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In mathematics Haboush's theorem, often still referred to as the Mumford conjecture, states that for any semisimple algebraic group ''G'' over a field ''K'', and for any linear representation ρ of ''G'' on a ''K''-vector space ''V'', given ''v'' ≠ 0 in ''V'' that is fixed by the action of ''G'', there is a ''G''-invariant polynomial ''F'' on ''V'', without constant term, such that :''F''(''v'') ≠ 0. The polynomial can be taken to be homogeneous, in other words an element of a symmetric power of the dual of ''V'', and if the characteristic is ''p''>0 the degree of the polynomial can be taken to be a power of ''p''. When ''K'' has characteristic 0 this was well known; in fact Weyl's theorem on the complete reducibility of the representations of ''G'' implies that ''F'' can even be taken to be linear. Mumford's conjecture about the extension to prime characteristic ''p'' was proved by W. J. , about a decade after the problem had been posed by David Mumford, in the introduction to the first edition of his book ''Geometric Invariant Theory''. ==Applications== Haboush's theorem can be used to generalize results of geometric invariant theory from characteristic 0, where they were already known, to characteristic ''p''>0. In particular Nagata's earlier results together with Haboush's theorem show that if a reductive group (over an algebraically closed field) acts on a finitely generated algebra then the fixed subalgebra is also finitely generated. Haboush's theorem implies that if ''G'' is a reductive algebraic group acting regularly on an affine algebraic variety, then disjoint closed invariant sets ''X'' and ''Y'' can be separated by an invariant function ''f'' (this means that ''f'' is 0 on ''X'' and 1 on ''Y''). C.S. Seshadri (1977) extended Haboush's theorem to reductive groups over schemes. It follows from the work of , Haboush, and Popov that the following conditions are equivalent for an affine algebraic group ''G'' over a field ''K'': *''G'' is reductive (its unipotent radical is trivial). *For any non-zero invariant vector in a rational representation of ''G'', there is an invariant homogeneous polynomial that does not vanish on it. *For any finitely generated ''K'' algebra on which ''G'' act rationally, the algebra of fixed elements is finitely generated. 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「Haboush's theorem」の詳細全文を読む スポンサード リンク
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