Milnor \(K\)-theory of smooth varieties
Let \(X\) be a smooth scheme of finite type over a field \(k\). Then one has an isomorphism between Bloch's higher Chow groups and motivic cohomology groups with coefficients in some abelian group \(A\): \[ H^{2q}_{{\mathcal M}}(X, A(q))\simeq \text{CH}^q(X,2q- p; A). \] In the case \(X= \text{Spec\,}F\), \(F\) a field and \(A= \mathbb{Z}\), then \(K^M_n(F)\simeq \text{CH}^n(F, n)\), where \(K^M_*\) is Milnor's K-theory. Generalizing results by \textit{M. Somekawa} [K-Theory 4, 105--119 (1990; Zbl 0721.14003)] and by \textit{W. Raskind} and \textit{M. Spiess} [Compos. Math. 121, 1--33 (2000; Zbl 0985.14003)], the author extends the setting of Milnor K-theory from the case of fields to that of a smooth projective variety in the following way: given a geometrically integral quasi-projective variety \(X\) which is smooth of dimension \(d\) over \(k\), one defines mixed K-groups \(K(k,{\mathcal C}H_0(X),\mathbb{G}_m)\) (where \(\mathbb{G}_m\) is the multiplicative group scheme). More generally: \(K_s(k,{\mathcal C}H_0(X_1),\dots,{\mathcal C}H_0(X_r)\); \(G_1,\dots, G_s)\) for smooth quasiprojective varieties \(X_1,\dots,X_r\) and semi-abelian varieties \(G_1,\dots,G_s\) over the field \(k\) is given in terms of the group \[ T= \bigoplus_{E/k\text{\,finite}} \text{CH}_0(X_1)_E\otimes \cdots\otimes \text{CH}_0(X_r)_E\otimes G_1(E)\otimes\cdots\otimes G_s(E) \] and a subgroup \(R\) of relations in \(T\) which are similar to the projection formula and the reciprocity law. Then one takes \(K_s(k,{\mathcal C}H_0(X),\mathbb{G}_m)\) to be the group \(K_s(k,{\mathcal C}H_0(X_1),\dots,{\mathcal C}H_0(X_r)\); \(G_1,\dots, G_s)\), in the case \(r= 1\) and \(G_1=\cdots = G_s=\mathbb{G}_m\), and \(K_s(k, \mathbb{G}_m)\) in the case \(r= 0\). The main result of the paper is the following theorem: Let \(X\) be a smooth quasiprojective variety of dimension \(d\) defined over a field \(k\) and let \(s\geq 0\) be an integer. Then there are isomorphism \[ \begin{aligned} H^d_{\text{Zar}}(X,{\mathcal K}^M_{d+s}) &\overset\simeq\rightarrow \text{CH}^{d+s}(X,s),\\ \text{CH}^{d+s}(X, s) &\overset\simeq\rightarrow K_s(k;{\mathcal C}H_0(X); \mathbb{G}_m), \end{aligned} \] where \({\mathcal K}^M_{d+s}\) is the sheafified version of Milnor K-groups.
scientific article; zbMATH DE number 2153594
scientific article