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On the structure of the group scheme $$\mathbb{Z}[\mathbb{Z}/p^ n]^ \times$$. (English) Zbl 0876.14031
Let $$A$$ be a ring and $$G$$ a finite group. It is an attractive problem to investigate the unit group of the group algebra $$A[G]$$. An important remark given by Serre [cf. T. Sekiguchi, F. Oort and N. Suwa, Ann. Sci. Éc. Norm. Supér., IV. Sér. 22, No. , 345-375 (1989; Zbl 0714.14024) chapter VI, 8-9] has not been paid regard to so much; he noticed that the unit group of $$K[G]$$ has a structure of algebraic group when $$K$$ is a field. In this article, we study the structure of group scheme $$U(G)$$, which represents the unit group of $$A[G]$$, where $$G$$ is a cyclic group of prime power order. It should be noted that a key of investigation is the group scheme $${\mathcal G}^{(\lambda)}$$, which plays an important role in the theory unifying the Kummer and Artin-Schreier-Witt theories. After a short review on Néron blow-ups of affine group schemes in section 1, we establish some formalisms on $$U(G)$$ in section 2. The structure of $$U(\mathbb{Z}/p^n)$$ is treated in section 3. We conclude the article, by giving a relation with $$U(\mathbb{Z}/p^n)$$ and the Kummer-Artin-Schreier-Witt theories. Our method can be applied without any difficulty to investigation of $$U(G)$$ for any finite commutative group $$G$$. We expect to describe detailed accounts in a paper in preparation.

##### MSC:
 14L15 Group schemes 20C05 Group rings of finite groups and their modules (group-theoretic aspects) 11R27 Units and factorization
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##### References:
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