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**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 \(k\).

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 \(k\).

Reviewer: Kevin Hutchinson (Dublin)