Tsutomu Sekiguchi, Yohei Toda
Department of Mathematics
Faculty of Science and Engineering
Chuo University, 1-13-27, Kasuga, Bunkyo-ku, Tokyo, 112-8551, JAPAN
Department of Mathematics
Faculty of Science and Engineering
Chuo University
1-13-27, Kasuga, Bunkyo-ku, Tokyo, 112-8551, JAPAN

Abstract. Our aim in this paper is to compute the first cohomology of some type of finite group schemes. L. G. Roberts gave the first cohomology of group schemes in certain conditions. We compute it by completely different way and under circumstances, by using the concept of cyclotomic twisted tori. The concept was introduced by Y. Koide and T. Sekiguchi, and they showed that such a twisted torus is isomorphic to a subgroup scheme in a Weil restriction of 1-dimensional algebraic torus given by the intersection of whole norm maps. Here we extend the isomorphism to a resolution of the cyclotomic twisted torus, consisting of Weil restriction of 1-dimensional algebraic tori and several norm maps. And we describe the endomorphism ring of a cyclotomic twisted torus. Moreover, we show that by using the resolution, one can compute that first cohomology of a cyclotomic twisted torus, and that one can describe the torsors of some type of finite group schemes by using the concept of cyclotomic twisted tori.

Received: July 18, 2013

AMS Subject Classification: 14L15

Key Words and Phrases: torsor, group scheme

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