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Motivic cohomology is an invariant of algebraic varieties and of more general schemes. It is a type of cohomology related to motives and includes the Chow ring of algebraic cycles as a special case. Some of the deepest problems in algebraic geometry and number theory are attempts to understand motivic cohomology.

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rdf:type	yago:WikicatCohomologyTheories yago:WikicatTopologicalMethodsOfAlgebraicGeometry yago:Ability105616246 yago:Abstraction100002137 yago:Cognition100023271 yago:Explanation105793000 yago:HigherCognitiveProcess105770664 yago:Know-how105616786 yago:Method105660268 yago:Process105701363 yago:PsychologicalFeature100023100 yago:Theory105989479 yago:Thinking105770926
rdfs:label	Cohomologie motivique (fr) モチヴィック・コホモロジー (ja) 모티브 코호몰로지 (ko) Motivic cohomology (en)
rdfs:comment	Motivic cohomology is an invariant of algebraic varieties and of more general schemes. It is a type of cohomology related to motives and includes the Chow ring of algebraic cycles as a special case. Some of the deepest problems in algebraic geometry and number theory are attempts to understand motivic cohomology. (en) Une cohomologie motivique est une théorie cohomologique en mathématiques dont l'existence a été conjecturée pour la première fois par Alexandre Grothendieck dans les années 1960. À l'époque, on la concevait comme construite sur les bases des (en) sur les cycles algébriques, en géométrie algébrique. Elle puise ses fondements en théorie des catégories, ce qui permet de déduire des conséquences à partir de ces conjectures. Grothendieck et Bombieri ont démontré la profondeur de cette approche en dérivant une (en) des conjectures de Weil de cette façon. Par contre, des preuves des conjectures standard n'ont pas été trouvées. (fr) モチヴィック・コホモロジー（英: motivic cohomology）とは、代数多様体などのスキームの不変量のひとつである。モチーフに関係する一種のコホモロジーであり、代数的サイクルのチャウ環を特別な場合として含んでいる。代数幾何学と数論における最も深い問題のいくつかはモチヴィック・コホモロジーを理解しようとする試みである。 (ja) 대수기하학에서 모티브 코호몰로지(영어: motivic cohomology)는 호몰로지 이론 중의 하나로서, 대수기하학의 연구 대상인 대수다양체 위에 정의할 수 있는 일종의 '범용 코호몰로지 이론'(universal cohomology theory)이다. 이 이론은 1960년대에, 수학자 알렉산더 그로텐디크에 의해서 처음으로 그 존재성이 예측되었다. 1960년대에 그로텐디크가 이 이론의 존재성을 예측할 당시에는, 이 이론은 아마도 대수기하학의 (the standard conjectures on algebraic cycles)이 증명되면 이론이 확립될 것이라고 생각하였다. 이러한 가설이 맞다는 가정하에서 적절한 범주론을 사용하여 그로텐디크와 엔리코 봄비에리는 (Weil conjecture)에 대한 색다른 증명을 얻어낼 수도 있었다. (베유 가설은, 피에르 들리뉴에 의해서 다른 방식으로 증명되었다.) 하지만, 불행하게도 현재까지도 이 '대수 사이클에 관한 표준 가설들'은 증명되지 않고 있다. (ko)
dcterms:subject	Cohomology theories Topological methods of algebraic geometry Homotopical algebra
Wikipage page ID	809314 (xsd:integer)
Wikipage revision ID	1032971484 (xsd:integer)
Link from a Wikipage to another Wikipage	Princeton University Press Motive (algebraic geometry) Multiplicative group Algebraic K-theory Atiyah-Hirzebruch spectral sequence Universal coefficient theorem Vladimir Voevodsky Derived category Presheaf with transfers Chow ring Cohomology theories Glossary of algebraic geometry Bloch's higher Chow groups Eric Friedlander Milnor K-theory Andrei Suslin Topological methods of algebraic geometry Long exact sequence Simplex Standard conjectures on algebraic cycles Stephen Lichtenbaum Clay Mathematics Monographs Étale cohomology A¹ homotopy theory Adams operation Triangulated category Nisnevich topology Alexander Beilinson Algebraic geometry Algebraic variety American Mathematical Society Daniel Quillen Faithful functor Field (mathematics) Number theory Regular function Ring (mathematics) Tensor product Abelian category Advances in Mathematics Homotopical algebra Coherent sheaf Cohomology Dirichlet's unit theorem Borel–Moore homology Poincare duality Spectral sequence Spencer Bloch Grothendieck group Chain complex Christophe Soulé Scheme (mathematics) Sheaf (mathematics) Vector bundle Zariski topology Exact sequence T-structure Smooth scheme Norm residue isomorphism theorem Proper scheme Values of L-functions Height pairing Hypercohomology Singular cohomology Birch–Swinnerton-Dyer conjecture Ext group dbr:Arxiv:1609.05516
Link from a Wikipage to an external page	http://www.arithgeo.ethz.ch/alpbach2011/Nori_TIFR%7Cvolume=%7Cpages=%7Carchive-url=https:/web.archive.org/web/20160922233016/http:/www.arithgeo.ethz.ch/alpbach2011/Nori_TIFR%7Carchive-date=22 https://web.archive.org/web/20190928160013/http:/www.icm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_20.pdf https://www.uni-due.de/~bm0032/publ/KthyMotI12.01.pdf http://math.rutgers.edu/~weibel/motiviclectures.html
sameAs	Motivic cohomology Motivic cohomology Motivic cohomology

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