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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 ([email protected]) 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).
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