Moving lemma for additive higher Chow groups
Let \(X\) be a smooth projective variety over a field \(k\) and let \(TH^q(X;n;m)\) be the additive higher Chow groups, as defined in [\textit{A. Krishna} and \textit{M. Levine}, J. Reine Angew. Math. 619, 75--140 (2008; Zbl 1158.14009)]. The main goal of this paper is to prove a moving lemma for these additive higher Chow groups. Additive higher Chow groups are expected to complement higher Chow groups for non reduced schemes, so as to obtain the right motivic cohomology groups. In particular, for a smooth projective variety \(X\) one expects an Atiyah-Hirzebruch spectral sequence NEWLINE\[NEWLINETH^{-q}(X, -p-q; m)\Rightarrow K^{\mathrm{nil}}_{-p-q}(X; m)NEWLINE\]NEWLINE where \(K^{\mathrm{nil}}_{-p-q}(X; m)\) is the homotopy fiber of the restriction map NEWLINE\[NEWLINEK(X\times\text{Spec}(k(t))\to K(X\times\text{Spec\,} k[t]/(t^{m+1})).NEWLINE\]NEWLINE The main result of this paper is the following moving lemma, which is the additive analogue of the moving lemma for higher Chow groups, as proved by Bloch and Levine.NEWLINENEWLINE Theorem 1. For a smooth projective variety \(X\) and a finite collection \({\mathcal W}\) of its locally closed algebraic sets, every additive higher Chow cycle is congruent to an admissible cycle intersecting properly all members of \({\mathcal W}\) times faces. In other words, the inclusion of complexes NEWLINE\[NEWLINETZ^q_{{\mathcal W}}(X,-; m)\to TZ^q(X,-; m)NEWLINE\]NEWLINE is a quasi-isomorphism.NEWLINENEWLINE As an application of their moving lemma the authors prove the following contravariant property for additive higher Chow groups.NEWLINENEWLINE Theorem 2. For a morphism \(f: X\to Y\) of quasi-projective varieties over a field \(k\), where \(Y\) is smooth and projective, there is a pull-back map NEWLINE\[NEWLINEf^*: TH^q(Y, n;m)\to T^q(X, n;m)NEWLINE\]NEWLINE satisfying the expected composition law.NEWLINENEWLINE The moving lemma can be applied to construct a triangulated category \({\mathcal D}{\mathcal M}(k;m)\) of mixed motives over \(k[t]/(t^{m+1})\), which extends the category constructed by \textit{M. Hanamura} [Invent. Math. 158, No. 1, 105--179 (2004; Zbl 1068.14022)].
scientific article; zbMATH DE number 6064711