\(\mathbb {A}^{1}\) -connectivity on Chow monoids versus rational equivalence of algebraic cycles

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• 作者：Vladimir Guletskiĭ
• 关键词：Algebraic cycles ; Chow monoids ; Group completion ; Nisnevich sheaves ; (Motivic) homotopy category ; Bousfield localization ; Loop functor ; Classifying spaces ; $${\mathbb {A}^{1}}$$ A 1 ; homotopy groups ; 14C25 ; 14F42 ; 18G55
• 刊名：European Journal of Mathematics
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：2
• 期：1
• 页码：169-195
• 全文大小：576 KB
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• 作者单位：Vladimir Guletskiĭ (1)
  
  1. Department of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool, L69 7ZL, England, UK
  
• 刊物类别：Algebraic Geometry;
• 刊物主题：Algebraic Geometry;
• 出版者：Springer International Publishing
• ISSN：2199-6768

文摘

Let k be a field of characteristic zero, and let X be a projective variety embedded into a projective space over k. For two natural numbers r and d let \(C_{r,d}(X)\) be the Chow scheme parametrizing effective cycles of dimension r and degree d on the variety X. Choosing an r-cycle of minimal degree gives rise to a chain of embeddings of Chow schemes, whose colimit is the connective Chow monoid \({C}_r^{\infty }(X)\) of r-cycles on X. Let be the classifying motivic space of this monoid. In the paper we establish an isomorphism between the Chow group of degree 0 dimension r algebraic cycles modulo rational equivalence on X, and the group of sections of the Nisnevich sheaf of \({\mathbb {A}^{1}}\)-path connected components of the loop space of at \({{\mathrm{Spec}}(k)}\). Equivalently, is isomorphic to the group of sections of the stabilized motivic fundamental group at \({{\mathrm{Spec}}(k)}\). Keywords Algebraic cycles Chow monoids Group completion Nisnevich sheaves (Motivic) homotopy category Bousfield localization Loop functor Classifying spaces \({\mathbb {A}^{1}}\)-homotopy groups