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One of the recurring themes of modern number theory seems to be a nagging feeling that almost all the things we know (and many more things we don’t know) can be expressed as elementary properties of {\zeta} and {L}-functions.

For example, the zeta function of a number field {K} is defined by the product

\displaystyle \zeta_K(s) = \prod_{\mathfrak{p} \text{ prime ideal of } \mathcal{O}_k} (1 - (N\mathfrak{p})^{-s})^{-1}.
If we expand this as a Dirichlet series

\displaystyle \zeta_K(s) = \sum_n a_n n^{-s}
the coefficients {a_n} give detailed information about the factorisation of ideals in {K}. For example, if {a_p=1} then {p} is totally ramified, whereas if it is equal to the degree of the field, {p} is totally split.

It is also well-known that the Riemann zeta function (the case {K={\mathbb Q}} above) controls the distribution of the prime numbers in {{\mathbb Z}}. By modifying this {\zeta} function we can obtain the Dirichlet L-functions which give us even more information. To define these, we start with a homomorphism {\chi: {\mathbb Z} \rightarrow {\mathbb Z}/q{\mathbb Z} \rightarrow {\mathbb C}^*} called a Dirichlet character, and `twist’ the {\zeta}-function using it:

\displaystyle L(\chi, s) = \sum_n \chi(n) n^{-s} = \prod_p (1 - \chi(p)p^{-s})^{-1}.
For example, the trivial character {\chi_1 = 1} recovers the original {\zeta}-function. Dirichlet famously used this, together with observations about the (lack of) zeroes of these (mostly) convergent power series at {Re(s)>1} and {s=1}, to deduce that any arithmetic progression which might plausibly contain infinitely many primes does. Note that if {q} is the least quotient {\chi} factors through, it is called the conductor of {\chi}.

It seems to be true that certainly for {E/{\mathbb Q}} an abelian extension, the above two objects are related, and in fact {\zeta_E} can be written as a product of Dirichlet L-functions. I don’t know enough class field theory to go into the general cases, but want to look at a specific case that seems quite interesting. Apparently the following result is true. Let {q} be a fixed prime, and {\chi} any quadratic (taking values {1} and {-1}) character with conductor {q}. Now consider the quadratic field {K = {\mathbb Q}(\sqrt{\chi(-1) q})}. We have the following identity:

\displaystyle \zeta_K(s) = \zeta(s) L(\chi, s).

Looks innocuous enough…. surely even if this is true, it can’t say much interesting.

Well, let’s unravel what it says:

\displaystyle \prod_{p \text{ ramified}} (1-p^{-s})^{-1} \prod_{p \text{ split}} (1-p^{-s})^{-2} \prod_{p \text{ inert}} (1-p^{-2s})^{-1} = \zeta(s) L(\chi, s).
So cancelling the familiar factors from {\zeta}, it says that

\displaystyle L(\chi, s) = \prod_{p \text{ split}} (1-p^{-s})^{-1} \prod_{p \text{ inert}} (1+p^{-s})^{-1}.
So comparing product expansions, it is equivalent to

\displaystyle \chi(p) = \begin{cases} 1 \text{ iff }p\text{ is split,}\\ 0 \text{ iff }p\text{ is ramified,}\\ -1 \text{ iff }p\text{ is inert.} \end{cases}.

Interesting. This splitting into cases reminds us of a more classical area of algebraic number theory. Write {K={\mathbb Q}(\sqrt{d})}, and assume {d \equiv 1 \mod 4}. Then the minimal polynomial is {X^2 + X + (d-1)/4}, which factorises modulo an odd prime {p} iff it does after we’ve multiplied by 4 and completed the square to get {(2X+1)^2 + d}, whose factorisation modulo {p}, together with the Kummer-Dedekind theorem for the splitting of primes in number fields whose ring of integers is primitively generated, gives a similar criterion in terms of the Legendre symbol:

\displaystyle \left(\frac{d}{p}\right) = \begin{cases} 1 \text{ iff }p\text{ is split,}\\ 0 \text{ iff }p\text{ is ramified,}\\ -1 \text{ iff }p\text{ is inert.} \end{cases}.
Now, take {\chi(n) = \left(\frac{n}{q}\right)} and, noting that {\chi(-1) = (-1)^{(q-1)/2}}, we see straight away, comparing the above two statements, that:

\displaystyle \left(\frac{p}{q}\right) = \left(\frac{d}{p}\right) = \left(\frac{(-1)^{(q-1)/2}}{p}\right)\left(\frac{q}{p}\right) = (-1)^{(p-1)(q-1)/4}\left(\frac{q}{p}\right).

So our identity in terms of {\zeta} and L-functions was actually a (fairly heavily) disguised statement of Gauss’ famous law of quadratic reciprocity! Of course, I haven’t given you a proof: I’m not personally sure exactly how to prove the identity about the {\zeta} and L-functions. Maybe I’ll let you know when I figure it out, but in the meantime there are plenty of proofs of quadratic reciprocity out there: http://www.rzuser.uni-heidelberg.de/~hb3/rchrono.html. I should also refer the interested reader to Kolowski’s opening chapter of `an introduction to the Langlands program’ which was certainly where I read about this.

In this post I shall try to look more closely at the relationship between categories of modules and the category of rings. In fact, we shall see that they form a relatively simple example of a stack, a categorical structure which seems to pop up in a lot of areas of mathematics. This example is a simplification of an example certainly appearing in Grothendieck’s FGA, but my hope is that by looking at a purely algebraic structure (where we look at fixed modules over fixed rings rather than sheaves of modules over schemes) we might gain some intuition about the kind of `gluing’ that goes on in algebraic geometry, as distinguished from gluing data over actual different bits of a space together.

Recall that for any ring {A}, there is a category {Mod_A} of modules with coefficients in {A}. We therefore might want to argue that there is a functor from {Rng} to {Cat} which takes {A \mapsto Mod_A}. However, this doesn’t quite work. What might happen to a morphism {f:A \rightarrow B}? It must be sent to a functor {f_*: Mod_A \rightarrow Mod_B}, and the obvious choice is to consider the functor {f_*: M \mapsto B \otimes_A M}, the extension of scalars along {f}. However, although they are canonically isomorphic, it is not true that we have actual equalities {A \otimes_A M = M} or that {C \otimes_B (B \otimes_A M) = C \otimes_A M}. We are therefore in the presence of a more subtle structure, not quite a category-valued functor, but almost one.

Let us look at it another way. Let {Mod} be the category of all modules. We define the objects of {Mod} to be all pairs {(A,M)} where {A} is a ring and {M} is an {A}-module, and the arrows to be pairs of maps {(f,\phi): (A,M) \rightarrow (A',M')} where {f:A' \rightarrow A} is a ring morphism (note reversed direction) and {\phi:M \rightarrow A \otimes_{A'} M'} a morphism of {A}-modules. We define the composite {(g,\psi) \circ (f,\phi)} by taking the ring homomorphism {f \circ g: A'' \rightarrow A} and the module homomorphism {M \rightarrow A \otimes_{A'} M' \rightarrow A \otimes_{A'} (A' \otimes_{A''} M'') \cong A \otimes_{A''} M''}, where the second map is {1 \otimes \psi}.

There is an obvious `projection’ functor {p: Mod \rightarrow Rng} which retains only the ring-related data (`projection onto the first co-ordinate’). This functor has the property that whenever there is an arrow {f:A' \rightarrow A} `downstairs’, we can associate with each {A'}-module {M'} the {A}-module {A \otimes_{A'} M'} together with the identity morphism {i} on {A \otimes_{A'} M'}, and this together with {f} itself gives an arrow {\phi: A \otimes_{A'} M \rightarrow M'} of {Mod} with {p\phi = f}.

This arrow {\phi} has a nice (but in this case rather obvious) universal property. It is probably worth the reader drawing a quick picture. Whenever there is a map {(g,\psi): (B,N) \rightarrow (A',M')} upstairs, and {h:A \rightarrow B} downstairs, we can find a unique map {\theta: N \rightarrow B \otimes_A A \otimes_{A'} M'} such that {i \circ \theta = \psi}. Indeed, we can just define {\theta: n \mapsto 1 \otimes 1 \otimes \psi(n)}, and conversely check that any map which factors so must be this one. This appears to be all slightly technical and silly, but what we have checked is that {\psi} is what is called a Cartesian arrow, and that therefore {p:Mod \rightarrow Rng} is an example of a fibred category.

The reason a fibred category is called what it is can be found by thinking about {p:Mod \rightarrow Rng} as a covering projection, so we imagine the `space of modules’ covering the `space of rings’. Each object {A} of {Rng} has an associated fibre, whose objects are {p^{-1}(A)} and whose arrows {p^{-1}(id_A)} (though this inverse image notation is rather dodgy: we aren’t talking about sets here!). In our case, the fibre over {A} is equivalent to the category {Mod_A} of {A}-modules, with {A}-module morphisms (we will probably need to identify {M} with {A \otimes_A M} and technical stuff like that, so it might be clumsy to say that the categories are actually the same or isomorphic). The definition of a fibred category insists that each `path’ (arrow) downstairs has at least one `lifting’ to a Cartesian arrow, a necessary constraint, if only to rule out bad behaviour on the fibres (isomorphic objects now have to behave reasonably similarly, etc.).

At the start of the post I mentioned it seemed like we almost have a functor. Indeed, it turns out that giving a fibred category {\mathcal{F} \rightarrow \mathcal{C}} is almost the same as giving what is called a pseudo-functor {\mathcal{C}^op \rightarrow Cat}, where we associate with each element {X} of {\mathcal{C}} its fibre {p^{-1}(X)} and for each arrow {f} we pick a `pullback’ for each object (analogous to our choice of {A \otimes_A M}) and use the pullbacks to define a functor {f^*} in the opposite direction. It is often useful to think of fibred categories as analogous to category-valued presheaves, but there is something quite elegant about the `covering space’ viewpoint which this can lack.

Now, let us return to our modules. The fibred category structure was essentially trivial – just verifying a few definitions. Here comes a much more interesting property: they are also a stack, in a suitable sense. Let’s see what this means.

Firstly, we put what is called a Grothendieck topology onto {Rng}, the underlying space. This should be thought of as a collection of atlases for objects in {Rng} (though our charts need not be isomorphisms but may be any map of our choosing). In our case, all the atlases will consist of a single chart, but in more serious algebraic-geometric situations (and in almost all topological situations) they may consist of many charts. We take the faithfully flat topology consisting of atlases containing a single faithfully flat map {A \rightarrow B}. One has to think of {B} covering {A} (again, there is a reversal of direction happening from standard covering space theory: instead of working with the category of rings one would normally work with a category of schemes – geometric spaces associated with rings whose arrows are reversed, but we don’t want to bother about that here). A faithfully flat map {A \rightarrow B} is one where the operation of tensoring, {M \mapsto B \otimes_A M}, can be performed and reversed without affecting the exactness of any sequence of {B}-modules or {A}-modules. For example, all morphisms of fields are faithfully flat (if you like, by dimension counting/picking bases).

Now we will define the notion of descent. In a general setting the idea of descent is to glue together various objects which are defined on all the different `charts’ of an atlas in the topology and suitably nicely isomorphic on overlapping regions, to get one big object on the whole space. However, in our case given our atlases consist of a single chart, this kind of intuition becomes a bit ridiculous. Indeed, I wonder to what extent the idea of `gluing’ really does descent theory justice. For we shall see that even with a single covering space, constructing these gluings is far from trivial.

Suppose {A \rightarrow B} is a ring homomorphism, and consider a {B}-module {N}. It would be nice to work out whether we can write {N} in the form {B \otimes_A M} for some {A}-module {M}, and indeed all the different ways in which this is possible. Since these correspond to the pullbacks in our fibred category above, this would be very nice. Descent theory gives us a very clever answer to this simple question in the case where {A \rightarrow B} is faithfully flat.

We define descent data to be an isomorphism {\psi: N \otimes_A B \rightarrow B \otimes_A N} of {B \otimes_A B}-modules satisfying a cocycle condition, which basically says that the three different ways to lift this isomorphism to one of {B \otimes B \otimes B}-modules (by putting {B} in various places) are suitably compatible. In terms of the gluing picture, this isomorphism is an algebraic analogue of demanding that when we take the objects we wish to glue and restrict them to the intersection of two of our charts, those objects must be isomorphic. On a normal topology, {U \cap U = U}, so the case of a cover by a single set isn’t very interesting, but in a Grothendieck topology the intersection corresponds to taking a pullback, which (after we have unwound some basic facts about schemes) corresponds to the pushout of {A \rightarrow B} along itself, which is {B \otimes_A B}. The thing about the third level corresponds to a technical compatibility condition on triple intersections from the general definition. We let {Mod_{A \rightarrow B}} be the category of {B}-modules with descent data.

So what is our picture? We have {B} covering {A}, and an object (module) over {B} equipped with some descent data. We hope that we can glue it all together (whatever that means) to get an object over our base space {A}. If we look a bit closer, things look encouraging.

Firstly, if reversing our operation of tensoring by {B} is really going to correspond to descent, then for {M} an {A}-module, we should have a natural descent datum on {B \otimes_A M}. In other words, we need {\psi: B \otimes M \otimes B \rightarrow B \otimes B \otimes M} a reasonably natural looking and `nice’ (satisfying the cocycle condition) isomorphism. That generated by {b \otimes m \otimes b' \mapsto b \otimes b' \otimes m} turns out to do the trick nicely. Thus we have a nice functor {\Phi: Mod_A \rightarrow Mod_{A \rightarrow B}}, our old friend `tensor by {B}‘, which also has the obvious action on morphisms.

We now need to look for a candidate inverse functor. Notice that {(1 \otimes m) \otimes 1 \mapsto 1 \otimes (1 \otimes m)} in the example above. This gives us (admittedely tenuous) motivation to try the following. Consider the set {N^\psi := \{n \in N: \psi(n\times 1) = 1\times n\}}, which I suggestively but perhaps wrongly think of as the `module fixed by {\psi}‘. It is easy to check that {N^\psi} gives us an {A}-module associated to our {B}-module with descent data, and can be extended to a functor {\Psi: Mod_{A \rightarrow B} \rightarrow Mod_A}.

Do these functors realise all our dreams of what descent could achieve? In our case, yes!

Theorem 1 (Faithfully flat descent) Let {A \rightarrow B} be faithfully flat. Then the functors {\Phi} and {\Psi} give an equivalence of categories

\displaystyle Mod_A \simeq Mod_{A \rightarrow B}.

I shall not prove it fully in this post, but refer an interested reader to Vistoli’s marvellous article on the topic. The proof of this proposition is essentially a dance around an exact sequence

\displaystyle 0 \rightarrow N \rightarrow F \otimes N \rightarrow F \otimes F \otimes N

In the proof, the `faithfully’ is essential for getting {\Psi \Phi \cong Id} and the `flat’ is essential for {\Phi \Psi \cong Id}, in both cases by hopping between exactness in {Mod_A} and {Mod_B}.

With faithfully flat descent established, we have that the fibred category {Mod \rightarrow Rng} is indeed a stack in the faithfully flat topology (a stack is simply a fibred category in which descent works). More significantly for algebraic geometry, one is also then well on the way to the fact that the category of quasi-coherent sheaves (a geometrisation of `modules’) over the category of schemes (a geometrisation of `rings’) is a stack with respect to the fpqc topology (made up of faithfully flat and quasi-compact atlases). This result taken alone could look dangerously like an `obvious’ geometric result, but in fact because of the deeply algebraic nature of Grothendieck topologies as fine as the fpqc topology, a large proportion of its content is nothing to do with gluing in a concrete sense, and much more to do with the subtle interactions within the underlying commutative algebra. I would guess it is also thence it gains much of its power for transferring algebraic data easily between different levels.

In this post I want to review some basic general properties of representable functors. It seems these `God-given’ machines play important roles in all sorts of different ways throughout algebraic geometry (and presumably other categorically founded areas of mathematics).

Firstly, recall that if we have a category {\mathcal{C}}, each object {U} has two associated representable functors, one covariant, the other contravariant: namely {h^A: B \mapsto \text{Hom}(A,B)} and {h_A: B \mapsto \text{Hom}(B,A)}, with arrows corresponding to post and pre composition by themselves respectively. A general set-valued functor on {\mathcal{C}} is called representable if it is isomorphic to such a functor (we say it is represented by {A}).

The most fundamental result about these functors is that they actually embed the category {\mathcal{C}} into the category Hom({\mathcal{C}}, Set). This is part of a story that culminates in the classical Yoneda Lemma, which says more: that the representable functors detect all the information contained in an arbitrary functor. Note that the sets of homomorphisms between two functors are just the sets of natural transformations.

Theorem 1 (Yoneda Lemma) Let {F: \mathcal{C} \rightarrow Set} be a covariant functor. Then for all objects {X} of {\mathcal{C}}, there is an isomorphism natural in {X}:

\displaystyle F(X) \cong \text{Hom}(h^X, F).

Let {G: \mathcal{C} \rightarrow Set} be a contravariant functor. Then for all objects {X} of {\mathcal{C}}, there is an isomorphism natural in {X}:

\displaystyle G(X) \cong \text{Hom}(h_X, G).

 

Note that setting {G=h_Y}, we get our embedding property:

\displaystyle \text{Hom}(X,Y) \cong \text{Hom}(h_X, h_Y).

In other words, the functor from {\mathcal{C}} into its category of set-valued presheaves is fully faithful.

The next property I want to discuss was pointed out to me recently as `almost tautological’, but seems to be responsible for much of what gives these functors their character.

Theorem 2 (Preservation of Limits) The functors {h^X} preserve all limits which already exist, and the {h_X} send existing colimits to limits.

Proof: Suppose {(L, p_D)} is a limit for a functor {G: \mathcal{D} \rightarrow \mathcal{C}}. Consider any cone {(A \rightarrow \text{Hom}(X,GD))_{D \in \mathcal{D}}} over {h^X \circ G}. Then any {a \in A} maps to a cone {(X \rightarrow GD)_{D \in \mathcal{D}}}, which corresponds simply to a map {f_a: X \rightarrow L} which together with the {p_D} factors our cone, since {L} is a limit in {\mathcal{C}}. Hence, we have a map {A \rightarrow \text{Hom}(X,L) = h^X(L)}. Since {A} was arbitrary, we deduce that {h^X(L)} is the limit of {h^X \circ G}, as was required. The contravariant case follows by considering it as covariant {\mathcal{C}^{op} \rightarrow \text{Set}}. \Box

Note that this is a very general statement. We now know that in particular the representable functors respect products, pullbacks, and equalisers.

Corollary 3 In an abelian category, the representable functors are left exact.

Proof: They preserve equalisers (or send coequalisers to equalisers). \Box

On a slightly deeper level, recall the adjoint functor theorem, which basically says that any suitably nice functor which preserves limits admits a left adjoint. In other words, we might hope that together with the functor which `forgets everything about {Y} except its relationships with {X}‘ we also have a functor which associates to each set {A} a universal object {Y_A} in {\mathcal{C}} which is `best’ at being described by something like {A}. I am not really sure how to push this idea further in general, but it can of course be remarked that in the category of {R}-modules (where the representable functors actually land back in the category {R}-modules: {h^X: R-Mod \rightarrow R-Mod \rightarrow Set}) we have such a left adjoint:

Proposition 4 The tensor product by {X}, {Y \mapsto X \otimes_R Y} is a left adjoint to the representable functor {h^X} (considered as taking values in {R}-modules).

It would be nice if in more general situations representable functors actually possessed concretely understandable left adjoints, especially if they shared properties with the tensor product. That they preserve limits certainly give us some hope that they might.

One final fact of a more geometrical flavour about representable functors. Let {X} be a topological space with an open cover {{U_i}}, and {Y} be some other topological space. If we are given continuous maps {f_i: U_i \rightarrow Y} with the property that {f_i = f_j} on {U_i \cap U_j} for all {i,j}, then since continuity is a local property, we can write down a unique function {f: X \rightarrow Y} which glues all the {f_i} together. Specifically, we just set {f(x) = f_i(x)} for some {i} where {x \in U_i}. This is well-defined because of the `agreement on intersections’ property, and it’s unique because {f} has to be a gluing.

From this discussion we conclude

Proposition 5 The representable presheaves (contravariant set-valued functors) {h_Y} on the category of topological spaces are in fact sheaves on the category of topological spaces (technically speaking, with the open cover topology).

In fact, it seems to be generally true that such results hold. In algebraic geometry however, a much less trivial phenomenon occurs. A discussion similar to the one above is enough to convince one that {h_Y} is a sheaf in the category of schemes with respect to the Zariski open cover topology. However, the Zariski topology is infamously deficient as a topology: open sets tend to be large and unwieldy, so one cannot take `small neighbourhoods’, one of the basic intuitions of classical topology and analysis, so algebraic geometers use the more general concept of `Grothendieck topologies’. These allow us to construct a much finer `topology’, in particular giving much more stringent sheaf conditions than those associated with merely the Zariski topology (having more sophisticated sheaf structures gives more information about the space – indeed, ├ętale cohomology – a technique for working out algebraic topology type invariants from algebraic spaces, relies on this idea).

With these more stringent sheaf conditions, many structures which were sheaves in the Zariski topology are ruled out. However, Grothendieck proved that representable functors survive (we can still glue morphisms of schemes even when defined over very fine covers).

Theorem 6 (Grothendieck) The representable presheaves on the category of schemes are sheaves in the fpqc topology. In particular, they are sheaves in the ├ętale topology.

Here is not the time or place to prove it, or indeed to define the terms above, but I hope this post gives a flavour of why representable functors are often rather special and why it’s likely I shall probably be mentioning them again before too long.