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On Suslin homology with integral coefficients in characteristic zero (with an appendix by Bruno Kahn). (English) Zbl 1477.14037

Let \(k\) be a field, and let \(\Delta^i\) be the \(i\)-th algebraic simplex over \(k\). Given a separated noetherian \(k\)-scheme \(X\), the Suslin homology groups of \(X\) (Definition 3.1) is defined to be the homotopy groups \[H^S_i(X)=\pi_i(C_\bullet(X))\] of the simplicial abelian group \(C_\bullet(X)\), where \(C_i(X)\) is the free abelian group generated by closed integral subschemes of \(\Delta^i\times X\) which are finite surjective over \(\Delta^i\). The main result of this paper (Theorem 4.7) is that when \(k\) is an algebraically closed field of characteristic \(0\) and \(X\) is separated of finite type over \(k\), the Suslin homology groups of \(X\) are of the following form, which the author calls mild (Definition 2.3): \[H^S_i(X)=V_i\oplus (\mathbb Q/\mathbb Z)^{s_i} \oplus Z_i\oplus T_i.\] Here
– \(V_i\) is a uniquely divisible group,
– \(s_i\) is a non-negative integer,
– \(Z_i\) is a finitely generated free abelian group, and
– \(T_i\) is a finite abelian group.
Descriptions about \(s_i\) and the rank of \(Z_i\), and the torsion part of \(H_i^S(X)\) are also given in terms of the \(\ell\)-adic étale cohomology.
The proof has two main ingredients. The first is a corollary of the classical Dold-Thom theorem due to Suslin-Voevodsky (of which the author includes a proof in the paper under review, Corollary 3.14), and it is used to show the result over \(k=\mathbb C\) (Theorem 3.17). The second is a variant of Jannsen’s rigidity theorem (Theorem 4.2), and it is used to extend the result to any algebraically closed field of characteristic \(0\) (Theorem 4.7).
Morphisms between Suslin homology groups are also studied in parallel (Theorem 4.8). At last, the author discusses conjectured behaviors of the pushforward map between Suslin homology groups induced by an equidimensional relative cycle of dimension \(0\) (Conjecture 5.1), as well as the behavior of Suslin homology groups in positive characteristics (Conjecture 5.2).
In the appendix, B. Khan gives a conceptual proof of the main theorem of the paper by considering Voevodsky’s big category of étale motives and the Betti realization functor building upon the work of Ayoub and Cicinski-Deglise. This in particular gives an affirmative answer to Conjecture 5.1. It also includes a short discussion in the case when \(k\) is of positive characteristic.

MSC:

14F42 Motivic cohomology; motivic homotopy theory
19E15 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)
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References:

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