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 $\operatorname{B}G$, right? WRONG! Here’s another thing you can do.

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

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 $\operatorname{PGL}_n$ over the ring $A:=\mathbf{Z}\left[\frac{1}{n}\right][\mu_n]$. Choosing a faithful linear representation (example: the adjoint) $\operatorname{PGL}_n\to\operatorname{GL} _N$ and forming the versal torsor $\operatorname{GL}_N\to\operatorname{GL}_N/\operatorname{PGL}_n$ gives me a scheme $X$ (the quotient) and a $\operatorname{PGL}_n$-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 $\operatorname{PGL}_n$-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.

As Harry Caray used to say, “Holy Cow!”