Holmstrom Suslin homology

Suslin homology

Following Levine, can define $H_n^{Sus}(X,A)$ for $X \in Sch_k$ and $A$ an abelian group.

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Suslin homology

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Suslin homology

AG (Algebraic geometry)

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Suslin homology

Mixed

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Suslin homology

Suslin: Exposé a la conference de K-théorie, Luminy 1987

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Suslin homology

Notes from André, section 15.4

Have algebraic standard simplex $\Delta^n$. These fit together to form a cosimplicial object (in $L(k)$, or in $cL(k)$). Details spelled out.

For any $X \in cL(k)$, the rule $n \mapsto c( \Delta^n, X)$ (finite correspondences) defines a simplicial abelian group, and the associated chain complex $C_*(X)(k)$ defines Suslin homology $H_i^S(X)$. Can think of this complex as zero-cycles on $X$ parameterised by the cosimplicial smooth variety $\Delta^{\bullet}$.

More generally, the rule $(Y \in L(k), n) \mapsto c( Y \times \Delta^n, X)$ defines a simplicial presheaf on $L(k)$ and we write $\underline{C}_*(X)$ for the associate chain complex of sheaves (concentrated in positive degrees). Can get a cochain complex by the usual definition $\underline{C}^{-n}(X) = \underline{C}_{n}(X)$

END of section. Ref to Suslin-Voevodsky.

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Suslin homology

André, section 18.5:

Theorem: For $X \in L(k)$, we have $H^S_i(X) = DM_{gm}(k)(\mathbb{Z}[i], M(X) )$.

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Suslin homology

See also Motivic cohomology

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Suslin homology

arXiv:0912.1168 Suslin’s singular homology and cohomology from arXiv Front: math.KT by Thomas H. Geisser We discuss Suslin’s singular homology and cohomology. In the first half we examine the p-part in characteristic p, and the situation over non-algebraically closed fields. In the second half we focus on finite base fields. We study finite generation properties, and give a modified definition which behaves like a homology theory: in degree zero it is a copy of Z for each connected component, in degree one it is related to the abelianized (tame) fundamental group, even for singular schemes, and it is expected to be finitely generated in general.

Schmidt and Spiess: “In this paper we show that the tamely ramified abelian coverings of smooth, quasiprojective varieties over finite fields can be described in terms of their 0th singular (Suslin) homology. This extends the unramified class field theory of Kato and Saito for smooth, projective varieties over finite fields to the quasiprojective case.”

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Suslin homology

Levine. In the printed version (K-theory handbook), it is page 446.

Suslin and Voevodsky: Singular homology of abstract algebraic varieties. They also have a few pages on the qfh and h topologies and their sheaves.

Geisser: Suslin’s singular homology and cohomology arXiv

nLab page on Suslin homology