Turkish Journal of Mathematics

Abstract: Let L be a local field and \tilde{L} the completion of the maximal unramified extension of L. In this short note, we prove \noindent (I) the sequences \noindent (1^\prime\prime) 0\rightarrow {\cal O}_L[[X₁,\ldots,X_n]]\rightarrow{\cal O}_\tilde{L}[[X₁,\ldots,X_n]] \stackrel{f_L-1}{\ra}{\cal O}_\tilde{L}[[X₁,\ldots,X_n]]\rightarrow 0 \noindent and \noindent (2^\prime\prime) 1\rightarrow {\cal O}_L[[X₁,\ldots,X_n]]^x\rightarrow{\cal O}_\tilde{L}[[X₁,\ldots,X_n]] ^x\stackrel{f_L-1}{\ra}{\cal O}_\tilde{L}[[X₁,\ldots,X_n]]\rightarrow 1 \noindent are exact, where q_L is the Frobenius automorphism over L applied on the coefficients of {\cal O}_\tilde{L}[[X₁,\ldots, X_n]], and q_L-1 respectively denotes the mapping a\mapstoa^{q L}-a for \noindent (1^\prime\prime) and e\mapsto\frac{e^fL}{e} for (2^\prime\prime); \noindent (II) {\cal C}^\circ(\tilde{L},h) being the group the Coleman power series of degree 0, the sequence 1\rightarrow {\cal C}^\circ(L, h)\rightarrow{\cal C}^\circ(\tilde{L},h)C^\circ(\tilde{L},h) \rightarrow 1 \noindent is exact, where {\cal C}^\circ(L,h)={\cal O}_L[[X]]^x\cap{\cal C}^\circ(\tilde{L},h)

Key Words: Formal power series, Local fields, Formal groups, Lubin-Tate theory, Coleman power series.