## 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 Much of what follows is based on unpublished notes from a class he taught on the theory of schemes.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 $X$ has an open covering by affine schemes $U_i = \text{Spec } A_i$ with each $A_i$ a UFD, then $X$ is factorial, but the converse is false. In fact, it may happen that a factorial scheme contains no affine open subschemes $U = \text{Spec } A$ with $A$ 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 $X$ 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 $X = \text{Spec } A$ is noetherian and integral as well, i.e. that $A$ 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 $X$ are domains then $X$ is the disjoint union of its irreducible components (because $X$ is noetherian!), which are themselves integral. So we are merely avoiding connectedness hypotheses and related complications.

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

1. $X$ is normal, and
2. for any open $U \subset X$ whose complement has codimension at least two, every line bundle on $U$ extends to a line bundle on $X$.

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 $X$ is normal if and only if $X$ is regular in codimension one and the following equivalent conditions hold:

1. no effective Cartier divisor on $X$ has an embedded point,
2. for some (equivalently, any) line bundle $\mathscr{L}$ and $j : U \to X$ the inclusion of an open set whose complement has codimension at least two, the canonical morphism $\mathscr{L} \to j_*j^*\mathscr{L}$ is an isomorphism, and
3. Serre’s condition $\text{S}_2$ holds, i.e. for any coherent sheaf $\mathscr{F}$ on $X$ whose support has codimension at least two, we have $\text{Ext}^1(\mathscr{F},\mathscr{O}_X) = 0$.

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 $X$ be a noetherian integral affine scheme. Then a rational function on $X$ 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 $\tfrac{f}{g}$ on $X = \text{Spec } A$ which is not regular, so $f \notin gA$. Then $I = (fA+gA)/gA \neq 0$: geometrically this is the ideal of functions on $V(g)$ which vanish on $V(f) \cap V(g)$, which is nonzero because $V(f)$ does not contain $V(g)$. Let $\mathfrak{p}$ be an associated point of the $A$-module $I$.

Since $I$ is an ideal in $A/gA$, we see that $\mathfrak{p}$ is an associated point of the Cartier divisor $V(g)$. It remains to show that $\tfrac{f}{g} \notin A_{\mathfrak{p}}$. Suppose $\tfrac{f}{g} = \tfrac{f'}{g'}$ in $A_{\mathfrak{p}}$, so that there exists $s \in A \setminus \mathfrak{p}$ with $sfg' = sf'g$. If $g' \notin \mathfrak{p}$, then the equation $sfg' = sf'g$ implies that $f$ vanishes in $(A/gA)_{\mathfrak{p}}$, so that $I_{\mathfrak{p}} = 0$. But $\mathfrak{p}$ is an associated point of $I$, which means it is an irreducible component of the support of $I$. This is a contradiction, so in fact $g' \in \mathfrak{p}$.

$\Box$

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 $Y \subset X$ be an effective Cartier divisor and $Z \subset Y$ an associated point of $Y$. We need to show that $Z$ is an irreducible component of $Y$, or equivalently that $Z$ has codimension one in $X$. Shrinking $X$ if necessary, we can take the Cartier divisor $Y = \text{Spec } A/fA$ to be principal, cut out by some nonzero function $f \in A$. Now $Z$ corresponds to a prime $\mathfrak{p} \in \text{Spec } A$ which is the annihilator of a nonzero function on $Y$, so there is some $g \in A \setminus fA$ such that

$\mathfrak{p} = \{ a \in A \ | \ ag \in fA \}$.

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

$\mathfrak{p}^{-1} = \{ x \in K \ | \ x\mathfrak{p} \subset A \}$,

where $K$ is the fraction field of $A$, and form the product $\mathfrak{p}\mathfrak{p}^{-1}$. The reader may object that we do not know that $\mathfrak{p}$ is invertible, but that can be seen as follows: since $\mathfrak{p} \subset \mathfrak{p}\mathfrak{p}^{-1} \subset A$ and $\mathfrak{p}$ is maximal, we must have $\mathfrak{p}\mathfrak{p}^{-1} = \mathfrak{p}$ or $\mathfrak{p}\mathfrak{p}^{-1} = A$. In the former case the elements of $\mathfrak{p}^{-1}$ are integral over $A$ because they stabilize the finitely generated $A$-submodule $\mathfrak{p} \subset K$, so $\mathfrak{p}^{-1} = A$ by our normality assumption. But now by the definition of $\mathfrak{p}$ we have $g/f \in \mathfrak{p}^{-1} = A$, i.e. $g \in fA$, a contradiction. Thus $\mathfrak{p}$ is indeed invertible, so that there exist some $x_1,\cdots,x_n \in \mathfrak{p}$ and $y_1,\cdots,y_n \in \mathfrak{p}^{-1}$ such that $\sum x_iy_i =1$. But then $x_iy_i \notin \mathfrak{p}$ for some $i$, so that $x_i^{-1} = (x_iy_i)^{-1}y_i \in \mathfrak{p}^{-1}$ has the property that $x_i^{-1}\mathfrak{p} = A$, i.e. $\mathfrak{p} = x_iA$.

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

Now let us prove that condition 1 implies condition 2 for $X$ regular in codimension one. The assertion is local, so it suffices to consider $\mathscr{L} = \mathscr{O}_X$. Thus we are trying to prove that $A \to \mathscr{O}_X(U)$ is an isomorphism. The map is injective because $X$ is integral, and for surjectivity suppose we are given $f \in \mathscr{O}_X(U)$. Then $f$ is regular along every subscheme of codimension one, which includes all associated points of effective Cartier divisors on $X$ by condition 1. Thus $f \in A$ by Lemma 1.

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

Finally, we check that condition 3 is equivalent to condition 2. For now $\mathscr{F}$ is any coherent sheaf. Given an open subscheme $U \subset X$, write $j : U \to X$ for the inclusion, and consider the short exact sequence

$0 \to \mathscr{O}_X \to j_*\mathscr{O}_U \to (j_*\mathscr{O}_U)/\mathscr{O}_X \to 0$.

This induces the long exact sequence

$\cdots \to \text{Hom}(\mathscr{F}|_U,\mathscr{O}_U) \to \text{Hom}(\mathscr{F},(j_*\mathscr{O}_U)/\mathscr{O}_X)$

$\to \text{Ext}^1(\mathscr{F},\mathscr{O}_X) \to \text{Ext}^1(\mathscr{F}|_U,\mathscr{O}_U) \to \cdots$.

Now let $U$ be the complement of the support of $\mathscr{F}$, so we obtain an isomorphism

$\text{Hom}(\mathscr{F},(j_*\mathscr{O}_U)/\mathscr{O}_X) \cong \text{Ext}^1(\mathscr{F},\mathscr{O}_X)$.

If condition 2 holds and the support of $\mathscr{F}$ 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 $Z = X \setminus U$ has codimension at least 2, we can take $\mathscr{F} = \mathscr{O}_Z$. The left side is identified with $\Gamma(Z,(j_*\mathscr{O}_U)/\mathscr{O}_X)$, and since $Z$ is affine the vanishing of the global sections implies $(j_*\mathscr{O}_U)/\mathscr{O}_X = 0$ as desired.

$\Box$.

Proof of Proposition 1. First assume that $X$ is locally factorial. We omit the standard argument that a unique factorization domain is normal. To prove condition 2, note that $X$ 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 $X$ or $U$ 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 $X$ 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 $A$ have the latter property. In fact, we will prove that every prime Weil divisor $Y \subset X$ is a Cartier divisor, which means that the ideal sheaf $\mathscr{I}_Y$ is a line bundle. Since the local ring of $X$ at $Y$ is a discrete valuation ring, the restriction of $\mathscr{I}_Y$ to the generic point of $Y$ is a trivial line bundle. Thus there is an open set $V \subset X$ with $V \cap Y$ dense in $Y$ such that $\mathscr{I}_Y$ is a line bundle over $V$. We have $\mathscr{I}_Y|_{X \setminus Y} = \mathscr{O}_{X \setminus Y}$, so that $\mathscr{I}_Y$ is a line bundle over $U = (X \setminus Y) \cup V$. Observe that the complement of $U$ has codimension at least two in $X$, so there is a line bundle $\mathscr{L}$ with $\mathscr{L}|_U \cong \mathscr{I}_Y|_U$. The map $\mathscr{L}|_U \to \mathscr{O}_U$ induces $\varphi : \mathscr{L} \to j_*\mathscr{O}_U \cong \mathscr{O}_X$, where $j : U \to X$ and the latter isomorphism uses condition 2 of Proposition 2. Since $X$ is integral $\varphi$ must be injective, and we claim that $\varphi$ maps $\mathscr{L}$ isomorphically onto $\mathscr{I}_Y$. To see that $\varphi$ lands in $\mathscr{I}_Y$, notice that $\mathscr{L} \to \mathscr{O}_Y$ factors through $\mathscr{L}|_Y \to \mathscr{O}_Y$, which vanishes because it is a generically zero map of line bundles on $Y$. The image $\varphi(\mathscr{L}) = \mathscr{I}_Z$ where $Z \subset X$ is a Cartier divisor, and we have just shown that $Y \subset Z$. Now $Y \cap U = Z \cap U$, and since $X \setminus U$ has codimension at least two this implies that $Y$ is the maximal reduced closed subscheme of $Z$, and that any component of $Z \setminus Y$ is an embedded point of $Z$. Thus $Y = Z$ by condition 1 of Proposition 2.

$\Box$

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

If $\mathscr{E}$ is a vector bundle on a scheme $S$, recall that the determinant $\text{det } \mathscr{E}$ is the line bundle on $S$ obtained by taking the top exterior power of $\mathscr{E}$ locally (the rank of $\mathscr{E}$ may vary but is at least locally constant). More generally, if $\mathscr{E}^{\bullet}$ is a bounded complex of vector bundles on $S$, we put

$\det \mathscr{E}^{\bullet} = \bigotimes_{n \in \mathbb{Z}} (\det \mathscr{E}^n)^{(-1)^n}$.

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 $\text{det}$ 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, $A$ will be a regular local ring of dimension $n$ with maximal ideal $\mathfrak{m}$ and residue field $k := A/\mathfrak{m}$.

Lemma 2 Any finitely generated $A$-module $M$ has a free resolution of length at most $n$.

Proof. We can construct a free resolution $F^{\bullet}$ of $M$, possibly of infinite length, in the following manner. Instead of picking an arbitrary set of generators of $M$, we lift a basis of $M/\mathfrak{m}M$: by Nakayama’s lemma this is the same as a minimal set of generators for $M$. This minimal set of generators yields an epimorphism $F^0 \to M$, where $F^0$ is a free module of the appropriate rank. Similarly, one can choose a minimal set of generators for the kernel of $F^0 \to M$, thereby obtaining $F^1 \to F^0$, and so on. It is not hard to see that this minimality condition is equivalent to the vanishing of all differentials in the complex $k \otimes_A F^{\bullet}$. But then we have

$k \otimes_A F^{-i} = H^{-i}(k \otimes_A F^{\bullet}) = \text{Tor}^A_i(k,M),$

so it suffices to show that $\text{Tor}^A_i(k,M) = 0$ for $i > n$, since this implies $F^{-i} = 0$ for $i > n$ (no need to apply Nakayama’s lemma: this is obvious because $F^{-i}$ is free).

For this, recall the existence of the Koszul resolution $K^{\bullet}$, which is a free resolution of the $A$-module $k$ of length $n$. Thus for $i > n$ we have

$\text{Tor}^A_i(k,M) = H^{-i}(K^{\bullet} \otimes M) = 0$.

$\Box$

Proof of the theorem. We reduce immediately to the local case, so $A$ is still a regular local ring and $X = \text{Spec } A$ as usual. First we must prove that $X$ is normal, for which it suffices to show that condition 3 of Proposition 2 holds (note that $A$ is a domain because its associated graded algebra is isomorphic to a polynomial ring, and in particular is a domain). So fix an $A$-module $M$ whose support has codimension at least two. This implies that we can find $f,g \in \mathfrak{m}$ satisfying $fM = gM = 0$ which are a regular sequence in the sense that if $I = fA + gA$, then the Koszul complex

$0 \to A \to A \oplus A \to A \to A/I \to 0$

is exact, where the first nonzero arrow sends $1 \mapsto (g,-f)$ and the second sends $(1,0) \mapsto f$, $(0,1) \mapsto g$. Consider the following segment of the long exact sequence gotten by applying $\text{Hom}(M,-)$:

$\cdots \to \text{Hom}(M,A) \to \text{Hom}(M,A/I) \to \text{Ext}^1(M,A) \to \text{Ext}^1(M,A) \to \cdots$.

Now $\text{Hom}(M,A) \to \text{Hom}(M,A/I)$ is an isomorphism because $IM = 0$, and $\text{Ext}^1(M,A) \to \text{Ext}^1(M,A)$ vanishes for the same reason. It follows that $\text{Ext}^1(M,A) = 0$ 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: $x \in X$ is the closed point, $U = X \setminus \{ x \}$, and $\mathscr{L}$ is a line bundle on $U$ which we want to extend to $X$. First, we claim we can extend $\mathscr{L}$ to a coherent sheaf on $X$. Indeed, if $j : U \to X$ is the inclusion, then $j_*\mathscr{L}$ is a quasi-coherent extension of $\mathscr{L}$, and since $X$ is noetherian $j_*\mathscr{L}$ is the filtered colimit of its coherent subsheaves. Thus there exists a coherent $\mathscr{F} \subset j_*\mathscr{L}$ such that $\mathscr{F}|_U = \mathscr{L}$. Now apply Lemma 2 to $\Gamma(X,\mathscr{F})$, so that we obtain a bounded free resolution of $\mathscr{F}$ and hence can make sense of $\text{det } \mathscr{F}$. This is a line bundle on $X$ with the property that

$(\text{det } \mathscr{F})|_U = \text{det}(\mathscr{F}|_U) = \text{det } \mathscr{L} = \mathscr{L}$.

$\Box$

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

Categories: algebraic geometry

## 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 $X$, from now on we will study (rank $1$) $\ell$-adic local systems on $X$, which can be considered as a “linearized” version of the problem. More precisely, since understanding the structure of the fundamental group $\pi_1(X,\overline{\eta})$ (here $\overline{\eta}$ is the geometric generic point, not that it really matters) is tantamount to understanding all étale covers of $X$, the Tannakian philosophy says that we should consider the monoidal category of finite-dimensional representations of $\pi_1(X,\overline{\eta})$, 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 $\ell$-adic local systems: characters of the Picard group $\text{Pic}(X)$ (more precisely, its profinite completion) correspond to “multiplicative” rank $1$ local systems on the Picard scheme $\text{Pic}_X$. This situation is very special to finite fields! A multiplicative local system on $\text{Pic}_X$ can be pulled back along the Abel-Jacobi map $X \to \text{Pic}_X$, given by $x \mapsto \mathscr{O}(x)$ on points, and global unramified class field theory says that this pullback establishes a bijection between multiplicative local systems on $\text{Pic}_X$ and rank $1$ local systems on $X$. 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 $\ell$ (traditionally one assumes $\ell$ is not equal to $p$, the characteristic of our finite ground field $\mathbb{F}_q$, but for us $\ell = p$ is fine). Let $\overline{\mathbb{Q}}_{\ell}$ be an algebraic closure of the $\ell$-adic numbers $\mathbb{Q}_{\ell}$. The totally disconnected topology on $\overline{\mathbb{Q}}_{\ell}$ 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 $\overline{\mathbb{Q}}_{\ell}$ is an algebraically closed field of characteristic zero.

Let $S$ be a connected scheme and $\overline{s} : \text{Spec } \Omega \to S$ a geometric point.

Definition An $\ell$-adic local system $\mathscr{F}$ on $S$ is a finite-dimensional continuous $\overline{\mathbb{Q}}_{\ell}$-representation of $\pi_1(S,\overline{s})$. The dimension of the representation is called the rank of $\mathscr{F}$.

As our terminology and notation suggest, $\ell$-adic local systems can be thought of as locally constant sheaves of $\overline{\mathbb{Q}}_{\ell}$-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 $\overline{\mathbb{Q}}_{\ell}$-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 $f : T \to S$ and $\overline{t} \in f^{-1}(\overline{s})$, we will write $f^*\mathscr{F}$ for the local system on $T$ obtained by restricting $\mathscr{F}$ along the homomorphism $\pi_1(T,\overline{t}) \to \pi_1(S,\overline{s})$ induced by $f$. The underlying $\overline{\mathbb{Q}}_{\ell}$-vector space of the representation $\mathscr{F}$ is denoted by $\mathscr{F}_{\overline{s}}$.

Now we come to the namesake of this post, for which we should assume that $S$ is defined over $\mathbb{F}_q$. This construction takes an $\ell$-adic local system $\mathscr{F}$ on $S$ and produces from it a function $t_{\mathscr{F}} : S(\mathbb{F}_q) \to \overline{\mathbb{Q}}_{\ell}$. Given $x : \text{Spec } \mathbb{F}_q \to S$,  there is an isomorphism $\pi_1(S,\overline{x}) \cong \pi_1(S,\overline{s})$ well-defined up to conjugation, so that after making this choice we can form the pullback $x^*\mathscr{F}$, a local system on $\text{Spec } \mathbb{F}_q$. Then we set

$t_{\mathscr{F}}(x) = \text{tr}(\text{Frob};x^*\mathscr{F})$,

the trace of the action of the Frobenius $a \mapsto a^q$, which does not depend on our choice because the trace is conjugation-invariant.

We will be interested in the case  where $\mathscr{F}$ has rank $1$, and then it is easier to describe $t_{\mathscr{F}}$. Namely, $x$ determines a canonical map $\widehat{\mathbb{Z}} = \pi_1(\mathbb{F}_q) \to \pi_1(S,\overline{s})^{\text{ab}}$, which we compose with $\mathscr{F}$ (now thought of a one-dimensional representation) to obtain a homomorphism $\widehat{\mathbb{Z}} \to \overline{\mathbb{Q}}_{\ell}^{\times}$. Evaluate this map at $1$ to obtain $t_{\mathscr{F}}(x)$.

Sometimes interesting classes of functions and sheaves match up under this correspondence. Let $G$ be a commutative algebraic group over $\mathbb{F}_q$: then one such class of functions is the group of characters of $G(\mathbb{F}_q)$, meaning one-dimensional representations $G(\mathbb{F}_q) \to \overline{\mathbb{Q}}_{\ell}^{\times}$. What sort of local system $\mathscr{F}$ on $G$ has the property that $t_{\mathscr{F}}$ is a character?

Definition A rank $1$ local system $\mathscr{F}$ on $G$ is called a character sheaf provided that $\mu^*\mathscr{F} \cong \mathscr{F} \boxtimes \mathscr{F}$, where $\mu : G \times G \to G$ is the multiplication map.

(Notation for those who haven’t seen it: if $\mathscr{F}$ is a sheaf on $S$ and $\mathscr{G}$ is a sheaf on $T$, then their external tensor product is $\mathscr{F} \boxtimes \mathscr{G} = p_S^*\mathscr{F} \otimes p_T^*\mathscr{G}$ where $p_S : S \times T \to S$ and $p_T : S \times T \to T$ 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 $\mathscr{F}$ be an $\ell$-adic local system on $S$. Then there is a canonical isomorphism $\mathscr{F} \to \text{Fr}_S^*\mathscr{F}$.

Proof. We reduce immediately to the case that $\mathscr{F}$ is a locally constant étale sheaf of finite sets, so there is some finite étale map $f : T \to S$ whose sheaf of local sections is $\mathscr{F}$. Write $\text{Fr}_S^*T$ for the fiber product of $\text{Fr}_S : S \to S$ and $f$, so it suffices to produce an isomorphism $T \to \text{Fr}_S^*T$ of $S$-schemes. Using the relation $f \circ \text{Fr}_T = \text{Fr}_S \circ f$, we obtain the desired map $g : T \to \text{Fr}_S^*T$ from $\text{Fr}_T$ and $f$. Since $f$ and $\text{Fr}_S^*T \to S$ are finite étale, so is $g$, and similarly $g$ is radicial (i.e. universally injective) because $\text{Fr}_T$ is. But a map which is both étale and radicial must be an open embedding, and $g$ is also finite, hence an isomorphism.

$\Box$

Now we come to the really interesting part. As usual $G$ is a commutative algebraic group over $\mathbb{F}_q$, which we now assume to be smooth and connected (the smoothness hypothesis is really not necessary).

Proposition Under the above assumptions, $\mathscr{F} \to t_{\mathscr{F}}$ is a bijection from character sheaves on $G$ to characters of $G(\mathbb{F}_q)$.

Proof. That $t_{\mathscr{F}}$ is a character follows from the easy identities $t_{f^*\mathscr{F}} = f^*t_{\mathscr{F}}$ and $t_{\mathscr{F} \otimes \mathscr{G}} = t_{\mathscr{F}} \cdot t_{\mathscr{G}}$. Suppose that we are given a character $\chi : G(\mathbb{F}_q) \to \overline{\mathbb{Q}}_{\ell}^{\times}$. The Lang isogeny $L : G \to G$ is a pointed finite Galois covering with group $G(\mathbb{F}_q)$, hence gives rise to a map $\pi_1(G,\overline{1}) \to G(\mathbb{F}_q)$, which we can restrict $\chi$ along to obtain a rank $1$ local system $\mathscr{F}(\chi)$ on $G$. The same identities show that $\mathscr{F}(\chi)$ 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 $t_{\mathscr{F}(\chi)} = \chi$. Given $x \in G(\mathbb{F}_q)$, we obtain a canonical map $\widehat{\mathbb{Z}} = \pi_1(\mathbb{F}_q) \to \pi_1(G,\overline{1})^{\text{ab}}$, whose value on $1 = \text{Frob}$ we will call $\text{Frob}_x$. By definition $t_{\mathscr{F}(\chi)}(x)$ is the value of $\mathscr{F}(\chi)$ on $\text{Frob}_x$, but $\mathscr{F}(\chi)$ factors through $\chi : G(\mathbb{F}_q) \to \overline{\mathbb{Q}}_{\ell}^{\times}$ by construction, so it suffices to show that the map $\pi_1(G,\overline{1}) \to G(\mathbb{F}_q)$ induced by $L$ sends $\text{Frob}_x$ to $x$. This means that when $\text{Frob}_x$ acts on the fiber $L^{-1}(\overline{1}) = G(\mathbb{F}_q)$, it just translates by $x$. Fixing $\overline{y} \in L^{-1}(\overline{x})$, we obtain an identification $L^{-1}(\overline{1}) \cong L^{-1}(\overline{x})$ since $L^{-1}(\overline{x}) = \overline{y}G(\mathbb{F}_q)$. Tracing through definitions, our claim follows from the calculation

$\text{Frob}_x \cdot \overline{y} = \text{Fr}_G(\overline{y}) = \overline{x} \overline{y}$.

Finally, we must prove that $\mathscr{F}(t_{\mathscr{F}}) = \mathscr{F}$. We claim that $L^*\mathscr{F}$ is trivial, or equivalently that $\mathscr{F}$ factors through the homomorphism $\pi_1(G,\overline{1}) \to G(\mathbb{F}_q)$ determined by $L$. Since this map sends $\text{Frob}_x \mapsto x$ the claim implies that $\mathscr{F}$ is determined by its values on $\text{Frob}_x$ for all $x \in G(\mathbb{F}_q)$, and we just proved that $t_{\mathscr{F}(t_{\mathscr{\chi}})} = t_{\mathscr{F}}$, from which it follows that $\mathscr{F}(t_{\mathscr{F}}) = \mathscr{F}$. As for the claim, observe that

$L^*\mathscr{F} = (i,\text{Fr}_G)^*\mu^*\mathscr{F} = (i,\text{Fr}_G)^*(\mathscr{F} \boxtimes \mathscr{F}) = i^*\mathscr{F} \otimes \text{Fr}_G^*\mathscr{F}$.

By the lemma $\text{Fr}_G^*\mathscr{F} = \mathscr{F}$, so we just have to check that $i^*\mathscr{F} \otimes \mathscr{F}$ is trivial. The latter sheaf is the pullback of $\mathscr{F}$ along $\mu \circ (i,\text{id}_G) : G \to G$, which is the trivial homomorphism, and since $t_{\mathscr{F}}(1) = 1$ we are done.

$\Box$

Next time we will consider $G = \text{Pic}_X$, 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 $\mathbb{F}_q$. 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 $\mathbb{F}_q$ arise as pullbacks of the Lang isogeny of the Jacobian along an Abel-Jacobi map.

Recall that any scheme $S$ over $\mathbb{F}_q$ has the Frobenius endomorphism $\text{Fr}_S$, which is defined to be the identity on the underlying set of $S$ and sends $f \mapsto f^q$ for local sections $f \in \mathscr{O}_S$. It is not hard to see that $\text{Fr}$ is natural in the sense that if $f : S \to T$ is a morphism of $\mathbb{F}_q$-schemes, we have $f \circ \text{Fr}_S = \text{Fr}_T \circ f$. A fancy way of saying this is that $\text{Fr}$ is an element of the Bernstein center of the category $\text{Sch}_{\mathbb{F}_q}$ of $\mathbb{F}_q$-schemes, i.e. an endomorphism of the identity functor on $\text{Sch}_{\mathbb{F}_q}$. In the sequel we will omit the subscript and simply write $\text{Fr}$.

Let $G$ be a group scheme over $\mathbb{F}_q$. We want to study the difference between the Frobenius endomorphism and the identity.

Definition The Lang map $L$ is the endomorphism of the underlying $\mathbb{F}_q$-scheme of $G$ given as the composition

$L : G \stackrel{(i,\text{Fr})}{\longrightarrow}G \times G \stackrel{m}{\longrightarrow} G$,

where $i$ and $m$ are the inversion and multiplication maps of $G$, respectively.

Thus, if $g \in G(S)$ for some $\mathbb{F}_q$-scheme $S$, we can write $L(g) = g^{-1}\text{Fr}_G(g)$. From this formula and the aforementioned fact that $\text{Fr}$ commutes with all morphisms, one deduces that if $G$ is commutative then $L$ is a group endomorphism.

Taking $S = \overline{\mathbb{F}}_q$, we see that the fiber of $L$ over $\overline{1} \in G(\overline{\mathbb{F}}_q)$ is precisely $G(\mathbb{F}_q)$.

Proposition Suppose $G$ is smooth (in particular, of finite type) over $\mathbb{F}_q$. Then the Lang map $L : G \to G$ is finite étale.

Proof. To prove that $L$ is étale, it suffices to show that the differential of $\overline{L} : \overline{G} \to \overline{G}$ is an isomorphism on the tangent space at each point of $\overline{G} = G \times_{\mathbb{F}_q} \overline{\mathbb{F}}_q$. Since $d\text{Fr} = 0$ we have $dL_{\overline{1}} = -\text{id}$, whence $L$ is étale at $\overline{1}$. Observe that $L$ intertwines two actions of $G$ on itself: right translations and $g \cdot h = h^{-1}g\text{Fr}(h)$. Since the action by right translations is transitive, it follows that $L$ 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 $L$ is finite.

$\Box$

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

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

$1 \to G(\mathbb{F}_q) \to G(\overline{\mathbb{F}}_q) \stackrel{L}{\to} G(\overline{\mathbb{F}}_q) \to 1$,

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

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

$\Box$

Corollary If $G$ is connected then any $G$-torsor $X$ is trivial.

Proof. Let us prove the equivalent statement $X(\mathbb{F}_q) \neq \varnothing$.The $\overline{G}$-torsor $\overline{X}$ clearly has a point $x \in \overline{X}(\overline{\mathbb{F}}_q) = X(\overline{\mathbb{F}}_q)$. Now one can find $g \in G(\overline{\mathbb{F}}_q)$ such that $g \cdot \text{Fr}(x) = x$, and we claim that if $h \in L^{-1}(g)$ (which exists according to the previous corollary) then $h \cdot x \in X(\mathbb{F}_q)$. Indeed,

$\text{Fr}(h \cdot x) = hL(h) \cdot \text{Fr}(x) = hg \cdot \text{Fr}(x) = h \cdot x$.

$\Box$

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

Corollary Let $X$ be a smooth, projective, and geometrically connected curve over $\mathbb{F}_q$. Then $X$ admits a degree $1$ zero-cycle.

Proof. Let $\text{Pic}_X$ denote the Picard functor introduced previously and $\widetilde{\text{Pic}}_X$ its fppf-sheafification, which we know to be representable by general results. Moreover, $\widetilde{\text{Pic}}_X^0 = \text{ker}(\widetilde{\text{Pic}}_X \stackrel{\text{deg}}{\to} \mathbb{Z})$ is smooth and connected (these facts are drawn from Chapter 6 of Néron Models by Bosch, Lütkebohmert, and Raynaud). Now $\widetilde{\text{Pic}}_X^1$ is a $\widetilde{\text{Pic}}_X$-torsor, so by the last corollary we have $\widetilde{\text{Pic}}_X^1(\mathbb{F}_q) \neq \varnothing$. But $\widetilde{\text{Pic}}_X^1(\mathbb{F}_q) = \text{Pic}^1(X)$ because $\text{Br}(\mathbb{F}_q) = 0$ (see the post on the Picard scheme for the exact sequence that implies this), so $X$ has a degree $1$ line bundle, or equivalently a degree $1$ zero-cycle.

$\Box$

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

Categories: algebraic geometry

## Adèlic groups and principal bundles

7 May 2013 1 comment

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 $G$-bundles on a curve can be uniformized by the adèlic points of $G$ (here $G$ is a connected smooth affine algebraic group satisfying some additional hypotheses which which hold, for instance, if the ground field is $\mathbb{C}$ or $G = \text{GL}_n$), so taking the multiplicative group $G = \mathbb{G}_m$ 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 $G$-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 $k$, let $X$ be a $k$-scheme, and $G$ an affine algebraic group over $k$, meaning an affine group scheme of finite type over $k$.

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

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

There are two basic constructions we perform with $G$-bundles. If $f : Y \to X$ is a morphism, then we can form the pullback bundle $f^*E = Y \times_X E$, which becomes a $G$-bundle when equipped with the obvious right action. Less obviously, for any scheme $Z$ with a left action of $G$, there is an associated bundle $E_Z = (E \times Z)/G$, where $E \times Z$ has the diagonal action

$(e,z) \cdot g = (e \cdot g,g^{-1} \cdot z)$.

One observes $E_Z$ is a scheme over $X$ which is isomorphic to $Z \times X \to X$ locally in the étale topology.

The latter construction allows us to produce a vector bundle on $X$ from a linear representation of $G$. It is a basic fact that, when applied to the standard representation of $\text{GL}_n$, this establishes an equivalence between $\text{GL}_n$-bundles and rank $n$ vector bundles. When $n = 1$, 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 $\mathbb{G}_m$-bundle.

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

Next, we explain the construction of the adèles. For this let $X$ be a smooth connected curve over $k$, and denote its field of rational functions by $K$. For each closed point $x \in X$, write $\mathcal{O}_x$ for the completed local ring of $X$ at $x$ (we will not use the non-completed local rings) with maximal ideal $\mathfrak{m}_x \subset \mathcal{O}_x$, residue field $k_x = \mathcal{O}_x/\mathfrak{m}_x$, and fraction field $K_x$. Alternatively, $K_x$ can be viewed as the completion of $K$ with respect to the $x$-adic absolute value. Recall that if we choose a local coordinate $\epsilon \in \mathfrak{m}_x \setminus \mathfrak{m}_x^2$ (a number theorist would say “uniformizer” or “prime element”) we get an isomorphism $\mathcal{O}_x \cong k_x[[\epsilon]]$ and hence $K_x \cong k_x((\epsilon))$.

Definition For any finite set $S \subset X$ of closed points, the $S$-adèles are the topological ring

$\mathbb{A}_S = \prod_{x \notin S} \mathcal{O}_x \times \prod_{x \in S} K_x$.

The full ring of adèles is the union $\mathbb{A} = \bigcup_S \mathbb{A}_S$ over all finite $S \subset X$, endowed with the colimit topology. The integral adèles are just $\mathbb{O} = \mathbb{A}_{\varnothing} = \prod_{x \in X} \mathcal{O}_x$.

Although the topology on $\mathbb{A}$ is important for many purposes, especially when $k$ is finite, we will not need it in what follows.

Now we begin to interpret these arithmetic constructions geometrically. Write $D_x = \text{Spec } \mathcal{O}_x$ and $D_x^o = \text{Spec } K_x$, which we view as the disk and punctured disk centered at $x$, respectively. If $G$ is an affine algebraic group over $k$ as before, then $G(D_x^o) = G(K_x)$ 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 $G(\mathbb{A})$ packages together all the loop groups $G(D_x^o)$ as $x$ varies through the closed points of $X$. It contains $G(K)$ and $G(\mathbb{O})$ as subgroups, which in particular act on $G(\mathbb{A})$ by translations. We declare that by convention $G(K)$ acts on the left and $G(\mathbb{O})$ acts on the right.

Definition Let $E$ be a $G$-bundle on $X$ and $x \in X$ a closed point. A (full) level structure on $E$ at $x$ is a trivialization of $E$ at the canonical morphism $D_x \to X$.

The reason for the parenthetical terminology “full” is that if $N \subset X$ is a finite subscheme (for example, an $n^{\text{th}}$-order neighborhood of $x$, of which $D_x$ is the colimit as $n$ varies) then a trivialization of $E$ at $N$ is called a structure of level $N$. 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 $X$ is a smooth connected curve over $k$.

Theorem (Uniformization) Let $G$ be a smooth affine algebraic group over $k$. Then there is a canonical bijection from $G(\mathbb{A})$ to the set of isomorphism classes of $G$-bundles on $X$ equipped with a generic trivialization (i.e. a trivialization at the generic point $\eta : \text{Spec } K \to X$) and full level structures at all closed points $x \in X$. Moreover, this bijection is equivariant for $G(K)$ and $G(\mathbb{O})$, which act on generic trivializations and full level structures respectively.

Proof. If we are given a $G$-bundle $E$ equipped with a generic level structure and full level structures at all closed points of $X$, we can construct an adèlic point of $G$ as follows. First observe that the given generic trivialization $\zeta : \text{Spec } K \times G \ \tilde{\to} \ \eta^*E$ “spreads out” to a trivialization $U \times G \ \tilde{\to} \ E|_U$ over a nonempty open set $U \subset X$ (this is where we use the hypothesis that $G$ is of finite type over $k$). Now the desired adèlic point is a Cech cocycle representative for $E$ with respect to the fpqc cover of $X$ consisting of all the formal disks $D_x$ together with $U$. More precisely, restricting the full level structures and the generic trivialization gives two trivializations of $E$ on every punctured disk $D_x^o$, and the difference of these two trivializations is an automorphism of the trivial $G$-bundle on $D_x^o$, or equivalently a point $g_x \in G(D_x^o)$ 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 $g_x \in G(D_x)$ for all $x \in U$, so as $x$ varies the $g_x$ give a well-defined element of $G(\mathbb{A})$.

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

One can inspect the constructions to see that they are mutually inverse and equivariant for $G(K)$ and $G(\mathbb{O})$.

$\Box$

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

Corollary If every $G$-bundle on $X$ admits a generic trivialization, then the bijection from the theorem descends to a $G(\mathbb{O})$-equivariant bijection from $G(K) \backslash G(\mathbb{A})$ to the set of isomorphism classes of $G$-bundles on $X$ equipped with full level structures at every closed point $x \in X$. If, in addition, every $G$-bundle on $X$ admits a trivialization at $\text{Spec } k(x) \to X$ for each closed point $x \in X$, then we obtain a bijection from $G(K) \backslash G(\mathbb{A})/G(\mathbb{O})$ to the set of isomorphism classes of $G$-bundles on $X$.

Proof. The only part that is not obvious is that if a $G$-bundle $E$ on $X$ trivializes over $\text{Spec } k(x)$, then it trivializes over $D_x$. But a trivialization is the same as a section, so by smoothness of $G$ this follows from Hensel’s lemma.

$\Box$

We conclude by pointing out when $G = \mathbb{G}_m$ is the multiplicative group, both sides of the final bijection have a group structure, and we actually obtain an isomorphism of groups $\mathbb{A}^{\times}/K^{\times}\mathbb{O}^{\times} \ \tilde{\to} \ \text{Pic}(X)$. The only additional content in this assertion is that tensoring line bundles corresponds to multiplying representative $\mathbb{G}_m$-valued Cech cocycles.

Next time, we’ll get into some specifics of the situation where $k$ 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 $X$ be a scheme over a field $k$. We can define a contravariant functor from the category $\text{Sch}_k$ of schemes over $k$ to the category $\text{Ab}$ of abelian groups by sending a $k$-scheme $S$ to the Picard group $\text{Pic}(X \times S)$. Here we can think of a line bundle on $X \times S$ as a family of line bundles on $X$ parameterized by $S$. This is the first functor which we might naïvely hope is representable: however, this is never the case if $X$ is nonempty. To see why, take $S = \mathbb{P}^1$, so $\mathbb{P}^1_X = X \times \mathbb{P}^1$ carries the nontrivial line bundle $\mathcal{O}(1)$.  But if we pull back along the standard Zariski covering $\mathbb{A}^1 \coprod \mathbb{A}^1 \to \mathbb{P}^1$, the bundle $\mathcal{O}(1)$ trivializes, so that in particular the induced map $\text{Pic}(\mathbb{P}^1_X) \to \text{Pic}(\mathbb{A}^1_X \coprod \mathbb{A}^1_X)$ is not injective. Since a representable functor is necessarily a sheaf with respect to the Zariski (Grothendieck) topology on $\text{Sch}_k$, it follows that $S \mapsto \text{Pic}(X \times S)$ is not representable.

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

Definition The Picard functor $\text{Pic}_X : \text{Sch}_k^{\text{op}} \to \text{Ab}$ is defined by the formula

$\text{Pic}_X(S) = \text{coker}(\text{Pic}(S) \to \text{Pic}(X \times S))$.

If $\text{Pic}_X$ is representable by a scheme, we call it the Picard scheme of $X$.

In particular we have $\text{Pic}_X(k) = \text{Pic}(X)$, so this definition is consistent with the notation from two posts ago (here and in the sequel, when $A$ is a $k$-algebra and $F$ is a functor on $k$-schemes we commit the standard abuse of notation $F(A) = F(\text{Spec } A)$). 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 $\text{Pic}_X$ is not representable but its fppf-sheafification is. The next theorem shows that $\text{Pic}_X$ suffices for our purposes, so we will not linger on this technical point.

Theorem If $X$ 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 $\mathbb{Z}$), then $\text{Pic}_X$ is represented by a scheme which is locally of finite type over $k$.

Proof. Theorem 3 in Section 8.2 of Néron Models says that the fppf-sheafification $\widetilde{\text{Pic}}_X$ of $\text{Pic}_X$ is representable. Our statement can be deduced from this and Proposition 4 in Section 8.1 as follows. The hypothesis that $\pi_*\mathcal{O}_{X \times S} = \mathcal{O}_S$ for any $k$-scheme $S$ is satisfied because $X$ is proper and geometrically integral. Thus the proposition yields an exact sequence

$0 \to \text{Pic}(S) \to \text{Pic}(X \times S) \to \widetilde{\text{Pic}}_X(S) \to \text{Br}(S) \to \text{Br}(X \times S)$,

where $\text{Br}$ is the Brauer group, i.e. the étale cohomology group $\text{Br}(S) = H^2(S,\mathbb{G}_m)$. We must prove that $\text{Br}(S) \to \text{Br}(X \times S)$ is injective: in fact, we will produce a retraction of this map. Let $x_1,\cdots,x_r \in X$ be closed points of degrees $d_1,\cdots,d_r$ respectively, and choose $m_1,\cdots,m_r \in \mathbb{Z}$ such that $\sum_i m_id_i = 1$. For each $1 \leq i \leq r$, the morphism $\text{Spec } k(x_i) \to X$ induces another $\text{Spec } k(x_i) \times S \to X \times S$, and we denote by $\varphi_i : \text{Br}(X \times S) \to \text{Br}(S)$ the composition of pullback along this map with the corestriction $\text{Br}(\text{Spec } k(x_i) \times S) \to \text{Br}(S)$. A standard property of restriction and corestriction maps says that the composition

$\text{Br}(S) \to \text{Br}(X \times S) \stackrel{\varphi_i}{\to} \text{Br}(S)$

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

$\Box$

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 $X$ is a projective and geometrically integral curve. Then there is a natural degree map $\text{Pic}_X \to \mathbb{Z}_k$, where $\mathbb{Z}_k$ is the constant group scheme over $k$ associated with the abstract group $\mathbb{Z}$. We write $\text{Pic}^d_X$ for the fiber over $d$, which parameterizes line bundles of degree $d$.

Proposition The kernel $\text{Pic}^0_X$ is a smooth, connected, commutative algebraic group over $k$, called the generalized Jacobian of $X$. If we assume also that $k$ is perfect, then the normalization $\widetilde{X} \to X$ induces an epimorphism $\text{Pic}^0_X \to \text{Pic}^0_{\widetilde{X}}$ whose kernel is a smooth connected affine algebraic group, and $\text{Pic}^0_{\widetilde{X}}$ is an abelian variety of dimension equal to the genus of $\widetilde{X}$.

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 $\text{ker}(\text{Pic}^0_X \to \text{Pic}^0_{\widetilde{X}})$, which encodes information about the singularities of $X$. 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 $k$ 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.

Categories: algebraic geometry

## 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 $X$ is a connected topological space and $x \in X$, then for any cover $f : Y \to X$ the topological fundamental group $\pi_1(X,x)$ acts on the fiber $f^{-1}(x)$. This construction extends to a functor from the category of covers of $X$ to $\pi_1(X,x)$-sets, which is an equivalence provided that $X$ admits a universal cover, e.g. if $X$ 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 $X$ over a field $k$ (all morphisms, products, etc. will be over $k$). The analogue of a topological finite cover is a finite étale morphism $f : Y \to X$, where finite means $f_*\mathcal{O}_Y$ is a finitely generated $\mathcal{O}_X$-module, and étale means the following. If $k$ is algebraically closed, we call $f$ étale provided that the differential $df_y : T_yY \to T_{f(y)}X$ is an isomorphism for every $y \in Y$. For arbitrary $k$, we say $f$ is étale provided that the map $\overline{f} : \overline{Y} \to \overline{X}$ obtained by extending scalars is étale (here $\overline{X} = X \times_k \overline{k}$, where $\overline{k}$ is an algebraic closure of $k$ and we are systematically confusing fields with their spectra). We should warn the reader here that this definition of étale is only appropriate when $X$ 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 $k = \mathbb{C}$ and consider the map $f : \mathbb{C} \setminus \{ 0 \} \to \mathbb{C} \setminus \{ 0 \}$ which sends $z \mapsto z^n$. 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 $f$ is generically $n$-to-$1$ on any such set. However, since the derivative of $f$ 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 $\text{exp} : \mathbb{C} \to \mathbb{C} \setminus \{ 0 \}$. 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 $X$ is surjective (recall that we are assuming that $X$ is connected).

So far our examples have been over $\mathbb{C}$, but there is another crucial feature which one does not notice when $k$ is algebraically closed: if $\ell / k$ is a finite separable extension of the ground field, then $\text{Spec } \ell \to \text{Spec } k$ is a finite étale morphism, and conversely any connected finite étale cover of $\text{Spec } k$ has this form. Since finite étale morphisms are stable under base change, any map of the form $X \times_k \ell \to X$ is finite étale.

Denote the category of all finite étale morphisms $Y \to X$ by $\text{fEt}_X$, with arrows in this category being given by commutative triangles in the usual way. Let $\Omega$ be a separably closed field containing $k$ and pick a morphism $\overline{x} : \text{Spec }\Omega \to X$, which we call a geometric point of $X$. Now if $Y \to X$ is a finite étale morphism, the fiber $Y_{\overline{x}} = (Y \times_X \Omega)(\Omega)$ is a finite set, and $Y \to Y_{\overline{x}}$ evidently extends to a functor $F_{\overline{x}} : \text{fEt}_X \to \text{fSet}$.

Definition The étale fundamental group of $X$ based at $\overline{x}$, denoted by $\pi_1(X,\overline{x})$, is the automorphism group of $F_{\overline{x}}$.

The group $\pi_1(X,\overline{x})$ has a natural profinite topology. Namely, a basis of open subgroups at the identity is given by the stabilizers of elements of $F_{\overline{x}}(Y) = Y_{\overline{x}}$ as $Y \to X$ varies through all of $\text{fEt}_X$, so that the action of $\pi_1(X,\overline{x})$ on each $F_{\overline{x}}(Y)$ is continuous. To summarize, $F_{\overline{x}}$ factors through the forgetful functor $\pi_1(X,\overline{x})\text{-fSet} \to \text{fSet}$, where $\pi_1(X,\overline{x})\text{-fSet}$ is the category of finite sets equipped with a continuous $\pi_1(X,\overline{x})$-action. Moreover, we have the following theorem, which confirms that we have found the correct analogue of the topological fundamental group.

Theorem The functor $F_{\overline{x}} : \text{fEt}_X \to \pi_1(X,\overline{x})\text{-fSet}$ is an equivalence.

From this one can deduce that if $k = \mathbb{C}$, 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 $Y \to X$ is a morphism and $\overline{y} \in Y_{\overline{x}}$, there is an induced continuous homomorphism $\pi_1(Y,\overline{y}) \to \pi_1(X,\overline{x})$. Also, if $\overline{x}' \in X(\Omega)$ is another geometric point, the groups $\pi_1(X,\overline{x})$ and $\pi_1(X,\overline{x}')$ are isomorphic, but noncanonically: the isomorphism is only determined up to conjugation. In particular, there is a canonical isomorphism $\pi_1(X,\overline{x})^{\text{ab}} \cong \pi_1(X,\overline{x}')^{\text{ab}}$, so after abelianization we may omit the basepoint from our notation and simply write $\pi_1(X)^{\text{ab}}$.

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 $X = \text{Spec } k$. Once we have chosen a separable closure $k^{\text{sep}}$ of $k$ there is a unique geometric point $\text{Spec } k^{\text{sep}} \to \text{Spec } k$, and we will simply write $\pi_1(k)$ for the fundamental group of $\text{Spec } k$ based there. It follows from the definitions and basic Galois theory that there is a canonical isomorphism $\pi_1(k) \cong \text{Gal}(k^{\text{sep}}/k)$.

Let’s return to the situation of the last post: let $X$ be a smooth, projective, and geometrically connected curve over $k$. Denote by $K$ its field of rational functions, $K^{\text{sep}}$ a fixed separable closure, and $K^{\text{nr}}$ the maximal unramified extension contained in $K^{\text{sep}}$. There is a natural choice of basepoint for $X$, namely the geometric generic point $\overline{\eta} : \text{Spec } K^{\text{sep}} \to X$.

Proposition There is a canonical isomorphism $\pi_1(X,\overline{\eta}) \cong \text{Gal}(K^{\text{nr}}/K)$.

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

Categories: Uncategorized

## 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 $X$ over a field $k$, which we will generally assume is either a finite field $\mathbb{F}_q$ or the complex numbers $\mathbb{C}$. Very broadly speaking, the goal is to understand all “covers” of $X$, by which we mean finite separable maps $Y \to X$ where $Y$ is another curve over $k$, 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 $X$, as we will explain at length. When $k = \mathbb{F}_q$, abelian covers correspond to finite-index subgroups of the Picard group of $X$ (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 $X$. The case $k = \mathbb{F}_q$ 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 $k = \mathbb{F}_q$, the basic correspondence can be realized using moduli of shtukas on $X$, 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 $\text{GL}_n$, and especially Drinfeld’s proof of the case $n = 2$ 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 $k = \mathbb{F}_q$ (the case $k = \mathbb{C}$ 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 $\pi_1(X)$ the étale fundamental group of $X$ based at the geometric generic point and $\pi_1(X)^{\text{ab}}$ its abelianization (as a profinite group). The structure morphism $X \to \text{Spec } \mathbb{F}_q$ induces a homomorphism $\pi_1(X) \to \widehat{\mathbb{Z}}$, and we write $W_X$ (respectively $W_X^{\text{ab}}$) for the Weil group of $X$, i.e. the preimage of $\mathbb{Z}$ in $\pi_1(X)$ (respectively $\pi_1(X)^{\text{ab}}$). It is not hard to see that the Weil group is a dense discrete subgroup of $\pi_1(X)$. Any closed point $x : \text{Spec } \mathbb{F}_{q^d} \to X$ induces a map $\mathbb{Z} \to W_X$, well-defined up to conjugation, and the image of $1$ is a conjugacy class in $W_X$ called the (arithmetic) Frobenius at $x$, which we denote by $\text{Fr}_x$. In particular, $\text{Fr}_x$ maps to a single element of $W_X^{\text{ab}}$, which we also denote by $\text{Fr}_x$.

The other object which appears in the theorem is the Picard group $\text{Pic}_X(\mathbb{F}_q)$ of isomorphism classes of line bundles on $X$ under tensor product. As the notation suggests, the Picard group is the group of rational points of the Picard group scheme $\text{Pic}_X$, which will be relevant later. For now, just observe that $\text{Pic}_X(\mathbb{F}_q)$ is generated by the line bundles $\mathcal{O}(x)$ as $x$ varies through the closed points of $X$. Now we can state the theorem.

Theorem (Unramified global class field theory) There is a unique map $\text{Pic}_X(\mathbb{F}_q) \to \pi_1(X)^{\text{ab}}$ which sends $\mathcal{O}(x) \mapsto \text{Fr}_x$ for each closed point $x \in X$. This map induces an isomorphism $\text{Pic}_X(\mathbb{F}_q) \cong W_X^{\text{ab}}$.

Note that the isomorphism $\text{Pic}_X(\mathbb{F}_q) \cong W_X^{\text{ab}}$ intertwines the degree map $\text{Pic}_X(\mathbb{F}_q) \to \mathbb{Z}$ with the natural map $W_X^{\text{ab}} \to \mathbb{Z}$. This is because if $x \in X$ is a degree $d$ point, then $\mathcal{O}(x)$ is a degree $d$ line bundle and $\text{Fr}_x$ induces the automorphism $a \mapsto a^{q^d}$ on $\overline{\mathbb{F}}_q$.

The uniqueness in the theorem is obvious, since the line bundles $\mathcal{O}(x)$ generate the Picard group. But the existence of this map is already a highly nontrivial statement: this says that if $\sum n_ix_i$ is a principal divisor on $X$, then $\prod \text{Fr}_{x_i}^{n_i}$ is trivial in $\pi_1(X)^{\text{ab}}$. This is an example of a reciprocity law in the sense of arithmetic class field theory.