Seminar on Étale Cohomology


 Charleen Jennings
 3 years ago
 Views:
Transcription
1 Seminar on Étale Cohomology Elena Lavanda and Pedro A. Castillejo SS 2015 Introduction The aim of this seminar is to give the audience an introduction to étale cohomology. In particular we will study the étale site and the category of sheaves over this site. Every talk contains examples that will lead to a deeper understanding of the theorems and of abstract definitions, via a concrete computation. We have followed Milne s notes in organising the program and we added materials from other references, as it is explained in the abstracts of the talks. In any case, the speakers may choose whatever references they like best, as long as they prove the result specified for their talks and present the corresponding examples. If anything is unclear, or you have problems in organising the time of the talk please contact the organisers, who will be happy to help you. Program of the talks 1.1 Introduction ( ) Give a soft introduction and motivate the study of étale cohomology. Details: Follow the introduction to étale cohomology presented by Milne. In particular, recall briefly cohomology theory from the point of view of algebraic topology. Quickly review sheaf cohomology and explain the inadequacy of Zariski topology. This motivates the definition of another cohomology theory, which will be the étale cohomology. Spend some words on étale topology, in particular on the definition of étale cohomology; then compare it with complex topology. State the comparison theorem and conclude with some applications of étale cohomology. If time permits, the speaker can add further motivations or applications of étale cohomolgy. [LEC, Section 1.1]. [Stacks, Étale Cohomology]. Dashtpeyma master s thesis, Chapter 6: 1
2 1.2 Étale morphisms and henselization ( ) Recall the notions of étale morphism and henselian rings. Details: First define an étale morphism of schemes by briefly recalling the notion of flat morphisms and unramified morphisms. Then give the definition of étale morphism for the specific case of nonsingular algebraic varieties. State proposition 2.1, prove corollary 2.2 and explain remark 2.3. Then prove proposition 2.9 and explain the example of fields. Conclude the part on étale morphism summarizing their properties. Dedicate the second part of the talk to the study of henselian rings ([LEC, 1.4 Interlude on Henselian rings]). Start by recalling definition 4.2, then state proposition 4.11 and explain definition State proposition 4.13 and prove that R h is Henselian by following the last paragraph of [Stacks, Prop ]. State proposition 4.15 and compute explicitely the henselization of a DVR by following the details of [mo105381] given by Qing Liu. Warning: here you have some work to do, please be careful and explain all the details of the example. Finally conclude stating corollary [LEC, Section 1.2 and the interlude on henselian rings of section 1.4]. [Stacks] Mathoverflow: Local ring for étale topology ( ) Describe the local ring for étale topology and compute it at a nonsingular point of a variety. Details: Explain the beginning of the subsection "The case of varieties". State proposition 4.1 and prove that O X, x is noetherian following [Art62, p. 86]. Now state theorem 4.4 and prove it in detail. Please pay attention in illustrating the matrix and in explaining the computation of the determinant. Recall proposition 4.13 and explain corollary Now state proposition 4.10 recalling the details of the 1dimensional case from the previous talk. State propositions 4.8 and 4.9, and if time permits, prove them. [LEC, Section 1.4]. [Art62]. 1.4 Sites ( ) Introduce the notion of sites and present some examples. Details: Start recalling the definition of sheaf given by [Mum88, Section 1.4, definition 3], with the two remarks that justify the abstract definition. Now follow [LEC] and define a site as in section 1.5. Explain the examples of the Zariski site of X, the étale site on X and develop the example of the étale site of P 1 (i.e. show what are the coverings in the étale site defined by P 1 ). Present 2
3 example 2.32 of [Vis05] and say some words of the fpqc topology after this example. Give definition 5.2 of [LEC] and explain the example that follows it adding the fppf and fpqc topology so that we have a more complete picture of possible sites over a scheme. [Mum65, pp ]. [LEC, Section 1.5]. [Mum88]. [Vis05]. 1.5 Galois coverings and fundamental group ( ) Give an overview on galois covering and the étale fundamental group. Details: Start with the definition of Galois covering given in [Con, Def ], and present example Then, with this motivation, define the étale fundamental group. Explain example and say some words about "the most interesting object in mathematics" in [LEC, p. 30]. Now continue with section 1.6 of Milne s notes, and recall the definition of a sheaf on the étale site. Conclude proving proposition 6.4. [LEC, Section 1.6]. [Con]. 1.6 Examples of sheaves on the étale site ( ) Present some examples of sheaves on the étale site. Details: For this talk it is sufficient to follow closely section 1.6 of Milne s notes. In particular, state without proof proposition 6.6, the criterion to be a sheaf. Explain the example of the structure sheaf without proving 6.8. Explain also the example of representable sheaves paying special attention to 6.10 (a), (b), and (c). Then present the examples of sheaves on Spec(k). Continue defining the stalks, including example 6.13 (specially part (b)). Define skyscraper sheaves, then go back and explain the notion of constant sheaves and finally conclude presenting locally constant sheaves and stating proposition [LEC, Section 1.6]. 1.7 Category of sheaves on the étale site ( ) Describe the category of sheaves on the étale site and explain Kummer sequence and ArtinSchreier sequence. 3
4 Details: For this talk the main reference is section 1.7 of Milne s notes. Skip the generalities on categories and start directly with the section of the category of sheaves. Defining a locally surjective morphism of sheaves, state proposition 7.6 and prove left exactness of the functor of global sections (proof of proposition 7.5). Prove proposition 7.8 and explain in detail example 7.9. Explain remark If time permits, spend some words on the construction of the sheafification. [LEC, Section 1.7]. [Mum65, pp ]. 1.8 Direct and inverse images of sheaves ( ) Define direct and inverse images of sheaves and the extension by zero. Details: For this talk the main reference is section 1.8 of Milne s notes. Start defining direct images of sheaves. Then prove lemma 8.1, explain the left exactness of the direct image and present a counterexample for the right exactness. Explain example 8.2 and state proposition 8.3 proving part (b). Explain corollary 8.4 and present one of the examples of 8.5, then state proposition 8.6. Define the inverse image, state proposition 8.7, example 8.8 and explain remark State that there are enough injectives (proposition 8.12). Then define extension by zero, state the adjunction formulae, state proposition 8.13 and prove proposition State corollary 8.18 explaining the definition of sheaf with support. [LEC, Section 1.8]. 1.9 Étale cohomology ( ) Define étale cohomology and explain its basic properties. Details: For this talk the main reference is section 1.9 of Milne s notes. Define and prove everything until theorem 9.7. omitting its proof. Then state corollary 9.8 and if time permits, prove it. Conclude discussing the geometrical aspect of the homotopy axiom. [LEC, Section 1.9] Higher direct image and Weildivisor exact sequence ( ) Define the higher direct image and prove the Weildivisor exact sequence. Details: This talk should be a summary of section 1.12 of Milne s notes and an introduction to section Start defining the higher direct image. Prove proposition 12.1, corollary 12.2, and explain examples 12.3, 12.4 and 12.5, with special attention to the nodal cubic. Now recall the first part of section 1.13, specially the exactness of the sequences of p. 84, and prepare the discussion in order to understand proposition 13.3 and prove it. Then conclude by proving proposition [LEC, Section 1.12 and 1.13]. 4
5 1.11 Cohomology of the multiplicative group scheme on a curve ( ) Compute the cohomology of G m on a curve. Details: Define C1 fields, note that these fields are called quasi algebraically closed in [LEC, p. 86]. Explain example 13.5 and state proposition Prove 13.7, which implies proving lemma 13.8 and explaining how does this relate with the Weildivisor exact sequence. Conclude explaining the discussion that follows the lemma, saying some words on global class field theory. [LEC, Section 1.13]. [CFT] Overview on main theorems of étale cohomology ( ) Overview on main theorems of étale cohomology. Details: This talk concludes the seminar and its aim is to present interesting results and applications of étale cohomology. The speaker is free to choose the results that he finds more interesting. Some examples could be: purity, proper and smooth base change, the comparison theorem, the Künneth formula, Poincaré duality and the Lefschetz fixedpoint formula. [LEC] Any other reference on étale cohomology. References [Art62] Michael Artin, Grothendieck topologies: notes on a seminar, Harvard University, Dept. of Mathematics, 1962, pp [CFT] J.S. Milne, Class Field Theory (v4.02), [Con] B. Conrad, Étale cohomology, url: CohEtale  09 / Elencj _ Etale / CONRAD % 20Etale % 20Cohomology. pdf. [LEC] J. Milne, Lectures on Etale Cohomology (v2.21), [Mum65] [Mum88] [Stacks] [Vis05] David Mumford, Picard groups of moduli problems, in: Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), New York: Harper & Row, 1965, pp David Mumford, The red book of varieties and schemes, Lecture notes in mathematics, Berlin: SpringerVerlag, The Stacks Project Authors, Stacks Project, columbia.edu, Angelo Vistoli, Grothendieck topologies, fibered categories and descent theory, in: Fundamental algebraic geometry, vol. 123, Math. Surveys Monogr. Amer. Math. Soc., 2005, pp
NOTES ON FLAT MORPHISMS AND THE FPQC TOPOLOGY
NOTES ON FLAT MORPHISMS AND THE FPQC TOPOLOGY RUNE HAUGSENG The aim of these notes is to define flat and faithfully flat morphisms and review some of their important properties, and to define the fpqc
More informationMODULI TOPOLOGY. 1. Grothendieck Topology
MODULI TOPOLOG Abstract. Notes from a seminar based on the section 3 of the paper: Picard groups of moduli problems (by Mumford). 1. Grothendieck Topology We can define a topology on any set S provided
More information1 Notations and Statement of the Main Results
An introduction to algebraic fundamental groups 1 Notations and Statement of the Main Results Throughout the talk, all schemes are locally Noetherian. All maps are of locally finite type. There two main
More informationTHE KEEL MORI THEOREM VIA STACKS
THE KEEL MORI THEOREM VIA STACKS BRIAN CONRAD 1. Introduction Let X be an Artin stack (always assumed to have quasicompact and separated diagonal over Spec Z; cf. [2, 1.3]). A coarse moduli space for
More informationElliptic curves, Néron models, and duality
Elliptic curves, Néron models, and duality Jean Gillibert Durham, Pure Maths Colloquium 26th February 2007 1 Elliptic curves and Weierstrass equations Let K be a field Definition: An elliptic curve over
More informationProgram of the DaFra seminar. Unramified and Tamely Ramified Geometric Class Field Theory
Program of the DaFra seminar Unramified and Tamely Ramified Geometric Class Field Theory Introduction Torsten Wedhorn Winter semester 2017/18 Gemetric class field theory gives a geometric formulation and
More informationRational sections and Serre s conjecture
FREIE UNIVERSITÄT BERLIN FORSCHUNGSSEMINAR SS 15 Rational sections and Serre s conjecture Lei Zhang March 20, 2015 Recall the following conjecture of Serre. INTRODUCTION Conjecture. Let K be a perfect
More informationSome remarks on Frobenius and Lefschetz in étale cohomology
Some remarks on obenius and Lefschetz in étale cohomology Gabriel Chênevert January 5, 2004 In this lecture I will discuss some more or less related issues revolving around the main idea relating (étale)
More informationMath 248B. Applications of base change for coherent cohomology
Math 248B. Applications of base change for coherent cohomology 1. Motivation Recall the following fundamental general theorem, the socalled cohomology and base change theorem: Theorem 1.1 (Grothendieck).
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationHomology and Cohomology of Stacks (Lecture 7)
Homology and Cohomology of Stacks (Lecture 7) February 19, 2014 In this course, we will need to discuss the ladic homology and cohomology of algebrogeometric objects of a more general nature than algebraic
More informationSHIMURA VARIETIES AND TAF
SHIMURA VARIETIES AND TAF PAUL VANKOUGHNETT 1. Introduction The primary source is chapter 6 of [?]. We ve spent a long time learning generalities about abelian varieties. In this talk (or two), we ll assemble
More informationConstruction of M B, M Dol, M DR
Construction of M B, M Dol, M DR Hendrik Orem Talbot Workshop, Spring 2011 Contents 1 Some Moduli Space Theory 1 1.1 Moduli of Sheaves: Semistability and Boundedness.............. 1 1.2 Geometric Invariant
More informationAzumaya Algebras. Dennis Presotto. November 4, Introduction: Central Simple Algebras
Azumaya Algebras Dennis Presotto November 4, 2015 1 Introduction: Central Simple Algebras Azumaya algebras are introduced as generalized or global versions of central simple algebras. So the first part
More informationPARABOLIC SHEAVES ON LOGARITHMIC SCHEMES
PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES Angelo Vistoli Scuola Normale Superiore Bordeaux, June 23, 2010 Joint work with Niels Borne Université de Lille 1 Let X be an algebraic variety over C, x 0 X. What
More informationWeil Conjectures (Deligne s Purity Theorem)
Weil Conjectures (Deligne s Purity Theorem) David Sherman, Ka Yu Tam June 7, 2017 Let κ = F q be a finite field of characteristic p > 0, and k be a fixed algebraic closure of κ. We fix a prime l p, and
More informationTHE SMOOTH BASE CHANGE THEOREM
THE SMOOTH BASE CHANGE THEOREM AARON LANDESMAN CONTENTS 1. Introduction 2 1.1. Statement of the smooth base change theorem 2 1.2. Topological smooth base change 4 1.3. A useful case of smooth base change
More informationwhere Σ is a finite discrete Gal(K sep /K)set unramified along U and F s is a finite Gal(k(s) sep /k(s))subset
Classification of quasifinite étale separated schemes As we saw in lecture, Zariski s Main Theorem provides a very visual picture of quasifinite étale separated schemes X over a henselian local ring
More informationThe moduli stack of vector bundles on a curve
The moduli stack of vector bundles on a curve Norbert Hoffmann norbert.hoffmann@fuberlin.de Abstract This expository text tries to explain briefly and not too technically the notions of stack and algebraic
More information1.6.1 What are Néron Models?
18 1. Abelian Varieties: 10/20/03 notes by W. Stein 1.6.1 What are Néron Models? Suppose E is an elliptic curve over Q. If is the minimal discriminant of E, then E has good reduction at p for all p, in
More informationRaynaud on F vector schemes and prolongation
Raynaud on F vector schemes and prolongation Melanie Matchett Wood November 7, 2010 1 Introduction and Motivation Given a finite, flat commutative group scheme G killed by p over R of mixed characteristic
More informationLectures on Galois Theory. Some steps of generalizations
= Introduction Lectures on Galois Theory. Some steps of generalizations Journée Galois UNICAMP 2011bis, ter Ubatuba?=== Content: Introduction I want to present you Galois theory in the more general frame
More information1.5.4 Every abelian variety is a quotient of a Jacobian
16 1. Abelian Varieties: 10/10/03 notes by W. Stein 1.5.4 Every abelian variety is a quotient of a Jacobian Over an infinite field, every abelin variety can be obtained as a quotient of a Jacobian variety.
More informationÉTALE COHOMOLOGY SEMINAR LECTURE 1
ÉTALE COHOMOLOGY SEMINAR LECTURE 1 EVAN JENKINS 1. Introduction The theory of étale cohomology springs from a simple question: is it possible to do algebraic topology on algebraic varieties (or, more generally,
More informationThe Picard Scheme and the Dual Abelian Variety
The Picard Scheme and the Dual Abelian Variety Gabriel DorfsmanHopkins May 3, 2015 Contents 1 Introduction 2 1.1 Representable Functors and their Applications to Moduli Problems............... 2 1.2 Conditions
More informationETALE COHOMOLOGY  PART 2. Draft Version as of March 15, 2004
ETALE COHOMOLOGY  PART 2 ANDREW ARCHIBALD AND DAVID SAVITT Draft Version as of March 15, 2004 Contents 1. Grothendieck Topologies 1 2. The Category of Sheaves on a Site 3 3. Operations on presheaves and
More informationSEMINAR: DERIVED CATEGORIES AND VARIATION OF GEOMETRIC INVARIANT THEORY QUOTIENTS
SEMINAR: DERIVED CATEGORIES AND VARIATION OF GEOMETRIC INVARIANT THEORY QUOTIENTS VICTORIA HOSKINS Abstract 1. Overview Bondal and Orlov s study of the behaviour of the bounded derived category D b (X)
More informationDERIVED CATEGORIES OF STACKS. Contents 1. Introduction 1 2. Conventions, notation, and abuse of language The lisseétale and the flatfppf sites
DERIVED CATEGORIES OF STACKS Contents 1. Introduction 1 2. Conventions, notation, and abuse of language 1 3. The lisseétale and the flatfppf sites 1 4. Derived categories of quasicoherent modules 5
More informationMotivic integration on Artin nstacks
Motivic integration on Artin nstacks Chetan Balwe Nov 13,2009 1 / 48 Prestacks (This treatment of stacks is due to B. Toën and G. Vezzosi.) Let S be a fixed base scheme. Let (Aff /S) be the category of
More informationPICARD GROUPS OF MODULI PROBLEMS II
PICARD GROUPS OF MODULI PROBLEMS II DANIEL LI 1. Recap Let s briefly recall what we did last time. I discussed the stack BG m, as classifying line bundles by analyzing the sense in which line bundles may
More informationLecture 3: Flat Morphisms
Lecture 3: Flat Morphisms September 29, 2014 1 A crash course on Properties of Schemes For more details on these properties, see [Hartshorne, II, 15]. 1.1 Open and Closed Subschemes If (X, O X ) is a
More informationOFER GABBER, QING LIU, AND DINO LORENZINI
PERIOD, INDEX, AND AN INVARIANT OF GROTHENDIECK FOR RELATIVE CURVES OFER GABBER, QING LIU, AND DINO LORENZINI 1. An invariant of Grothendieck for relative curves Let S be a noetherian regular connected
More informationLecture 7: Etale Fundamental Group  Examples
Lecture 7: Etale Fundamental Group  Examples October 15, 2014 In this lecture our only goal is to give lots of examples of etale fundamental groups so that the reader gets some feel for them. Some of
More informationA tale of Algebra and Geometry
A tale of Algebra and Geometry Dan Abramovich Brown University University of Pisa June 4, 2018 Abramovich (Brown) A tale of Algebra and Geometry June 4, 2018 1 / 12 Intersection theory on algebraic stacks
More informationRESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES RIMS A Note on an Anabelian Open Basis for a Smooth Variety. Yuichiro HOSHI.
RIMS1898 A Note on an Anabelian Open Basis for a Smooth Variety By Yuichiro HOSHI January 2019 RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES KYOTO UNIVERSITY, Kyoto, Japan A Note on an Anabelian Open Basis
More informationSchemes of Dimension 2: Obstructions in Non Abelian Cohomology
Pure Mathematical Sciences, Vol. 6, 2017, no. 1, 3945 HIKARI Ltd, www.mhikari.com https://doi.org/10.12988/pms.2017.711 Schemes of Dimension 2: Obstructions in Non Abelian Cohomology Bénaouda Djamai
More informationSeminar on RapoportZink spaces
Prof. Dr. U. Görtz SS 2017 Seminar on RapoportZink spaces In this seminar, we want to understand (part of) the book [RZ] by Rapoport and Zink. More precisely, we will study the definition and properties
More informationAPPENDIX 2: AN INTRODUCTION TO ÉTALE COHOMOLOGY AND THE BRAUER GROUP
APPENDIX 2: AN INTRODUCTION TO ÉTALE COHOMOLOGY AND THE BRAUER GROUP In this appendix we review some basic facts about étale cohomology, give the definition of the (cohomological) Brauer group, and discuss
More informationThe Néron Ogg Shafarevich criterion Erik Visse
The Néron Ogg Shafarevich criterion Erik Visse February 17, 2017 These are notes from the seminar on abelian varieties and good reductions held in Amsterdam late 2016 and early 2017. The website for the
More informationCHEAT SHEET: PROPERTIES OF MORPHISMS OF SCHEMES
CHEAT SHEET: PROPERTIES OF MORPHISMS OF SCHEMES BRIAN OSSERMAN The purpose of this cheat sheet is to provide an easy reference for definitions of various properties of morphisms of schemes, and basic results
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationOn log flat descent. Luc Illusie, Chikara Nakayama, and Takeshi Tsuji
On log flat descent Luc Illusie, Chikara Nakayama, and Takeshi Tsuji Abstract We prove the log flat descent of log étaleness, log smoothness, and log flatness for log schemes. Contents 1. Review of log
More informationThe proétale topology for schemes Oberseminar of the AG Schmidt, Wintersemester 2017
The proétale topology for schemes Oberseminar of the AG Schmidt, Wintersemester 2017 Pavel Sechin Time and place: Tuesday, 11:0013:00, Room SR3, Mathematikon, Start: 24.10.2017 The étale topology is
More informationTopics in Algebraic Geometry
Topics in Algebraic Geometry Nikitas Nikandros, 3928675, Utrecht University n.nikandros@students.uu.nl March 2, 2016 1 Introduction and motivation In this talk i will give an incomplete and at sometimes
More informationON TAME STACKS IN POSITIVE CHARACTERISTIC
ON TAME STACKS IN POSITIVE CHARACTERISTIC DAN ABRAMOVICH, MARTIN OLSSON, AND ANGELO VISTOLI Contents 1. Linearly reductive finite group schemes 1 2. Tame stacks 13 3. Twisted stable maps 20 4. Reduction
More informationAPPLICATIONS OF LOCAL COHOMOLOGY
APPLICATIONS OF LOCAL COHOMOLOGY TAKUMI MURAYAMA Abstract. Local cohomology was discovered in the 960s as a tool to study sheaves and their cohomology in algebraic geometry, but have since seen wide use
More informationON THE ISOMORPHISM BETWEEN THE DUALIZING SHEAF AND THE CANONICAL SHEAF
ON THE ISOMORPHISM BETWEEN THE DUALIZING SHEAF AND THE CANONICAL SHEAF MATTHEW H. BAKER AND JÁNOS A. CSIRIK Abstract. We give a new proof of the isomorphism between the dualizing sheaf and the canonical
More information1 Replete topoi. X = Shv proét (X) X is locally weakly contractible (next lecture) X is replete. D(X ) is left complete. K D(X ) we have R lim
Reference: [BS] Bhatt, Scholze, The proétale topology for schemes In this lecture we consider replete topoi This is a nice class of topoi that include the proétale topos, and whose derived categories
More informationDESCENT THEORY (JOE RABINOFF S EXPOSITION)
DESCENT THEORY (JOE RABINOFF S EXPOSITION) RAVI VAKIL 1. FEBRUARY 21 Background: EGA IV.2. Descent theory = notions that are local in the fpqc topology. (Remark: we aren t assuming finite presentation,
More informationALGEBRAIC GROUPS JEROEN SIJSLING
ALGEBRAIC GROUPS JEROEN SIJSLING The goal of these notes is to introduce and motivate some notions from the theory of group schemes. For the sake of simplicity, we restrict to algebraic groups (as defined
More informationForschungsseminar: Brauer groups and Artin stacks
Universität DuisburgEssen, Düsseldorf SS 07 Forschungsseminar: Brauer groups and Artin stacks Organisation: Jochen Heinloth, Marc Levine, Stefan Schröer Place and time: Thursdays, 1416 Uhr ct, T03 R03
More informationNotes on pdivisible Groups
Notes on pdivisible Groups March 24, 2006 This is a note for the talk in STAGE in MIT. The content is basically following the paper [T]. 1 Preliminaries and Notations Notation 1.1. Let R be a complete
More informationConstructible Derived Category
Constructible Derived Category Dongkwan Kim September 29, 2015 1 Category of Sheaves In this talk we mainly deal with sheaves of Cvector spaces. For a topological space X, we denote by Sh(X) the abelian
More informationGKSEMINAR SS2015: SHEAF COHOMOLOGY
GKSEMINAR SS2015: SHEAF COHOMOLOGY FLORIAN BECK, JENS EBERHARDT, NATALIE PETERNELL Contents 1. Introduction 1 2. Talks 1 2.1. Introduction: Jordan curve theorem 1 2.2. Derived categories 2 2.3. Derived
More information1 Existence of the Néron model
Néron models Setting: S a Dedekind domain, K its field of fractions, A/K an abelian variety. A model of A/S is a flat, separable Sscheme of finite type X with X K = A. The nicest possible model over S
More informationLecture 4: Abelian varieties (algebraic theory)
Lecture 4: Abelian varieties (algebraic theory) This lecture covers the basic theory of abelian varieties over arbitrary fields. I begin with the basic results such as commutativity and the structure of
More informationCANONICAL EXTENSIONS OF NÉRON MODELS OF JACOBIANS
CANONICAL EXTENSIONS OF NÉRON MODELS OF JACOBIANS BRYDEN CAIS Abstract. Let A be the Néron model of an abelian variety A K over the fraction field K of a discrete valuation ring R. Due to work of MazurMessing,
More informationHochschild homology and Grothendieck Duality
Hochschild homology and Grothendieck Duality Leovigildo Alonso Tarrío Universidade de Santiago de Compostela Purdue University July, 1, 2009 Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality
More informationMINIMAL MODELS FOR ELLIPTIC CURVES
MINIMAL MODELS FOR ELLIPTIC CURVES BRIAN CONRAD 1. Introduction In the 1960 s, the efforts of many mathematicians (Kodaira, Néron, Raynaud, Tate, Lichtenbaum, Shafarevich, Lipman, and DeligneMumford)
More informationARITHMETICALLY COHENMACAULAY BUNDLES ON HYPERSURFACES
ARITHMETICALLY COHENMACAULAY BUNDLES ON HYPERSURFACES N. MOHAN KUMAR, A. P. RAO, AND G. V. RAVINDRA Abstract. We prove that any rank two arithmetically Cohen Macaulay vector bundle on a general hypersurface
More informationNOTES ON PROCESI BUNDLES AND THE SYMPLECTIC MCKAY EQUIVALENCE
NOTES ON PROCESI BUNDLES AND THE SYMPLECTIC MCKAY EQUIVALENCE GUFANG ZHAO Contents 1. Introduction 1 2. What is a Procesi bundle 2 3. Derived equivalences from exceptional objects 4 4. Splitting of the
More informationOF AZUMAYA ALGEBRAS OVER HENSEL PAIRS
SK 1 OF AZUMAYA ALGEBRAS OVER HENSEL PAIRS ROOZBEH HAZRAT Abstract. Let A be an Azumaya algebra of constant rank n over a Hensel pair (R, I) where R is a semilocal ring with n invertible in R. Then the
More informationCOMPACTIFICATION OF TAME DELIGNE MUMFORD STACKS
COMPACTIFICATION OF TAME DELIGNE MUMFORD STACKS DAVID RYDH Abstract. The main result of this paper is that separated Deligne Mumford stacks in characteristic zero can be compactified. In arbitrary characteristic,
More informationCHAPTER 0 PRELIMINARY MATERIAL. Paul Vojta. University of California, Berkeley. 18 February 1998
CHAPTER 0 PRELIMINARY MATERIAL Paul Vojta University of California, Berkeley 18 February 1998 This chapter gives some preliminary material on number theory and algebraic geometry. Section 1 gives basic
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 43 AND 44
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 43 AND 44 RAVI VAKIL CONTENTS 1. Flat implies constant Euler characteristic 1 2. Proof of Important Theorem on constancy of Euler characteristic in flat families
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 47 AND 48
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 47 AND 48 RAVI VAKIL CONTENTS 1. The local criterion for flatness 1 2. Basepointfree, ample, very ample 2 3. Every ample on a proper has a tensor power that
More informationAN INTRODUCTION TO AFFINE SCHEMES
AN INTRODUCTION TO AFFINE SCHEMES BROOKE ULLERY Abstract. This paper gives a basic introduction to modern algebraic geometry. The goal of this paper is to present the basic concepts of algebraic geometry,
More informationMODULI SPACES AND DEFORMATION THEORY, CLASS 1. Contents 1. Preliminaries 1 2. Motivation for moduli spaces Deeper into that example 4
MODULI SPACES AND DEFORMATION THEORY, CLASS 1 RAVI VAKIL Contents 1. Preliminaries 1 2. Motivation for moduli spaces 3 2.1. Deeper into that example 4 1. Preliminaries On the off chance that any of you
More information3. Lecture 3. Y Z[1/p]Hom (Sch/k) (Y, X).
3. Lecture 3 3.1. Freely generate qfhsheaves. We recall that if F is a homotopy invariant presheaf with transfers in the sense of the last lecture, then we have a well defined pairing F(X) H 0 (X/S) F(S)
More informationTranscendence theory in positive characteristic
Prof. Dr. Gebhard Böckle, Dr. Patrik Hubschmid Working group seminar WS 2012/13 Transcendence theory in positive characteristic Wednesdays from 9:15 to 10:45, INF 368, room 248 In this seminar we will
More informationON A VANISHING THEOREM OF S. SAITO AND K. SATO. JeanBaptiste Teyssier
ON A VANISHING THEOREM OF S. SAITO AND K. SATO by JeanBaptiste Teyssier Introduction This text is an expanded version of a talk given for the Winter research seminar Chow groups of zero cycles over padic
More informationGOOD MODULI SPACES FOR ARTIN STACKS. Contents
GOOD MODULI SPACES FOR ARTIN STACKS JAROD ALPER Abstract. We develop the theory of associating moduli spaces with nice geometric properties to arbitrary Artin stacks generalizing Mumford s geometric invariant
More informationInfinite root stacks of logarithmic schemes
Infinite root stacks of logarithmic schemes Angelo Vistoli Scuola Normale Superiore, Pisa Joint work with Mattia Talpo, Max Planck Institute Brown University, May 2, 2014 1 Let X be a smooth projective
More informationladic Representations
ladic Representations S. M.C. 26 October 2016 Our goal today is to understand ladic Galois representations a bit better, mostly by relating them to representations appearing in geometry. First we ll
More informationSERRE FINITENESS AND SERRE VANISHING FOR NONCOMMUTATIVE P 1 BUNDLES ADAM NYMAN
SERRE FINITENESS AND SERRE VANISHING FOR NONCOMMUTATIVE P 1 BUNDLES ADAM NYMAN Abstract. Suppose X is a smooth projective scheme of finite type over a field K, E is a locally free O X bimodule of rank
More informationLecture 9  Faithfully Flat Descent
Lecture 9  Faithfully Flat Descent October 15, 2014 1 Descent of morphisms In this lecture we study the concept of faithfully flat descent, which is the notion that to obtain an object on a scheme X,
More informationLogarithmic geometry and moduli
Logarithmic geometry and moduli Lectures at the Sophus Lie Center Dan Abramovich Brown University June 1617, 2014 Abramovich (Brown) Logarithmic geometry and moduli June 1617, 2014 1 / 1 Heros: Olsson
More informationNOTES ON THE CONSTRUCTION OF THE MODULI SPACE OF CURVES
NOTES ON THE CONSTRUCTION OF THE MODULI SPACE OF CURVES DAN EDIDIN The purpose of these notes is to discuss the problem of moduli for curves of genus g 3 1 and outline the construction of the (coarse)
More informationAFFINE PUSHFORWARD AND SMOOTH PULLBACK FOR PERVERSE SHEAVES
AFFINE PUSHFORWARD AND SMOOTH PULLBACK FOR PERVERSE SHEAVES YEHAO ZHOU Conventions In this lecture note, a variety means a separated algebraic variety over complex numbers, and sheaves are Clinear. 1.
More informationTensor Categories and Representation Theory
Tensor Categories and Representation Theory Seminar at the HU Berlin, Summer 2017 Thomas Krämer Date and Venue: Friday, 1517 h, Room 1.115 / RUD 25 Prerequisites: Inscription: At some Fridays we may start
More informationRealizing Families of Landweber Exact Theories
Realizing Families of Landweber Exact Theories Paul Goerss Department of Mathematics Northwestern University Summary The purpose of this talk is to give a precise statement of 1 The HopkinsMiller Theorem
More informationHodge Theory of Maps
Hodge Theory of Maps Migliorini and de Cataldo June 24, 2010 1 Migliorini 1  Hodge Theory of Maps The existence of a Kähler form give strong topological constraints via Hodge theory. Can we get similar
More informationWhat are stacks and why should you care?
What are stacks and why should you care? Milan Lopuhaä October 12, 2017 Todays goal is twofold: I want to tell you why you would want to study stacks in the first place, and I want to define what a stack
More informationA padic GEOMETRIC LANGLANDS CORRESPONDENCE FOR GL 1
A padic GEOMETRIC LANGLANDS CORRESPONDENCE FOR GL 1 ALEXANDER G.M. PAULIN Abstract. The (de Rham) geometric Langlands correspondence for GL n asserts that to an irreducible rank n integrable connection
More informationArtin Approximation and Proper Base Change
Artin Approximation and Proper Base Change Akshay Venkatesh November 9, 2016 1 Proper base change theorem We re going to talk through the proof of the Proper Base Change Theorem: Theorem 1.1. Let f : X
More informationTHE HILBERT STACK. T is proper, flat, and of finite presentation. Let HS mono
THE HILBERT STACK JACK HALL AND DAVID RYDH ABSTRACT. Let π : X S be a morphism of algebraic stacks that is locally of finite presentation with affine stabilizers. We prove that there is an algebraic Sstack
More informationIntroduction and preliminaries Wouter Zomervrucht, Februari 26, 2014
Introduction and preliminaries Wouter Zomervrucht, Februari 26, 204. Introduction Theorem. Serre duality). Let k be a field, X a smooth projective scheme over k of relative dimension n, and F a locally
More informationSPEAKER: JOHN BERGDALL
November 24, 2014 HODGE TATE AND DE RHAM REPRESENTATIONS SPEAKER: JOHN BERGDALL My goal today is to just go over some results regarding HodgeTate and de Rham representations. We always let K/Q p be a
More informationDERIVED EQUIVALENCES AND GORENSTEIN PROJECTIVE DIMENSION
DERIVED EQUIVALENCES AND GORENSTEIN PROJECTIVE DIMENSION HIROTAKA KOGA Abstract. In this note, we introduce the notion of complexes of finite Gorenstein projective dimension and show that a derived equivalence
More information1. THE CONSTRUCTIBLE DERIVED CATEGORY
1. THE ONSTRUTIBLE DERIVED ATEGORY DONU ARAPURA Given a family of varieties, we want to be able to describe the cohomology in a suitably flexible way. We describe with the basic homological framework.
More informationON THE FUNDAMENTAL GROUPS OF LOG CONFIGURATION SCHEMES
Math. J. Okayama Univ. 51 (2009), 1 26 ON THE FUNDAMENTAL GROUPS OF LOG CONFIGURATION SCHEMES Yuichiro HOSHI Abstract. In the present paper, we study the cuspidalization problem for the fundamental group
More informationRigid Geometry and Applications II. Kazuhiro Fujiwara & Fumiharu Kato
Rigid Geometry and Applications II Kazuhiro Fujiwara & Fumiharu Kato Birational Geometry from Zariski s viewpoint S U D S : coherent (= quasicompact and quasiseparated) (analog. compact Hausdorff) U
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43 RAVI VAKIL CONTENTS 1. Facts we ll soon know about curves 1 1. FACTS WE LL SOON KNOW ABOUT CURVES We almost know enough to say a lot of interesting things about
More informationTHE MOTIVE OF THE FANO SURFACE OF LINES. 1. Introduction
THE MOTIVE OF THE FANO SURFACE OF LINES HUMBERTO A. DIAZ Abstract. The purpose of this note is to prove that the motive of the Fano surface of lines on a smooth cubic threefold is finitedimensional in
More informationPreliminary Exam Topics Sarah Mayes
Preliminary Exam Topics Sarah Mayes 1. Sheaves Definition of a sheaf Definition of stalks of a sheaf Definition and universal property of sheaf associated to a presheaf [Hartshorne, II.1.2] Definition
More information6. Lecture cdh and Nisnevich topologies. These are Grothendieck topologies which play an important role in SuslinVoevodsky s approach to not
6. Lecture 6 6.1. cdh and Nisnevich topologies. These are Grothendieck topologies which play an important role in SuslinVoevodsky s approach to not only motivic cohomology, but also to MorelVoevodsky
More informationCHAPTER 1. Étale cohomology
CHAPTER 1 Étale cohomology This chapter summarizes the theory of the étale topology on schemes, culminating in the results on ladic cohomology that are needed in the construction of Galois representations
More informationAN INTRODUCTION TO MODULI SPACES OF CURVES CONTENTS
AN INTRODUCTION TO MODULI SPACES OF CURVES MAARTEN HOEVE ABSTRACT. Notes for a talk in the seminar on modular forms and moduli spaces in Leiden on October 24, 2007. CONTENTS 1. Introduction 1 1.1. References
More informationContributors. Preface
Contents Contributors Preface v xv 1 Kähler Manifolds by E. Cattani 1 1.1 Complex Manifolds........................... 2 1.1.1 Definition and Examples.................... 2 1.1.2 Holomorphic Vector Bundles..................
More informationDescent on the étale site Wouter Zomervrucht, October 14, 2014
Descent on the étale site Wouter Zomervrucht, October 14, 2014 We treat two eatures o the étale site: descent o morphisms and descent o quasicoherent sheaves. All will also be true on the larger pp and
More information