Versal torsors are awesome

I was just at Bar Ilan University for Louis Rowen’s great birthday conference. Included among my fellow speakers were Danny Krashen and Alexander Merkurjev.
They told me about something I should have seen long ago: versal torsors. Among other things, versal torsors give a way to produce universal bounds on symbol lengths for central division algebras with bounded index. This blew my mind. Let me show you how it works.

Look at what happened to my classifying space!

Let $G$ be a linear algebraic group. Suppose I want to understand $G$ -torsors. As a card-carrying algebraic geometer and stack aficionado, I should just “use”
the classifying stack $\mathrm{B}G$ , right? WRONG! Here’s another thing you can do.

Choose a faithful representation $G\hookrightarrow\mathrm{GL}_n$ (over whatever base we’re using) and form the quotient $B=\mathrm{GL}_n/G$ . By the definition of quotients, there is an induced $G$ -torsor $T\to B$ (i.e., $T=\mathrm{GL}_n$ !!). This is called a versal torsor. Why versal? For any field $K$ over the base we get an exact sequence of pointed sets

$G(K)\to\mathrm{GL}_n(K)\to B(K)\to\mathrm{H}^1(\mathrm{Spec}(K),G)\to\mathrm{H}^1(\mathrm{Spec}(K), \mathrm{GL}_n(K)).$

Hilbert’s Theorem 90 tells us that the last set is a singleton. We conclude the following amazing fact: for any field $K$ over the base and any $G$ -torsor $\Gamma$ over $K$ , there is a $K$ -point of $B$ such that the fiber of $T$ over that point is isomorphic to $\Gamma$ .

I don’t know why this freaks me out so much; perhaps it has to do with my perception that there are a lot of $G$ -torsors over arbitrary fields, whereas this tells us that they are all points of a single variety. (Corollary: the essential dimension of $G$ is finite. When you put it that way, it doesn’t sound so outrageous.)

Now that they gave me this thing, what can I do with it?

Here’s an amazing application of versal torsors. Let $G$ be the group $\mathrm{PGL}_n$ over the ring $A:=\mathbf{Z}\left[\frac{1}{n}\right][\mu_n]$ . Choosing a faithful linear representation (example: the adjoint) $\mathrm{PGL}_n\to\mathrm{GL _N$ and forming the versal torsor $\mathrm{GL}_N\to\mathrm{GL}_N/\mathrm{PGL}_n$ gives me a scheme $X$ (the quotient) and a $\mathrm{PGL}_n$ -torsor (let’s call it $T\to X$ ) with the amazing property that for any field $K$ in which $n$ is invertible and containing a primitive $n$ th root of unity and any $\mathrm{PGL}_n$ -torsor $Y$ over $K$ , there is a $K$ -point of $X$ such that the fiber of $T$ over that point is isomorphic to $Y$ . (I’m just restating the above.)

Here’s something we can do with this. Given a central division algebra $D$ over such a field $K$ with index $n$ , the Merkurjev-Suslin theorem says that there is a sequence of symbol algebras $(a_1,b_1),\ldots,(a_m,b_m)$ over $K$ such that $D$ is Brauer-equivalent to the tensor product $E=\bigotimes_i (a_i,b_i)$ (that is, there are matrix algebras over $D$ and $E$ that are isomorphic). The question is: how many symbols do we need?

The surprising answer is: there is a universal bound on the number of symbols we need! Here’s how to prove this. Let $\alpha$ be the Brauer class of the versal torsor over $X$ . Merkurjev-Suslin gives us an expression for $\alpha$ restricted to the generic point of $X$ in terms of symbols. Standard spreading out arguments make this expression valid in an open neighborhood (i.e., the symbols extend to Azumaya algebras in a neighborhood). Take the complement of this open subset; this is some closed subset $X'\subset X$ . Repeat this at a generic point of $X'$ . Since any locally-closed stratification of $X$ is finite ( $X$ being Noetherian), this process will terminate. Take the largest number of symbols you used and that’s your bound! Every algebra of degree $n$ over any field factors through $X$ , so it will factor through a piece of the stratification, and we can use the symbol decomposition on that stratum.