{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,3]],"date-time":"2026-04-03T09:31:08Z","timestamp":1775208668430,"version":"3.50.1"},"reference-count":25,"publisher":"Cambridge University Press (CUP)","issue":"6","license":[{"start":{"date-parts":[[2012,10,12]],"date-time":"2012-10-12T00:00:00Z","timestamp":1350000000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Compositio Math."],"published-print":{"date-parts":[[2012,11]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    In this article we prove that Kontsevich\u2019s category\n                    <jats:italic>NC<\/jats:italic>\n                    <jats:sub>num<\/jats:sub>\n                    (\n                    <jats:italic>k<\/jats:italic>\n                    )\n                    <jats:sub>\n                      <jats:italic>F<\/jats:italic>\n                    <\/jats:sub>\n                    of noncommutative numerical motives is equivalent to the one constructed by the authors in [Marcolli and Tabuada,\n                    <jats:italic>Noncommutative motives, numerical equivalence, and semisimplicity<\/jats:italic>\n                    , Amer. J. Math., to appear, available at arXiv:1105.2950]. As a consequence, we conclude that\n                    <jats:italic>NC<\/jats:italic>\n                    <jats:sub>num<\/jats:sub>\n                    (\n                    <jats:italic>k<\/jats:italic>\n                    )\n                    <jats:sub>\n                      <jats:italic>F<\/jats:italic>\n                    <\/jats:sub>\n                    is abelian semi-simple as conjectured by Kontsevich.\n                  <\/jats:p>","DOI":"10.1112\/s0010437x12000383","type":"journal-article","created":{"date-parts":[[2012,10,12]],"date-time":"2012-10-12T04:42:29Z","timestamp":1350016949000},"page":"1811-1820","source":"Crossref","is-referenced-by-count":10,"title":["Kontsevich\u2019s noncommutative numerical motives"],"prefix":"10.1017","volume":"148","author":[{"given":"Matilde","family":"Marcolli","sequence":"first","affiliation":[]},{"given":"Gon\u00e7alo","family":"Tabuada","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2012,10,12]]},"reference":[{"key":"S0010437X12000383_ref25","unstructured":"[Tab11] Tabuada G. , Chow motives versus noncommutative motives, J. Noncommut. Geom., to appear, arXiv:1103.0200."},{"key":"S0010437X12000383_ref23","first-page":"3309","article-title":"Invariants additifs de dg-cat\u00e9gories","volume":"53","author":"Tabuada","year":"2005","journal-title":"Int.\u00a0Math.\u00a0Res.\u00a0Not."},{"key":"S0010437X12000383_ref18","unstructured":"[MT11a] Marcolli M. and Tabuada G. , Noncommutative motives, numerical equivalence, and semi-simplicity, Amer. J. Math., to appear, available at arXiv:1105.2950."},{"key":"S0010437X12000383_ref16","unstructured":"[Kon10] Kontsevich M. , Mixed noncommutative motives, Talk at the FRG Workshop on Homological Mirror Symmetry, University of Miami, 2010, notes available at www-math.mit.edu\/auroux\/frg\/miami10-notes."},{"key":"S0010437X12000383_ref15","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-8176-4747-6_7"},{"key":"S0010437X12000383_ref6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.17323\/1609-4514-2003-3-1-1-36","article-title":"Generators and representability of functors in commutative and noncommutative geometry","volume":"3","author":"Bondal","year":"2003","journal-title":"Mosc. Math. J."},{"key":"S0010437X12000383_ref24","unstructured":"[Tab10] Tabuada G. , A guided tour through the garden of noncommutative motives, Extended notes of a survey talk on noncommutative motives given at the 3era Escuela de Inverno Luis Santal\u00f3-CIMPA: Topics in Noncommutative Geometry, Buenos Aires, July 26 to August 6, 2010, Clay Mathematics Proceedings, vol. 16, to appear, available at arXiv:1108.3787."},{"key":"S0010437X12000383_ref2","first-page":"125","article-title":"Erratum: Nilpotence, radicaux et structures mono\u00efdales","volume":"108","author":"Andr\u00e9","year":"2002","journal-title":"Rend.\u00a0Semin.\u00a0Mat. Univ.\u00a0Padova"},{"key":"S0010437X12000383_ref7","doi-asserted-by":"publisher","DOI":"10.1017\/is011011005jkt169"},{"key":"S0010437X12000383_ref9","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2003.05.001"},{"key":"S0010437X12000383_ref20","unstructured":"[MT11c] Marcolli M. and Tabuada G. , Unconditional motivic Galois groups and Voevodsky\u2019s nilpotence conjecture in the noncommutative world, Preprint (2011), arXiv:1112.5422."},{"key":"S0010437X12000383_ref12","first-page":"151","volume-title":"International Congress of Mathematicians 2006 (Madrid), vol.\u00a0II","author":"Keller","year":"2006"},{"key":"S0010437X12000383_ref22","unstructured":"[Shk07] Shklyarov D. , On Serre duality for compact homologically smooth DG algebras, Preprint (2007), arXiv:math\/0702590."},{"key":"S0010437X12000383_ref1","first-page":"107","article-title":"Nilpotence, radicaux et structures mono\u00efdales","volume":"108","author":"Andr\u00e9","year":"2002","journal-title":"Rend.\u00a0Semin.\u00a0Mat. Univ.\u00a0Padova"},{"key":"S0010437X12000383_ref11","first-page":"461","volume-title":"Proceedings of the International Congress of Mathematicians 2010 (Hyderabad, India), vol. II","author":"Kaledin","year":"2010"},{"key":"S0010437X12000383_ref8","unstructured":"[Dri02] Drinfeld V. , DG categories, Talk in the University of Chicago geometric Langlands seminar, 2002, notes available at www.math.utexas.edu\/users\/benzvi\/GRASP\/lectures\/Langlands.html."},{"key":"S0010437X12000383_ref17","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-10-00664-8"},{"key":"S0010437X12000383_ref3","doi-asserted-by":"publisher","DOI":"10.1007\/BF01681436"},{"key":"S0010437X12000383_ref14","unstructured":"[Kon05] Kontsevich M. , Noncommutative motives, Talk at the Institute for Advanced Study on the occasion of the 61st birthday of Pierre Deligne, October 2005, video available at http:\/\/video.ias.edu\/Geometry-and-Arithmetic."},{"key":"S0010437X12000383_ref21","doi-asserted-by":"publisher","DOI":"10.1515\/9781400837212"},{"key":"S0010437X12000383_ref4","first-page":"1183","article-title":"Representable functors, Serre functors, and mutations","volume":"53","author":"Bondal","year":"1989","journal-title":"Izv. Akad. Nauk SSSR Ser. Mat."},{"key":"S0010437X12000383_ref5","first-page":"669","article-title":"Framed triangulated categories","volume":"181","author":"Bondal","year":"1990","journal-title":"Mat. Sb."},{"key":"S0010437X12000383_ref19","unstructured":"[MT11b] Marcolli M. and Tabuada G. , Noncommutative numerical motives, Tannakian structures, and motivic Galois groups, Preprint (2011), arXiv:1110.2438."},{"key":"S0010437X12000383_ref10","volume-title":"Model categories","author":"Hovey","year":"1999"},{"key":"S0010437X12000383_ref13","unstructured":"[Kon98] Kontsevich M. , Triangulated categories and geometry, Course at the \u00c9cole Normale Sup\u00e9rieure, Paris, 1998, notes available at www.math.uchicago.edu\/mitya\/langlands.html."}],"container-title":["Compositio Mathematica"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0010437X12000383","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,3]],"date-time":"2026-04-03T08:35:39Z","timestamp":1775205339000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0010437X12000383\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,10,12]]},"references-count":25,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2012,11]]}},"alternative-id":["S0010437X12000383"],"URL":"https:\/\/doi.org\/10.1112\/s0010437x12000383","relation":{},"ISSN":["0010-437X","1570-5846"],"issn-type":[{"value":"0010-437X","type":"print"},{"value":"1570-5846","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,10,12]]}}}