Hermitian vector bundles and extension groups on arithmetic schemes. I. Geometry of numbers

2007/01/12 by Jean-Benoît Bost, Jean-Benoit Bost, Bost, Jean-Benoit +2 · 2 citations
Mathematics · #11H55 #14F05 #14G40 (Primary) 11H31 #18G15 (Secondary) #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT #msc:11H31 #msc:11H55 #msc:14F05 #msc:14G40 #msc:18G15

paper · pdf · doi:10.48550/arxiv.math/0701343

arxiv created 2007/01/12 · openalex publication_date 2007/01/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We define and investigate extension groups in the context of Arakelov geometry. The 'arithmetic extension groups' we introduce are extensions by groups of analytic types of the usual extension groups attached to Ø_X-modules over an arithmetic scheme X. In this paper, we focus on the first arithmetic extension group - the elements of which may be described in terms of admissible short exact sequences of hermitian vector bundles over X - and we especially consider the case when X is an 'arithmetic curve', namely the spectrum \Spec Ø_K of the ring of integers in some number field K. Then the study of arithmetic extensions over X is related to old and new problems concerning lattices and the geometry of numbers.

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