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In algebraic geometry, the homogeneous coordinate ring ''R'' of an algebraic variety ''V'' given as a subvariety of projective space of a given dimension ''N'' is by definition the quotient ring :''R'' = ''K''(''X''1, ''X''2, ..., ''X''''N'' )/''I'' where ''I'' is the homogeneous ideal defining ''V'', ''K'' is the algebraically closed field over which ''V'' is defined, and :''K''(''X''1, ''X''2, ..., ''X''''N'' ) is the polynomial ring in ''N'' + 1 variables ''X''i. The polynomial ring is therefore the homogeneous coordinate ring of the projective space itself, and the variables are the homogeneous coordinates, for a given choice of basis (in the vector space underlying the projective space). The choice of basis means this definition is not intrinsic, but it can be made so by using the symmetric algebra. ==Formulation== Since ''V'' is assumed to be a variety, and so an irreducible algebraic set, the ideal ''I'' can be chosen to be a prime ideal, and so ''R'' is an integral domain. The same definition can be used for general homogeneous ideals, but the resulting coordinate rings may then contain non-zero nilpotent elements and other divisors of zero. From the point of view of scheme theory these cases may be dealt with on the same footing by means of the Proj construction. The correspondence between homogeneous ideals ''I'' and varieties is bijective for ideals not containing the ideal ''J'' generated by all the ''X''''i'', which corresponds to the empty set because not all homogeneous coordinates can vanish at a point of projective space. This correspondence is known as projective Nullstellensatz. 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「Homogeneous coordinate ring」の詳細全文を読む スポンサード リンク
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