Irreducible components of rigid spaces

Conrad, Brian

doi:10.5802/aif.1681

Cited by 133 publications

(155 citation statements)

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“…Transitivity with respect to further extension of the base field and the exact "pullback" functor Coh(X) → Coh(k ⊗ k X) on categories of coherent sheaves are defined as in the quasi-separated case. Finally, Lemma A.2.4 carries over (with the same proof) in the pseudo-separated case, and all appearances of "quasiseparated" in [11] may be replaced with "pseudo-separated" (see the end of [11, §3.1…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

“…The rigid-analytic input for what follows is a proper flat map f : X → S between rigid spaces, and we assume that H 0 (X s , O Xs ) = k(s) for all s ∈ S (a condition that is satisfied if each fiber X s is geometrically reduced and geometrically connected in the sense of [11]). The rigid-analytic theorem on cohomology and base change (whose proof goes as in the algebraic case, with the help of [31]) implies that the natural map O S → f * O X is an isomorphism and that this property persists after any base change on S and (for quasi-separated or pseudo-separated S) any extension on the base field; we say O S = f * O X universally.…”

Section: Picard Groupsmentioning

confidence: 99%

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Relative ampleness in rigid geometry

Conrad¹

2006

Ann. inst. Fourier

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Section: Annales De L'institut Fouriermentioning

confidence: 99%

Section: Picard Groupsmentioning

confidence: 99%

Relative ampleness in rigid geometry

Conrad¹

2006

Ann. inst. Fourier

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“…Since Supp Z M n is a priori closed, Corollary 2.2.6 of [Con99] implies that Supp Z M n contains an entire irreducible component of Z , say Z 0 . By Proposition 4.1.3, the image of Z 0 is Zariski-open in W , so we may choose an arithmetic weight λ 0 ∈ w * Z 0 .…”

Section: The Support Of Overconvergent Cohomology Modulesmentioning

confidence: 99%

Universal eigenvarieties, trianguline Galois representations, and p-adic Langlands functoriality

Hansen

Newton

2015

Journal Für Die Reine Und Angewandte Mathematik

146

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Using the overconvergent cohomology modules introduced by Ash and Stevens, we construct eigenvarieties associated with reductive groups and establish some basic geometric properties of these spaces, building on work of Ash-Stevens, Urban, and others. We also formulate a precise modularity conjecture linking trianguline Galois representations with overconvergent cohomology classes. In the course of giving evidence for this conjecture, we establish several new instances of p-adic Langlands functoriality. Our main technical innovations are a family of universal coefficients spectral sequences for overconvergent cohomology and a generalization of Chenevier's interpolation theorem.

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“…Conrad in [6] introduces the notion of normalization of a special formal scheme, and Nicaise in [28] recalls the definition of irreducibility in formalism. Let X be a special formal R-scheme, and n : X → X a normalization map (which is a finite morphism of special formal R-schemes).…”

Section: 3mentioning

confidence: 99%

A Short Survey on the Integral Identity Conjecture and Theories of Motivic Integration

Thuong

2016

Acta Math Vietnam

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Abstract. In Kontsevich-Soibelman's theory of motivic Donaldson-Thomas invariants for three-dimensional noncommutative Calabi-Yau varieties, the integral identity conjecture plays a crucial role as it involves the existence of these invariants. A purpose of this note is to show how the conjecture arises. Because of the integral identity's nature we shall give a quick tour on theories of motivic integration, from which lead to a proof of the conjecture for ground fields algebraically closed of characteristic zero. IntroductionHistorically, Thomas in his thesis and his paper [31] introduced an invariant for a 3-dimensional Calabi-Yau manifold M as a counting invariant of coherent sheaves on M , which analogizes the Casson invariant on a real 3-dimensional manifold. This kind of invariant was then named after him and his advisor Donaldson. According to [16], the moduli space M of coherent sheaves on M can be locally presented as the critical locus of a holomorphic Chern-Simons functional f . By this, a Donaldson-Thomas invariant for M can be described as an integral of the Behrend function over the moduli space M. In view of [1], the value of the Behrend function is defined in terms of the Euler characteristic of the Milnor fiber F f of the Chern-Simons functional f , from which the Donaldson-Thomas invariant arises. In [17], replacing the Milnor fiber by a so-called motivic Milnor fiber (defined by Denef-Loeser [8] using motivic integration) Kontsevich and Soibelman study motivic Donaldson-Thomas invariants for 3-dimensional Calabi-Yau manifolds. The theory of Kontsevich and Soibelman allows, under appropriate realizations (e.g., a cohomology functor), to obtain refinements of (classical) Donaldson-Thomas invariants. The motivic Donaldson-Thomas invariants are also realized physically as "BPS invariants". Among 3-dimensional Calabi-Yau categories, the derived category of coherent sheaves on a compact (or local) 3-dimensional Calabi-Yau manifolds is a central object.Let us now give a brief review due to [17] and [18] on the direct elements in KontsevichSoibelman's theory concerning the integral identity conjecture. Let C be an ind-constructible triangulated A ∞ -category over a field k. For any strict sector V in R 2 , we consider a collection of full subcategories C V of C. Then one can construct the motivic Hall algebra H(C) as a graded associative algebra admitting for any strict sector V an element A Hall V invertible in a completion H(C V ) of H(C V ). More precisely, in terms of a countable decomposition Ob(C) = i∈I Y i into 2010 Mathematics Subject Classification. Primary 03C60, 14B20, 14E18, 14G22, 32S45, 11S80. Key words and phrases. motivic integration, formal schemes, rigid varieties, volume Poincaré series, resolution of singularity, integral identity conjecture, definable sets. where 1 S is the identity function interpreted as a counting measure (see [17, Section 6.1]). One requires that A Hall V must satisfy the factorization property that A Hall V = A Hall V 1 · A Hall V 2 , where V = V 1 V 2 and th...

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Irreducible components of rigid spaces

Cited by 133 publications

References 13 publications

Relative ampleness in rigid geometry

Relative ampleness in rigid geometry

Universal eigenvarieties, trianguline Galois representations, and p-adic Langlands functoriality

A Short Survey on the Integral Identity Conjecture and Theories of Motivic Integration

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