Fujiwara’s theorem for equivariant correspondences

Olsson, Martin

doi:10.1090/s1056-3911-2014-00628-7

Cited by 6 publications

(4 citation statements)

References 19 publications

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“…(i) Behrend defined convergent complexes with respect to arithmetic Frobenius elements ([2], 6.2.3), and our definition is for geometric Frobenius, and it is essentially the same as Behrend's definition, except we work with ι-mixed Weil complexes (which means all Weil complexes, by (2.8.1)) for an arbitrary isomorphism ι : Q ℓ → C, while [2] works with pure or mixed lisse-étale sheaves with integer weights. Also our definition is a little different from that in [27]; the condition there is weaker.…”

Section: Convergent Complexes and Finitenessmentioning

confidence: 98%

“…If one wants to use Poincaré duality to get a functional equation for the zeta function, ( [27], 5.17) and ([24], 9.1.2) suggest that we should assume X 0 to be proper smooth and of finite diagonal. Under these assumptions, one gets the expected functional equation for the zeta function, as well as the independence of ℓ for the coarse moduli space, which is proper but possibly singular.…”

Section: 4mentioning

confidence: 99%

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L-series of Artin stacks over finite fields

Sun

2012

ANT

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We develop the notion of stratifiability in the context of derived categories and the six operations for stacks. Then we reprove the Lefschetz trace formula for stacks, and give the meromorphic continuation of L-series (in particular, zeta functions) of ‫ކ‬ q -stacks. We also give an upper bound for the weights of the cohomology groups of stacks, and an "independence of " result for a certain class of quotient stacks.

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Section: Convergent Complexes and Finitenessmentioning

confidence: 98%

Section: 4mentioning

confidence: 99%

L-series of Artin stacks over finite fields

Sun

2012

ANT

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“…Proof. Combining Theorem 5.1 and Corollary 5.8 of [Ols1], we have that Rπ * Q ℓ is acyclic in non-zero degrees, so we only need to show that R 0 π * Q ℓ,X = Q ℓ,X which follows easily from the fact that the topological spaces of X and X are homeomorphic.…”

Section: 2mentioning

confidence: 98%

Albanese and Picard 1-Motives in Positive Characteristic

Mannisto

2013

Preprint

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We investigate how to define 1-motives M 1 (c) (X), M 2d−1 (c) (X) for X a variety over a perfect field k of positive characteristic, such that the ℓ-adic realizations of these 1-motives are canonically isomorphic to H 1 (c) (X k , Z ℓ (1)) and H 2d−1 (c) (X k , Z ℓ (d)) respectively. This is the analogue in positive characteristic of previous results of Barbieri-Viale and Srinivas, except that we only consider the ℓ-adic realization but also consider compactly supported cohomology. The 1-motives M 1 (X) and M 1 c (X) can be defined by standard techniques, and indeed this case is probably well known. But the 1-motives M 2d−1 (X) and M 2d−1 c (X) require stronger tools, namely a strong version of de Jong's alterations theorem and some cycle class theory on smooth Deligne-Mumford stacks which may be of independent interest. Unfortunately, we only succeed in defining M 2d−1 (X) and M 2d−1 c (X) when X is a variety over an algebraically closed field, and only up to isogeny. Nevertheless, as a corollary to our definition of these 1-motives, for a variety X over a finite field k we deduce independence of ℓ for the cohomology groupsPETER MANNISTO Definition 1.3. [Del74, Définition 10.1.1] Let k be a field. The category of (free) 1-motives over k, denoted M 1 (k), is the category of 2-term complexesof commutative group schemes over k, where L is an étale-locally constant sheaf such that L(k s ) is a free finitely generated abelian group, and G is a semi-abelian variety over k. The morphisms in M 1 (k) are the morphisms of complexes of sheaves.In [Del74, Sect. 10], Deligne constructs realization functors T ℓ : M 1 (k) → Rep Gal(k s /k) (Z ℓ ) (for ℓ = p), and if k ⊆ C, a realization functor from 1-motives to mixed Hodge structures. Deligne conjectured in [Del74, 10.4.1] that certain mixed Hodge structures associated to a separated finite type scheme over C arise naturally from 1-motives; in particular, he conjectured that the mixed Hodge structures H 1 (X, Z(1)) and H 2d−1 (X, Z(d))/torsion occur as the Hodge realizations of 1-motives M 1 (X) and M 2d−1 (X), respectively, defined purely algebraically. This special case of Deligne's conjecture was solved in [BVS01], and much more general cases were handled in [BRS03] and [BVK12].For étale cohomology, one can make the following conjecture, which is an ℓ-adic analogue of this special case of Deligne's conjectures on 1-motives. Note that we restrict ourselves to the case where k is perfect; as noted in [Ram04, p. 3], it is not clear that this conjecture should be true for non-perfect fields.Conjecture 1.4. Fix a perfect field k of characteristic p ≥ 0, and let Sch/k denote the category of separated finite type k-schemes. Then there exist functors, with the property that for all primes ℓ = p,In addition, let Sch d /k ⊂ Sch/k be the full subcategory of d-dimensional separated finite type k-schemes. Then there exist functorswith the property that for all primes ℓ = p,1.5. In [BVS01], Barbieri-Viale and Srinivas solve this conjecture in the case where char k = 0, for non-compactl...

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“…We can replace D b c (X, Ω) by D b c (X, Q ℓ ), then by D b c (X, Q ℓ ) (using (3.5.9 ii, 3.5.12)), and finally by D b c (X , Q ℓ ) (using (3.4.3)). From ([19], 5.17) we know that there is a canonical isomorphismf ! ≃ f * on D − c (X , Q ℓ ).…”

mentioning

confidence: 99%