Standard Conjectures on Algebraic Cycles
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High Quality Content by WIKIPEDIA articles! In mathematics, the standard conjectures about algebraic cycles is a term applied to a package of several conjectures describing the relationship of algebraic cycles and Weil cohomology theories. One of the original applications envisaged by Alexander Grothendieck was to prove that his construction of pure motives gave an abelian category that is semisimple. Moreover, as he pointed out, the standard conjectures also imply the hardest part of the Weil conjectures, namely the "Riemann hypothesis" conjecture that remained open at the end of the 1960s an...
High Quality Content by WIKIPEDIA articles! In mathematics, the standard conjectures about algebraic cycles is a term applied to a package of several conjectures describing the relationship of algebraic cycles and Weil cohomology theories. One of the original applications envisaged by Alexander Grothendieck was to prove that his construction of pure motives gave an abelian category that is semisimple. Moreover, as he pointed out, the standard conjectures also imply the hardest part of the Weil conjectures, namely the "Riemann hypothesis" conjecture that remained open at the end of the 1960s and was proved later by Pierre Deligne; for details on the link between Weil and standard conjectures, see Kleiman (1968). The standard conjectures remain open problems, so that their application gives only conditional proofs of results. In quite a few cases, including that of the Weil conjectures, other methods have been found to prove such results unconditionally.
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