scholarly journals Tensor Structure on Smooth Motives

2011 ◽  
Vol 9 (1) ◽  
pp. 57-101 ◽  
Author(s):  
Anandam Banerjee
Keyword(s):  
Characteristic Zero ◽  
Full Subcategory ◽  
Resolution Of Singularities ◽  
Homotopy Category ◽  
Tensor Structure ◽  
Perfect Field ◽  
Base Scheme

AbstractRecently, Bondarko constructed a DG category of motives, whose homotopy category is equivalent to Voevodsky's category of effective geometric motives, assuming resolution of singularities. Soon after, Levine extended this idea to construct a DG category whose homotopy category is equivalent to the full subcategory of motives over a base-scheme S generated by the motives of smooth projective S-schemes, assuming that S is itself smooth over a perfect field. In both constructions, the tensor structure requires ℚ-coefficients. In this article, it is shown how to provide a tensor structure on the homotopy category mentioned above, when S is semi-local and essentially smooth over a field of characteristic zero. This is done by defining a pseudo-tensor structure on the DG category of motives constructed by Levine, which induces a tensor structure on its homotopy category.

scholarly journals On the Functoriality of the Slice Filtration

2013 ◽  
Vol 11 (1) ◽  
pp. 55-71 ◽  
Author(s):  
Pablo Pelaez
Keyword(s):  
Finite Type ◽  
Characteristic Zero ◽  
Resolution Of Singularities ◽  
Homotopy Category ◽  
Stable Homotopy ◽  
Canonical Structure ◽  
Motivic Cohomology ◽  
Ring Spectrum ◽  
Stable Homotopy Category ◽  
Slice Filtration

AbstractLet k be a field with resolution of singularities, and X a separated k-scheme of finite type with structure map g. We show that the slice filtration in the motivic stable homotopy category commutes with pullback along g. Restricting the field further to the case of characteristic zero, we are able to compute the slices of Weibel's homotopy invariant K-theory [24] extending the result of Levine [10], and also the zero slice of the sphere spectrum extending the result of Levine [10] and Voevodsky [23]. We also show that the zero slice of the sphere spectrum is a strict cofibrant ring spectrum HZXsf which is stable under pullback and that all the slices have a canonical structure of strict modules over HZXsf. If we consider rational coefficients and assume that X is geometrically unibranch then relying on the work of Cisinski and Déglise [4], we deduce that the zero slice of the sphere spectrum is given by Voevodsky's rational motivic cohomology spectrum HZX ⊗ ℚ and that the slices have transfers. This proves several conjectures of Voevodsky [22, conjectures 1, 7, 10, 11] in characteristic zero.

A STRENGTHENING OF RESOLUTION OF SINGULARITIES IN CHARACTERISTIC ZERO

2003 ◽  
Vol 86 (2) ◽  
pp. 327-357 ◽  
Author(s):  
A. BRAVO ◽  
O. VILLAMAYOR U.
Keyword(s):  
Characteristic Zero ◽  
Resolution Of Singularities ◽  
Closed Subscheme ◽  
Subject Classification ◽  
Invertible Sheaf ◽  
Simple Manner ◽  
Birational Morphism ◽  
Exceptional Locus

Let $X$ be a closed subscheme embedded in a scheme $W$, smooth over a field ${\bf k}$ of characteristic zero, and let ${\mathcal I} (X)$ be the sheaf of ideals defining $X$. Assume that the set of regular points of $X$ is dense in $X$. We prove that there exists a proper, birational morphism, $\pi : W_r \longrightarrow W$, obtained as a composition of monoidal transformations, so that if $X_r \subset W_r$ denotes the strict transform of $X \subset W$ then:(1) the morphism $\pi : W_r \longrightarrow W$ is an embedded desingularization of $X$ (as in Hironaka's Theorem);(2) the total transform of ${\mathcal I} (X)$ in ${\mathcal O}_{W_r}$ factors as a product of an invertible sheaf of ideals ${\mathcal L}$ supported on the exceptional locus, and the sheaf of ideals defining the strict transform of $X$ (that is, ${\mathcal I}(X){\mathcal O}_{W_r} = {\mathcal L} \cdot {\mathcal I}(X_r)$).Thus (2) asserts that we can obtain, in a simple manner, the equations defining the desingularization of $X$.2000 Mathematical Subject Classification: 14E15.

scholarly journals ℤ[1/p]-motivic resolution of singularities

2011 ◽  
Vol 147 (5) ◽  
pp. 1434-1446 ◽  
Author(s):  
M. V. Bondarko
Keyword(s):  
Preceding Paper ◽  
Resolution Of Singularities ◽  
Cohomology Theory ◽  
Spectral Sequences ◽  
Perfect Field ◽  
Chow Motives ◽  
Unramified Cohomology ◽  
Weight Structure ◽  
Weight Structures

AbstractThe main goal of this paper is to deduce (from a recent resolution of singularities result of Gabber) the following fact: (effective) Chow motives with ℤ[1/p]-coefficients over a perfect field k of characteristic p generate the category DMeffgm[1/p] (of effective geometric Voevodsky’s motives with ℤ[1/p]-coefficients). It follows that DMeffgm[1/p] can be endowed with a Chow weight structure wChow whose heart is Choweff[1/p] (weight structures were introduced in a preceding paper, where the existence of wChow for DMeffgmℚ was also proved). As shown in previous papers, this statement immediately yields the existence of a conservative weight complex functor DMeffgm[1/p]→Kb (Choweff [1/p]) (which induces an isomorphism on K0-groups), as well as the existence of canonical and functorial (Chow)-weight spectral sequences and weight filtrations for any cohomology theory on DMeffgm[1/p] . We also mention a certain Chow t-structure for DMeff−[1/p] and relate it with unramified cohomology.

scholarly journals The First Monsky-Washnitzer Cohomology Group

1972 ◽  
Vol 48 ◽  
pp. 99-128
Author(s):  
David Meredith
Keyword(s):  
Finite Type ◽  
Characteristic Zero ◽  
Cohomology Group ◽  
Valuation Ring ◽  
Residue Field ◽  
Discrete Valuation Ring ◽  
Perfect Field ◽  
Quotient Field ◽  
Complete Discrete Valuation Ring

Throughout this paper, k is a perfect field of characteristic p > 0, R is a complete discrete valuation ring with residue field k and quotient field of characteristic zero, and Z is a connected smooth prescheme of finite type over k.

Model Structures for the 𝔸1-homotopy Category

2019 ◽  
pp. 175-212
Author(s):  
Christian Haesemeyer ◽  
Charles A. Weibel
Keyword(s):  
Full Subcategory ◽  
Homotopy Category ◽  
Symmetric Power ◽  
Model Structure ◽  
Model Categories ◽  
Simplicial Presheaves ◽  
Basic Ideas ◽  
Quillen Model ◽  
Model Structures ◽  
Bousfield Localization

This chapter provides the 𝔸1-local projective model structure on the categories of simplicial presheaves and simplicial presheaves with transfers. These model categories, written as Δ‎opPshv(Sm)𝔸1 and Δ‎op PST(Sm)𝔸1, are first defined. Their respective homotopy categories are Ho(Sm) and the full subcategory DM eff nis ≤0 of DM eff nis. Afterward, this chapter introduces the notions of radditive presheaves and ̅Δ‎-closed classes, and develops their basic properties. The theory of ̅Δ‎-closed classes is needed because the extension of symmetric power functors to simplicial radditive presheaves is not a left adjoint. This chapter uses many of the basic ideas of Quillen model categories, which is a category equipped with three classes of morphisms satisfying five axioms. In addition, much of the material in this chapter is based upon the technique of Bousfield localization.

Cohomology Operations

2019 ◽  
pp. 213-231
Author(s):  
Christian Haesemeyer ◽  
Charles A. Weibel
Keyword(s):  
Homotopy Category ◽  
Perfect Field ◽  
Motivic Cohomology ◽  
Steenrod Operations ◽  
Axiomatic Treatment ◽  
Cohomology Operations

This chapter concerns cohomology operations. Although motivic cohomology was originally defined for smooth varieties over the perfect field 𝑘, it is more useful to view it as a functor defined on the pointed 𝔸1-homotopy category Ho ·, discussed previously in chapter 12. After defining cohomology operations and giving a few examples, the chapter turns to an axiomatic treatment of the motivic Steenrod operations. The motivic Milnor operations are then presented. Thereafter, this chapter demonstrates that the sequence of Milnor operations 𝑄𝑖 is exact on the reduced cohomology of the suspension Σ‎𝔛 attached to a Rost variety 𝑋, using the degree map 𝑡𝒩. It concludes with Voevodsky's motivic degree theorem.

Exceptional divisors that are not uniruled belong to the image of the Nash map

2011 ◽  
Vol 11 (2) ◽  
pp. 273-287 ◽  
Author(s):  
Monique Lejeune-Jalabert ◽  
Ana J. Reguera
Keyword(s):  
Irreducible Component ◽  
Characteristic Zero ◽  
Exceptional Divisor ◽  
Resolution Of Singularities ◽  
Closed Field ◽  
Algebraically Closed Field

AbstractWe prove that, ifXis a variety over an uncountable algebraically closed fieldkof characteristic zero, then any irreducible exceptional divisorEon a resolution of singularities ofXwhich is not uniruled, belongs to the image of the Nash map, i.e. corresponds to an irreducible component of the space of arcs$X_\infty^{\mathrm{Sing}}$onXcentred in SingX. This reduces the Nash problem of arcs to understanding which uniruled essential divisors are in the image of the Nash map, more generally, how to determine the uniruled essential divisors from the space of arcs.

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