Degenerated fibers of Severi-Brauer schemes of orders
type: Item
instance of: scholarly article
title: Degenerated fibers of Severi-Brauer schemes of orders
zbMATH Open document ID: 0909.16012
author: Emmanuelle Frossard
published in: Journal of Algebra
publication date: 1999-04-06T00:00:00Z
review text: Let \(R\) be a Dedekind domain with field of fractions \(K\) and \(A\) a central simple algebra over \(K\). If \(\Lambda\) is a hereditary \(R\)-order of \(A\), then Artin canonically associated to it its so-called Severi-Brauer scheme \(X\), by letting \(X(R')\) consist for any commutative \(R\)-algebra \(R'\) of all left ideals \(I'\) of \(\Lambda'=\Lambda\otimes R'\), such that \(\Lambda'/I'\) is projective over \(R'\) and \(I'\) has rank \(n\) over \(R'\). In this case, the generic fibre \(X_K\) of \(X\) over \(\text{Spec}(R)\) is the classical Severi-Brauer variety associated to the central simple algebra \(A\). On the other hand, for any closed point \(p\) of \(\text{Spec}(R)\) the special fibre \(X_p\) is the scheme of left ideals of rank \(n\) of \(\Lambda/p\Lambda\). If \(X_p\) is non-degenerate, i.e., if \(\Lambda\) is unramified over \(p\), then \(\Lambda/p\Lambda\) is a central simple algebra over \(k(p)\), the residue field of \(p\), and \(X_p\) is a Severi-Brauer variety as well. The principal purpose of this paper is to study the Chow groups of dimension 0 of the special fibres of Artin's models, i.e., the connected component containing the generic fibre of the previously defined Severi-Brauer schemes. The main result is as follows. Let \(R\) be a complete discrete valuation ring with maximal ideal \(pR\) and field of fractions \(K\), and with perfect residue field. Let \(X\) be an Artin model over \(\text{Spec}(R)\) associated to a central simple algebra \(A\). If (a) the index of \(A\) has no square factors and \(A\) is not ramified or (b) \(A\) is totally ramified (of arbitrary index), then \(A_0(X) =0\). Of course, this applies to the case where \(k\) has cohomological dimension (at most) 1, for in this case \(A\) is totally ramified, as the skew residue field of \(A\) is trivial.
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 14C25
zbMATH DE Number: 1106615
zbMATH Keywords: hereditary orders
zbMATH Keywords: central simple algebras
zbMATH Keywords: Severi-Brauer schemes
zbMATH Keywords: Severi-Brauer varieties
zbMATH Keywords: Chow groups
zbMATH Keywords: Artin models
MaRDI profile type: Publication
full work available at URL: jabr.1997.7141
OpenAlex ID: W2055678103
cites work: Q4742905
cites work: Q3704013
cites work: Q4166837
cites work: Q4378058
cites work: Q3328651
cites work: Q4061092
DOI: 10.1006/JABR.1997.7141
description: scientific article; zbMATH DE number 1106615