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In mathematics, specifically algebraic topology, the cohomology ring of a topological space ''X'' is a ring formed from the cohomology groups of ''X'' together with the cup product serving as the ring multiplication. Here 'cohomology' is usually understood as singular cohomology, but the ring structure is also present in other theories such as de Rham cohomology. It is also functorial: for a continuous mapping of spaces one obtains a ring homomorphism on cohomology rings, which is contravariant. Specifically, given a sequence of cohomology groups ''H''''k''(''X'';''R'') on ''X'' with coefficients in a commutative ring ''R'' (typically ''R'' is Z''n'', Z, Q, R, or C) one can define the cup product, which takes the form : The cup product gives a multiplication on the direct sum of the cohomology groups : A numerical invariant derived from the cohomology ring is the cup-length, which means the maximum number of graded elements of degree ≥ 1 that when multiplied give a non-zero result. For example a complex projective space has cup-length equal to its complex dimension. == Examples == * where . * where . *By the Künneth formula, the mod 2 cohomology ring of ''n'' products of is a polynomial ring in ''n'' variables with coefficients in . 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「cohomology ring」の詳細全文を読む スポンサード リンク
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