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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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