manuscripta mathematica

, Volume 130, Issue 3, pp 387–409

# Singularities of generic characteristic polynomials and smooth finite splittings of Azumaya algebras over surfaces

Open Access
Article

## Abstract

Let k be an algebraically closed field. Let P(X11, . . . , Xnn, T) be the characteristic polynomial of the generic matrix (Xij) over k. We determine its singular locus as well as the singular locus of its Galois splitting. If X is a smooth quasi-projective surface over k and A an Azumaya algebra on X of degree n, using a method suggested by M. Artin, we construct finite smooth splittings for A of degree n over X whose Galois closures are smooth.

### Mathematics Subject Classification (2000)

Primary 16H05 Secondary 14F22

## Notes

### Acknowledgments

The authors acknowledge support from NSF DMS-0653382 and from Max-Planck Institut für Mathematik, Bonn. We thank Jean-Louis Colliot-Thélène, Aise Johan de Jong, and David Saltman for several discussions.

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