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

Research paper by Vladimir Guletskiĭ

Indexed on: 04 Jan '16Published on: 04 Jan '16Published in: European Journal of Mathematics



Abstract

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 Open image in new window be the classifying motivic space of this monoid. In the paper we establish an isomorphism between the Chow group Open image in new window 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 Open image in new window at \({{\mathrm{Spec}}(k)}\). Equivalently, Open image in new window is isomorphic to the group of sections of the stabilized motivic fundamental group Open image in new window at \({{\mathrm{Spec}}(k)}\).