FI-modules over Noetherian rings

New paper on the arXiv, joint with Tom Church, Benson Farb, and UW grad student Rohit Nagpal.  In our first paper on FI-modules (which I blogged about earlier this year) a crucial tool was the fact that the category of FI-modules over a field of characteristic 0 is Noetherian; that is, a submodule of a finitely generated FI-module is again finitely generated.  But we didn’t know how to prove this over a more general ring, which limited the application of some of our results.

In the new paper, we show that the category of FI-modules is Noetherian over an arbitrary Noetherian ring.  Sample consequence:  if M is a manifold and Conf^n M is the configuration space of ordered n-tuples of distinct points on M, then we show that

dim_k H_i(Conf^n M, k)

is a polynomial function of n, for all n greater than some threshold.  In our previous paper, we could prove this only when k had characteristic 0; now it works for mod p cohomology as well.  We also discuss some of the results of Andy Putman’s paper on stable cohomology of congruence subgroups — it is a bad thing that I somehow haven’t blogged about this awesome paper! — showing how, at the expense of losing stable ranges, you can remove from his results some of the restrictions on the characteristic of the coefficient field.

Philosophically, this paper makes me happy because it brings me closer to what I wanted to do in the first place — talk about the representation theory of symmetric groups “for general n” without giving names to representations.  In characteristic 0, this desire might seem a bit perverse, given the rich and beautiful story of the bijection between irreducible representations and partitions.  But in characteristic p, the representation theory of S_n is much harder to describe.  So it is pleasing to be able to talk, in a principled way, about what one might call “representation stability” in this context; I think that when V is a finitely generated FI-module over a finite field k it makes sense to say that the representations V_n of k[S_n] are “the same” for n large enough, even though I don’t have as nice a description of the isomorphism classes of such representations.

 

 

 

 

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.

FI-modules and representation stability, II

Here are some sequences of vector spaces.  In each case, the sequence is indexed by n, and all other variables are understood to be constant.  So suppose V_n is the space

• H^i(Conf^n M, Q) for M a connected oriented manifold of dimension at least 2.
• The (j_1, .. j_r)-multidegree piece of the diagonal coinvariant algebra on r sets of n variables.
• H^i(M_{g,n},Q), the cohomology of the moduli space of curves of genus g with n marked points.
• The tautological subring of the above.
• The space of degree-d polynomials on the rank variety parametrizing nxn matrices of rank at most r.

By a character polynomial we mean a polynomial with integral coefficients in variables X_1, X_2, X_3, … .  We interpret these symbols (and thus character polynomials) as class functions on the symmetric group by S_n by taking

X_i(s) = number of i-cycles in s

for each permutation s.

Then we show that, in each of the examples above, there’s a character polynomial P such that the character of the action of S_n on V_n is given by P, for all sufficiently large n.  This is one way in which one can say that a sequence of representations of larger and larger symmetric groups are “all the same.”  In particular, by plugging in the identity we find that dim V_n is a polynomial in n, for n large enough.

For many of these examples, almost nothing is known about dimensions of individual spaces!  So a strong regularity theorem like this is perhaps surprising.  Even more surprising (to us at any rate) is that theorems like this require only very meager input from whatever context generate the vector spaces.  You get this stability (and many others) almost for free.

More about how it all works tomorrow!

FI-modules and representation stability, I

Tom Church, Benson Farb and I have just posted a new paper, “FI-modules:  a new approach to representation stability,” on the arXiv.  This paper has occupied a big chunk of our attention for about a year, so I’m very pleased to be able to release it!

Here is the gist.  Sometimes life hands you a sequence of vector spaces.  Sometimes these vector spaces even come with maps from one to the next.  And when you are very lucky, those maps become isomorphisms far enough along in the sequence; because at that point you can describe the entire picture with a finite amount of information, all the vector spaces after a certain point being canonically the same.  In this case we typically say we have found a stability result for the sequence.

But sometimes life is not so nice.  Say for instance we study the cohomology groups of configuration spaces of points of n distinct ordered points on some nice manifold M.  As one does.  In fact, let’s fix an index i and a coefficient field k and let V_n be the vector space H^i(Conf^n M, k.)

(In the imaginary world where there are people who memorize every word posted on this blog, those people would remember that I also sometimes use Conf^n M to refer to the space parametrizing unordered n-tuples of distinct points.  But now we are ordered.  This is important.)

For instance, you can let M be the complex plane, in which case we’re just computing the cohomology of the pure braid group.  Or, to put it another way, the cohomology of the hyperplane complement you get by deleting the hyperplanes (x_i-x_j) from C^n.

This cohomology was worked out in full by my emeritus colleagues Peter Orlik and Louis Solomon.  But let’s stick to something much easier; what about the H^1?  That’s just generated by the classes of the hyperplanes we cut out, which form a basis for the cohomology group.  And now you see a problem.  If V_n is H^1(Conf^n C, k), then the sequence {V_n} can’t be stable, because the dimensions of the spaces grow with n; to be precise,

dim V_n = (1/2)n(n-1).

But all isn’t lost.  As Tom and Benson explained last year in their much-discussed 2010 paper, “Representation stability and homological stability,” the right way to proceed is to think of V_n not as a mere vector space but as a representation of the symmetric group on n letters, which acts on Conf^n by permuting the n points.  And as representations, the V_n are in a very real sense all the same!  Each one is

“the representation of the symmetric group given by the action on unordered pairs of distinct letters.”

Of course one has to make precise what one means when one says “V_m and V_n are the same symmetric group representation”, when they are after all representations of different groups.  Church and Farb do exactly this, and show that in many examples (including the pure braid group) some naturally occuring sequences do satisfy their condition, which they call “representation stability.”

So what’s in the new paper?  In a sense, we start from the beginning, defining representation stability in a new way (or rather, defining a new thing and showing that it agrees with the Church-Farb definition in cases of interest.)  And this new definition makes everything much cleaner and dramatically expands the range of examples where we can prove stability.  This post is already a little long, so I think I’ll start a new one with a list of examples at the top.