2014/11/18 by Vladimir Guletskiĭ, Guletskii, Vladimir
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Polynomial and algebraic computation #Commutative Algebra and Its Applications
paper · pdf · doi:10.48550/arxiv.1411.4896
Let k be a field of characteristic zero, and let X be a projective\nvariety embedded into a projective space over k. For two natural numbers r\nand d let Cr,d(X) be the Chow scheme parametrizing effective cycles of\ndimension r and degree d on the variety X. An effective r-cycle of\nminimal degree on X gives rise to a chain of embeddings of Cr,d(X) into\nCr,d+1(X), whose colimit is the connective Chow monoid Cr\∞ (X)\nof r-cycles on X. Let BCr\∞ (X) be the motivic classifying space\nof this monoid. In the paper we establish an isomorphism between the Chow group\nCHr(X)0 of degree 0 dimension r algebraic cycles modulo rational\nequivalence on X, and the group of sections of the sheaf of mathbb\nA1-path connected components of the loop space of BCr\∞ (X) at\nSpec(k). Equivalently, CHr(X)0 is isomorphic to the group of sections of\nthe S1 wedge mathbb A1-fundamental group \Π 1S1 wedge mathbb\nA1(BCr\∞ (X)) at Spec(k).\n