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.
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