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491
Smashing Subcategories And The Telescope Conjecture  An Algebraic Approach
 Invent. Math
, 1998
"... . We prove a modified version of Ravenel's telescope conjecture. It is shown that every smashing subcategory of the stable homotopy category is generated by a set of maps between finite spectra. This result is based on a new characterization of smashing subcategories, which leads in addition to ..."
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Cited by 33 (8 self)
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. We prove a modified version of Ravenel's telescope conjecture. It is shown that every smashing subcategory of the stable homotopy category is generated by a set of maps between finite spectra. This result is based on a new characterization of smashing subcategories, which leads in addition to a classification of these subcategories in terms of the category of finite spectra. The approach presented here is purely algebraic; it is based on an analysis of pureinjective objects in a compactly generated triangulated category, and covers therefore also situations arising in algebraic geometry and representation theory. Introduction Smashing subcategories naturally arise in the stable homotopy category S from localization functors l : S ! S which induce for every spectrum X a natural isomorphism l(X) ' X l(S) between the localization of X and the smash product of X with the localization of the sphere spectrum S. In fact, a localization functor has this property if and only if it preserv...
Homotopical Algebraic Geometry I: Topos theory
, 2002
"... This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos. For this, we use Scategories (i.e. simplicially enriched categories) as models for certain kind of ..."
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Cited by 29 (18 self)
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This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos. For this, we use Scategories (i.e. simplicially enriched categories) as models for certain kind of ∞categories, and we develop the notions of Stopologies, Ssites and stacks over them. We prove in particular, that for an Scategory T endowed with an Stopology, there exists a model
Compact moduli of plane curves
, 2004
"... We construct a compactification Md of the moduli space of plane curves of degree d. We regard a plane curve C ⊂ P 2 as a surfacedivisor pair (P 2, C) and define Md as a moduli space of pairs (X, D) where X is a degeneration of the plane. We show that, if d is not divisible by 3, the stack Md is smo ..."
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Cited by 27 (6 self)
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We construct a compactification Md of the moduli space of plane curves of degree d. We regard a plane curve C ⊂ P 2 as a surfacedivisor pair (P 2, C) and define Md as a moduli space of pairs (X, D) where X is a degeneration of the plane. We show that, if d is not divisible by 3, the stack Md is smooth and the degenerate surfaces X can be described explicitly.
Hypercohomology spectra and Thomason’s descent theorem
 IN ALGEBRAIC KTHEORY, FIELDS INSTITUTE COMMUNICATIONS
, 1997
"... The celebrated LichtenbaumQuillen conjectures predict that for a sufficiently nice scheme and given prime ℓ, the ℓadic algebraic Kgroups of X are closely related to the ℓadic étale cohomology groups of X. More precisely, one version of the conjectures asserts that there is a descent spectral se ..."
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Cited by 27 (3 self)
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The celebrated LichtenbaumQuillen conjectures predict that for a sufficiently nice scheme and given prime ℓ, the ℓadic algebraic Kgroups of X are closely related to the ℓadic étale cohomology groups of X. More precisely, one version of the conjectures asserts that there is a descent spectral sequence of AtiyahHirzebruch type H p q ét (X; Zℓ ()) ⇒ πq−p(KX 2 ℓ) but with the convergence only valid in sufficiently high degrees. Here the coefficient sheaves are Tate twists of the ℓadic integers, and are to be interpreted as zero if q is odd. Throughout this paper, étale cohomology is continuous étale cohomology [19], and the indicated abutment of the spectral sequence consists of the homotopy groups of the Bousfield ℓadic completion of the spectrum KX, not the naive ℓadic completion of the Kgroups. In a remarkable paper [42], Thomason proved the LichtenbaumQuillen conjectures for a certain localized form of Ktheory socalled “Bottperiodic” Ktheory. The first step was the development of an elaborate theory of hypercohomology spectra H ·(X; E) associated to étale presheaves of spectra E or more generally, to presheaves of spectra on a Grothendieck site. These hypercohomology spectra are by their very construction naturally Supported by a grant from the National Science Foundation 1 equipped with a suitable descent spectral sequence, and there is a natural map E(X)−→H · (X; E). In particular, the Ktheory of a scheme X maps to its associated hypercohomology spectrum H · ét(X; K). Fix a prime ℓ, which we will assume is odd in order to simplify the discussion.Let L(−) denote Bousfield localization with respect to complex Ktheory, and let ˆ L(−) denote its ℓadic completion. The main theorem of [42] can be stated as follows. Theorem 0.1 Let X be a separated noetherian regular scheme of finite Krull dimension, with sufficiently nice residue fields of characteristic different from ℓ. Then the natural map KX−→H · ét(X; K) induces a weak equivalence
Finiteness theorems in geometric classfield theory, Enseign
 Math
, 1981
"... Mit dem Zugriff auf den vorliegenden Inhalt gelten die Nutzungsbedingungen als akzeptiert. Die angebotenen Dokumente stehen für nichtkommerzielle Zwecke in Lehre, Forschung und für die private Nutzung frei zur Verfügung. Einzelne Dateien oder Ausdrucke aus diesem Angebot können zusammen mit diesen ..."
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Cited by 26 (3 self)
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Mit dem Zugriff auf den vorliegenden Inhalt gelten die Nutzungsbedingungen als akzeptiert. Die angebotenen Dokumente stehen für nichtkommerzielle Zwecke in Lehre, Forschung und für die private Nutzung frei zur Verfügung. Einzelne Dateien oder Ausdrucke aus diesem Angebot können zusammen mit diesen Nutzungsbedingungen und unter deren Einhaltung weitergegeben werden.