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On Suslin’s Singular Homology and Cohomology



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223 Documenta Math On Suslin s Singular Homology and Cohomology Dedicated to A A Suslin on his 60th birthday Thomas Geisser1 Received December 10 2009 Revised April 4 2010 Abstract We study properties of Suslin homology and cohomology over non algebraically closed base fields and their p part in characteristic p In the second half we focus on finite fields and consider finite generation questions and connections to tamely ramified class field theory 2010 Mathematics Subject Classification 19E15 14F42 14G15 Keywords and Phrases Suslin homology motivic homology algebraic cycles albanese map 1 Introduction Suslin and Voevodsky defined Suslin homology also called singular homology HiS X A of a scheme of finite type over a field k with coefficients in an abelian group A as Tori Cork X A Here Cork i X is the free abelian group generated by integral subschemes Z of i X which are finite and surjective over i and the differentials are given by alternating sums of pull back maps along face maps Suslin cohomology HSi X A is defined to be ExtiAb Cork X A Suslin and Voevodsky showed in 22 that over a separably closed field in which m is invertible one has i HSi X Z m X Z m Het 1 see 2 for the case of fields of characteristic p In the first half of this paper we study both the situation that m is a power of the characteristic of k and that k is not algebraically closed In the second 1 Supported in part by NSF grant No 0901021 Documenta Mathematica Extra Volume Suslin 2010 223 249 224 Thomas Geisser half we focus on finite base fields and discuss a modified version of Suslin homology which is closely related to etale cohomology on the one hand but is also expected to be finitely generated Moreover its zeroth homology is Z 0 X and its first homology is expected to be an integral model of the abelianized tame fundamental group We start by discussing the p part of Suslin homology over an algebraically closed field of characteristic p We show that assuming resolution of singularities



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