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Sheaves and Functions Modulo p : Lectures on the Woods Hole Trace Formula

Author: Lenny Taelman
Publisher: Cambridge : Cambridge University Press, 2015.
Series: London Mathematical Society lecture note series, no. 429.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
The Woods Hole trace formula is a Lefschetz fixed-point theorem for coherent cohomology on algebraic varieties. It leads to a version of the sheaves-functions dictionary of Deligne, relating characteristic-p-valued functions on the rational points of varieties over finite fields to coherent modules equipped with a Frobenius structure. This book begins with a short introduction to the homological theory of crystals  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
(OCoLC)922155193
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Lenny Taelman
ISBN: 9781316480687 1316480682 9781316575093 1316575098 9781316574768 1316574768
OCLC Number: 932462465
Notes: Title from publisher's bibliographic system (viewed on 15 Dec 2015).
Description: 1 online resource (vi, 125 pages).
Contents: Cover --
Series information --
Title page --
Copyright information --
Table of contents --
Introduction --
Chapter 1 tau -sheaves, crystals, and their trace functions --
1. Coherent tau-sheaves --
2. The trace function of a coherent tau-sheaf --
3. Nilpotent coherent tau-sheaves --
4. Crystals --
5. Pointwise criteria --
Notes --
Exercises --
Chatper 2 Functors between categories of crystals --
1. Pullback --
2. Tensor product --
3. Direct images --
4. Extension by zero --
5. Pushforward with proper support --
Notes --
Exercises --
Chapter 3 The Woods Hole trace formula --
1. The Grothendieck group of crystals --
2. The sheaves-functions dictionary --
3. A quotient of the Grothendieck group of crystals --
4. Proof of the Localization Theorem --
Notes --
Exercises --
Chapter 4 Elementary applications --
1. Congruences between number of points --
2. Chevalley-Warning --
3. Polynomial sums --
4. Hasse invariant --
Notes --
Exercises --
Chapter 5 Crystals with coefficients --
1. tau-sheaves and crystals with coefficients --
2. Crystals with finite coefficients --
3. Traces of crystals with finite field coefficients --
Notes --
Chapter 6 Cohomology of symmetric powers of curves --
1. Symmetric tensors and exterior tensor powers --
2. Some sheaves on the symmetric powers of a scheme --
3. Cohomology of exterior symmetric tensor powers --
4. Crystals on symmetric powers --
Notes --
Exercises --
Chapter 7 Trace formula for L-functions --
1. L-functions of tau-sheaves and crystals --
2. Computation of L-functions via symmetric powers --
3. Proof of the trace formula for L-functions --
Notes --
Exercises --
Chapter 8 Special values of L-functions --
1. Example: the Goss zeta function of a scheme over F[sub(q)][t] --
2 Extensions of tau-sheaves --
3. Determinants and Fitting ideals --
4. Special values of L-functions of tau-sheaves. 5. An example failing the semi-simplicity hypothesis --
Notes --
Exercises --
Appendix A The trace formula for a transversalendomorphism --
1. Extensions --
1.1. Extension groups and cup product --
1.2. Base-change of extensions --
1.3. Künneth formula --
2. Grothendieck-Serre duality --
2.1. Duality in the derived category --
2.2. Duality for proper smooth variety --
2.3. Duality for proper smooth map --
2.4. Künneth and duality for a coordinate projection --
2.5. Duality for regular closed immersion --
2.6. Compatibility --
2.7. Transversal base change --
3. A local computation --
4. Duality for the graph of a morphism --
5. Woods Hole trace formula --
Bibliography --
Index.
Series Title: London Mathematical Society lecture note series, no. 429.
Other Titles: Sheaves & Functions Modulo 'p'
Responsibility: Lenny Taelman.

Abstract:

This graduate-level introduction describes a version of the sheaves-functions dictionary of Deligne, relating characteristic-p-valued functions on the rational points of varieties over finite fields  Read more...

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