Milnor \(K\)-theory of smooth varieties
type: Item
instance of: scholarly article
title: Milnor \(K\)-theory of smooth varieties
zbMATH Open document ID: 1083.19001
author: Reza Akhtar
published in: \(K\)-Theory
publication date: 2005-04-07T00:00:00Z
review text: 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…
reviewed by: Claudio Pedrini
Mathematics Subject Classification ID: 19D45
Mathematics Subject Classification ID: 14C15
Mathematics Subject Classification ID: 14C25
zbMATH DE Number: 2153594
zbMATH Keywords: Milnor K-group
zbMATH Keywords: higher Chow group
zbMATH Keywords: mixed K-group
zbMATH Keywords: zero-dimensional cycles
MaRDI profile type: Publication
full work available at URL: s10977-004-9124-y
OpenAlex ID: W2020754674
cites work: Algebraic cycles and higher K-theory
cites work: Q3328651
cites work: The \(K\)-theory of fields in characteristic \(p\)
cites work: Riemann-Roch theorems for higher algebraic K-theory
cites work: Algebraic \(K\)-theory and quadratic forms. With an appendix by J. Tate
cites work: Milnor \(K\)-theory of rings, higher Chow groups and applications
cites work: Higher algebraic K-theory: I
cites work: Q4949646
cites work: On Milnor K-groups attached to semi-Abelian varieties
cites work: Milnor \(K\)-theory is the simplest part of algebraic \(K\)-theory
DOI: 10.1007/S10977-004-9124-Y
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description: scientific article; zbMATH DE number 2153594