$K$-motives of algebraic varieties

Garkusha, Grigory; Panin, Ivan

doi:10.4310/hha.2012.v14.n2.a13

Cited by 13 publications

(41 citation statements)

References 31 publications

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“…For example, Dundas-Röndigs-Østvaer [DRO1,DRO2] use enriched category theory to give a model for the Morel-Voevodsky category SH(k). In [GP1,GP2,GP3] enrichments of smooth algebraic varieties over symmetric spectra have been used in order to develop the theory of "K-motives" and solve a problem for the motivic spectral sequence.…”

Section: Introductionmentioning

confidence: 99%

Derived categories for Grothendieck categories of enriched functors

Garkusha¹,

Jones²

2019

Model Theory of Modules, Algebras And Categories

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The derived category D[C , V ] of the Grothendieck category of enriched functors [C , V ], where V is a closed symmetric monoidal Grothendieck category and C is a small Vcategory, is studied. We prove that if the derived category D(V ) of V is a compactly generated triangulated category with certain reasonable assumptions on compact generators or K-injective resolutions, then the derived category D[C , V ] is also compactly generated triangulated. Moreover, an explicit description of these generators is given.

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Section: Introductionmentioning

confidence: 99%

Derived categories for Grothendieck categories of enriched functors

Garkusha¹,

Jones²

2019

Model Theory of Modules, Algebras And Categories

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“…Another application is given in Theorem 3.8 for the bivariant motivic spectral sequence in the sense of [2].…”

Section: Introductionmentioning

confidence: 99%

“…It is Grayson's motivic specttral sequence [5] applied to bivariant algebaraic K-theory of smooth algebraic varieties (see [2,Section 7] for details and definitions). The spectral sequence is strongly convergent and the following relation is true (over perfect fields) by [2, 7.8]:…”

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confidence: 99%

Comparing motives of smooth algebraic varieties

Garkusha

2018

Comptes Rendus. Mathématique

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“…The Voevodsky triangulated category of motives DM eff − (k) [16] provides a natural framework to study motivic cohomology. In [2] the authors constructed a triangulated category of K-motives providing a natural framework for Grayson's motivic spectral sequence [5]…”

Section: Introductionmentioning

confidence: 99%

“…We work in the framework of strict V -spectral categories introduced in the paper (Definition 2.5). The main feature of such a spectral category O is that it is connective and Nisnevich excisive in the sense of [2], and π 0 O-(pre)sheaves, where π 0 O is a ringoid associated to O, share lots of common properties with (pre)sheaves with transfers (or Cor-(pre)sheaves) in the sense of Voevodsky [15].…”

Section: Introductionmentioning

confidence: 99%

The triangulated category of K-motives

Garkusha¹,

Panin²

2014

J K-Theor

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For any perfect field k a triangulated category of K-motives DK eff − (k) is constructed in the style of Voevodsky's construction of the category DM eff − (k). To each smooth k-variety X the K-motive M K (X) is associated in the category DK eff − (k) and K n (X) = Hom DK eff − (k) (M K (X) [n], M K (pt)), n ∈ Z, where pt = Spec(k) and K(X) is Quillen's K-theory of X.where pt = Spec(k) and K(X ) is Quillen's K-theory of X . Thus Quillen's K-theory is represented by the K-motive of the point.The spectral category K is of great utility in authors' paper [3], in which they solve some problems related to the motivic spectral sequence. In fact, the problems were the main motivation for constructing the spectral category K and developing the machinery of K-motives.Throughout the paper we denote by Sm/k the category of smooth separated schemes of finite type over the base field k. PRELIMINARIESWe work in the framework of spectral categories and modules over them in the sense of Schwede-Shipley [12]. We start with preparations.Recall that symmetric spectra have two sorts of homotopy groups which we shall refer to as naive and true homotopy groups respectively following terminology of [11]. Precisely, the kth naive homotopy group of a symmetric spectrum X is defined as the colimit π k (X ) = colim n π k+n X n .Denote by γX a stably fibrant model of X in Sp Σ . The k-th true homotopy group of X is given bythe naive homotopy groups of the symmetric spectrum γX . Naive and true homotopy groups of X can be considerably different in general (see, e.g., [6,11]). The true homotopy groups detect stable equivalences, and are thus more important than the naive homotopy groups. There is an important class of semistable symmetric spectra within whichπ *isomorphisms coincide with π * -isomorphisms. Recall that a symmetric spectrum is semistable if some (hence any) stably fibrant replacement is a π * -isomorphism. Suspension spectra, Eilenberg-Mac Lane spectra, Ω-spectra or Ω-spectra from some point X n on are examples of semistable symmetric spectra (see [11]). So Waldhausen's algebraic K-theory symmetric spectrum, which we shall use later, is semistable. Semistability is preserved under suspension, loop, wedges and shift.A symmetric spectrum X is n-connected if the true homotopy groups of X are trivial for k n. The spectrum X is connective if it is (−1)-connected, i.e., its true homotopy groups vanish in negative dimensions. X is bounded below if π i (X ) = 0 for i ≪ 0. Definition 2.1. (1) Following [12] a spectral category is a category O which is enriched over the category Sp Σ of symmetric spectra (with respect to smash product, i.e., the monoidal closed structure of [6, 2.2.10]). In other words, for every pair of objects o, o ′ ∈ O there is a morphism symmetric spectrum O(o, o ′ ), for every object o of O there is a map from the sphere spectrum S to O(o, o) (the "identity element" of o), and for each triple of objects there is an associative and unital composition map of symmetric spectra O(o ′ , o ′′ ) ∧ O(o, o ′ ) → O(o, o ′′ )....

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$K$-motives of algebraic varieties

Cited by 13 publications

References 31 publications

Derived categories for Grothendieck categories of enriched functors

Derived categories for Grothendieck categories of enriched functors

Comparing motives of smooth algebraic varieties

The triangulated category of K-motives

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