On the existence of maximal orders

TL;DR

This paper extends the theory of maximal orders to more general ground rings, removes separability constraints in certain algebraic contexts, and demonstrates their realization as endomorphism rings of abelian varieties within each isogeny class.

Contribution

It generalizes the existence of maximal orders beyond semi-simple algebras, relaxes separability assumptions, and connects maximal orders to abelian varieties.

Findings

Maximal orders exist in broader algebraic settings.

Separable algebra conditions can be relaxed to Japanese rings.

Maximal orders can be realized as endomorphism rings of abelian varieties.

Abstract

We generalize the existence of maximal orders in a semi-simple algebra for general ground rings. We also improve several statements in Chapter 5 and 6 of Reiner's book concerning separable algebras by removing the separability condition, provided the ground ring is only assumed to be Japanese, a very mild condition. Finally, we show the existence of maximal orders as endomorphism rings of abelian varieties in each isogeny class.