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## Forms and Galois cohomology

Yesterday we gave a brief and abstract description of Galois descent, the punchline of which was that Galois descent could abstractly be described as a natural equivalence

$\displaystyle C(k) \cong C(L)^G$

where $f : k \to L$ is a Galois extension, $G = \text{Aut}(L)$ is the Galois group of $L$ (thinking of $L$ as an object of the category of field extensions of $k$ at all times), $C(k)$ is a category of “objects over $k$,” and $C(l)$ is a category of “objects over $L$.”

In fact this description is probably only correct if $k \to L$ is a finite Galois extension; if $k \to L$ is infinite it should probably be modified by requiring that every function of $G$ that occurs (e.g. in the definition of homotopy fixed points) is continuous with respect to the natural profinite topology on $G$. To avoid this difficulty we’ll stick to the case that $k \to L$ is a finite extension.

Today we’ll recover from this abstract description the somewhat more concrete punchline that $k$-forms $c_k \in C(k)$ of an object $c_L \in C(L)$ can be classified by Galois cohomology $H^1(BG, \text{Aut}(c_L))$, and we’ll give some examples.

## Projective representations are homotopy fixed points

Yesterday we described how a (finite-dimensional) projective representation $\rho : G \to PGL_n(k)$ of a group $G$ functorially gives rise to a $k$-linear action of $G$ on $\text{Mod}(M_n(k)) \cong \text{Mod}(k)$ such that the Schur class $s(\rho) \in H^2(BG, k^{\times})$ classifies this action.

Today we’ll go in the other direction. Given an action of $G$ on $\text{Mod}(k)$ explicitly described by a 2-cocycle $\eta \in Z^2(BG, k^{\times})$, we’ll recover the category of $\eta$-projective representations, or equivalently the category of modules over the twisted group algebra $k \rtimes_{\eta} G$, by taking the homotopy fixed points of this action. We’ll end with another puzzle.

## Projective representations give categorical representations

Today we’ll resolve half the puzzle of why the cohomology group $H^2(BG, k^{\times})$ appears both when classifying projective representations of a group $G$ over a field $k$ and when classifying $k$-linear actions of $G$ on the category $\text{Mod}(k)$ of $k$-vector spaces by describing a functor from the former to the latter.

(There is a second half that goes in the other direction.)

## Projective representations

Three days ago we stated the following puzzle: we can compute that isomorphism classes of $k$-linear actions of a group $G$ on the category $C = \text{Mod}(k)$ of vector spaces over a field $k$ correspond to elements of the cohomology group

$\displaystyle H^2(BG, k^{\times})$.

This is the same group that appears in the classification of projective representations $G \to PGL(V)$ of $G$ over $k$, and we asked whether this was a coincidence.

Before answering the puzzle, in this post we’ll provide some relevant background information on projective representations.

## Fixed points of group actions on categories

Previously we described what it means for a group $G$ to act on a category $C$ (although we needed to slightly correct our initial definition). Today, as the next step in our attempt to understand Galois descent, we’ll describe what the fixed points of such a group action are.

John Baez likes to describe (vertical) categorification as replacing equalities with isomorphisms, which we saw on full display in the previous post: we replaced the equality $F(g) F(h) = F(gh)$ with isomorphisms $\eta(g, h) : F(g) F(h) \cong F(gh)$, and as a result we found 2-cocycles lurking in this story.

I prefer to describe categorification as replacing properties with structures, in the nLab sense. That is, the real import of what we just did is to replace the property (of a function between groups, say) that $F(g) F(h) = F(gh)$ with the structure of a family of isomorphisms between $F(g) F(h)$ and $F(gh)$. The use of the term “structure” emphasizes, as we also saw in the previous post, that unlike properties, structures need not be unique.

Accordingly, it’s not surprising that being a fixed point of a group action on a category is also a structure and not a property. Suppose $F : G \to \text{Aut}(C)$ is a group action as in the previous post, and $c \in C$ is an object. The structure of a fixed point, or more precisely a homotopy fixed point, is the data of a family of isomorphisms

$\displaystyle \alpha(g) : c \cong F(g) c$

which satisfy the compatibility condition that the two composites

$\displaystyle c \xrightarrow{\alpha(g)} F(g) c \xrightarrow{F(g)(\alpha(h))} F(g) F(h) c \xrightarrow{\eta(g, h)(c)} F(gh) c$

and

$\displaystyle c \xrightarrow{\alpha(gh)} F(gh) c$

are equal, as well as the unit condition that

$\displaystyle \alpha(e) = \varepsilon(c) : c \to F(e) c$

where $\varepsilon$ is the unit isomorphism $\text{id}_C \cong F(e)$. This is, in a sense we’ll make precise below, a 1-cocycle condition, but this time with nontrivial (local) coefficients.

Curiously, when the action $F$ is trivial (meaning both that $F(g) = \text{id}_C$ and that $\eta(g, h) = e \in Z(C)^{\times}$), this reduces to the definition of a group action of $G$ on $c \in C$ in the usual sense. In general, we can think of homotopy fixed point structure as a “twisted” version of a group action on $c \in C$ where the twist is provided by the group action on $C$.

## Group actions on categories

Yesterday we decided that it might be interesting to describe various categories as “fixed points” of Galois actions on various other categories, whatever that means: for example, perhaps real Lie algebras are the “fixed points” of a Galois action on complex Lie algebras. To formalize this we need a notion of group actions on categories and fixed points of such group actions.

So let $G$ be a group and $C$ be a category. For starters, we should probably ask for a functor $F(g) : C \to C$ for each $g \in G$. Next, we might naively ask for an equality of functors

$\displaystyle F(g) F(h) = F(gh) : C \to C$

but this is too strict: functors themselves live in a category (of functors and natural transformations), and so we should instead ask for natural isomorphisms

$\displaystyle \eta(g, h) : F(g) F(h) \cong F(gh)$.

These natural isomorphisms should further satisfy the following compatibility condition: there are two ways to use them to write down an isomorphism $F(g) F(h) F(k) \cong F(ghk)$, and these should agree. More explicitly, the composite

$\displaystyle F(g) F(h) F(k) \xrightarrow{F(g) \eta(h, k)} F(g) F(hk) \xrightarrow{\eta(g, hk)} F(ghk)$

should be equal to the composite

$\displaystyle F(g) F(h) F(k) \xrightarrow{\eta(g, h) F(k)} F(gh) F(k) \xrightarrow{\eta(gh, k)} F(ghk)$.

(There’s also some stuff going on with units which I believe we can ignore here. I think we can just require that $F(e) = \text{id}_C$ on the nose and nothing will go too horribly wrong.)

These natural isomorphisms $\eta(g, h)$ can be regarded as a natural generalization of 2-cocycles, and the condition above as a natural generalization of a cocycle condition. Below the fold we’ll describe this and other aspects of this definition in more detail, and we’ll end with two puzzles about the relationship between this story and group cohomology.

## Topological Diophantine equations

The problem of finding solutions to Diophantine equations can be recast in the following abstract form. Let $R$ be a commutative ring, which in the most classical case might be a number field like $\mathbb{Q}$ or the ring of integers in a number field like $\mathbb{Z}$. Suppose we want to find solutions, over $R$, of a system of polynomial equations

$\displaystyle f_1 = \dots = f_m = 0, f_i \in R[x_1, \dots x_n]$.

Then it’s not hard to see that this problem is equivalent to the problem of finding $R$-algebra homomorphisms from $S = R[x_1, \dots x_n]/(f_1, \dots f_m)$ to $R$. This is equivalent to the problem of finding left inverses to the morphism

$\displaystyle R \to S$

of commutative rings making $S$ an $R$-algebra, or more geometrically equivalent to the problem of finding right inverses, or sections, of the corresponding map

$\displaystyle \text{Spec } S \to \text{Spec } R$

of affine schemes. Allowing $\text{Spec } S$ to be a more general scheme over $\text{Spec } R$ can also capture more general Diophantine problems.

The problem of finding sections of a morphism – call it the section problem – is a problem that can be stated in any category, and the goal of this post is to say some things about the corresponding problem for spaces. That is, rather than try to find sections of a map between affine schemes, we’ll try to find sections of a map $f : E \to B$ between spaces; this amounts, very roughly speaking, to solving a “topological Diophantine equation.” The notation here is meant to evoke a particularly interesting special case, namely that of fiber bundles.

We’ll try to justify the section problem for spaces both as an interesting problem in and of itself, capable of encoding many other nontrivial problems in topology, and as a possible source of intuition about Diophantine equations. In particular we’ll discuss what might qualify as topological analogues of the Hasse principle and the Brauer-Manin obstruction.