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Uniqueness of indecomposable idempotents in algebras with involution

Valentijn Karemaker, Akio Tamagawa, Chia-Fu Yu

Abstract

We prove uniqueness of a decomposition of $1$ into indecomposable Hermitian idempotents in an order of a finite-dimensional $\mathbb{Q}$-algebra with positive involution, by generalising a result of Eichler on unique decomposition of lattices. We use this result to prove that polarised abelian varieties over any field admit a unique decomposition into indecomposable polarised abelian subvarieties, a result previously shown by Debarre and Serre with different methods and over algebraically closed fields. We prove that an analogous uniqueness result holds true for arbitrary polarised integral Hodge structures, and derive a consequence for their automorphism groups.

Uniqueness of indecomposable idempotents in algebras with involution

Abstract

We prove uniqueness of a decomposition of into indecomposable Hermitian idempotents in an order of a finite-dimensional -algebra with positive involution, by generalising a result of Eichler on unique decomposition of lattices. We use this result to prove that polarised abelian varieties over any field admit a unique decomposition into indecomposable polarised abelian subvarieties, a result previously shown by Debarre and Serre with different methods and over algebraically closed fields. We prove that an analogous uniqueness result holds true for arbitrary polarised integral Hodge structures, and derive a consequence for their automorphism groups.
Paper Structure (10 sections, 13 theorems, 30 equations)

This paper contains 10 sections, 13 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Traces, involutions, semi-simplicity, non-degeneracy and positive-definiteness
  3. Various traces of finite-dimensional algebras
  4. Relations between semi-simplicity, non-degeneracy and positive-definiteness
  5. Uniqueness of orthogonal decompositions of positive-definite Hermitian lattices
  6. Proof of Theorem \ref{['thm:UniDecIdem']}
  7. Proof of Theorem \ref{['thm:1.3']}
  8. Proof of Theorem \ref{['thm:1.5']}
  9. Polarised Hodge structures
  10. Uniqueness of orthogonal decompositions

Key Result

Theorem 1.1

Assume that $\mathop{\rm Tr}\nolimits_{R^0/\mathbb{Q}}(x x^*)>0$ for any $x\neq 0\in R^0$. If then $\{i_\nu\}_{\nu}$ is uniquely determined.

Theorems & Definitions (28)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Example 2.1
  • Proposition 2.3
  • proof
  • ...and 18 more