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/lp/de-gruyter/arakelov-theory-of-noncommutative-arithmetic-surfaces-5Z9yaeehDS
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- Publisher
- de Gruyter
- Copyright
- © Walter de Gruyter Berlin · New York 2010
- ISSN
- 0075-4102
- eISSN
- 1435-5345
- DOI
- 10.1515/CRELLE.2010.036
- Publisher site
- See Article on Publisher Site
Abstract
The purpose of the present article is to initiate Arakelov theory of noncommutative arithmetic surfaces. Roughly speaking, a noncommutative arithmetic surface is a noncommutative projective scheme of cohomological dimension 1 of finite type over Spec ℤ. An important example is the category of coherent right 𝒪-modules, where 𝒪 is a coherent sheaf of 𝒪 X -algebras and 𝒪 X is the structure sheaf of a commutative arithmetic surface X . Since smooth hermitian metrics are not available in our noncommutative setting, we have to adapt the definition of arithmetic vector bundles on noncommutative arithmetic surfaces. Namely, we consider pairs (ℰ, β ) consisting of a coherent sheaf ℰ and an automorphism β of the real sheaf ℰ ℝ induced by ℰ. We define the intersection of two such objects using the determinant of the cohomology and prove a Riemann-Roch theorem for arithmetic line bundles on noncommutative arithmetic surfaces.
Journal
Journal für die reine und angewandte Mathematik (Crelle's Journal) – de Gruyter
Published: May 1, 2010