Cohomology of locally-closed semi-algebraic subsets
TL;DR: In this article, it was shown that these vector spaces are finite dimensional continuous representations of the Galois group of ksep/k, and satisfy the usual long exact sequence and Kunneth formula.
Abstract: Let k be a non-Archimedean field, let l be a prime number distinct from the characteristic of the residue field of k. If χ is a separated k-scheme of finite type, Berkovich’s theory of germs allows to define etale l-adic cohomology groups with compact support of locally closed semi-algebraic subsets of χan. We prove that these vector spaces are finite dimensional continuous representations of the Galois group of ksep/k, and satisfy the usual long exact sequence and Kunneth formula. This has been recently used by E. Hrushovski and F. Loeser in a paper about the monodromy of the Milnor fibration. In this statement, the main difficulty is the finiteness result, whose proof relies on a cohomological finiteness result for affinoid spaces, recently proved by V. Berkovich.
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TL;DR: In this paper, a geometric interpretation of Block and Gabbottsche's refined tropical curve counting invariants in terms of virtual $\chi-y$-specializations of motivic measures of semialgebraic sets in relative Hilbert schemes is proposed.
Abstract: We propose a geometric interpretation of Block and G\"ottsche's refined tropical curve counting invariants in terms of virtual $\chi_{-y}$-specializations of motivic measures of semialgebraic sets in relative Hilbert schemes. We prove that this interpretation is correct for linear series of genus 1, and in arbitrary genus after specializing from $\chi_{-y}$ to Euler characteristic.
23 citations
TL;DR: In this article, a geometric interpretation of Block and Gottsche's refined tropical curve counting invariants in terms of virtual χ − y specializations of motivic measures of semialgebraic sets in relative Hilbert schemes is proposed.
Abstract: We propose a geometric interpretation of Block and Gottsche’s refined tropical curve counting invariants in terms of virtual χ − y specializations of motivic measures of semialgebraic sets in relative Hilbert schemes. We prove that this interpretation is correct for linear series of genus 1, and in arbitrary genus after specializing from χ − y –genus to Euler characteristic.
15 citations
Cites methods from "Cohomology of locally-closed semi-a..."
...We associate a semialgebraic set in the relative compactified Jacobian or relative Hilbert scheme of points to each tropical curve, and relate combinatorially defined tropical multiplicities to motivic invariants of these semialgebraic sets, using the theory of motivic integration of Hrushovski and Kazhdan [21], together with recent results of Hrushovski and Loeser [22] and Martin [33] on the `–adic cohomology of locally closed semialgebraic sets....
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...1 in [33], using the additivity of the trace with respect to semialgebraic decompositions in X ....
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...14 in [33], using Berkovich’s finiteness result in [6, Theorem 1....
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TL;DR: In this article, a ring morphism from the Grothendieck ring of semi-algebraic sets over a field of characteristic zero containing all roots of unity was constructed, which fits inside a commutative diagram involving Hrushovski and Kazhdan's motivic integration.
Abstract: Let k be a field of characteristic zero containing all roots of unity and $$K=k(( t))$$
. We build a ring morphism from the Grothendieck ring of semi-algebraic sets over K to the Grothendieck ring of motives of rigid analytic varieties over K. It extends the morphism sending the class of an algebraic variety over K to its cohomological motive with compact support. We show that it fits inside a commutative diagram involving Hrushovski and Kazhdan’s motivic integration and Ayoub’s equivalence between motives of rigid analytic varieties over K and quasi-unipotent motives over k; we also show that it satisfies a form of duality. This allows us to answer a question by Ayoub, Ivorra and Sebag about the analytic Milnor fiber.
3 citations
Posted Content•
TL;DR: In this paper, a ring morphism from the Grothendieck group of semi-algebraic sets over $K$ to the Grohendiecker group of motives of rigid analytic varieties over $k$ is presented.
Abstract: Let $k$ be a field of characteristic zero containing all roots of unity and $K=k((t))$. We build a ring morphism from the Grothendieck group of semi-algebraic sets over $K$ to the Grothendieck group of motives of rigid analytic varieties over $K$. It extend the morphism sending the class of an algebraic variety over $K$ to its cohomological motive with compact support. We show that it fits inside a commutative diagram involving Hrushovski and Kazhdan's motivic integration and Ayoub's equivalence between motives of rigid analytic varieties over $K$ and quasi-unipotent motives over $k$ ; we also show that it satisfies a form of duality. This allows us to answer a question by Ayoub, Ivorra and Sebag about the analytic Milnor fiber.
1 citations
27 May 2016
TL;DR: In this paper, Grothendieck's Lefschetz trace formula for varieties over valued fields is compared to a trace formula based on non-archimedean geometry and motivic integration.
Abstract: We present some recent trace formulas for varieties over valued fields which can be seen as analogues of Grothendieck’s Lefschetz trace formula for varieties over finite fields. This involves motivic integration and non-archimedean geometry.
1 citations
Cites background from "Cohomology of locally-closed semi-a..."
...Using further results from [24], one proves the existence of a unique morphism EUét : K(VFK) −→ K(μ̂−Mod) (5....
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...Furthermore, Florent Martin proved in [24] that they are finite dimension Q -vector spaces and that they are zero for i > 2d....
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References
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Book•
08 Jul 1990
TL;DR: The spectrum of a commutative Banach ring is described in this paper, and the dimension of a Banach algebra is defined in terms of the spectrum of an analytic space.
Abstract: The spectrum of a commutative Banach ring Affinoid spaces Analytic spaces Analytic curves Analytic groups and buildings The homotopy type of certain analytic spaces Spectral theory Perturbation theory The dimension of a Banach algebra.
980 citations
"Cohomology of locally-closed semi-a..." refers background in this paper
...One can associate to it a k-analytic space X an [Ber90]....
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TL;DR: In this article, the authors present a set of conditions générales d'utilisation, i.e., the copie ou impression of a fichier do not contenir the présente mention de copyright.
Abstract: © Publications mathématiques de l’I.H.É.S., 1993, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
543 citations
"Cohomology of locally-closed semi-a..." refers background or methods in this paper
...Using [Ber93] one can define l-adic cohomology groups H c(X ,Ql) which have good properties if l is different from char(k̃) (in particular, they are finite dimensional vector spaces when k is algebraically closed)....
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...We want to stress out that in the previous sections, instead of working with the étale cohomology developed by Berkovich in [Ber93], we could also have used the theory of adic spaces and its étale cohomology theory, developed by R....
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...If U is a semi-algebraic subset of X , using the theory of k-germs developed in [Ber93], it...
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Book•
01 Jan 1996
TL;DR: In this article, the etale cohomology of rigid analytic varieties and analytic adic spaces is studied in the context of the analysis of adic space and etale site of a rigid analytic variety.
Abstract: Summary of the results on the etale cohomology of rigid analytic varieties - Adic spaces - The etale site of a rigid analytic variety and an adic space - Comparison theorems - Base change theorems - Cohomology with compact support - Finiteness - Poincare Duality - Partially proper sites of rigid analytic varieties and analytic adic spaces
395 citations
TL;DR: In this paper, the vanishing cycles functor from the category of etale sheaves on the generic fibre Xη of X (which is a k-analytic space) to the categories of etales on the closed fibre Xs of X, which is a scheme over the residue field of k, was constructed and studied.
Abstract: Let k be a non-Archimedean field, and let X be a formal scheme locally finitely presented over the ring of integers k◦ (see §1). In this work we construct and study the vanishing cycles functor from the category of etale sheaves on the generic fibre Xη of X (which is a k-analytic space) to the category of etale sheaves on the closed fibre Xs of X (which is a scheme over the residue field of k). We prove that if X is the formal completion X of a scheme X finitely presented over k◦ along the closed fibre, then the vanishing cycles sheaves of X are canonically isomorphic to those of X (as defined in [SGA7], Exp. XIII). In particular, the vanishing cycles sheaves of X depend only on X , and any morphism φ : Ŷ → X induces a homomorphism from the pullback of the vanishing cycles sheaves of X under φs : Ys → Xs to those of Y. Furthermore, we prove that, for each X , one can find a nontrivial ideal of k◦ such that if two morphisms φ, ψ : Ŷ → X coincide modulo this ideal, then the homomorphisms between the vanishing cycles sheaves induced by φ and ψ coincide. These facts were conjectured by P. Deligne. In §1 we associate with a formal scheme X locally finitely presented over k◦ a k-analytic space Xη (in the sense of [Ber1] and [Ber2]). In §2 we find that the morphism φη : Yη → Xη, which is induced by an etale morphism of formal schemes φ : Y → X, possesses a certain property. Morphisms of k-analytic spaces with this property are called quasi-etale, and they give rise to a quasi-etale site Xqet of a k-analytic space X. There is a canonical morphism of sites Xqet → Xet, where Xet is the etale site introduced in [Ber2]. We show that the inverse image functor identifies the category of etale sheaves X et with a full subcategory of
237 citations