math 216: foundations of algebraic geometry - Stanford University

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$math 216: foundations of algebraic geometry - Stanford University$

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$RICHARD SCHOEN - SPECTRAL GEOMETRY (MATH 286 ...$

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19.1. Motivating example: blowing up the origin in the plane 415 19.2. Blowing up, by universal property 417 19.3. The blow-up exists, and is projective 421 19.4. Examples and computations 425 Chapter 20. Čech cohomology of quasicoherent sheaves 433 20.1. (Desired) properties of cohomology 433 20.2. Definitions and proofs of key properties 438 20.3. Cohomology of line bundles on projective space 443 20.4. Riemann-Roch, degrees of coherent sheaves, arithmetic genus, and Serre duality 445 20.5. Hilbert polynomials, genus, and Hilbert functions 452 20.6. ⋆ Serre’s cohomological characterization of ampleness 458 20.7. Higher direct image sheaves 460 20.8. ⋆ “Proper pushforwards of coherents are coherent”, and Chow’s lemma 464 Chapter 21. Application: Curves 469 21.1. A criterion for a morphism to be a closed embedding 469 21.2. A series of crucial tools 471 21.3. Curves of genus 0 474 21.4. Classical geometry arising from curves of positive genus 475 21.5. Hyperelliptic curves 477 21.6. Curves of genus 2 481 21.7. Curves of genus 3 482 21.8. Curves of genus 4 and 5 483 21.9. Curves of genus 1 486 21.10. Elliptic curves are group varieties 493 21.11. Counterexamples and pathologies from elliptic curves 497 Chapter 22. ⋆ Application: A glimpse of intersection theory 501 22.1. Intersecting n line bundles with an n-dimensional variety 501 22.2. Intersection theory on a surface 505 22.3. ⋆⋆ Nakai and Kleiman’s criteria for ampleness 509 Chapter 23. Differentials 513 23.1. Motivation and game plan 513 23.2. Definitions and first properties 514 23.3. Examples 526 23.4. Nonsingularity and k-smoothness revisited 529 23.5. The Riemann-Hurwitz Formula 534 Part VI. More 541 Chapter 24. Derived functors 543 24.1. The Tor functors 543 24.2. Derived functors in general 547 24.3. Derived functors and spectral sequences 549 24.4. ⋆ Derived functor cohomology of O-modules 554 24.5. ⋆ Čech cohomology and derived functor cohomology agree 556 Chapter 25. Flatness 563
25.1. Introduction 563 25.2. Easier facts 565 25.3. Flatness through Tor 569 25.4. Ideal-theoretic criteria for flatness 571 25.5. Topological aspects of flatness 577 25.6. Local criteria for flatness 581 25.7. Flatness implies constant Euler characteristic 584 25.8. Cohomology and base change: Statements and applications 587 25.9. ⋆ Proofs of cohomology and base change theorems 591 25.10. ⋆ Flatness and completion 597 Chapter 26. Smooth, étale, unramified [in progress] 599 26.1. Some motivation 599 26.2. Definitions and easy consequences 600 26.3. Harder facts: Left-exactness of the relative cotangent and conormal sequences in the presence of smo 26.4. Generic smoothness results 605 26.5. Bertini’s Theorem 608 26.6. ⋆⋆ Formally unramified, smooth, and étale 611 Chapter 27. Formal functions draft [in proress] 613 Chapter 28. Regular sequences [in progress] 615 28.1. Regular sequence related facts from earlier 615 28.2. Results from [Gr-d] 618 28.3. What we still want for the Serre duality chapter 619 28.4. Older notes on adjunction 619 28.5. Cohen-Macaulay rings and schemes (should go at start) 620 Chapter 29. Twenty-seven lines 625 29.1. Introduction 625 29.2. Preliminary facts 626 29.3. Every smooth cubic surface (over k) has 27 lines 627 29.4. Every smooth cubic surface (over k) is a blown up plane 630 Chapter 30. ⋆ Proof of Serre duality 633 30.1. Introduction 633 30.2. Serre duality holds for projective space 634 30.3. Ext groups and Ext sheaves for O-modules 635 30.4. Serre duality for projective k-schemes 639 30.5. The adjunction formula for the dualizing sheaf, and ωX = det ΩX 643 Bibliography 645 Index 647
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62 Math 216: Foundations of Algebra
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Part II Schemes
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CHAPTER 6 Some properties of scheme
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April 4, 2012 draft 143 6.1.2. Dime
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April 4, 2012 draft 145 6.3.1. Prop
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April 4, 2012 draft 147 6.3.5. Unim
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April 4, 2012 draft 149 irreducible
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April 4, 2012 draft 151 6.4.5. Thus
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April 4, 2012 draft 153 that Z[ √
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The next property makes (A) more st
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April 4, 2012 draft 157 6.5.G. EXER
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We prove Theorem 6.5.6 in a series
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Part III Morphisms of schemes
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CHAPTER 9 Closed embeddings and rel
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April 4, 2012 draft 209 closed. (ii
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April 4, 2012 draft 211 FIGURE 9.1.
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April 4, 2012 draft 213 P3 k . In f
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April 4, 2012 draft 215 (perhaps no
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April 4, 2012 draft 217 FIGURE 9.3.
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April 4, 2012 draft 219 Example 4.
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Part IV Harder properties of scheme
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Part V Quasicoherent sheaves
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CHAPTER 17 Pushforwards and pullbac
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April 4, 2012 draft 377 The similar
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April 4, 2012 draft 379 (π ∗ G )
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April 4, 2012 draft 381 pullback of
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April 4, 2012 draft 383 Every time
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April 4, 2012 draft 385 of π : P 1
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April 4, 2012 draft 387 We next wa
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April 4, 2012 draft 391 [EGA, III1.
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CHAPTER 18 Relative versions of Spe
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April 4, 2012 draft 399 18.1.D. EXE
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April 4, 2012 draft 401 Show that t
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finite type quasicoherent sheaves.
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April 4, 2012 draft 407 quasicohere
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April 4, 2012 draft 409 namely X).
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April 4, 2012 draft 411 18.4.A. EXE
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April 4, 2012 draft 413 (b) Suppose
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CHAPTER 19 ⋆ Blowing up a scheme
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April 4, 2012 draft 417 which lets
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April 4, 2012 draft 419 Spec A. (A
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Part VI More
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544 we hope M ⊗A N ′ → M ⊗A
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546 Proof. Given a short exact sequ
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548 products to products (a consequ
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550 24.3.C. EXERCISE. Show that you
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552 Suppose · · · → C p−1
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554 Exercise 24.3.D describes what
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556 — Z U (V) = Z if V ⊂ U, and
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558 Then we take cohomology in the
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560 Prove this by endowing X with t
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CHAPTER 25 Flatness The concept of
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• In §25.2, we discuss some of t
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Motivated by Proposition 25.2.3, th
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the same idea as the proof of Theor
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(b) If M ′ and M ′′ are both
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is everything. As M is finitely pre
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25.4.K. EXERCISE. Suppose (with the
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y 2 . Then show that t is not a zer
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(c) Recall the Going-Up Theorem, de
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25.6 Local criteria for flatness (T
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t is a non-zerodivisor on B. Then M
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Proof. The question is local on the
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In §17.4.8, we produced a proper n
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25.8.1. Semicontinuity theorem. —
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pullback: given a fibered square (2
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we will give a new proof of Theorem
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25.9.C. EXERCISE. (a) Describe a ma
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25.9.K. EXERCISE. Put all the piece
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CHAPTER 29 Twenty-seven lines 29.1
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(29.2.0.1) if and only if (29.2.0.2
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involving precisely one x or y surv
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29.4.B. EXERCISE. Show that the bir
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CHAPTER 30 ⋆ Proof of Serre duali
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30.2.A. EXERCISE. Suppose F = O(m).
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and consider the spectral sequence
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where in the middle term, the “a
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By Exercise 30.3.H again, then stro
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30.4.B. EXERCISE (STRONG SERRE DUAL
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Bibliography [ACGH] E. Arbarello, M
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0-ring, 11 A-scheme, 147 A[[x]], 11
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Cohen-Seidenberg Lying Over Theorem
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generic fiber, 575 generic point, 1
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Noetherian module, 114 Noetherian r
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Serre duality (strong form), 634 Se
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