The Lefschetz-Riemann-Roch formula

Donovan, Peter

doi:10.24033/bsmf.1680

Cited by 40 publications

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“…. , υ 7 As the computations are similar for all υ ∈ V, we only do this explicitly for υ 3 . We have that p a (C υ3 ) = g(υ 3 ) = 0, so it remains only to compute C 2 υ3 .…”

Section: Computations Of Jumps For G =mentioning

confidence: 99%

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Galois actions on Néron models of Jacobians

Halle¹

2010

Ann. inst. Fourier

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We study group actions on regular models of curves. If X is a smooth curve defined over the fraction field K of a complete discrete valuation ring R, every tamely ramified extension K ′ /K with Galois group G induces a G-action on the extension X K ′ of X to K ′ . In this paper we study the extension of this G-action to certain regular models of X K ′ . In particular, we are interested in the induced action on the cohomology groups of the structure sheaf of the special fiber of such a regular model. We obtain a formula for the Brauer trace of the endomorphism induced by a group element on the alternating sum of the cohomology groups. Inspired by this global study, we also consider similar group actions on the cohomology of the structure sheaf of the exceptional locus of a tame cyclic quotient singularity, and obtain an explicit polynomial formula for the Brauer trace of the endomorphism induced by a group element on the alternating sum of the cohomology groups.We apply these results to study a natural filtration of the special fiber of the Néron model of the Jacobian of X by closed, unipotent subgroup schemes. We show that the jumps in this filtration only depend on the fiber type of the special fiber of the minimal regular model with strict normal crossings for X over Spec(R), and in particular are independent of the residue characteristic. Furthermore, we obtain information about where these jumps occur. We also compute the jumps for each of the finitely many possible fiber type for curves of genus 1 and 2.

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“…. , υ 7 As the computations are similar for all υ ∈ V, we only do this explicitly for υ 3 . We have that p a (C υ3 ) = g(υ 3 ) = 0, so it remains only to compute C 2 υ3 .…”

Section: Computations Of Jumps For G =mentioning

confidence: 99%

“…We apply the so called Lefschetz-Riemann-Roch formula [7,Corollary 5.5], in order to get a formula for the Brauer trace of the endomorphism induced by each g ∈ G on the formal difference…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

Galois actions on Néron models of Jacobians

Halle¹

2010

Ann. inst. Fourier

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“…Combined with the Lefschetz-Riemann-Roch theorem, this yields a coherent trace formula which generalizes to possibly singular, possibly nonprojective schemes with non-finite algebraic group action the Atiyah-Bott type Woods Hole fixed point formula of [1,4,6,16].…”

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confidence: 99%

Lefschetz-Riemann-Roch theorem and coherent trace formula

Thomason

1986

Invent Math

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For actions of an algebraic group scheme G on a reasonable scheme X, this paper defines a equivariant topological G-theory which compares to the algebraic K-theory of G-modules. There is a Lefschetz-Riemann-Roch theorem for the comparison map from equivariant algebraic to topological G-theory. This is an algebraic analogue of the Atiyah-Segal G-index theorem of [2], but holds for generally non-compact algebraic groups, and for possibly singular varieties in finite characteristics. It generalizes the Lefschetz-Riemann-Roch theorem for singular varieties of Baum, Fulton, and Quart [4], in that the group G need not be finite, and in that the schemes may be non-projective and in mixed characteristic.An analogue of the Segal localization or concentration theorem of [19] is proved. Combined with the Lefschetz-Riemann-Roch theorem, this yields a coherent trace formula which generalizes to possibly singular, possibly nonprojective schemes with non-finite algebraic group action the Atiyah-Bott type Woods Hole fixed point formula of [1,4,6,16].For example, let T be a torus over a field k. Let X be a scheme proper over k on which T acts. Let j: X--,Z be a T-equivariant closed immersion into a smooth T-scheme Z proper over k. Let ~ be a coherent T sheaf on X. Then the two virtual representations of T over k are equalHere X r is the fixed point scheme, @ denotes total tensor product in the derived category, and hence is the alternating sum of all the Tor's, and the division signifies multiplication by the inverse in a localization of K(T,Z r) by inverting all non-zero virtual representations over k. The key point in the proofs of these results is the strong link between algebraic geometry and algebraic topology given by my theorem in [24] 4.16, * Partially supported by NSF

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“…The cohomological formula (4.10) specializes through this map to the formula of Donovan ( [2] and [5]). …”

Section: R(d)->s~1r(d)- W(k)mentioning

confidence: 99%

“…Our main result is a localization theorem for the equivariant -functor, Ky Namely, the inclusion i : X° --> X induces a map classical type, among others the Woods Hole fixed point formula and those of [2], [4], [5], [9]. It should be mentioned that the localization theorem is inspired by a similar topological theorem, see [1] for reference.…”

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confidence: 99%

Diagonalizably linearized coherent sheaves

Nielsen¹

1974

Bul. Soc. Math. France

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The Lefschetz-Riemann-Roch formula

Cited by 40 publications

References 1 publication

Galois actions on Néron models of Jacobians

Galois actions on Néron models of Jacobians

Lefschetz-Riemann-Roch theorem and coherent trace formula

Diagonalizably linearized coherent sheaves

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