Mixed motives over k[t]/(t^m+1)

Abstract

Link to this article: http://journals.cambridge.org/abstract_S1474748011000181 How to cite this article: Amalendu Krishna and Jinhyun Park (2012). Mixed motives over k[t]/(t m+1 ).Abstract For a perfect field k, we use the techniques of Bondal-Kapranov and Hanamura to construct a tensor triangulated category of mixed motives over the truncated polynomial ring k[t]/(t m+1 ). The extension groups in this category are given by Bloch's higher Chow groups and the additive higher Chow groups. The main new ingredient is the moving lemma for additive higher Chow groups by the authors and its refinements.Possibly motivated by [8], Goncharov [11] used his idea of k-scissors congruence to define Euclidean scissors congruence groups to get a Lie coalgebra Q • (k ) in the category of Q -modules. He conjectured that if k is algebraically closed, the category of finite dimensional graded comodules over Q • (k ) should be equivalent to a subcategory of the conjectured category of mixed Tate motives over k (see [11, Conjecture 1.3]). However, a complete construction of even the category of these mixed Tate motives was not known so far.Our aim in this paper is to give a complete construction of a bigger category of mixed motives over any given truncated polynomial ring k m = k[t]/(t m+1 ) (note that k 0 = k). We expect that this category has all the expected properties. In particular, the Ext groups should give the K-theory of the infinitesimal thickenings of smooth projective varieties. Although we are unable to prove this last property, there are strong indications that this should indeed be true as we shall see shortly. Some more results in this direction will appear in [18].Before we describe the main result, we fix some terminologies. Let SmProj/k denote the category of smooth projective varieties over k. The category of all quasi-projective schemes over k will be denoted by Sch/k. The subcategory with only proper morphisms will be denoted by Sch /k. Let DM(k) denote (the integral version of) Hanamura's triangulated category of mixed motives over k (in [14]). For an X ∈ Sch/k, let TCH r log (X, n; m) denote the log additive higher Chow groups as in [16]. Note that for X smooth and projective, these are just the additive higher Chow groups TCH r (X, n; m) in [16,22]. We refer to § 4 for the review of these groups. Let CH r (X, n) denote the higher Chow groups of X. Finally, recall from [24] that for a scheme X, the higher K i (X)-groups of perfect complexes on X have gamma filtrations which induce Adams operations on K i (X) Q . The rth eigenspace for this operation is denoted by K (r) i (X). Theorem 1.1. For m 0, there exists a tensor triangulated category DM(k; m) such that the following results hold.(1) There are natural functorsthat are faithful (but not full) and(2) There exists the motive functor with the modulus m augmentation,such thatwhere Z(r) and (−)(r) are Tate objects and Tate twists.We also expect Goncharov's category [11] to be a subcategory of MT M(k; m) defined in Definition 6.10.We now give a brief outline o...

Zero-cycles with modulus and relative $K$-theory

“…Theorem 1.3 (3) says that the known isomorphism between (relative) 0-cycles and Milnor K-theory (see [10] and [24] for regular semi-local rings and [43] and [52] for fields) also holds for truncated polynomial rings over regular semi-local rings. This provides a concrete evidence that the additive higher Chow groups should be the motivic cohomology groups if one could extend Voevodsky's theory of motives to so-called fat points (infinitesimal extensions of spectra of fields), see [33].…”

Zero-cycles with modulus and relative K-theory

“…Theorem 1.3 (4) says that this isomorphism also holds for truncated polynomial rings over such rings. This provides a concrete evidence that if one could extend Voevodsky's theory of motives to the theory of 'non-A 1 -invariant' motives over so-called fat points (infinitesimal extensions of spectra of fields), then the underlying motivic cohomology groups must be the additive higher Chow groups (see [35]).…”

De Rham–Witt sheaves via algebraic cycles

“…Let be the (finite) collection , where is an irreducible component of and is a face. Under the moving lemma [KP16, Theorem 4.10], by the argument of [KP17a, Lemmas 3.5 and 3.10], there exists a finite collection of locally closed subsets of such that for in the sense of [KP12b, Definition 5.3], we have: is well-defined; for all ; for all and all irreducible components of . …”