Ben Williams

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This paper has been accepted for publication in the Journal of $K$–theory.

The main result of this paper is a computation of the motivic cohomology of varieties of $n \times m$-matrices of of rank $m$, including both the ring structure and the action of the reduced power operations. The argument proceeds by a comparison of the general linear group-scheme with a Tate suspension of a space which is $\mathbb{A}^1$-equivalent to projective $n-1$-space with a disjoint basepoint.

Joint with Ben Antieau

Here are the slides I gave on this topic at UBC in October 2012

We introduce and solve a period-index problem for the Brauer group of a topological space. The period-index problem is to relate the order of a class in the Brauer group to the degrees of Azumaya algebras representing it. For any space of dimension $d$, we give upper bounds on the index depending only on $d$ and the order of the class. By the Oka principle, this also solves the period-index problem for the analytic Brauer group of any Stein space that has the homotopy type of a finite CW-complex. Our methods use twisted topological $K$-theory, which was first introduced by Donovan and Karoubi. We also study the cohomology of the projective unitary groups to give cohomological obstructions to a class being represented by an Azumaya algebra of degree $n$. Applying this to the finite skeleta of the Eilenberg-MacLane space $K(Z/\ell,2)$, where $\ell$ is a prime, we construct a sequence of spaces with an order $\ell$ class in $\operatorname{Br}$, but whose indices tend to infinity.

This paper has been accepted for publication in Algebraic & Geometric Topology.

We derive a version of the Rothenberg-Steenrod, fiber-to-base, spectral sequence for cohomology theories represented in model categories of simplicial presheaves. We then apply this spectral sequence to calculate the equivariant motivic cohomology of the general linear group with a general $\mathbb{G}_m$–action, this coincides with the equivariant higher Chow groups. Some of the equivariant motivic cohomology of a Stiefel variety with a general $\mathbb{G}_m$–action is deduced as a corollary.

Joint with Ben Antieau

We completely solve the topological period-index problem for finite CW complexes of cohomological dimension at most 6. While for odd primes, the solution agrees with what is expected in algebraic geometry, the solution at the prime 2 does not, which suggests that something interesting is happening: either some restriction on the homotopy type of complex 3-folds is intervening, or the period–index conjecture is false at the prime 2 in dimension 3. At present, we cannot conclude which of these possibilities occurs.

Joint with Ben Antieau

We use the Serre–Godeaux varieties of finite groups, projective representation theory, the twisted Atiyah–Segal completion theorem, and our previous work on the topological period–index problem to compute the étale index of Brauer classes alpha in some specific examples. In particular, these computations show that the étale index of alpha differs from the period of alpha in general. As an application, we compute the index of unramified classes in the function fields of high-dimensional Serre–Godeaux varieties in terms of projective representation theory.

Joint with Ben Antieau

Here are the slides I gave on this topic at USC in September 2012,

And here are the (shorter) slides I gave on this topic at the JMM in San Diego in January 2013.

We show that, in general, over a regular noetherian affine scheme, there exist Brauer classes such that the division algebra over the generic point has no maximal orders that are Azumaya. We produce examples in dimensions 6. Despite the algebraic nature of the result, our proof relies on the algebraic topology of classifying spaces of algebraic groups.

This is mostly restated more correctly and intelligibly in papers above, the two papers dealing with the motivic cohomology of Stiefel Manifolds.

Chapters not appearing yet in any publicly available preprint make use of the fiber-to-base spectral sequence for motivic cohomology in order to calculate the higher Herzog–Kühl equations. It is full of embarassing errors and misprints, and is included here only for completeness.

The purpose of these notes is to prove a theorem which asserts that a certain kind of equivariant $\mathbb{A}^{\!1}$ weak equivalence may be detected by looking at fixed-point loci. The $\mathbb{A}^{\!1}$ weak equivalence in question is defined as an $\mathbb{A}^{\!1}$ localization of a local model structure on the category of simplicial presheaves on the category of smooth G-schemes, local with respect to an `intermediate' Nisnevich topology, whence the title. The topology is intermediate in that it has fewer covers than the most naive equivariant Nisnevich topology, where the covers are all Nisnevich covers that are compatible with the group action, and more covers than the `strong' Nisnevich topology where the covers are Nisnevich maps which are equivariant and are equivariantly split. The intermediate covers are the Nisnevich covers that remain Nisnevich covers after taking fixed-point loci. This topology admits other definitions, however, one of which, in terms of isotropy groups is also given.
The proof that $\mathbb{A}^{\!1}$weak equivalence can be detected on fixed-point loci is in two parts. First this is proved for local weak equivalence with respect to the topology, which is proved by finding a conservative set of points which are also compatible with the taking of fixed-point loci. The second step is to pass to the $\mathbb{A}^{\!1}$ localization, which rests on the two observations that the taking of fixed points commutes with the Sing-functor of Morel and Voevodsky, and that the taking of fixed point loci preserve fibrations in local flasque model structures.