The local symbol complex of a reciprocity functor.(English)Zbl 1353.19003

A theory of reciprocity functors and associated $$K$$-groups has recently been developed by F. Ivorra and K. Rülling [“K-groups of reciprocity functors”, Preprint, to appear in J. Alg. Geom., arxiv:1209.1217], building on ideas of B. Kahn and K. Somekawa. Let $$F$$ be a perfect field. Let $$\mathrm{Reg}^{\leq 1}\mathrm{Cor}$$ be the category whose objects are regular $$F$$-schemes of dimension at most $$1$$ which are separated and of finite type over a field $$k$$ which is finitely-generated over $$F$$. A reciprocity functor $$\mathcal{M}$$ is a presheaf of abelian groups on $$\mathrm{Reg}^{\leq 1}\mathrm{Cor}$$ satisfying certain additional properties which guarantee the existence of local symbols associated to points on smooth projective geometrically connected curves over $$k$$ yielding a generalized reciprocity law for the curve. Examples include smooth commutative algebraic groups over $$S=\mathrm{Spec} F$$, homotopy invariant Nisnevich sheaves with transfers and Rost’s cycle modules.
Given a collection of reciprocity functors $$\mathcal{M}_1,\ldots, \mathcal{M}_r$$, Ivorra and Rülling associate to it a kind of product $$T(\mathcal{M}_1,\ldots,\mathcal{M}_r)$$, also a reciprocity functor, which they call the $$K$$-group of $$\mathcal{M}_1,\ldots, \mathcal{M}_r$$. When $$k$$ is algebraically closed and $$C$$ is a smooth complete curve over $$k$$ with generic point $$\eta_C$$ a reciprocity functor $$\mathcal{M}$$, and the associated reciprocity law, gives rise to a complex of abelian groups: $\left(\mathcal{M}\otimes^{M} \mathbb{G}_m\right)(\eta_C)\to \oplus_{P\in C}\mathcal{M}(k)\to \mathcal{M}(k)$ where $$\otimes^{M}$$ is the tensor product of Mackey functors. The main theorem of the article under review (Theorem 3.11) is that the homology of this complex is naturally isomorphic to the $$K$$-group $$T(\mathcal{M}, \mathrm{CH}_0(C)^0)( k)$$ when this latter group satisfies a pair of technical conditions (3.3 and 3.10 in the article). The author shows that the conditions of Theorem 3.11 hold in the following cases: (i) $$\mathcal{M}=T(\mathcal{F}_1,\ldots,\mathcal{F}_r)$$ where $$\mathcal{F}_1,\ldots,\mathcal{F}_r$$ are homotopy invariant Nisnevich sheaves with transfers and (ii) $$\mathcal{M}= T(\mathbb{G}_a,\mathcal{M}_1,\ldots,\mathcal{M}_r)$$ where $$\mathcal{M}_1,\ldots,\mathcal{M}_r$$ are arbitrary reciprocity functors. In a final section the author treats some cases in which the main result extends to a non-algebraically closed field $$k$$.

MSC:

 19D45 Higher symbols, Milnor $$K$$-theory 14C25 Algebraic cycles
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