DOCSLIB.ORG

Notes for Math 274 - Stacks

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Notes for Math 274 - Stacks Notes for Math 274 - Stacks Contents 0 How these notes came to exist 4 1 Motivation: non-representable functors 5 2 Grothendieck topologies 9 3 Sheaves and Topoi 11 Limits ....................................... 13 4 Continuous functors between sites 15 5 When f ∗ commutes with finite limits 18 Faithfullyflatdescent .............................. 19 6 Representable functors are fppf sheaves 21 7 Descent 25 Descentingeneral ................................ 25 Descent for morphisms of sheaves/schemes . ... 26 Descentforsheavesinasite ........................... 27 Descentforsheavesofmodules . 29 Descent for quasi-coherent sheaves . .. 30 8 Descent for Mg, g ≥ 2 33 9 Separated schemes: a warmup for algebraic spaces 35 10 Properties of Sheaves and Morphisms 39 11 Algebraic spaces 43 Quotientsbyfreeactionsoffinitegroups . 43 Aquotientbyanon-freeaction . 46 Quotientsbyrelations .............................. 46 12 Properties of Algebraic Spaces. Etale´ Relations. 48 13 Affine/(Finite Etale´ Relation) = Affine, Part I 53 14 Affine/(Finite Etale´ Relation) = Affine, Part II 59 15 Quasi-coherent Sheaves on Algebraic Spaces 61 Pullbacks of quasi-coherent sheaves . 62 2 16 Relative Spec 65 17 Separated, quasi-finite, locally finite type =⇒ quasi-affine 69 18 Chow’s Lemma. 74 19 Sheaf Cohomology 76 HigherDirectImagesofProperMaps. 77 Coherenceofhigherdirectimages . 80 Underlying topological space of an algebraic space . .. 82 21 Fibered categories 84 22 The 2-Yoneda lemma 89 Presheaves and categories fibered in sets . .. 90 23 Split fibered categories 92 24 Stacks 96 25 Groupoids. Stackification. 99 26 Quotients by group actions 103 27 Algebraic Stacks 106 28 More about Algebraic Stacks; Examples 109 29 Hilb and Quot 112 30 Sheaf cohomology and Torsors 116 31 Gerbes 122 32 Properties of algebraic stacks 129 33 X Deligne-Mumford ⇔ ∆X formally unramified 132 Artin’sTheorem ................................. 137 35 The Lisse-´etale site on an algebraic stack 139 36 Quasi-coherent sheaves on algebraic stacks 143 37 Push-forward of Quasi-coherent sheaves 146 3 38 Keel-Mori 148 Coarsemodulispaces............................... 148 Proof:I ...................................... 150 Proof:II...................................... 153 Proof:III ..................................... 156 Proof:IV ..................................... 160 42 Cohomological descent 163 MoreCohomologicalDescent . 166 44 Brauer Groups and Gabber’s Theorem 169 Appendix 172 ∗ ⋆ A1 Verification of the adjunctions f ⊢ f∗ and f ⊢ f∗ ............ 172 A2 Extendingproperties ............................ 172 A3 EffectiveDescentClasses. 172 A4 DescentforAlgebraicSpaces. 173 A5 2-Categories................................. 174 E Exercises and solutions 178 Exerciseset1 ................................... 178 Exerciseset2 ................................... 180 Exerciseset3 ................................... 182 Exerciseset4 ................................... 184 Exerciseset5 ................................... 184 Exerciseset6 ................................... 184 Exerciseset7 ................................... 184 References 185 4 0 How these notes came to exist 0 How these notes came to exist In Spring 2007, Martin Olsson taught “Math 274—Topics in Algebra—Stacks” at UC Berkeley. Anton Geraschenko LATEXed these notes in class,1 and edited them with other people in the class. They still get modified sometimes. They should be available at http://math.berkeley.edu/~anton/index.php?m1=writings The most recent version of the source files are in in the SVN repository svn://sheafy.net/courses/stacks_sp2007 You can make commits to the repository with the username “guest” and the empty password, but I would rather you email me (geraschenko@gmail.com) so I can set up a user for you. – When something doesn’t make sense to me, I mark it with three big, eye-catching stars [[⋆⋆⋆ like this]]. If you can clear any of these up for me, let me know. – If you have notes that I’m missing or if you have a correct/clear explanation for something which is incorrect/unclear, let me know (either tell me what you’d like to modify, give me some notes to go on, or update the tex yourself and send me a copy). Real (mathematical) errors should be fixed because it would be immoral to let them propagate (er . that is, sit there), and typographical errors hardly take any time to fix, so you shouldn’t be shy about telling me about them. 1With the exception of two lectures (originally 30 and 31, but the content has been mixed up with nearby lectures), which were reconstructed from the notes of Tony Varilly, Ed Carter, and Anne Shiu, along with the usual conversation that went into editing. 1 Motivation: non-representable functors 5 1 Motivation: non-representable functors You should do homework (especially if you’re enrolled). You’ll have to do a lot of work, even if you already know a lot. We’ll try to organize a discussion section. The prerequisite is schemes. The references are good; you should look at them. Vistoli’s notes on Grothendieck topologies are good; Knutson’s book is good, so we’ll try to put it on reserve in the library. In a (Nov. 5 1959) letter from Grothendieck to Serre, Grothendieck talks about moduli spaces and says that he keeps running up against the problem that objects have automorphisms. A main point for today is that many interesting functors are not representable. If op X is a scheme, then we get a functor hX : Sch → Set given by Y 7→ Hom(Y,X). op Lemma 1.1 (Yoneda). The functor h− : Sch → Fun(Sch , Set) is fully faithful. op Definition 1.2. A functor F : Sch → Set is representable if F ≃ hX for some X. When you represent a functor F , you give the scheme X together with the natural isomorphism F ≃ hX . When you do this, X is unique up to unique isomorphism. ⋄ n n n ′ Example 1.3. A : Y 7→ Γ(Y, OY ) . On morphisms, A takes g : Y → Y to the ∗ n ′ n pull-back map g : Γ(Y, OY ) → Γ(Y , OY ′ ) . Let X = Spec Z[x1,...,xn]. Then ∼ ∼ n n Hom(Y,X) = Homalg Z[x1,...,xn], Γ(Y, OY ) = Γ(Y, OY ) = A (Y ) is a natural isomorphism, so X represents An. ⋄ Example 1.4. An r {0} should be a sub-functor of An. We define it by Y 7→ n (y1,...,yn) ∈ Γ(Y, OY ) |for every p ∈ Y , the images of the yi are not all zero in k(p) . In the homework, you will show that this functor is representable. ⋄ e Definition 1.5. An elliptic curve over a scheme Y is a diagram E u / Y where e f is a section of f and the fibers of f are genus 1 curves. ⋄ Example 1.6. Let M1,1 be the functor defined by Y 7→ {isoclasses of elliptic curves ′ over Y }. If g : Y → Y is a morphism of schemes, then we define M1,1(g): M1,1(Y ) → ′ ∗ M1,1(Y ) to be the usual pull-back g . M (g)(E)= g∗E / E g∗E / E 1,1 O = V · ✤ ④④ g∗f f ∃! ✤ ④④ e ✤ ④④e◦g ④④ ′ g / Y ′ / Y Y Y id M g ′ ∗ The section Y → g E is induced by idY and e ◦ g by the universal property of pull- backs. ⋄ Proposition 1.7. M1,1 is not representable. 6 1 Motivation: non-representable functors The intuitive reason that M1,1 is not representable is the following lemma. It says that you cannot have any kind of twisting of bundles. Lemma 1.8. If F is a representable functor, with s1,s2 ∈ F (Y ) and a covering Y = Ui such that s1|Ui = s2|Ui for all i, then s1 = s2. ∼ Proof.S We have that F = hX for some X, and s1 and s2 are given by morphisms Y → X that they agree on a cover of Y . Since morphisms glue, s1 = s2. Unfortunately, to show that M1,1 is not representable via this Lemma, you need to generalize your notion of covering (to ´etale covers). We’ll see these later. For now we’ll give another proof. Proof of 1.7. Assume there is a scheme M and an isomorphism M1,1 ≃ hM . Let k be an algebraically closed field of characteristic not 2. Consider R = k[λ]λ(1−λ), so 1 2 Spec R is Ak with 0 and 1 removed. Let E ⊆ PR be the closed subscheme defined by 2 y z = x(x − z)(x − λz), so the fibers Eλ of the natural map E → Spec R are genus 1 curves. We define e : Spec R → D(z) ∼= Spec R[x, y] by the map R[x, y] → R, x, y 7→ 0 and observe that the image of e lies in E, and e is a section of E → Spec R. Observe that there is an action of S3 on R generated by σ0 : λ 7→ 1/λ and σ1 : λ 7→ 8 (λ2−λ+1)3 1/(1 − λ). Let j =2 λ2(λ−1)2 ∈ k(λ). Lemma 1.9. The fixed points k(λ)S3 are exactly the elements of k(j). Proof. First check that j ∈ k(λ)S3 . Then we have that k(j) ⊆ k(λ)S3 ⊆ k(λ). By Galois theory, the second extension is degree 6, and the total extension is degree 6, so the first two fields are equal. Lemma There are two steps remaining in the proof. (1) Let Eη be the generic fiber of E → Spec R. It is given by a map φ : Spec k(λ) → M. We claim that this map has to factor through the obvious map g : Spec k(λ) → Spec k(j). M φ ! E / E˜ ! ψ < ❉b η φ ②② ❉ ψ · ②② ②② ❉ ②② ❉ g g η = Spec k(λ) / Spec k(j) Spec k(λ) / Spec k(j) As we see from the diagram on the left, a factorization of φ is the same thing as a morphism Hom(g, M): φ 7→ ψ. By the natural isomorphism hM ≃ M1,1, this is the same as a morphism M1,1(g): E˜ 7→ E, where E˜ is the elliptic curve over k(j) defined by ψ, as shown in the diagram on the right. Thus, we have Eη = E˜ ×Spec k(j) Spec k(λ).
Load more

Recommended publications
  • 6. Galois Descent for Quasi-Coherent Sheaves 19 7
    DESCENT 0238 Contents 1. Introduction 2 2. Descent data for quasi-coherent sheaves 2 3. Descent for modules 4 4. Descent for universally injective morphisms 9 4.1. Category-theoretic preliminaries 10 4.3. Universally injective morphisms 11 4.14. Descent for modules and their morphisms 12 4.23. Descent for properties of modules 15 5. Fpqc descent of quasi-coherent sheaves 17 6. Galois descent for quasi-coherent sheaves 19 7. Descent of finiteness properties of modules 21 8. Quasi-coherent sheaves and topologies 23 9. Parasitic modules 31 10. Fpqc coverings are universal effective epimorphisms 32 11. Descent of finiteness and smoothness properties of morphisms 36 12. Local properties of schemes 39 13. Properties of schemes local in the fppf topology 40 14. Properties of schemes local in the syntomic topology 42 15. Properties of schemes local in the smooth topology 42 16. Variants on descending properties 43 17. Germs of schemes 44 18. Local properties of germs 44 19. Properties of morphisms local on the target 45 20. Properties of morphisms local in the fpqc topology on the target 47 21. Properties of morphisms local in the fppf topology on the target 54 22. Application of fpqc descent of properties of morphisms 54 23. Properties of morphisms local on the source 57 24. Properties of morphisms local in the fpqc topology on the source 58 25. Properties of morphisms local in the fppf topology on the source 58 26. Properties of morphisms local in the syntomic topology on the source 59 27. Properties of morphisms local in the smooth topology on the source 59 28.
    [Show full text]
  • Simplicial Presheaves and Derived Algebraic Geometry. Bertrand Toen
    Simplicial presheaves and derived algebraic geometry. Bertrand Toen To cite this version: Bertrand Toen. Simplicial presheaves and derived algebraic geometry.. Simplicial Methods for Operads and Algebraic Geometry, Springer Basel, pp.119-185, 2010, Advanced Courses in Mathematics - CRM Barcelona, 978-3-0348-0051-8. 10.1007/978-3-0348-0052-5. hal-00772850v1 HAL Id: hal-00772850 https://hal.archives-ouvertes.fr/hal-00772850v1 Submitted on 11 Jan 2013 (v1), last revised 18 Feb 2013 (v2) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Simplicial presheaves and derived algebraic geometry Bertrand To¨en Laboratoire Emile Picard Universit´ePaul Sabatier, Bat 1R2 31062 Toulouse Cedex , France email: bertrand.toen@math.univ-toulouse.fr May 2009 2 Contents 1 Motivations and objectives 7 1.1 The notion of moduli spaces . 7 1.2 Construction of moduli spaces: one example . 8 1.3 Conclusions . 12 2 Simplicial presheaves as stacks 13 2.1 Review of the model category of simplicial presheaves . 13 2.2 Basic examples . 19 3 Algebraic stacks 29 3.1 Schemes and algebraic n-stacks . 30 3.2 Some examples . 33 3.3 Coarse moduli spaces and homotopy sheaves .
    [Show full text]
  • SHEAVES on ARTIN STACKS 1.1. in the Book ([LM-B])
    SHEAVES ON ARTIN STACKS MARTIN OLSSON Abstract. We develop a theory of quasi–coherent and constructible sheaves on algebraic stacks correcting a mistake in the recent book of Laumon and Moret-Bailly. We study basic cohomological properties of such sheaves, and prove stack–theoretic versions of Grothendieck’s Fundamental Theorem for proper morphisms, Grothendieck’s Existence Theorem, Zariski’s Connectedness Theorem, as well as finiteness Theorems for proper pushforwards of coherent and constructible sheaves. We also explain how to define a derived pullback functor which enables one to carry through the construction of a cotangent complex for a morphism of algebraic stacks due to Laumon and Moret–Bailly. 1.1. In the book ([LM-B]) the lisse-´etale topos of an algebraic stack was introduced, and a theory of quasi–coherent and constructible sheaves in this topology was developed. Unfor- tunately, it was since observed by Gabber and Behrend (independently) that the lisse-´etale topos is not functorial as asserted in (loc. cit.), and hence the development of the theory of sheaves in this book is not satisfactory “as is”. In addition, since the publication of the book ([LM-B]), several new results have been obtained such as finiteness of coherent and ´etale cohomology ([Fa], [Ol]) and various other consequences of Chow’s Lemma ([Ol]). The purpose of this paper is to explain how one can modify the arguments of ([LM-B]) to obtain good theories of quasi–coherent and constructible sheaves on algebraic stacks, and in addition we provide an account of the theory of sheaves which also includes the more recent results mentioned above.
    [Show full text]
  • Brauer Groups and Etale Cohomology in Derived Algebraic Geometry
    Brauer groups and etale´ cohomology in derived algebraic geometry Benjamin Antieau∗ and David Gepner October 2, 2012 Abstract In this paper, we study Azumaya algebras and Brauer groups in derived algebraic geometry. We establish various fundamental facts about Brauer groups in this setting, and we provide a com- putational tool, which we use to compute the Brauer group in several examples. In particular, we show that the Brauer group of the sphere spectrum vanishes, and we use this to prove two uniqueness theorems for the stable homotopy category. Our key technical results include the lo- cal geometricity, in the sense of Artin n-stacks, of the moduli space of perfect modules over a smooth and proper algebra, the ´etale local triviality of Azumaya algebras over connective derived schemes, and a local to global principle for the algebraicity of stacks of stable categories. Key Words Commutative ring spectra, derived algebraic geometry, moduli spaces, Azumaya algebras, and Brauer groups. Mathematics Subject Classification 2010 Primary: 14F22, 18G55. Secondary: 14D20, 18E30. Contents 1 Introduction 2 1.1 Setting......................................... 2 1.2 Summary ....................................... 5 1.3 Acknowledgments ................................. 9 arXiv:1210.0290v1 [math.AG] 1 Oct 2012 2 Ring and module theory 9 2.1 Ringsandmodules ................................. 9 2.2 Compactobjectsandgenerators . ...... 9 2.3 Topologieson affineconnectivederivedschemes . 12 2.4 Tor-amplitude................................... 13 2.5 Vanishingloci................................... 15 3 Module categories and their module categories 16 3.1 R-linearcategories .................................. 16 3.2 Smoothandproperalgebras . .... 20 3.3 Azumayaalgebras................................. 22 ∗This material is based upon work supported in part by the NSF under Grant No. DMS-0901373. 1 4 Sheaves 23 4.1 Stacksofalgebraandmodulecategories .
    [Show full text]
  • Derived Algebraic Geometry
    EMS Surv. Math. Sci. 1 (2014), 153–240 EMS Surveys in DOI 10.4171/EMSS/4 Mathematical Sciences c European Mathematical Society Derived algebraic geometry Bertrand Toën Abstract. This text is a survey of derived algebraic geometry. It covers a variety of general notions and results from the subject with a view on the recent developments at the interface with deformation quantization. Mathematics Subject Classification (2010). 14A20, 18G55, 13D10. Keywords. Derived scheme, derived moduli, derived stack. Contents 1 Selected pieces of history . 159 2 The notion of a derived scheme . 170 2.1 Elements of the language of 1-categories . 173 2.2 Derived schemes . 184 3 Derived schemes, derived moduli, and derived stacks . 189 3.1 Some characteristic properties of derived schemes . 189 3.2 Derived moduli problems and derived schemes . 194 3.3 Derived moduli problems and derived Artin stacks . 198 3.4 Derived geometry in other contexts . 203 4 The formal geometry of derived stacks . 205 4.1 Cotangent complexes and obstruction theory . 205 4.2 The idea of formal descent . 207 4.3 Tangent dg-lie algebras . 209 4.4 Derived loop spaces and algebraic de Rham theory . 211 5 Symplectic, Poisson and Lagrangian structures in the derived setting . 215 5.1 Forms and closed forms on derived stacks . 215 Bertrand Toën, Université de Montpellier 2, Case courrier 051, Bât 9, Place Eugène Bataillon, Montpellier Cedex 5, France E-mail: bertrand.toen@um2.fr 154 Bertrand Toën 5.2 Symplectic and Lagrangian structures . 220 5.3 Existence results . 223 5.4 Polyvectors and shifted Poisson structures .
    [Show full text]
  • Lectures on Etale Cohomology
    Lectures on Etale Cohomology J.S. Milne Version 2.10 May 20, 2008 These are the notes for a course taught at the University of Michigan in 1989 and 1998. In comparison with my book, the emphasis is on heuristic arguments rather than formal proofs and on varieties rather than schemes. The notes also discuss the proof of the Weil conjectures (Grothendieck and Deligne). BibTeX information @misc{milneLEC, author={Milne, James S.}, title={Lectures on Etale Cohomology (v2.10)}, year={2008}, note={Available at www.jmilne.org/math/}, pages={196} } v2.01 (August 9, 1998). First version on the web; 197 pages. v2.10 (May 20, 2008). Fixed many minor errors; changed TeX style; 196 pages Please send comments and corrections to me at math0 at jmilne.org. The photograph is of the Dasler Pinnacles, New Zealand. Copyright c 1998, 2008, J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Contents Contents 3 I Basic Theory 7 1 Introduction . 7 2 Etale Morphisms . 16 3 The Etale Fundamental Group . 25 4 The Local Ring for the Etale Topology . 31 5 Sites . 38 6 Sheaves for the Etale Topology . 41 7 The Category of Sheaves on Xet........................ 49 8 Direct and Inverse Images of Sheaves. 57 9 Cohomology: Definition and the Basic Properties. 63 10 Cechˇ Cohomology. 68 11 Principal Homogeneous Spaces and H 1.................... 74 12 Higher Direct Images; the Leray Spectral Sequence. 79 13 The Weil-Divisor Exact Sequence and the Cohomology of Gm. 82 14 The Cohomology of Curves.
    [Show full text]
  • ´Etale Cohomology I I H´Et(X,F ) := H (X´Et,F )
    Etale´ cohomology Prof. Dr. Uwe Jannsen Summer Term 2015 Inhaltsverzeichnis 1 Introduction 1 2 Grothendieck topologies/Sites 2 3 Constructions for presheaves and sheaves 4 4 The abelian categories of sheaves and presheaves 14 4.A Representable functors, limits, and colimits 17 4.B Filtered categories 30 5 Cohomology on sites 32 6 Spectral sequences 37 7 The ´etalesite 49 8 The ´etalesite of a field 57 9 Henselian rings 62 10 Examples of ´etalesheaves 72 11 The decomposition theorem 80 12 Cechˇ cohomology 86 13 Comparison of sites 94 14 Descent theory and the multiplicative group 98 15 Schemes of dimension 1 105 1 Introduction In mathematics, one often looks for invariants which characterize or classify the regarded objects. Often such invariants are given by cohomology groups. This is a long standing approach in topology, where one considers singular cohomology groups Hi(X; Q) of a topological space X, which are defined by explicit `cycles' and `boun- daries'. These suffice to determine the genus g of a (compact) Riemann surface: If X looks topologically like a sphere with g handles: g=1 g=2 1 then dimQ H (X; Q) = 2g. These cohomology groups can also be obtained as sheaf cohomo- logy (of a constant sheaf). Riemann surfaces can also be regarded as complex algebraic curves, i.e., as algebraic curves over a field C of complex numbers. For any algebraic varieties X over any field k (or any scheme) one can consider sheaf cohomology with respect to the Zariski topology. This is useful for coherent sheaves, for example for the Grothendieck-Serre duality and the Riemann-Roch theorem.
    [Show full text]
  • 18.726 Algebraic 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 (K.S. Kedlaya, MIT, Spring 2009) E´ tale cohomology (updated 13 May 09) In this lecture, we give a hint of the theory of ´etale cohomology. Standard references: Milne, E´ tale Cohomology (he also has some more accessible lecture notes online at jmilne.org); Tamme, Introduction to E´ tale Cohomology; Freitag-Kiehl, E´ tale Cohomology and the Weil Conjectures. You might want to read Hartshorne Appendix C first for an overview. Also, note that there is a “rogue” volume of SGA called SGA 4 1/2, written mostly by Deligne after the fact, which gives a surprisingly legible (albeit in French) account of this stuff. Since this is the last lecture of the course, I would like to take the opportunity to thank you, the participants, for all the hard work you put in on the problem sets, and especially for all your feedback on the notes. If you have further questions about algebraic geometry, from the general to the specific, I would be happy to discuss them! 1 Motivation: the Weil conjectures Let X be a variety over a finite field Fq. Weil predicted that the zeta function of X, defined as an Euler product deg(x→Fq ) −1 ζX (T ) = �(1 T ) − x over the closed points of X, could always be interpreted as the power series expansion of a rational function of T ; this analogizes the analytic continuation of the Riemann zeta function.
    [Show full text]
  • Tame Stacks and Log Flat Torsors
    Tame stacks and log flat torsors Jean Gillibert March 2012 Abstract We compare tame actions in the category of schemes with torsors in the category of log schemes endowed with the log flat topology. We prove that actions underlying log flat torsors are tame. Conversely, starting from a tame cover of a regular scheme that is a fppf torsor on the complement of a divisor with normal crossings, it is possible to build a log flat torsor that dominates this cover. In brief, the theory of log flat torsors gives a canonical approach to the problem of extending torsors into tame covers. 1 Introduction Let X be a scheme, and let G be a finite locally free group scheme over X. Let Y an X-scheme on which G acts. What does it means for the action of G on Y to be tame ? A large amount of work has been done on this question in the past few years. Let us first mention the foundational paper by Chinburg, Erez, Pappas and Taylor [CEPT96], where a notion of tame action was introduced. This notion was used in many subsequent papers. In the present text, we refer to it as CEPT-tameness. More recently, and independently, Abramovich, Olsson and Vistoli have introduced in arXiv:1203.6870v1 [math.AG] 30 Mar 2012 [AOV08] the notion of tame stack. Sophie Marques proved in [Mar12] that, if the quotient Y/G exists in the category of schemes, and if the morphism Y → Y/G is flat, then these two notions of tameness coincide, namely, the action of G of Y is CEPT-tame if and only if the quotient stack [Y/G] is tame (see Remark 3.7).
    [Show full text]
  • Contents 1 Introduction (Martin Olsson)
    STAX Contents 1 Introduction (Martin Olsson) 1 2 Background: schemes as functors (Andrew Niles) 2 2.1 Category theory . .2 2.2 Schemes . .3 2.3 Sheaves . .3 3 Sites and sheaves (Slater Stich) 4 4 Fibered categories (Aaron Mazel-Gee) 6 4.1 Definitions and basic facts . .6 4.2 The 2-Yoneda lemma . .7 4.3 Categories fibered in groupoids . .8 4.3.1 ... coming from co/groupoid objects . .8 4.3.2 ... and 2-categorical fiber product thereof . .9 5 Faithfully flat descent (Andrew Dudzik) 10 6 Algebraic spaces (Yuhao Huang) 12 7 Coarse moduli spaces, part 1 (Shelly Manber) 13 8 Quasicoherent sheaves of modules on algebraic spaces (Jason Ferguson) 15 9 Algebraic stacks (Doosung Park) 17 10 Quasicoherent sheaves on algebraic stacks (Katrina Honigs) 19 10.1 Stacks and algebraic stacks . 19 10.2 Quasicoherent sheaves on stacks . 20 11 Mg is a Deligne-Mumford stack (Martin Olsson) 20 12 Coarse moduli spaces and the Keel-Mori theorem (Piotr Achinger) 22 12.1 Moduli problems . 22 12.2 The Keel-Mori theorem . 23 12.3 Tameness and quasicoherent sheaves . 24 13 Gerbes (Peter Mannisto) 24 13.1 Torsors . 24 13.2 Gerbes . 25 1 Introduction (Martin Olsson) Motivation for stacks comes from: • algebraic topology; 1 • moduli theory; • not working with one arm tied behind your back. In algebraic topology, we might like to talk about classifying spaces. But one quickly learns that algebraic varieties have such rigid structure that we can't make the same constructions. In topology, cohomology is a representable functor: for any abelian group A and nonnegative number n, there is a space K(A; n) such that Hn(X; A) ' [X; K(A; n)] (where the square brackets denote homotopy classes of maps).
    [Show full text]
  • Geometric Langlands in Prime Characteristic Tsao-Hsien Chen
    Geometric Langlands in Prime Characteristic by Tsao-Hsien Chen Bachelor of Science, National Taiwan University (2004) Submitted to the Department of Mathematics ARCHIVES in partial fulfillment of the requirements for the degree of M4\13SAGHU_- - 1T " Doctor of Philosophy at the 1 I MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2012 © Massachusetts Institute of Technology 2012. All rights reserved. A uthor ............................................... Department of Mathematics May 1, 2012 A Certified by...... .7. Roman Bezrukavnikov Professor of Mathematics, MIT Thesis Supervisor Accepted by .... 7, Bjorn Poonen Chairman, Department Committee on Graduate Theses 2 Geometric Langlands in Prime Characteristic by Tsao-Hsien Chen Submitted to the Department of Mathematics on May 1, 2012, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract Let C be a smooth projective curve over an algebraically closed field k of sufficiently large characteristic. Let G be a semisimple algebraic group over k and let GV be its Langlands dual group over k. Denote by BunG the moduli stack of G-bundles on C and LocSysGv the moduli stack of Gv-local systems on C. Let DBunG be the sheaf of crystalline differential operator algebra on BunG. In this thesis I construct an equivalence between the derived category D(QCoh(LocSys~v)) of quasi-coherent sheaves on some open subset LocSysov C LocSysGv and derived category D(DOunG mod) of modules over some localization DBunG of DBunG. This generalizes the work of Bezrukavnikov-Braverman in the GL, case. Thesis Supervisor: Roman Bezrukavnikov Title: Professor of Mathematics, MIT 3 4 Acknowledgments I would like to thank my advisor, Roman Bezrukavnikov, for his guidance, support and encouragement for the past five years.
    [Show full text]
  • Lectures on Etale Cohomology
    Lectures on Etale Cohomology J.S. Milne Version 2.10 May 20, 2008 These are the notes for a course taught at the University of Michigan in 1989 and 1998. In comparison with my book, the emphasis is on heuristic arguments rather than formal proofs and on varieties rather than schemes. The notes also discuss the proof of the Weil conjectures (Grothendieck and Deligne). BibTeX information @misc{milneLEC, author={Milne, James S.}, title={Lectures on Etale Cohomology (v2.10)}, year={2008}, note={Available at www.jmilne.org/math/}, pages={196} } v2.01 (August 9, 1998). First version on the web; 197 pages. v2.10 (May 20, 2008). Fixed many minor errors; changed TeX style; 196 pages Please send comments and corrections to me at math0 at jmilne.org. The photograph is of the Dasler Pinnacles, New Zealand. Copyright c 1998, 2008, J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Contents Contents 3 I Basic Theory 7 1 Introduction . 7 2 Etale Morphisms . 16 3 The Etale Fundamental Group . 25 4 The Local Ring for the Etale Topology . 31 5 Sites . 38 6 Sheaves for the Etale Topology . 41 7 The Category of Sheaves on Xet........................ 49 8 Direct and Inverse Images of Sheaves. 57 9 Cohomology: Definition and the Basic Properties. 63 10 Cechˇ Cohomology. 68 11 Principal Homogeneous Spaces and H 1.................... 74 12 Higher Direct Images; the Leray Spectral Sequence. 79 13 The Weil-Divisor Exact Sequence and the Cohomology of Gm. 82 14 The Cohomology of Curves.
    [Show full text]