Documenta Math. 223On Suslin’s Singular Homology and CohomologyDedicated to A. A. Suslin on his 60th birthdayThomas Geisser1Received: December 10, 2009Revised: April 4, 2010Abstract. We study properties of Suslin homology and cohomologyover non-algebraically closed base fields, and their p-part in charac-teristic p. In the second half we focus on finite fields, and considerfinite generation questions and connections to tamely ramified classfield theory.2010 Mathematics Subject Classification: 19E15, 14F42, 14G15Keywords and Phrases: Suslin homology, motivic homology, algebraiccycles, albanese map1 IntroductionSuslin and Voevodsky defined Suslin homology (also called sing ular homol-ogy) HSi(X, A) of a scheme of finite type over a field k with coefficients inan abelian group A as Tori(Cork(∆∗, X), A). Here Cork(∆i, X) is the freeabelian group generated by integral subschemes Z of ∆i× X which are finiteand surjective over ∆i, and the differentials are given by alternating sums ofpull-back maps along face maps. Suslin cohomology HiS(X, A) is defined tobe ExtiAb(Cork(∆∗, X), A). Suslin and Voevodsky showed in [22] that over aseparably closed field in which m is invertible, one hasHiS(X, Z/m)∼=Hiet(X, Z/m) (1)(see [2] for the case of fields of characteris tic p).In the first ha lf of this paper, we study both the s ituation that m is a powerof the characteristic of k, and that k is no t algebraically close d. In the sec ond1Supported in part by NSF grant No.0901021Documenta Mathematica · Extra Volume Suslin (2010) 223–249224 Thomas Geisserhalf, we focus on finite base fields and discuss a modified version of Suslinhomology, which is closely related to etale cohomology on the one hand, but isalso expected to be finitely generated. Moreover, its zeroth homology is Zπ0(X),and its first homology is exp ected to be an integral model of the abelianizedtame fundamental group.We start by dis cussing the p-part of Suslin homology over an algebraica llyclosed field of characteristic p. We show that, as suming resolution of singular-ities, the groups HSi(X, Z/pr) are finite abelian groups, and vanish outside therange 0 ≤ i ≤ dim X. Thus Suslin cohomology with finite coefficients is etalecohomology away from the characteristic, but better behaved than etale coho-mology at the characteristic (for example, H1et(A1, Z/p) is not finite). Moreover,Suslin homolog y is a birational invariant in the following strong sense: If X hasa resolution of singularities p : X′→ X which is an isomorphism outside of theopen subset U, then HSi(U, Z/pr)∼=HSi(X, Z/pr). It was pointed out to us byN.Otsub o that this can be applied to generalize a theorem of Spiess-Szamuely[20] to include p-torsion:Theorem 1.1 Let X be a smooth, connected, quasi-projective variety over analgebraically closed field and assume resolut ion of singularities. Then the al-banese mapalbX: HS0(X, Z)0→ AlbX(k)from the degree-0-part of Suslin homology to the k-valued points of the Albanesevariety induces an isomorphism on torsion groups.Next we examine the situation over non-alg e braically closed fields. We redefineSuslin homology and cohomology by imposing Galois descent. Concretely, ifGkis the absolute Galois group of k, then we define Galois-Suslin homology tobeHGSi(X, A) = H−iRΓ(Gk, Cor¯k(∆∗¯k,¯X) × A),and Galois-Suslin cohomology to beHiGS(X, A) = ExtiGk(Cor¯k(∆∗¯k,¯X), A).Ideally one would like to define Galois-Suslin homology via Galois homology,but we are not awa re of such a theory. With ra tional coefficients, the newlydefined gro ups agree with the original groups. On the other hand, with finitecoefficients prime to the characteristic, the proof o f (1) in [22] carries overto show that HiGS(X, Z/m)∼=Hiet(X, Z/m). As a corollary, we obtain anisomorphism between HGS0(X, Z/m) and the ab elianized fundamental groupπab1(X)/m for any separated X of finite type over a finite field and m invertible.The second half of the paper focus es on the case of a finite base field. We workunder the assumption of resolution of singularities in order to see the picture ofthe properties which ca n expected. The critical re ader can view our statementsas theor ems for schemes of dimension at most three, and conjectures in gen-eral. A theorem of Jannsen-Saito [11] can be generalized to show that SuslinDocumenta Mathematica · Extra Volume Suslin (2010) 223–249On Suslin’s Singular Homology and Cohomology 225homology and cohomology with finite coefficients for any X over a finite field isfinite. Rationally, HS0(X, Q)∼=H0S(X, Q)∼=Qπ0(X). Most other properties ar eequivalent to the following Conjecture P0considered in [7]: For X smooth andproper over a finite field, CH0(X, i) is torsion for i 6= 0. This is a pa rticularcase of Parshin’s conjecture that Ki(X) is torsion for i 6= 0. Conjecture P0is equivalent to the vanishing of HSi(X, Q) for i 6= 0 and all smooth X. Forarbitrary X of dimension d, Conjecture P0implies the vanishing of HSi(X, Q)outside of the range 0 ≤ i ≤ d and its finite dimensionality in this range. Com-bining the torsion and rational case, we show that HSi(X, Z) and HiS(X, Z) arefinitely generated for all X if and only if Conjecture P0holds.Over a finite field and with integral coefficie nts, it is more natural to imposedescent by the Weil group G generated by the Frobenius endomorphism ϕinstead of the Galois group [14, 3, 4, 7]. We define arithmetic homologyHari(X, A) = TorGi(Cor¯k(∆∗¯k,¯X), A)and arithmetic cohomologyHiar(X, Z) = ExtiG(Cor¯k(∆∗¯k,¯X), Z).We show that Har0(X, Z)∼=H0ar(X, Z)∼=Zπ0(X)and that arithmetic homologyand cohomology lie in lo ng e xact sequences with Ga lois-Suslin homology andcohomology, respectively. They are finitely generated abelian groups if andonly if Conjecture P0holds.The difference between arithmetic and Suslin homology is measured by athird theory, which we call Kato-Suslin homology, and which is defined asHKSi(X, A) = Hi((Cor¯k(∆∗¯k,¯X) ⊗ A)G). By definition there is a long exactsequence· · · → HSi(X, A) → Hari+1(X, A) → HKSi+1(X, A) → HSi−1(X, A) → · · · .It follows that HKS0(X, Z) = Zπ0(X)for any X. As a generalizatio n of theintegral version [7] of Kato’s conjecture [12], we proposeConjecture 1.2 The groups HKSi(X, Z) vanish for all smooth X and i > 0.Equivalently, there are short exact sequences0 → HSi+1(¯X, Z)G→ HSi(X, Z) → HSi(¯X, Z)G→ 0for all i ≥ 0 and all smooth X. We