Degenerated fibers of Severi-Brauer schemes of orders
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.
scientific article; zbMATH DE number 1106615