Noncommutative ampleness from finite endomorphisms

TL;DR

This paper investigates the properties of bimodules and associated algebras defined by finite endomorphisms on projective schemes, revealing differences from automorphism cases, including non-noetherian structures and one-sided ampleness.

Contribution

It introduces the concept of noncommutative ampleness for finite endomorphisms and explores their algebraic properties, contrasting with automorphism scenarios.

Findings

One-sided ampleness can occur with finite endomorphisms.

Rings and bimodule algebras associated with such endomorphisms are not noetherian.

Finite endomorphisms exhibit different ampleness properties than automorphisms.

Abstract

Let $X$ be a projective integral scheme with endomorphism $\sigma$, where $\sigma$ is finite, but not an automorphism. We examine noncommutative ampleness of bimodules defined by $\sigma$. In contrast to the automorphism case, one-sided ampleness is possible. We also find that rings and bimodule algebras associated with $\sigma$ are not noetherian.