Solutions to the exercises from the book "Albert algebras over commutative rings"

TL;DR

This work extends the theory of Albert algebras over rings by (i) proving that Albert algebras are $i$-exceptional over every non-zero ring, (ii) developing descent criteria from fields to Dedekind domains for cubic Jordan and Albert algebras, and (iii) locating tight, small-degree separable extensions that split or reduce Freudenthal algebras. The methods combine permanence principles for polynomial laws, descent along faithfully flat extensions, and structural tools from the Tits constructions and Rost invariants to obtain explicit splitting and reduction results for Freudenthal algebras across ranks 1, 3, 9, 15, and 27. The results illuminate how higher Jordan and Freudenthal structures behave under base change and yield concrete links to associated algebraic groups of types $\mathsf{A}_2$ and $\mathsf{F}_4$, with potential arithmetic and geometric applications. Collectively, the sections sharpen our understanding of when and how Albert and Freudenthal algebras become split or reduced under controlled extensions, and clarify descent phenomena over Dedekind domains.

Abstract

This document presents the solutions to the exercises in the book "Albert algebras over commutative rings" published by Cambridge University Press, 2024, as well as errata and addenda.

Theorems & Definitions (58)

• Theorem A1.1
• Theorem A1.2
• proof
• Definition A1.3
• Lemma A1.4
• proof
• Lemma A1.5
• proof
• Lemma A1.6
• proof