Additive higher Chow groups of schemes

Krishna, Amalendu; Levine, Marc

doi:10.1515/crelle.2008.041

Cited by 42 publications

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“…As seen in [16], the additive higher Chow groups are expected to give rise to Atiyah-Hirzebruch spectral sequence…”

Section: ) Suppose K Is Algebraically Closed Of Characteristic Zeromentioning

confidence: 94%

“…For an X ∈ Sch/k, let TCH r log (X, n; m) denote the log additive higher Chow groups as in [16]. 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.…”

Section: How To Cite This Article: Amalendu Krishna and Jinhyun Park mentioning

confidence: 99%

“…If only one of Z 1 and Z 2 is an additive admissible cycle, then Z 1 Z 2 is an additive admissible cycle by [16,Lemma 4.2]. By convention, the product of two additive admissible cycles is zero.…”

Section: External Productmentioning

confidence: 99%

“…The extension groups in this category are given by Bloch's higher Chow groups and the additive higher Chow groups. For an X ∈ Sch/k, let TCH r log (X, n; m) denote the log additive higher Chow groups as in [16]. 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]).…”

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

Krishna

Park

2012

J. Inst. Math. Jussieu

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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...

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“…As seen in [16], the additive higher Chow groups are expected to give rise to Atiyah-Hirzebruch spectral sequence…”

Section: ) Suppose K Is Algebraically Closed Of Characteristic Zeromentioning

confidence: 94%

Section: How To Cite This Article: Amalendu Krishna and Jinhyun Park mentioning

confidence: 99%

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confidence: 99%

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

Krishna

Park

2012

J. Inst. Math. Jussieu

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“…For infinitesimal motivic cohomology in the context of additive Chow groups we refer the reader to [5,18,16,19]. …”

Section: Outlinementioning

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Additive polylogarithms and their functional equations

Ünver

2010

Math. Ann.

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Let k[ε] 2 := k[ε]/(ε 2 ). The single valued real analytic n-polylogarithm L n : C → R is fundamental in the study of weight n motivic cohomology over a field k, of characteristic 0. In this paper, we extend the construction in Ünver (Algebra Number Theory 3:1-34, 2009) to define additive n-polylogarithms li n :k[ε] 2 → k and prove that they satisfy functional equations analogous to those of L n . Under a mild hypothesis, we show that these functions descend to an analog of the nth Bloch group B n (k[ε] 2 ) defined by Goncharov (Adv Math 114:197-318, 1995). We hope that these functions will be useful in the study of weight n motivic cohomology over k[ε] 2 .

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Pure weight perfect Modules on divisorial schemes

Hiranouchi¹,

Mochizuki²

2010

Deformation Spaces

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In this paper, we introduce a notion of weight r pseudo-coherent Modules associated to a regular closed immersion i : Y ֒→ X of codimension r, and prove that there is a canonical derived Morita equivalence between the DG-category of perfect complexes on a divisorial scheme X whose cohomological support are in Y and the DG-category of bounded complexes of weight r pseudo-coherent O X -Modules supported on Y . The theorem implies that there is the canonical isomorphism between the Bass-Thomason-Trobaugh nonconnected K-theory [TT90], [Sch06] (resp. the Keller-Weibel cyclic homology [Kel98], [Wei96]) for the immersion and the Schlichting nonconnected K-theory [Sch04] associated to (resp. that of) the exact category of weight r pseudo-coherent Modules. For the connected K-theory case, this result is just Exercise 5.7 in [TT90]. As its application, we will decide on a generator of the topological filtration on the non-connected K-theory (resp. cyclic homology theory) for affine Cohen-Macaulay schemes. * Supported by the JSPS Fellowships for Young Scientists. † This research is supported by JSPS core-to-core program 18005 pseudo-coherent O X -Modules K S (Wt(X on Y )) (resp. HC(Wt(X on Y ))). That is, we have isomorphismsfor each q ∈ Z. For the connected K-theory this result is nothing other than Exercise 5.7 in [TT90]. For Grothendieck groups (q = 0), there is a detailed proof if X is the spectrum of a Cohen-Macaulay local ring and Y is the closed point of X ([RS03], Prop. 2). For K-theory, as mentioned in Exercise 5.7, this problem is related with the works [Ger74], [Gra76] and [Lev88]. Namely the problem about describing the homotopy fiber of K B (X) → K B (X Y ) (or rather than K Q (X) → K Q (X Y )) by using the K-theory of a certain exact category. As described in [Ger74], there is an example due to Deligne which suggests difficulty of the problem for a general closed immersion. Conversely, the example indicate that for an appropriate scheme X, there is a good class of pseudo-coherent O X -Modules. That is, Modules of pure weight. This concept is intimately related to Weibel's K-dimensional conjecture [Wei80] (see Conj. 6.4), Gersten's conjecture [Ger73] and its consequences. These subjects will be treated in [HM08], [Moc08]. Notice that there are different notions of pure weight by Grayson [Gra95] and Walker [Wal00] and these two notions are compatible in a particular situation [Wal96]. In a future work, the authors hope to compare the Walker weight with the Thomason-Trobaugh one by utilizing the (equidimensional ) bivariant algebraic K-theory [GW00]. Now we explain the structure of the paper. In §2, we describe to our motivational picture. After reviewing the fundamental facts in §3, we will define the notion of weight and state the main theorem in §4. The proof of the main theorem will be given in §5. Finally we will give applications of the main theorem in §6.Convention. Throughout this paper, we use the letter X to denote a scheme. A complex means a chain complex whose boundary morphism is increase level of te...

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Additive higher Chow groups of schemes

Cited by 42 publications

References 30 publications

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

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

Additive polylogarithms and their functional equations

Pure weight perfect Modules on divisorial schemes

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Additive higher Chow groups of schemes

Cited by 42 publications

References 30 publications

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

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

Additive polylogarithms and their functional equations

Pure weight perfect Modules on divisorial schemes

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

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