Thread by @littmath

Today I am turning 32 — probably the last time my age will be a fifth power! In celebration, here’s a thread on one of my favorite techniques in mathematics: “spreading out.” It’s one of the sources of the many mysterious connections between geometry and arithmetic. 1/n

Let me start with a simple example, due to Serre. Suppose f: ℂ^n—>ℂ^n is a polynomial map such that f ⚬ f = id. That is, f is it’s own inverse.

Then I claim that f has a fixed point. 2/n

That is, there’s some point x such that f(x)=x. For example, if n=1 and

f(x)=1-x

then x=1/2 is a fixed point. 3/n

Let’s prove an auxiliary claim first. Suppose S is a set with an odd number of elements, and

g: S—>S

is any map such that g ⚬ g = id. Then I claim g has a fixed point. 4/n

Indeed, the set of points that g does *not* fix has even size, because it’s partitioned into pairs {x, g(x)}! So any point left over is a fixed point.

So if F is a finite field with an odd number of elements, a polynomial map

g: F^n—>F^n

with g ⚬ g=id has a fixed point. 5/n

We actually didn’t use that g is polynomial, just that F has odd size. But this statement looks a lot like the statement we wanted over the complex numbers — though the proof crucially uses that the field is finite.

Our idea will be to reduce to this case by “spreading out.” 6/n

Ok, so let’s go back to our map

f: ℂ^n—>ℂ^n.

This map is (by assumption) given by n polynomials in n variables. The key observation is that whatever these polynomials are, they only have finitely many coefficients, say {a_1, ..., a_m}. 7/n

In particular, if R is the subring of ℂ generated over ℤ by {a_1, ..., a_m}, then f actually induced a polynomial map

R^n—>R^n

such that f ⚬ f = id.

And R has interesting arithmetic. This step — using finiteness to get to an arithmetic situation — is “spreading out.” 8/n

In particular, if 𝔪 is a maximal ideal of R, we can reduce our new *arithmetic* f mod 𝔪. And now we get a map

(R/𝔪)^n—>(R/𝔪)^n which is its own inverse.

But for most 𝔪, R/𝔪 will have odd size! So this last map will have a fixed point. 9/n

So what we’ve shown is that for any 𝔪 with |R/𝔪| odd, f has a fixed point mod 𝔪. And it turns out, with a tiny bit of algebraic geometry, that’s enough to conclude f has a fixed point over ℂ. QED. 10/n

Ok, on to example #2: Malcev’s theorem. Before stating the theorem, I need to introduce two properties of a group G.

(1) A group G is *linear* if there is an injective homomorphism

G—>GL_n(K)

for some field K. That is, G admits a faithful K-representation. 11/n

(2) A group G is *residually finite* if for any non-identity g ∈ G, there exists a finite group H and a homomorphism

f: G—>H

such that f(g) is not the identity.

(For experts, this is the same as the natural map from G to its profinite completion being injective.) 12/n

Malcev’s theorem states that any finitely generated linear group is residually finite.

The proof is via “spreading out”! 13/n

Because G is linear by assumption, there is a field K and an injective homomorphism

ρ: G—>GL_n(K).

Now choose some finite set of generators {g_1,...,g_m} of G. 14/n

The finite set of matrices ρ(g_1), ..., ρ(g_m) have finitely many entries; let R be the subring of K generated by those entries. Because the g_i generate G, ρ actually comes from an injective representation

γ: G—>GL_n(R)!

15/n

Now let 𝔪 be a maximal ideal of R. Because R is finitely generated R/𝔪^s is finite for any s. Hence

γ_s: G—>GL_n(R)—>GL_n(R/𝔪^s)

is a homomorphism to a finite group. 16/n

But as s ranges over all positive integers, the intersection of the ideals 𝔪^s is the zero ideal! So the γ_s are jointly injective.

In other words, for any non-identity g ∈ G, *some* γ_s will send it to something nontrivial. And that’s exactly what we wanted to prove. 17/n

There are a lot of other examples. For example, remember Maschke’s theorem, which says that any complex representation of a finite group is a direct sum of irreducibles? 18/n

You might ask which *infinite* groups this is true for. Some examples include SL_n(ℤ) if n>2; again the proof is by “spreading out”!

(Maybe I’ll do another thread on this if there’s interest.) 19/n

Another incredible example is Mori’s famous “bend and break” technique, which lets one produce rational curves on certain complex varieties via reduction mod p. 20/n

Deligne-Illusie’s awesome proof of Hodge-to-de Rham degeneration and Kodaira vanishing uses related techniques (in fact both of these examples heavily exploit the Frobenius map, which I haven’t talked about here). 21/n

(I first used spreading out in my PhD thesis, where I methods related to Deligne-Illusie to prove other vanishing results. It was the beginning of a beautiful friendship.) 22/n

And there’s lots of open conjectures about spreading out. For example, the Grothendieck-Katz p-curvature conjecture gives a criterion for understanding when a polynomial ODE has algebraic solutions in terms of its reductions mod p. 23/n

Anyway, thanks for getting this far through my spreading out my love for spreading out! 24/n, n=24.