## Spec is representable

Saw Matt Baker at the Joint Meetings and he told me about this crazy paper he just posted, “Matroids over Hyperfields.”   A hyperring is just like a ring except addition is multivalued; given elements x and y of R, x+y is a subset of R which you can think of as “the possible outcomes of summing x and y.”  A hyperfield is a hyperring in which every nonzero element has a multiplicative inverse.

Here’s an example familiar to tropical geometers:  let T be the hyperfield whose elements are $\mathbf{R} \bigcup -\infty$, whose multiplication law is real addition, and whose addition law is

a + b = max(a,b) if a <> b

a + b = {c: c < a} if a=b

In other words, each element of T can be thought of as the valuation of an otherwise unspecified element of a field with a non-archimedean valuation, and then the addition law answers the question “what is ord(x+y) if ord(x) = a and ord(y) = b”?

This may sounds at first like an almost aggressively useless generalization, but no!  The main point of Matt’s paper is that it makes sense to talk about a matroid with coefficients in a hyperfield, and that lots of well-studied flavors of matroids can be written as “matroids over F” for a suitable hyperfield F; in this way, a lot of different stories about different matroid theories get unified and cleaned up.

In fact, a matroid itself turns out to be the same thing as a matroid over K, where K is the Krasner hyperfield:  just two elements 0 and 1, with the multiplication law you expect, and addition given by

0 + 0 = 0

0 + 1 = 1

1 + 1 = {0,1}

One thing I like about K is that it repairs the problem (if you see it as a problem) that the category of fields has no terminal object.  K is terminal in the category of hyperfields; any hyperfield (and in particular any field) has a unique map to K which sends 0 to 0 and everything else to 1.

More generally, as Matt observes, if R is a commutative ring, a homomorphism f from R to K is nothing other than a prime ideal of R — namely, f^{-1}(0).  So once you relax a little and accept the category of hyperfield, the functor Spec: Rings -> Sets is representable!  I enjoy that.

Update:  David Goss points out that this observation about Spec and the Krasner hyperfield is due to Connes and Consani in “The hyperring of adèle classes” JNT 131, (2011) 159-194, p.161.  In fact, for any scheme X of finite type over Z, the underlying Zariski set of X is naturally identified with Hom(Spec(K),X); so Spec(K) functions as a kind of generic point that’s agnostic to characteristic.

## FI-modules and representation stability, III

So how does this paper work?  The main idea is quite simple.  Let’s come back to the example of V_n = H^i(Conf^n M,Q), with i fixed and n ranging over nonnegative integers.  Then we have a sequence of vector spaces

V_0, V_1, V_2, …

But more than a sequence.  You have a map Conf^{n+1} -> Conf^n which is “forget the n+1 st point” — which functorially hands you a map V_n -> V_{n+1}.  So you have a diagram

V_0 -> V_1 -> V_2 -> …..

But in fact you have even more than this!  There’s no reason you have to forget just the n+1 st point.  You have tons of maps from Conf^n to Conf^m for all m <= n; one for each m-element subset of 1..n.  And there are lots of natural identifications between the compositions of these maps.  When you keep track of all the maps at your disposal, what you find is that the vector spaces V_n have a very rigid structure.

Definition:  FI is the category of finite sets with injections.  An FI-module over a ring R is a functor from FI to R-modules.

So V is an FI-module over Q!  (The vector space V_n is revealed as the image of the finite set [1..n] under the functor V.) And the main work of our paper is the study of the category of FI-modules, which sheds a great deal of light on representation stability.  For instance, we show that an FI-module over Q yields a representation-stable sequence in Church-Farb’s original sense if and only if it is finitely generated in the natural sense.  Moreover, the category of FI-modules over Q is Noetherian, in the sense that subobjects of finitely generated FI-modules are again finitely generation.  (The Noetherianness was proven independently by Snowden in a different form.)  Theorems like this very easy to show that tons of examples in nature (like the ones in the previous post) yield representation-stable sequences.  The work is all in the definitions and basic properties; once you have that, proving stability in particular examples is often a matter of a few lines.  For instance, you get a fairly instant proof of Murnaghan’s theorem on stability of Kronecker products; from this point of view, this becomes a theorem about the finite generation of a single object in an abelian category, rather than a theorem about a list of coefficients eventually setting down to constancy.

Sometimes there is more structure still.  Suppose, for example, that the manifold M above has nonempty boundary.  Then there are not only maps from Conf^{n+1} to Conf^n, but maps going the other way; you can add a new point in a little neighborhood of a boundary component.  (This is familiar from the configuration space of the complex plane, where you add new points at “the west pole” in the infinite negative real direction.)  These maps don’t quite compose on the nose, but they’re OK up to homotopy, and so the cohomology groups acquire a system of maps going both up and down.  It turns out that the right structure to describe such systems is given by the category of finite sets with partial injections; i.e. a map from A to B is an isomorphism from a subset of A to a subset of B.  We call this category FI#, and we call a functor from FI# to R-modules an FI#-module over R.

When your vector spaces carry an FI#-module structure you can really go to town.  It turns out that all the “eventuallies” disappear; when M is an open manifold, the dimension of H^i(Conf^n M) is a polynomial in n on the nose, for all n.  What’s more, if you want to show finite generation for FI#-modules, it suffices to show that dim V_n is bounded by some polynomial in n.  Once it’s less than a polynomial, it is a polynomial!  This stuff, unlike some other results in our paper, works in any characteristic and in fact is even fine with integral coefficients.