Limits of Rank 4 Azumaya Algebras and Applications to Desingularization

Proceedings Mathematical Sciences

T E Venkata Balaji¹ 

[1] Chennai Mathematical Institute, G. N. Chetty Road, T. Nagar, Chennai 00 0, India$$

关键词: Azumaya algebra;    Clifford algebra;    desingularization;    moduli space;    semi-regular quadratic form;    simple module;    vector bundle.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

It is shown that the schematic image of the scheme of Azumaya algebra structures on a vector bundle of rank 4 over any base scheme is separated, of finite type, smooth of relative dimension 13 and geometrically irreducible over that base and that this construction base-changes well. This fully generalizes Seshadri's theorem in [16] that the variety of specializations of (2 × 2)-matrix algebras is smooth in characteristic ≠ 2. As an application, a construction of Seshadri in [16] is shown in a characteristic-free way to desingularize the moduli space of rank 2 even degree semi-stable vector bundles on a complete curve. As another application, a construction of Nori over $mathbb{Z}$ (Appendix, [16]) is extended to the case of a normal domain which is a universally Japanese (Nagata) ring and is shown to desingularize the Artin moduli space [1] of invariants of several matrices in rank 2. This desingularization is shown to have a good specialization property if the Artin moduli space has geometrically reduced fibers – for example this happens over $mathbb{Z}$. Essential use is made of Kneser's concept [8] of `semi-regular quadratic module'. For any free quadratic module of odd rank, a formula linking the half-discriminant and the values of the quadratic form on its radical is derived.

【 授权许可】

Unknown