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Math 256B Notes Math 256B Notes James McIvor April 16, 2012 Abstract These are my notes for Martin Olsson's Math 256B: Algebraic Ge- ometry course, taught in the Spring of 2012, to be updated continually throughout the semester. In a few places, I have added an example or fleshed out a few details myself. I apologize for any inaccuracies this may have introduced. Please email any corrections, questions, or suggestions to mcivor at math.berkeley.edu. Contents 0 Plan for the course 4 1 Wednesday, January 18: Differentials 4 1.1 An Algebraic Description of the Cotangent Space . .4 1.2 Relative Spec . .5 1.3 Dual Numbers . .6 2 Friday, January 20th: More on Differentials 7 2.1 Relationship with Leibniz Rule . .7 2.2 Relation with the Diagonal . .8 2.3 K¨ahlerDifferentials . 10 3 Monday, Jan. 23rd: Globalizing Differentials 12 3.1 Definition of the Sheaf of Differentials . 12 3.2 Relating the Global and Affine Constructions . 13 3.3 Relation with Infinitesimal Thickenings . 15 4 Wednesday, Jan. 25th: Exact Sequences Involving the Cotangent Sheaf 16 1 4.1 Functoriality of ΩX=Y ..................... 16 4.2 First Exact Sequence . 18 4.3 Second Exact Sequence . 20 n 5 Friday, Jan. 27th: Differentials on P 22 5.1 Preliminaries . 22 5.2 A Concrete Example . 22 1 5.3 Functorial Characterization of ΩX=Y ............. 23 1 1 6 Monday, January 30th: Proof of the Computation of Ω n 24 PS =S 7 Wednesday, February 1st: Conclusion of the Computation 1 of Ω n 28 PS =S 7.1 Review of Last Lecture's Construction . 28 7.2 Conclusion of the Proof . 29 7.3 An Example . 31 7.4 Concluding Remarks . 31 8 Friday, February 3rd: Curves 32 8.1 Separating Points and Tangent Vectors . 32 8.2 Motivating the Terminology . 32 8.3 A Preparatory Lemma . 33 9 Monday, February 6th: Proof of the Closed Embedding Criteria 34 9.1 Proof of Proposition 8.1 . 34 9.2 Towards Cohomology . 36 10 Wednesday, Feb. 8th: Cohomology of Coherent Sheaves on a Curve 37 10.1 Black Box: Cohomology of Coherent Sheaves on Curves . 37 10.2 Further Comments; Euler Characteristic . 39 10.3 Statement of Riemann-Roch; Reduction to the More Fa- miliar Version . 40 11 Friday, Feb. 10th: Proof and Consequences of Riemann- Roch 41 11.1 Preparatory Remarks . 41 11.2 Proof of the Riemann-Roch Theorem . 42 11.3 Consequences . 43 12 Monday, Feb. 13th: Applying Riemann-Roch to the Study of Curves 44 n 12.1 Using Riemann-Roch to Detect Closed Immersions to P .. 44 12.2 Examples . 45 13 Wednesday, Feb. 15th: \Where is the Equation?" 46 14 Friday, Feb. 17th: Ramification and Degree of a Mor- phism 49 15 Wednesday, Feb. 22nd: Hyperelliptic Curves 51 15.1 Canonical Embeddings . 51 15.2 Hyperelliptic Curves . 52 16 Friday, Feb. 24th: Riemann-Hurwitz Formula 53 16.1 Preparatory Lemmata . 53 16.2 Riemann-Hurwitz Formula . 55 2 17 Monday, Feb. 27th: Homological Algebra I 56 17.1 Motivation . 56 17.1.1 Algebraic topology . 56 17.1.2 Algebraic geometry { a.k.a. the Black Box . 56 17.2 Complexes, cohomology and snakes . 57 17.3 Homotopy . 59 18 Wednesday, Feb. 29th: Homological Algebra II 61 18.1 Additive and abelian categories . 61 18.2 δ-functors . 63 18.3 Left- and right-exact functors . 65 18.4 Universality of cohomology . 66 19 Friday, March 2nd: Homological Algebra III 67 19.1 Derived functors { a construction proposal . 67 19.1.1 Derived functors { resolution of technical issues . 69 19.2 Derived functors { conclusion . 71 19.3 Two useful lemmas . 72 20 Monday, March 5th: Categories of Sheaves Have Enough Injectives 72 21 Wednesday, March 7th: Flasque Sheaves 75 22 Friday, March 9th: Grothendieck's Vanishing Theorem 77 22.1 Statement and Preliminary Comments . 77 22.2 Idea of the Proof; Some lemmas . 78 23 Monday, March 12th: More Lemmas 79 24 Wednesday, March 14th: Proof of the Vanishing Theorem 80 24.1 Base step . 80 24.2 Reduction to X irreducible . 81 24.3 Reduction to the case where F = ZU ............. 81 24.4 Conclusion of the Proof . 83 25 Friday, March 16th: Spectral Sequences - Introduction 83 25.1 Motivating Examples . 83 25.2 Definition . 84 25.3 Spectral Sequence of a Filtered Complex . 85 26 Monday, March 19th: Spectral Sequence of a Double Com- plex 86 26.1 Double Complex and Total Complex . 86 26.2 Application: Derived Functors on the Category of Complexes 88 27 Wednesday, March 21st: Spectral Sequence of an Open Cover 89 28 Friday, March 23rd: Cech^ Cohomology Agrees with De- rived Functor Cohomology in Good Cases 92 3 29 Monday, April 2nd: Serre's Affineness Criterion 96 30 Wednesday, April 4th: Cohomology of Projective Space 99 31 Friday, April 6th: Conclusion of Proof and Applications 101 32 Monday, April 9th: Finite Generation of Cohomology 103 33 Wednesday, April 11th: The Hilbert Polynomial 106 0 Plan for the course 1. Differentials 2. Curve Theory (Assuming Cohomology) 3. Build Cohomology Theory 4. Surfaces (Time Permitting) Professor Olsson's office hours are Wednesdays 9-11AM in 879 Evans Hall. 1 Wednesday, January 18: Differentials 1.1 An Algebraic Description of the Cotangent Space We want to construct the relative cotangent sheaf associated to a mor- phism f : X ! Y . The motivation is as follows. A differential, or du- ally, a tangent vector, should be something like the data of a point in a scheme, together with an “infinitesimal direction vector" at that point. Algebraically, this can be described by taking the morphism Spec k ! Spec k[]=(2) and mapping it into our scheme. These two schemes have the same topological space, and the nilpotency in k[]=(2) is meant to capture our intuition of the “infinitesimal”. More generally, we can take any “infinitesimal thickening", i.e., a closed immersion T0 ,! T defined by a square-zero ideal sheaf J ⊂ OT and map that into X. Of course, we want this notion to be relative, so we should really map this thicken- ing into the morphism f, which amounts to filling in the dotted arrow in following diagram T0 / X _ > T / Y which should be thought of as representing the following picture in the case when Y = Spec k picture omitted/pending 2 Example 1.1. Take Y = Spec k, T = Spec k and T0 = Spec k[]=( ). Consider the set of morphisms filling in following diagram: 4 x2X Spec k / X or, equivalently, k o OX;x 8 O O 7!0 z Spec k[]=(2) / Spec k k[]=(2) o k Commutativity of the square forces the composite k ! k to be the identity, so the maximal ideal m ⊂ OX;x maps to zero in k, hence along the dotted arrow, m gets mapped into (), giving a new diagram a k o O =m2 O X;xO 7!0 b y k[]=(2) o k 2 2 But k ⊕ m=m =∼ OX;x=m via the map (α; β) 7! b(α) + β. More- over, k ⊕ m=m2 can be given the structure of a k-algebra by defining the multiplication (α; β) · (α0; β0) = (αα0; αβ0 + α0β): Also, as k-algebras, k[]=(2) =∼ k ⊕ k · , so in fact we are looking for dotted arrows filling in the following diagram of k-algebras: a k o k ⊕ m=m2 O O 7!0 b y k[]=(2) o k Such arrows are determined by the map m=m2 ! k. Thus the set of arrows filling in diagram 1 is in bijection with (m=m2)_. This justifies the definition of the tangent space of X over k at x as (m=m2)_. Greater generality can be achieved by working with X as a scheme over some arbitrary scheme Y , and also by replacing the ring of dual numbers k[]=(2) by a more general construction. Before doing so, we need the notion of relative Spec. 1.2 Relative Spec Given a scheme X and a quasicoherent sheaf of algebras A on X, we show how to construct a scheme Spec (A). Loosely speaking, the given sheaf X A includes both the data of a ring over each affine open of X, and the gluing data for these rings. We take Spec of each ring A(U) for opens U of X, and glue them according to the specifications of the sheaf A. Alternatively, Spec (A) is the representing object for the functor X F : (Schemes=X)op ! Set f ∗ which sends T ! X to the set of OT -algebra morphisms f A!OT . Two things need to be checked, namely that this functor is representable at all, 5 and that the scheme obtained by gluing described above actually is the representing object. Here's why it's locally representable: If X and T are affine, say X = Spec B, and T = Spec R, with A = A~ ∗ ∼ for some B-algebra A, then f A = (A⊗B R) , by definition of the pullback sheaf. By the equivalence of categories between quasicoherent sheaves of ∗ OT -modules and R-modules, the set of OT -algebra maps f A!OT is in natural bijection with the set of R-algebra maps A ⊗B R ! R, so we are looking for dotted maps in the following diagram A ⊗B R / R O w; ww ww ww ww R w But the vertical map on the left fits into a fibered square as follows A / A ⊗B R / R O O w; ww ww ww ww B / R w Moreover, to give a map of R-algebras A ⊗B R ! R is equivalent, by the universal property of the fibered square, to giving a map A ! R along the top: ( A / A ⊗B R / R O O w; ww ww ww ww B / R w ∗ This shows that in this case, the set of maps of OT -algebras f A!OT is in bijection with the maps of R-algebras A ! R, and hence that Spec A represents the functor F .
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