K-theoretic exceptional collections at roots of unity

TL;DR

This paper explores the use of cyclotomic specializations in equivariant K-theory to establish congruences for invariants of exceptional objects in derived categories, with applications to varieties like Grassmannians and quadrics.

Contribution

It introduces a novel approach using cyclotomic specializations to derive congruences for invariants of exceptional objects in derived categories of specific varieties.

Findings

Rank of exceptional objects on certain products of projective spaces is congruent to ±1 modulo a prime p.

Derived congruences apply to varieties including Grassmannians and smooth quadrics.

New K-theoretic methods connect cyclotomic specializations to categorical invariants.

Abstract

Using cyclotomic specializations of the equivariant $K$-theory with respect to a torus action we derive congruences for discrete invariants of exceptional objects in derived categories of coherent sheaves on a class of varieties that includes Grassmannians and smooth quadrics. For example, we prove that if $X={\Bbb P}^{n_1-1}\times...\times{\Bbb P}^{n_k-1}$, where $n_i$'s are powers of a fixed prime number $p$, then the rank of an exceptional object on $X$ is congruent to $\pm 1$ modulo $p$.