A Schur–Zassenhaus Theorem for association schemes

Davenport-Hasse theorem and cyclotomic association schemes

Definition. Let q be a prime power and e be a divisor of q − 1. Fix a generator α of the multiplicative group of GF (q). Then 〈α〉 is a subgroup of index e and its cosets are 〈α〉α, i = 0, . . . , e− 1. Define R0 = {(x, x)|x ∈ GF (q)} Ri = {(x, y)|x, y ∈ GF (q), x− y ∈ 〈αe〉αi−1}, (1 ≤ i ≤ e) R = {Ri|0 ≤ i ≤ e} Then (GF (q),R) forms an association scheme and is called the cyclotomic scheme of clas...

An Assmus-Mattson theorem for codes over commutative association schemes

Mnëv’s Universality Theorem for Schemes

We prove a scheme-theoretic version of Mnëv’s Universality Theorem, suitable to be used in the proof of Murphy’s Law in Algebraic Geometry [Va, Main Thm. 1.1]. Somewhat more precisely, we show that any singularity type of finite type over Z appears on some incidence scheme of points and lines, subject to some particular further constraints. This paper is dedicated to Joe Harris on the occasion ...

Detection Theorem for Finite Group Schemes

The classical detection theorem for finite groups (due to D. Quillen [Q] and B. Venkov [Q-V]) tells that if π is a finite group and Λ is a Z/p[π]-algebra then a cohomology class z ∈ H(π,Λ) is nilpotent iff for every elementary abelian psubgroup i : π0 →֒ π the restriction i (z) ∈ H(π0,Λ) of z to π0 is nilpotent. This theorem is very useful for the identification of the support variety of the gro...

Watts Theorem for Schemes: Preliminary Version

We describe obstructions to a direct-limit preserving right-exact functor between categories of quasi-coherent sheaves on schemes being isomorphic to tensoring with a bimodule. When the domain scheme is affine, all obstructions vanish and we recover Watts Theorem. We use our description of these obstructions to prove that if a direct-limit preserving right-exact functor F from a smooth curve is...