TAILIEUCHUNG - Đề tài "Cyclic homology, cdhcohomology and negative K-theory"

We prove a blow-up formula for cyclic homology which we use to show that infinitesimal K-theory satisfies cdh-descent. Combining that result with some computations of the cdh-cohomology of the sheaf of regular functions, we verify a conjecture of Weibel predicting the vanishing of algebraic K-theory of a scheme in degrees less than minus the dimension of the scheme, for schemes essentially of finite type over a field of characteristic zero. Introduction The negative algebraic K-theory of a singular variety is related to its geometry. . | Annals of Mathematics Cyclic homology cdh-cohomology and negative K-theory By G. Corti nas C. Haesemeyer M. Schlichting and C. Weibel Annals of Mathematics 167 2008 549 573 Cyclic homology cdh-cohomology and negative K-theory By G. Cortinas C. HAESEMEyER M. Schlichting and C. Weibel Abstract We prove a blow-up formula for cyclic homology which we use to show that infinitesimal K-theory satisfies cdh-descent. Combining that result with some computations of the cdh-cohomology of the sheaf of regular functions we verify a conjecture of Weibel predicting the vanishing of algebraic K-theory of a scheme in degrees less than minus the dimension of the scheme for schemes essentially of finite type over a field of characteristic zero. Introduction The negative algebraic K-theory of a singular variety is related to its geometry. This observation goes back to the classic study by Bass and Murthy 1 which implicitly calculated the negative K-theory of a curve X. By definition the group K-n X describes a subgroup of the Grothendieck group K0 Y of vector bundles on Y X X A1 0 ra. The following conjecture was made in 1980 based upon the Bass-Murthy calculations and appeared in 38 . Recall that if F is any contravariant functor on schemes a scheme X is called F-regular if F X F X X Ar is an isomorphism for all r 0. K-dimension Conjecture . Let X be a Noetherian scheme of dimension d. Then Km X 0 for m d and X is K-d-regular. In this paper we give a proof of this conjecture for X essentially of finite type over a field F of characteristic 0 see Theorem . We remark that this conjecture is still open in characteristic p 0 except for curves and surfaces Cortinas research was partially supported by the Ramon y Cajal fellowship by ANPCyT grant PICT 03-12330 and by MEC grant MTM00958. Haesemeyer s research was partially supported by the Bell Companies Fellowship and RTN Network HPRN-CT-2002-00287. Schlichting s research was partially supported by RTN Network HPRN-CT-2002-00287.

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