , we will write for the local system on obtained by restricting along the homomorphism induced by . The underlying -vector space of the representation is denoted by .

Now we come to the namesake of this post, for which we should assume that is defined over . This construction takes an -adic local system on and produces from it a function . Given , there is an isomorphism well-defined up to conjugation, so that after making this choice we can form the pullback , a local system on . Then we set

,

the trace of the action of the Frobenius , which does not depend on our choice because the trace is conjugation-invariant.

We will be interested in the case where has rank , and then it is easier to describe . Namely, determines a canonical map , which we compose with (now thought of a one-dimensional representation) to obtain a homomorphism . Evaluate this map at to obtain .

Sometimes interesting classes of functions and sheaves match up under this correspondence. Let be a commutative algebraic group over : then one such class of functions is the group of characters of , meaning one-dimensional representations . What sort of local system on has the property that is a character?

**Definition** A rank local system on is called a *character sheaf* provided that , where is the multiplication map.

(Notation for those who haven’t seen it: if is a sheaf on and is a sheaf on , then their *external tensor product* is where and are the projections.)

Sometimes character sheaves are called multiplicative local systems. The latter terminology is arguably better, since “character sheaf” has other meanings. This is analogous to how “character” can refer not only to a one-dimensional representation but also to the trace function associated to a higher-dimensional representation.

Before we prove the main result of this post, we need a lemma.

**Lemma** Let be an -adic local system on . Then there is a canonical isomorphism .

*Proof**.* We reduce immediately to the case that is a locally constant étale sheaf of finite sets, so there is some finite étale map whose sheaf of local sections is . Write for the fiber product of and , so it suffices to produce an isomorphism of -schemes. Using the relation , we obtain the desired map from and . Since and are finite étale, so is , and similarly is radicial (i.e. universally injective) because is. But a map which is both étale and radicial must be an open embedding, and is also finite, hence an isomorphism.

Now we come to the really interesting part. As usual is a commutative algebraic group over , which we now assume to be smooth and connected (the smoothness hypothesis is really not necessary).

**Proposition** Under the above assumptions, is a bijection from character sheaves on to characters of .

*Proof.* That is a character follows from the easy identities and . Suppose that we are given a character . The Lang isogeny is a pointed finite Galois covering with group , hence gives rise to a map , which we can restrict along to obtain a rank local system on . The same identities show that is a character sheaf if one argues in the opposite direction.

It remains to show that these constructions are mutually inverse, and first we’ll check that . Given , we obtain a canonical map , whose value on we will call . By definition is the value of on , but factors through by construction, so it suffices to show that the map induced by sends to . This means that when acts on the fiber , it just translates by . Fixing , we obtain an identification since . Tracing through definitions, our claim follows from the calculation

.

Finally, we must prove that . We claim that is trivial, or equivalently that factors through the homomorphism determined by . Since this map sends the claim implies that is determined by its values on for all , and we just proved that , from which it follows that . As for the claim, observe that

.

By the lemma , so we just have to check that is trivial. The latter sheaf is the pullback of along , which is the trivial homomorphism, and since we are done.

Next time we will consider , a disconnected group for which the proposition is almost true.

## The Lang isogeny

Sorry for the long wait: I had almost abandoned the project when I spoke to someone at a conference who claimed he was actually reading these posts. This took me by surprise and has inspired me to continue writing.

Today we specialize to the case of a finite ground field . Our goal is to prove Lang’s theorem on the surjectivity of his eponymous map and give some applications. This map plays a vital role in geometric class field theory: all geometrically connected étale covers of a smooth projective curve over arise as pullbacks of the Lang isogeny of the Jacobian along an Abel-Jacobi map.

Recall that any scheme over has the *Frobenius endomorphism* , which is defined to be the identity on the underlying set of and sends for local sections . It is not hard to see that is natural in the sense that if is a morphism of -schemes, we have . A fancy way of saying this is that is an element of the Bernstein center of the category of -schemes, i.e. an endomorphism of the identity functor on . In the sequel we will omit the subscript and simply write .

Let be a group scheme over . We want to study the difference between the Frobenius endomorphism and the identity.

**Definition** The *Lang map* is the endomorphism of the underlying -scheme of given as the composition

,

where and are the inversion and multiplication maps of , respectively.

Thus, if for some -scheme , we can write . From this formula and the aforementioned fact that commutes with all morphisms, one deduces that if is commutative then is a group endomorphism.

Taking , we see that the fiber of over is precisely .

**Proposition** Suppose is smooth (in particular, of finite type) over . Then the Lang map is finite étale.

*Proof.* To prove that is étale, it suffices to show that the differential of is an isomorphism on the tangent space at each point of . Since we have , whence is étale at . Observe that intertwines two actions of on itself: right translations and . Since the action by right translations is transitive, it follows that is étale everywhere.

Now any étale morphism is finite over some nonempty open set in the target, as can be seen by writing it locally as a standard étale morphism. Another routine application of equivariance shows that is finite.

This justifies the terminology “Lang isogeny,” at least when is commutative. We record below a couple of important consequences of this result.

**Corollary** (Lang) If is connected then the Lang map is surjective. In particular, fits into a short exact sequence of pointed sets

,

which is a short exact sequence of groups in case is commutative.

*Proof.* Since is finite étale it is both closed and open, so for connected it is surjective. The rest is a combination of previous remarks.

**Corollary** If is connected then any -torsor is trivial.

*Proof*. Let us prove the equivalent statement .The -torsor clearly has a point . Now one can find such that , and we claim that if (which exists according to the previous corollary) then . Indeed,

.

Finally, we give an application that was promised in a previous post.

**Corollary** Let be a smooth, projective, and geometrically connected curve over . Then admits a degree zero-cycle.

*Proof.* Let denote the Picard functor introduced previously and its fppf-sheafification, which we know to be representable by general results. Moreover, is smooth and connected (these facts are drawn from Chapter 6 of *Néron Models* by Bosch, Lütkebohmert, and Raynaud). Now is a -torsor, so by the last corollary we have . But because (see the post on the Picard scheme for the exact sequence that implies this), so has a degree line bundle, or equivalently a degree zero-cycle.

Next time we’ll discuss -adic local systems and Grothendieck’s functions-sheaves correspondence. The Lang isogeny will be important in the passage from functions to sheaves.

## Adèlic groups and principal bundles

Today we will establish a result which, in some form, goes back to Weil: the “uniformization” by the idèles of the Picard group of a curve. More generally, the moduli of -bundles on a curve can be uniformized by the adèlic points of (here is a connected smooth affine algebraic group satisfying some additional hypotheses which which hold, for instance, if the ground field is or ), so taking the multiplicative group we recover the previous sentence. This fact is remarkable because it connects the geometry of principal bundles on our curve with the arithmetic of its ring of adèles. There is even a “stacky” version of the theorem, which we will not treat here, but can be found in a slightly altered form as Theorem 5.1.1 in Sorger’s Lectures on moduli of principal -bundles over algebraic curves.

First let us define the geometric objects appearing in the theorem, which are of basic interest in algebraic geometry and representation theory. Fix a ground field , let be a -scheme, and an affine algebraic group over , meaning an affine group scheme of finite type over .

**Definition** A *(principal)* *-bundle* on is a scheme over equipped with a right action of which, locally in the étale topology, is isomorphic to the *trivial bundle* .

Morphisms of principal bundles are simply -equivariant -morphisms, which are automatically isomorphisms by local triviality.

There are two basic constructions we perform with -bundles. If is a morphism, then we can form the pullback bundle , which becomes a -bundle when equipped with the obvious right action. Less obviously, for any scheme with a left action of , there is an associated bundle , where has the diagonal action

.

One observes is a scheme over which is isomorphic to locally in the étale topology.

The latter construction allows us to produce a vector bundle on from a linear representation of . It is a basic fact that, when applied to the standard representation of , this establishes an equivalence between -bundles and rank vector bundles. When , which is the case we are interested in for the purposes of class field theory, the inverse is easy to describe: just remove the image of the zero section from a line bundle to obtain a -bundle.

If is an -point of , then by a *trivialization* of at we mean an isomorphism of with the trivial -bundle on , or equivalently a section of the projection . It is not hard to see that the group of automorphisms of the trivial -bundle on is , and in particular acts simply transitively on the set of trivializations of at , provided that this set is nonempty.

Next, we explain the construction of the adèles. For this let be a smooth connected curve over , and denote its field of rational functions by . For each closed point , write for the completed local ring of at (we will not use the non-completed local rings) with maximal ideal , residue field , and fraction field . Alternatively, can be viewed as the completion of with respect to the -adic absolute value. Recall that if we choose a local coordinate (a number theorist would say “uniformizer” or “prime element”) we get an isomorphism and hence .

**Definition** For any finite set of closed points, the *-adèles* are the topological ring

.

The full ring of *adèles* is the union over all finite , endowed with the colimit topology. The *integral adèles* are just .

Although the topology on is important for many purposes, especially when is finite, we will not need it in what follows.

Now we begin to interpret these arithmetic constructions geometrically. Write and , which we view as the disk and punctured disk centered at , respectively. If is an affine algebraic group over as before, then is analoguous to a topological loop group, and sometimes this terminology is still employed in algebraic geometry. In our global situation, the group of adèlic points packages together all the loop groups as varies through the closed points of . It contains and as subgroups, which in particular act on by translations. We declare that by convention acts on the left and acts on the right.

**Definition** Let be a -bundle on and a closed point. A *(full) level structure* on at is a trivialization of at the canonical morphism .

The reason for the parenthetical terminology “full” is that if is a finite subscheme (for example, an -order neighborhood of , of which is the colimit as varies) then a trivialization of at is called a *structure of level* . We will not need the latter notion in this post.

Finally we come to the promised result. The proof is actually quite instructive, which is one reason we chose to state it in this generality. Recall our standing assumption that is a smooth connected curve over .

**Theorem** (Uniformization) Let be a smooth affine algebraic group over . Then there is a canonical bijection from to the set of isomorphism classes of -bundles on equipped with a generic trivialization (i.e. a trivialization at the generic point ) and full level structures at all closed points . Moreover, this bijection is equivariant for and , which act on generic trivializations and full level structures respectively.

*Proof.* If we are given a -bundle equipped with a generic level structure and full level structures at all closed points of , we can construct an adèlic point of as follows. First observe that the given generic trivialization “spreads out” to a trivialization over a nonempty open set (this is where we use the hypothesis that is of finite type over ). Now the desired adèlic point is a Cech cocycle representative for with respect to the fpqc cover of consisting of all the formal disks together with . More precisely, restricting the full level structures and the generic trivialization gives two trivializations of on every punctured disk , and the difference of these two trivializations is an automorphism of the trivial -bundle on , or equivalently a point of the loop group (we should really be careful about the order we take this difference, but let’s avoid writing lots of formulas). Notice that for all , so as varies the give a well-defined element of .

Conversely, given an adèlic point , we continue to think of it as a Cech cocycle and construct the desired -bundle by “gluing,” or rather fpqc descent. Let be the finite set of closed points such that and write , so that and the for form an fpqc cover of . Then the for can be viewed as transition maps along which we glue the trivial bundles on the to the trivial bundle on . The resulting -bundle is *a priori* only locally trivial in the fpqc topology, but then the étale local triviality follows by smoothness of . Of course comes equipped with a trivialization over , hence a generic trivialization, as well as full level structures at each . Now restricting the trivialization of over to for each gives a full level structure there, but the level structure we really want is obtained by composing this one with , thought of as an automorphism of the trivial bundle on .

One can inspect the constructions to see that they are mutually inverse and equivariant for and .

Our final result, which is the punchline of this post, is an immediate consequence of the theorem.

**Corollary** If every -bundle on admits a generic trivialization, then the bijection from the theorem descends to a -equivariant bijection from to the set of isomorphism classes of -bundles on equipped with full level structures at every closed point . If, in addition, every -bundle on admits a trivialization at for each closed point , then we obtain a bijection from to the set of isomorphism classes of -bundles on .

*Proof.* The only part that is not obvious is that if a -bundle on trivializes over , then it trivializes over . But a trivialization is the same as a section, so by smoothness of this follows from Hensel’s lemma.

We conclude by pointing out when is the multiplicative group, both sides of the final bijection have a group structure, and we actually obtain an isomorphism of groups . The only additional content in this assertion is that tensoring line bundles corresponds to multiplying representative -valued Cech cocycles.

Next time, we’ll get into some specifics of the situation where is finite, and in particular explain how the hypotheses of the corollary then hold quite often. We will probably not use the results in this post very much moving forward, but they are important for establishing the connection between our approach and harmonic analysis on adèlic groups.

## The Picard scheme of a projective curve

Today we study the objects appearing on the opposite side of the class field theory isomorphism from the fundamental group, namely line bundles on our curve. Ultimately we are interested in the Picard group, which is the abstract abelian group of isomorphism classes of line bundles, but in order to actually prove the main theorem we must reformulate it in a still more geometrical way. To this end, we view the Picard group as a “moduli problem,” meaning we allow line bundles on the curve to vary in algebraic families and try to find some space which parameterizes them. It turns out that in our situation, general theorems of Grothendieck guarantee that this space is actually a scheme, and a pretty reasonable one at that. Be warned that this is a special and pleasant state of affairs, as many moduli problems of interest are not represented by a scheme: in particular, this is not the case for vector bundles of higher rank.

This post will be more technical than usual, so readers who are mostly interested in applications to class field theory should feel free to skim over the details. For background material on Grothendieck topologies and descent, see Vistoli’s article in *FGA Explained* or Chapter 6 of *Néron Models* by Bosch, Lütkebohmert, and Raynaud. General theorems about the Picard scheme can be found in Kleiman’s article in the former book or Chapters 8 and 9 in the latter.

Let be a scheme over a field . We can define a contravariant functor from the category of schemes over to the category of abelian groups by sending a -scheme to the Picard group . Here we can think of a line bundle on as a family of line bundles on parameterized by . This is the first functor which we might naïvely hope is representable: however, this is never the case if is nonempty. To see why, take , so carries the nontrivial line bundle . But if we pull back along the standard Zariski covering , the bundle trivializes, so that in particular the induced map is not injective. Since a representable functor is necessarily a sheaf with respect to the Zariski (Grothendieck) topology on , it follows that is not representable.

This leads us to the following definition, in which we simply declare the bundles which arise by pullback along to be trivial.

**Definition** The* Picard functor* is defined by the formula

.

If is representable by a scheme, we call it the *Picard scheme of* .

In particular we have , so this definition is consistent with the notation from two posts ago (here and in the sequel, when is a -algebra and is a functor on -schemes we commit the standard abuse of notation ). Note that if the Picard scheme exists, it is a commutative group scheme by construction.

Ours is not the most refined version of the Picard functor, since there are situations where is not representable but its fppf-sheafification is. The next theorem shows that suffices for our purposes, so we will not linger on this technical point.

**Theorem** If is proper, geometrically integral, and admits a degree 1 zero-cycle (i.e. a finite collection of closed points whose degrees generate the unit ideal in ), then is represented by a scheme which is locally of finite type over .

*Proof.* Theorem 3 in Section 8.2 of *Néron Models* says that the fppf-sheafification of is representable. Our statement can be deduced from this and Proposition 4 in Section 8.1 as follows. The hypothesis that for any -scheme is satisfied because is proper and geometrically integral. Thus the proposition yields an exact sequence

,

where is the Brauer group, i.e. the étale cohomology group . We must prove that is injective: in fact, we will produce a retraction of this map. Let be closed points of degrees respectively, and choose such that . For each , the morphism induces another , and we denote by the composition of pullback along this map with the corestriction . A standard property of restriction and corestriction maps says that the composition

is multiplication by , so is the desired retraction (thanks to Olivier Benoist and Jason Starr for explaining this to me over at MO).

We will prove soon that a projective and geometrically integral curve over a finite field admits a degree 1 zero-cycle. In fact, this is true for an arbitrary geometrically integral variety over a finite field: the general statement can be deduced from the Lang-Weil estimates, but our proof for projective curves will be less advanced and hopefully more understandable.

Finally, let us give some more refined information about the Picard scheme when is a projective and geometrically integral curve. Then there is a natural degree map , where is the constant group scheme over associated with the abstract group . We write for the fiber over , which parameterizes line bundles of degree .

**Proposition** The kernel is a smooth, connected, commutative algebraic group over , called the *generalized Jacobian of *. If we assume also that is perfect, then the normalization induces an epimorphism whose kernel is a smooth connected affine algebraic group, and is an abelian variety of dimension equal to the genus of .

This is essentially Corollary 11 in Section 9.2 of *Néron Models, *except for the assertion about dimension, which is Theorem 1(b) in Section 8.4. There the interested reader can also find more details regarding the linear algebraic group , which encodes information about the singularities of . For instance, this group is a split extension of a unipotent group by a torus: the former comes from cusps and the latter from nodes (we remind the reader that here is perfect).

Even if we are only interested *a priori* in smooth curves, these generalized Jacobians of singular curves enter naturally when considering ramified covers. However, we will focus on the unramified case for a while, and there we really only need to consider smooth curves.

Since this post has already gotten out of hand, we’ll put off the discussion of idèles until next time. While not logically necessary, this will establish the connection with the more classical arithmetic formulation of class field theory.

## Étale morphisms and the fundamental group

In this post I’m going to give the definition and basic properties of the étale fundamental group, whose abelianization appears on one side of class field theory. This is the object which is related *a priori* to our goal of understanding (abelian) covers of a curve: it precisely encodes the symmetries of all such covers. My initial goal of proving unramified global class field theory in the next post was too ambitious: I’ll slow down the pace a bit and spend the next four or five posts covering foundational material, then move on to the main theorems. This post is quite long, and to prevent it from being even longer I have not attempted to give any proofs or even the most general statements. Instead I refer the reader to SGA1, which is the standard reference, or chapters 4 and 5 of Szamuely’s excellent book *Galois Groups and Fundamental Groups*.

The idea is to introduce an algebro-geometric analogue of the topological fundamental group, but the conventional approach to the latter using loops is inappropriate in this algebraic setting. Instead, we find our inspiration in the following observation: if is a connected topological space and , then for any cover the topological fundamental group acts on the fiber . This construction extends to a functor from the category of covers of to -sets, which is an equivalence provided that admits a universal cover, e.g. if is a manifold.

Let’s try to imitate this idea in the world of algebraic geometry. For most of today we can work with any connected smooth variety over a field (all morphisms, products, etc. will be over ). The analogue of a topological finite cover is a finite étale morphism , where *finite* means is a finitely generated -module, and *étale* means the following. If is algebraically closed, we call étale provided that the differential is an isomorphism for every . For arbitrary , we say is étale provided that the map obtained by extending scalars is étale (here , where is an algebraic closure of and we are systematically confusing fields with their spectra). We should warn the reader here that this definition of étale is only appropriate when is smooth, but since this is the only case which concerns us we will not attempt to give more general definitions.

Why étale morphisms? We want a notion which behaves like a local diffeomorphism of smooth manifolds, and that condition is equivalent to inducing an isomorphism on tangent spaces at each point by the inverse function theorem. In algebraic geometry the Zariski topology is too coarse for actual local isomorphisms to be of any use, but étale morphisms are the appropriate replacement. To see why, let and consider the map which sends . This map is a cover in the topological sense with respect to the analytic (i.e. metric) topology, but is not even locally injective in the Zariski topology, since Zariski open sets are cofinite and is generically -to- on any such set. However, since the derivative of is everywhere nonzero, this map is étale.

The finiteness hypothesis is easier to explain: an étale morphism automatically has finite fibers, so for example, there is no algebraic version of . But (Zariski) open embeddings are étale, and we do not want to allow any of these other than isomorphisms. Note that since finite morphisms are closed and étale morphisms are open, any finite étale morphism into is surjective (recall that we are assuming that is connected).

So far our examples have been over , but there is another crucial feature which one does not notice when is algebraically closed: if is a finite separable extension of the ground field, then is a finite étale morphism, and conversely any connected finite étale cover of has this form. Since finite étale morphisms are stable under base change, any map of the form is finite étale.

Denote the category of all finite étale morphisms by , with arrows in this category being given by commutative triangles in the usual way. Let be a separably closed field containing and pick a morphism , which we call a *geometric point of* . Now if is a finite étale morphism, the fiber is a finite set, and evidently extends to a functor .

**Definition** The *étale** fundamental group of* *based at* , denoted by , is the automorphism group of .

The group has a natural profinite topology. Namely, a basis of open subgroups at the identity is given by the stabilizers of elements of as varies through all of , so that the action of on each is continuous. To summarize, factors through the forgetful functor , where is the category of finite sets equipped with a continuous -action. Moreover, we have the following theorem, which confirms that we have found the correct analogue of the topological fundamental group.

**Theorem** *The functor* * is an equivalence.*

From this one can deduce that if , the étale fundamental group is the profinite completion of the topological fundamental group based at the same point.

It is not hard to see that the étale fundamental group is functorial in the sense that if is a morphism and , there is an induced continuous homomorphism . Also, if is another geometric point, the groups and are isomorphic, but noncanonically: the isomorphism is only determined up to conjugation. In particular, there is a canonical isomorphism , so after abelianization we may omit the basepoint from our notation and simply write .

Finally, we explain the relationship between the étale fundamental group and Galois groups in the cases which interest us. This will establish the connection to the arithmetic language in which class field theory was originally developed. First, consider the case . Once we have chosen a separable closure of there is a unique geometric point , and we will simply write for the fundamental group of based there. It follows from the definitions and basic Galois theory that there is a canonical isomorphism .

Let’s return to the situation of the last post: let be a smooth, projective, and geometrically connected curve over . Denote by its field of rational functions, a fixed separable closure, and the maximal unramified extension contained in . There is a natural choice of basepoint for , namely the geometric generic point .

**Proposition** *There is a canonical isomorphism **.*

Next time, we’ll discuss the Picard scheme and how it relates to the adèles in number theory.

## Geometric class field theory

Today, after a long absence from the blogosphere, I’m starting a series of posts on geometric class field theory. My goal is to make the presentation so geometrical that it is easily comprehensible to readers with backgrounds in algebraic geometry but not number theory. Of course, the story is enriched by the analogy with number fields, and I will frequently draw attention to this analogy, but it will be unnecessary for both the statements and the proofs of the main results.

The main character is a smooth, projective, and (geometrically) connected curve over a field , which we will generally assume is either a finite field or the complex numbers . Very broadly speaking, the goal is to understand all “covers” of , by which we mean finite separable maps where is another curve over , but this is far too ambitious for us. We will focus our attention on abelian covers, which are the connected Galois covers whose automorphism group is abelian (recall that a connected cover is called *Galois* if its automorphism group acts transitively on the geometric generic fiber, or equivalently has cardinality equal to the degree of the cover). Then there is a correspondence involving moduli of line bundles on , as we will explain at length. When , abelian covers correspond to finite-index subgroups of the Picard group of (with level structure in the ramified case).

This is very much like the number theorist’s goal of understanding (abelian) extensions of a number field. Indeed, we are doing the same for the field of rational functions on . The case was developed classically along the same lines as arithmetic class field theory, and to my knowledge it was Deligne who first gave a purely geometric proof in the sixties.

Here is a more precise outline of the plan. Our short-term goal will be to prove the main theorem of class field theory in the unramified setting. After that, we will move on to local class field theory, which we will approach using a geometric version of Lubin-Tate theory. The natural next step is to return to and prove the the general ramified case of global class field theory. Along the way we will explain how, in the case , the basic correspondence can be realized using moduli of shtukas on , and how this relates to Drinfeld modules and explicit class field theory. Finally, in the distant future we might say some words about the higher rank case, which is the geometric Langlands correspondence for , and especially Drinfeld’s proof of the case in positive characteristic.

So that this post is not entirely devoid of content, let’s go ahead and state the main theorem of unramified global geometric class field theory when (the case is slightly harder to formulate, but we’ll get to it). Next time we’ll give (some) definitions and explain how our statement relates to more classical formulations, and probably move on to the proof two posts from now.

We will denote by the étale fundamental group of based at the geometric generic point and its abelianization (as a profinite group). The structure morphism induces a homomorphism , and we write (respectively ) for the *Weil group* of , i.e. the preimage of in (respectively ). It is not hard to see that the Weil group is a dense discrete subgroup of . Any closed point induces a map , well-defined up to conjugation, and the image of is a conjugacy class in called the *(arithmetic) Frobenius at *, which we denote by . In particular, maps to a single element of , which we also denote by .

The other object which appears in the theorem is the Picard group of isomorphism classes of line bundles on under tensor product. As the notation suggests, the Picard group is the group of rational points of the Picard group scheme , which will be relevant later. For now, just observe that is generated by the line bundles as varies through the closed points of . Now we can state the theorem.

**Theorem** (Unramified global class field theory) There is a unique map which sends for each closed point . This map induces an isomorphism .

Note that the isomorphism intertwines the degree map with the natural map . This is because if is a degree point, then is a degree line bundle and induces the automorphism on .

The uniqueness in the theorem is obvious, since the line bundles generate the Picard group. But the existence of this map is already a highly nontrivial statement: this says that if is a principal divisor on , then is trivial in . This is an example of a reciprocity law in the sense of arithmetic class field theory.