×

Algebraic \(K\)-theory, algebraic cycles and arithmetic geometry. (English) Zbl 1115.19003

Friedlander, Eric M. (ed.) et al., Handbook of \(K\)-theory. Vol. 1 and 2. Berlin: Springer (ISBN 3-540-23019-X/hbk). 351-428 (2005).
This article is part of a collection of papers by the name “Handbook of \(K\)-Theory”. As such it contains a survey of the relations between algebraic \(K\)-theory and the theory of algebraic cycles with applications to arithmetical geometry. Recent results by Suslin, Voevodski, Beilinson, Levine, Friedlander, etc. have greatly contributed to clearify the situation: for smooth varieties there is an isomorphism between Bloch higher Chow groups and motivic cohomology as well as an Atiyah-Hirzebruch type spectral sequence \[ E^{p,q}_2= H^p(X, \mathbb{Z}(- q/2))\Rightarrow K_{-p-q}(X), \] where \(H^*\) are the motivic cohomology groups. Furthermore \[ H^{2j-i}(X, \mathbb{Z}(i))\simeq K^{(j)}_i(X), \] where \(K^{(j)}_i(X)\) are the eigenspaces of the Adams operations.
This article contains a description of the above results as well as a quite precise discussion about the following conjecture and its consequence in the theory of motives (Beilinson-Soule conjecture):
For \(X\) a regular (noetherian, separated) scheme of finite type over a field \(k\), \(H^i(X, \mathbb{Z}(n))= 0\) for \(n\geq 0\) and \(i< 0\).
The article also contains a chapter on the theory of motives, and a very accurate description of the applications to arithmetical geometry. The author discusses Lichtenbaum’s and Soule’s conjectures on the orders of poles and special values of the zeta function \[ \zeta(X, s)=\coprod(1- N(x)^{-s})^{-1}, \] where: \(X\) is a scheme of finite type over \(\mathbb{Z}\) (or over a ring of algebraic integers), \(x\in X_{(0)}\), the set of closed points of \(X\) and \(n(x)= |k(x)|\), the cardinality of the residue field at \(x\).
In the case \(X\) is a smooth projective variety over a finite field \(k\), Tate has formulated his famous conjecture: \[ \text{ord}_{s= n}\zeta(X, s)=- \dim_{\mathbb{Q}}A^n_{\text{num}}(X, \mathbb{Q}), \] where \(n\geq 0\), \(A^n_{\text{num}}(X, \mathbb{Q})\) is the group (with \(\mathbb{Q}\) coefficients) of codimension \(n\) cycles on \(X\) which are numerically equivalent to \(0\). Also according to Beilinson, this group should coincide with \(A^n_{\text{rat}}(X,\mathbb{Q})\), where \(A^*_{\text{rat}}\) is the subgroup of cycles rationally equivalent to \(0\). A chapter of the article is dedicated to a motivic reformulation of the above conjectures by Tate and Beilinson. Finally the article contains the proof of some unconditional results.
In particular: if \(X\) is a smooth projective variety over a finite field \(k\) such that its motive \(h(X)\) belongs to \({\mathcal A}_{\text{ab}}\) and \(\dim X\leq 3\) then the Tate-Beilinson conjecture holds for \(X\). Here \({\mathcal A}_{\text{ab}}\) denotes the thick subcategory of the category of Chow motives (with \(\mathbb{Q}\) coefficients) over \(k\), generated by Artin motives and motives of abelian varieties.
For the entire collection see [Zbl 1070.19002].

MSC:

19E15 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)
14C25 Algebraic cycles
PDFBibTeX XMLCite