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On the deformations of Witt groups to tori. II. (English) Zbl 0735.14033
To construct the unified Kummer-Artin-Schreier-Witt theory, which combines the theories of Kummer and Artin-Schreier-Witt, we must construct the deformations of Witt groups to tori. For the one dimensional case, such deformations are discussed by W. C. Waterhouse and B. Weisfeiler [J. Algebra 66, 550-568 (1980; Zbl 0452.14013)], the author and F. Oort [Algebraic and topological theories, Kinokuniya Co. Ltd., 283-298 (1985)] and the author, F. Oort and N. Suwa [Ann. Sci. Éc. Norm. Supér., IV. Sér. 22, No. 3, 345-375 (1989; Zbl 0714.14024)]. In this paper, the author gives two methods for constructing such deformations of Witt groups in the higher dimensional cases. One is to take unit groups of suitable ring schemes, and the other is to compute the extensions of lower dimensional deformations of Witt groups. Moreover, by using these methods, the author discusses the details of such deformations in the two dimensional case.

MSC:
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
11E81 Algebraic theory of quadratic forms; Witt groups and rings
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