Exterior powers of F-zips

TL;DR

This paper develops the theory of exterior powers of F-zips over fields of positive characteristic, introduces a new invariant related to morphism dimensions, and demonstrates that this invariant alone cannot classify F-zips.

Contribution

It defines exterior powers of F-zips, explores their properties, and investigates a new invariant based on morphism spaces, showing it is insufficient for classification.

Findings

The new invariant can be computed for given F-zip types.

The invariant decomposes into two finite-dimensional subspaces.

The invariant does not fully classify F-zips.

Abstract

An F-zip over a field of positive characteristic is a vector space together with two filtrations whose subquotients are related in a certain way. We will define the category of F-zips and some basic constructions in it, especially exterior powers. If the ground field is algebraically closed, one can give a classification of F-Zips in terms of combinatorics. However, the way constructions and concepts in the category of F-zips manifest themselves in terms of the classifying invariant, is yet to be fully understood. The theory of F-crystals suggests that another invariant might be useful in trying to improve the understanding of F-zips. Given an F-zip, we calculate for every $1$-dimensional F-zip (of which there is essentially one for every integer $d$) $\mathbf{1}(d)$ and every $r\in\mathbb{Z}$ the dimension of the space of F-zip morphisms from $\mathbf{1}(d)$ into the $r$-th exterior power of the given F-zip. To make sense of this however, we will have to canonically decompose these spaces of morphisms each into two subspaces that are finite-dimensional over the prime field and its prime field respectively. One result will then be a way to calculate these numbers for a given isomorphism type. Our main result however, is a negative one: The invariant does not classify F-zips.