## Voevodsky’s motives and Weil reciprocity.(English)Zbl 1309.19009

Duke Math. J. 162, No. 14, 2751-2796 (2013); corrigendum ibid. 164, No. 10, 2093-2098 (2015).
M. Somekawa [K-Theory 4, No. 2, 105–119 (1990; Zbl 0721.14003)] defined $$K$$-groups $$K(k;G_1,\dots,G_n)$$, where $$G_1,\dots, G_n$$ are semi-abelian varieties over the field $$k$$, and stated that, in the conjectured abelian category $${\mathcal M}{\mathcal M}$$ of mixed motives over $$k$$, there should be an isomorphism of the form $K(k; G_1,\dots, G_n)\simeq \text{Ext}^n_{{\mathcal M}{\mathcal M}}(\mathbb{Z},G_1[-1]\otimes\cdots\otimes G_n[-1]),$ where $$G_1[-1],\dots, G_n[-1]$$ are the corresponding 1-motives.
In this paper the authors prove a “real” isomorphism from a group, which is a generalization of $$K(k;G_1,\dots, G_n)$$, and an Hom in the existing category $${\mathbf D}{\mathbf M}^{\text{eff}}_-$$, defined by Voevodsky, of effective motivic complexes. More precisely $k(k,{\mathcal F}_1,\dots,{\mathcal F}_n)\simeq\text{Hom}_{{\mathbf D}{\mathbf M}^{\text{eff}_-}}(\mathbb{Z},{\mathcal F}_1[0],\dots,{\mathcal F}-n[0]),$ where $${\mathcal F}_1,\dots,{\mathcal F}_n$$ are homotopy invariant sheaves with transfers. In the case $${\mathcal F}_1={\mathcal F}_2=\cdots={\mathcal F}_n= {\mathbf G}_m$$ the left-hand side is isomorphic to the usual Milnor $$K$$-group $$K^M_n(k)$$, while the right-hand side is motivic cohomology. Therefore, for a perfect field $$k$$, the above formula yields the Suslin-Voevodsky isomorphism for motivic cohomology $$K^M_n(k)\simeq H^n_M(k,\mathbb{Z}(n))$$.
The above result has the following application to algebraic cycles
Theorem 1. Let $$X_1,\dots, X_n$$ be quasi-projective schemes over a field $$k$$ of characteristic 0 and let $$X= X_1\times X_2\times\cdots\times X_n$$. For every $$r\geq 0$$ there is an isomorphism $K(k;\underline{\mathrm{CH}}_0(X_1,\dots, \underline{\mathrm{CH}}_0(X_n),\,{\mathbf G}_M,\dots,{\mathbf G}_M)\simeq \mathrm{CH}_{-r}(X,r)),$ where $$\underline{\mathrm{CH}}_0(X_i)$$, for $$1\leq i\leq n$$, are the homotopy invariant Nisnevich sheaves with transvers $$U\to \mathrm{CH}_0(X_i\times_k k(U))$$ and $$\mathrm{CH}_i(X,j)$$, for $$i,j\in\mathbb{Z}$$, are Bloch’s higher Chow groups.
Special cases of Theorem 1 were previously known, when the $$X_i$$’s are smooth projective and $$r= 0$$ or $$n=1$$.

### MSC:

 19E15 Algebraic cycles and motivic cohomology ($$K$$-theoretic aspects) 14F42 Motivic cohomology; motivic homotopy theory 19D45 Higher symbols, Milnor $$K$$-theory 19F15 Symbols and arithmetic ($$K$$-theoretic aspects)

Zbl 0721.14003
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### References:

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