Generalizing the fundamental theorem of Galois theory

I just finished a new paper with Jeremiah Heller, available on the arXiv here. By taking the compositum of some beautiful work of Po Hu and Marc Levine, we’re able to — in a certain sense — generalize Galois theory.

Do you remember the Galois correspondence? I remember the Galois correspondence. I had the good fortune to learn about Galois theory (the study of how group theory controls field extensions) and the fundamental group of a topological space simultaneously during my undergrad. If you replace the word “Galois group” with “fundamental group” and “field extension” with “covering space” you get essentially isomorphic theories. A decade of reflection has hardly diminished my enthusiasm for this consilience. (Feel free to weigh in on whether this qualifies as consilience. I think it does, and I think consilience is precisely what we get excited about when we connect disparate fields of mathematics.)

So I’m very excited about this new work. The technical details are all in the paper, and I’d like to give a feel for the constructions and results here. We start with a field $k$ and finite Galois extension $L/k$. Galois theory says there is a bijection between subgroups $H\le G=\mathrm{Gal}(L/k)$ and subextensions $L/E/k$: $H$ corresponds to $L^H$, the fixed field of $H$, and $L/E/k$ corresponds to automorphisms of $L$ fixing $E$. For the sake of generalization, I’d like to phrase this theory a little strangely: there’s a full and faithful embedding $c_{L/k}$ of the category of $G$-orbits into smooth $k$-schemes taking $G/H$ to $\mathrm{Spec}(L^H)$.

Formal nonsense allows us to lift $c_{L/k}$ to a functor $c_{L/k}^{\ast }$ on simplicial presheaf objects in these categories. This maneuver is justified by the following facts. First, we can model $G$-spaces as simplicial presheaves on $G$-orbits. Secondly, simplicial presheaves on smooth schemes is precisely the Morel-Voevodsky category of “motivic spaces.” Motivic spaces are an enlargement of the category of smooth $k$-schemes that permits certain constructions and allows us to define a meaningful notion of homotopy in which the affine line plays the role of the unit interval. This gives us “motivic homotopy theory,” and it turns out that $c_{L/k}^{\ast }$ plays nicely with the notions of $G$-equivariant and motivic homotopy. (For the cognoscenti: $c_{L/k}^{\ast }$ is a left Quillen functor for appropriate model categories.) Once you have the right perspective, it’s fairly straightforward to see that $c_{L/k}^{\ast }$ induces a full and faithful functor on associated homotopy categories, just like the Galois correspondence! In this sense, $G$-equivariant homotopy theory is a “sub-theory” of motivic homotopy theory over $k$.

Jeremiah and I were interested in whether the above embedding result stabilized. Whenever we work in a (suitably nice) symmetric monoidal model category, we have the right to stabilize with respect to a specific (cofibrant) object. The objects of the associated stable category are called spectra, and this category enjoys the nice property that the stabilization object is now invertible with respect to the monoidal structure. In equivariant homotopy, it turns out it’s very useful to stabilize with respect to the regular representation sphere $S^G$, i.e., the $|G|$-fold smash product of $S^1$ with the natural $G$-action. In motivic homotopy theory, you should stabilize with respect to ${\mathbf{P}}^1$, the projective line. Performing these stabilizations gives you desirable transfer maps and Thom isomorphisms.

Po Hu shows that $c_{L/k}^{\ast }$ takes $S^G$ to a smash invertible object in the stable motivic homotopy category. As such, we can extend $c_{L/k}^{\ast }$ to a (left Quillen) functor between the associated stable categories. We’re then immediately permitted to ask whether $c_{L/k}^{\ast }$ is still full and faithful on homotopy categories, i.e., is $c_{L/k}^{\ast }$ still a “Galois correspondence.” The answer, in general, is no, and we can observe this by studying the effect of $c_{L/k}^{\ast }$ on the ring of endomorphisms of the sphere spectrum. This is a map $A(G)\to GW(k)$ from the Burnside ring of $G$ to the Grothendieck-Witt ring of $k$, and work of Marc Hoyois implies that it agrees with the classical “trace form” homomorphism introduced by Dress. Plying some standard techniques from the arithmetic theory of quadratic forms, Jeremiah and I prove that Dress’s map is an isomorphism if and only if either $k$ is quadratically closed and $L=k$, or $k$ is euclidean (formally real with only two square classes) and $L=k[i]$. The prototypical examples of such field extensions are $L=k=\mathbf{C}$ and $k=\mathbf{R}$, $L=\mathbf{C}$, but there are other more exotic examples, e.g., $k$ could be the so-called real constructible numbers (the intersection of the quadratic closure of $\mathbf{Q}$ with $\mathbf{R}$). [Side note: these are exactly the real numbers whose absolute value can be constructed with ruler and compass!]

What we gain from this is the knowledge that $L/k$ must be of a very special form in order for $c_{L/k}^{\ast }$ to be full and faithful on stable homotopy categories. Marc Levine studied the case $L=k$ algebraically closed of characteristic zero and proved that $c_{L/L}^{\ast }$ is indeed full and faithful under these hypotheses. Here the Galois group $G=e$ is trivial (stable $e$-equivariant homotopy is just stable homotopy) and $c_{L/L}^{\ast }$ is induced by the well-known “constant presheaf” functor. Jeremiah and I extend Levine’s theorem, proving that when $k$ is real closed and $L=k[i]$ (which is necessarily the algebraic closure of $k$), $c_{L/k}^{\ast }$ is full and faithful. In particular, we now know that $c_{L/k}^{\ast }$ is full and faithful when $L=k=\mathbf{C}$ and when $k=\mathbf{R}$, $L=\mathbf{C}$. Could there be quadratically closed fields $L$ for which our theorem on $c_{L/L}^{\ast }$ still holds? What about when $k$ is the real constructible numbers and $L=k[i]$? We don’t think so, and conjecture as much in the paper. (Note that our necessary conditions only depend on ${\pi }_0$ of the sphere spectrum. Perhaps a better understanding of ${\pi }_n$ for larger $n$ can eliminate more cases?)

We’ve arrived, then, at our generalization of Galois theory: the Galois correspondence once again induces a full and faithful functor, but this time on the level of stable equivariant and motivic homotopy categories (but only under very restrictive hypotheses on the fields). Now I’ll be the first to admit that the Galois theory of the extension $\mathbf{C}/\mathbf{R}$ is not particularly interesting, but stable $G$-equivariant homotopy theory for $G$ the cyclic group of order two? There’s a lot of amazing stuff in there, and everything has a full faithful copy in the stable motivic homotopy category of $\mathbf{R}$!

I’ll send you to the paper if you want to see how we proved these results, but the key tools were stable equivariant Betti realization, the motivic Adams spectral sequence, and Morel’s splitting of the rationalized motivic sphere spectrum. (Our methods are in the same spirit as Levine’s, but independent. He uses the slice spectral sequence, whose convergence properties are poorly understood when $k$ has infinite cohomological dimension [e.g. $k=\mathbf{R}$]. In particular, we reproduce his full faithfulness theorem for the constant presheaf functor via new methods.)

So where to next? First, it would be interesting to leverage this theorem in order to produce motivic computations of equivariant invariants. Computations in the $G$-equivariant setting are notoriously difficult (even for $G$ cyclic of order two), while there’s been a lot of recent success in producing stable motivic computations.

Far more speculatively, one could try to extend these techniques to profinite Galois extensions. The algebraic (or separable) closure of a field is rarely a finite field extension. In fact, our theorem covers exactly these cases. So we’ll have to work with profinite Galois groups if we want to study other algebraic closures. This would likely involve a lot of foundational work on profinite equivariant homotopy theory. Recent conversations with people who know a thing or two about profinite equivariance have me thinking you might need to look at discontinuous $G$-actions — yikes! — at least if you still want a full faithfulness theorem to hold. Keep in mind that you might now want your “Burnside ring” to contain torsion, the generic situation with $GW(k)$! Anyway, it should be fun to think about.