Algebraic elements in division algebras over function fields of curves
type: Item
instance of: scholarly article
title: Algebraic elements in division algebras over function fields of curves
zbMATH Open document ID: 0596.12012
DOI: 10.1007/BF02776077
author: Michel Van den Bergh
author: Jan Van Geel
published in: Israel Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: Let \(C\) be a curve over the field \(k\) with function field \(K\) and \(D\) a division algebra with center \(K\). Consider a maximal order \({\mathcal O}_{\Lambda}\) in \(D\), i.e. a sub-\({\mathcal O}_ C\)-sheaf of maximal orders of the constant sheaf \(\mathbb D\). As in the commutative case, one may consider the cohomology spaces of coherent \({\mathcal O}_{\Lambda}\)-modules, and relate them by an analogue of the Riemann-Roch theorem. One has two different notions of genus for \(D\), \(g_ D\) and \(g_{\Lambda}\), where \(g_{\Lambda}\) depends on the choice of the maximal order \({\mathcal O}_{\Lambda}\). Some natural questions arise: -- Under which circumstances does \(g_{\Lambda}\) depend only on \(D\)? -- Does \(D\) contain \(k\)-algebraic sub-division algebras? Several partial answers to these questions are given, assuming additional conditions on \(k\) or on the genus of \(K\) over \(k\). Furthermore, some very instructive examples and counterexamples are provided.
reviewed by: Ernst-Ulrich Gekeler
Mathematics Subject Classification ID: 11R52
Mathematics Subject Classification ID: 12F20
Mathematics Subject Classification ID: 12E15
Mathematics Subject Classification ID: 14F10
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 16Kxx
Mathematics Subject Classification ID: 14H05
zbMATH DE Number: 3959573
zbMATH Keywords: division algebra over function field
zbMATH Keywords: sheaf of differentials
zbMATH Keywords: maximal order
zbMATH Keywords: Riemann-Roch theorem
zbMATH Keywords: genus
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02776077
OpenAlex ID: W1972274844
description: scientific article; zbMATH DE number 3959573