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In mathematics, the Frobenius endomorphism is defined in any commutative ring ''R'' that has characteristic ''p'', where ''p'' is a prime number. Namely, the mapping φ that takes ''r'' in ''R'' to ''r''''p'' is a ring endomorphism of ''R''. The image of φ is then ''R''''p'', the subring of ''R'' consisting of ''p''-th powers. In some important cases, for example finite fields, φ is surjective. Otherwise φ is an endomorphism but not a ring ''automorphism''. The terminology of geometric Frobenius arises by applying the spectrum of a ring construction to φ. This gives a mapping :φ *: Spec(''R''''p'') → Spec(''R'') of affine schemes. Even in cases where ''R''''p'' = ''R'' this is not the identity, unless ''R'' is the prime field. Mappings created by fibre product with φ *, i.e. base changes, tend in scheme theory to be called ''geometric Frobenius''. The reason for a careful terminology is that the Frobenius automorphism in Galois groups, or defined by transport of structure, is often the inverse mapping of the geometric Frobenius. As in the case of a cyclic group in which a generator is also the inverse of a generator, there are in many situations two possible definitions of Frobenius, and without a consistent convention some problem of a minus sign may appear. ==References== * , p. 5 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Arithmetic and geometric Frobenius」の詳細全文を読む スポンサード リンク
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