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April 4, 2012 draft 63 Note that we
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April 4, 2012 draft 65 Gluability a
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April 4, 2012 draft 67 3.2.11. ⋆
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April 4, 2012 draft 69 commutes. (N
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April 4, 2012 draft 71 Similarly, k
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April 4, 2012 draft 73 s ′ p at a
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April 4, 2012 draft 75 As a reality
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April 4, 2012 draft 77 cokernel is
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April 4, 2012 draft 79 check that a
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April 4, 2012 draft 81 3.6.D. EXERC
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April 4, 2012 draft 83 We require t
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April 4, 2012 draft 85 classical to
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CHAPTER 4 Toward affine schemes: th
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April 4, 2012 draft 91 4.1.2. Aside
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April 4, 2012 draft 93 — this is
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April 4, 2012 draft 95 the (0) idea
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April 4, 2012 draft 97 consisting o
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April 4, 2012 draft 99 Another exam
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A n F 2 (2) A n F 3 April 4, 2012 d
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April 4, 2012 draft 103 make that p
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April 4, 2012 draft 105 topology on
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April 4, 2012 draft 107 We will use
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April 4, 2012 draft 109 4.6.H. EXER
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April 4, 2012 draft 111 FIGURE 4.8.
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April 4, 2012 draft 113 • If A is
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April 4, 2012 draft 115 [(x)] [(x
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CHAPTER 5 The structure sheaf, and
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a 1 f l 1 1 a 2 f l 2 2 (a) ∈ Af
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April 4, 2012 draft 121 (We will la
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Spec C[x, y]): April 4, 2012 draft
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April 4, 2012 draft 125 (nonempty)
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April 4, 2012 draft 127 It is not i
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April 4, 2012 draft 129 FIGURE 5.6.
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April 4, 2012 draft 131 5.4.E. EXER
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April 4, 2012 draft 133 properties
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April 4, 2012 draft 135 (Everything
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April 4, 2012 draft 137 5.5.G. EASY
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k can be replaced by any ring A as
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CHAPTER 7 Morphisms of schemes 7.1
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April 4, 2012 draft 165 7.2.C. EASY
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April 4, 2012 draft 167 back to fun
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sense of the following sentence:
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April 4, 2012 draft 171 can be inte
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April 4, 2012 draft 173 corresponds
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April 4, 2012 draft 175 function fi
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C q April 4, 2012 draft 177 y p slo
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April 4, 2012 draft 179 showing tha
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April 4, 2012 draft 181 {(f1, . . .
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April 4, 2012 draft 183 (b) Define
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CHAPTER 8 Useful classes of morphis
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April 4, 2012 draft 187 of A, but n
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[q] April 4, 2012 draft 189 [p] Spe
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April 4, 2012 draft 191 8.3.1. Quas
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Notice that the maps i (Ai) si Apr
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April 4, 2012 draft 195 0 FIGURE 8.
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April 4, 2012 draft 197 of Spec A,
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April 4, 2012 draft 199 are also co
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April 4, 2012 draft 201 Chevalley
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April 4, 2012 draft 203 8.4.O. EXER
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April 4, 2012 draft 205 in S• = A
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CHAPTER 10 Fibered products of sche
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April 4, 2012 draft 225 have to che
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April 4, 2012 draft 227 Until the e
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April 4, 2012 draft 229 these funct
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April 4, 2012 draft 231 base change
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April 4, 2012 draft 233 f : X → Z
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(i) Then the preimage of 1 is two p
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April 4, 2012 draft 237 10.3.G. EXE
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April 4, 2012 draft 239 A and B int
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April 4, 2012 draft 241 (b) Show th
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April 4, 2012 draft 243 10.5.11. Co
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April 4, 2012 draft 245 (b) If k is
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April 4, 2012 draft 247 10.6.2. Imp
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April 4, 2012 draft 249 will need t
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April 4, 2012 draft 251 to the Galo
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CHAPTER 11 Separated and proper mor
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X ×Y X April 4, 2012 draft 255 FIG
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April 4, 2012 draft 257 Figure 11.2
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April 4, 2012 draft 259 is a fiber
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X ×Z Y Γf pr1 X April 4, 2012 dra
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April 4, 2012 draft 263 then i : V
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April 4, 2012 draft 265 f1 FIGURE 1
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April 4, 2012 draft 267 11.3.1. Def
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April 4, 2012 draft 269 Suppose f
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CHAPTER 12 Dimension 12.1 Dimension
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April 4, 2012 draft 275 n. (See Exe
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April 4, 2012 draft 277 12.1.7. A f
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April 4, 2012 draft 279 preimages o
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April 4, 2012 draft 281 12.2.9. The
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April 4, 2012 draft 283 We end with
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April 4, 2012 draft 285 12.3.2. Kru
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April 4, 2012 draft 287 over some f
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April 4, 2012 draft 289 Another gen
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April 4, 2012 draft 291 (As a speci
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trim and extend (12.4.1.2) to the f
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April 4, 2012 draft 295 must eventu
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CHAPTER 13 Nonsingularity (“smoot
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April 4, 2012 draft 299 13.1.B. IMP
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April 4, 2012 draft 301 domain, any
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April 4, 2012 draft 303 13.2.3. Smo
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April 4, 2012 draft 305 If the degr
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April 4, 2012 draft 307 In particul
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April 4, 2012 draft 309 13.3.9. The
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(a) (A, m) is regular. (b) m is pri
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April 4, 2012 draft 313 An integral
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April 4, 2012 draft 315 13.4.13. Re
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April 4, 2012 draft 317 13.5.A. EXE
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April 4, 2012 draft 319 the valuati
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April 4, 2012 draft 321 Proof. Appl
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April 4, 2012 draft 323 13.7.4. A f
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CHAPTER 14 Quasicoherent and cohere
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April 4, 2012 draft 329 understandi
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April 4, 2012 draft 331 open set ha
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April 4, 2012 draft 333 14.2.A. UNI
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April 4, 2012 draft 335 Because qua
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April 4, 2012 draft 337 14.3.D. VER
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April 4, 2012 draft 339 14.4 Quasic
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April 4, 2012 draft 341 are linearl
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April 4, 2012 draft 343 14.5.G. EXE
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April 4, 2012 draft 345 The proof i
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April 4, 2012 draft 347 bundle case
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April 4, 2012 draft 349 Proposition
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April 4, 2012 draft 351 up is the r
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CHAPTER 15 Line bundles: Invertible
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April 4, 2012 draft 355 15.1.C. ESS
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April 4, 2012 draft 357 trivial). C
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April 4, 2012 draft 359 (a) Show th
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April 4, 2012 draft 361 The next tw
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April 4, 2012 draft 363 affine spac
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CHAPTER 16 Quasicoherent sheaves on
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April 4, 2012 draft 367 16.2.A. EXE
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April 4, 2012 draft 369 every point
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they are an adjoint pair. QCoh Proj
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April 4, 2012 draft 373 16.4.5. The
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CHAPTER 20 Čech cohomology of quas
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April 4, 2012 draft 435 • Hi (Pn
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April 4, 2012 draft 437 sheaf on Pr
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April 4, 2012 draft 439 Define Hi U
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April 4, 2012 draft 441 Suppose fir
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April 4, 2012 draft 443 Z on the co
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April 4, 2012 draft 445 The “1 ne
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April 4, 2012 draft 447 (This remar
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April 4, 2012 draft 449 20.4.5. The
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April 4, 2012 draft 451 Oηi-module
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April 4, 2012 draft 453 The Hilbert
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April 4, 2012 draft 455 (b) If pX(m
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April 4, 2012 draft 457 We now revi
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April 4, 2012 draft 459 I as the id
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April 4, 2012 draft 461 (2) If 0
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April 4, 2012 draft 463 (b) (cohomo
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April 4, 2012 draft 465 The special
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the composition is projective). Apr
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CHAPTER 21 Application: Curves We n
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April 4, 2012 draft 471 I next clai
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is exact as L is locally free) to g
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April 4, 2012 draft 475 these two s
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April 4, 2012 draft 477 α, β, γ,
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April 4, 2012 draft 479 21.5.A. EXE
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April 4, 2012 draft 481 induced by
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April 4, 2012 draft 483 4 with 3 se
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On to genus 5! April 4, 2012 draft
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given by April 4, 2012 draft 487 L
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April 4, 2012 draft 489 equalities
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April 4, 2012 draft 491 Choose proj
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April 4, 2012 draft 493 In seemingl
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April 4, 2012 draft 495 We extend t
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April 4, 2012 draft 497 21.10.F. IM
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April 4, 2012 draft 499 there are f
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CHAPTER 22 ⋆ Application: A glimp
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April 4, 2012 draft 503 (There are
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April 4, 2012 draft 505 22.1.7. ⋆
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April 4, 2012 draft 507 proper nonp
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April 4, 2012 draft 509 22.2.S. EXE
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vanishing, Theorem 20.1.3(ii)). Thi
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CHAPTER 23 Differentials 23.1 Motiv
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April 4, 2012 draft 515 23.2.A. TRI
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April 4, 2012 draft 517 Then the fi
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April 4, 2012 draft 519 Finally, we
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April 4, 2012 draft 521 23.2.H. EXE
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April 4, 2012 draft 523 Ω S −1
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April 4, 2012 draft 525 We can now
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(a) Show that dx y April 4, 2012 dr
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April 4, 2012 draft 529 Under this
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April 4, 2012 draft 531 projective
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April 4, 2012 draft 533 23.4.G. EXE
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April 4, 2012 draft 535 set is nonz
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April 4, 2012 draft 537 Here are so
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April 4, 2012 draft 539 23.5.N. ⋆
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CHAPTER 24 Derived functors In this
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Here we use the freeness of A⊕ni
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(24.1.2.1), build a map — a three
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it is divisible (i.e. for every q
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Before we give the proof (in §24.3
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24.3.E. EXERCISE. Describe an injec
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24.4.A. EXERCISE (PUSHFORWARD OF IN
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precisely, choose a finite affine o
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0 (where it has cohomology F (X)),
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which is i RkΓ(Ui, F ) (using the
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564 the result behaves well in flat
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566 (Torsion-freeness was defined i
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568 25.2.K. FLAT MAPS SEND ASSOCIAT
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570 recalling key properties of Tor
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572 follows by considering the Tor1
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574 a ↦→ w + y, b ↦→ x + z
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576 y 2 = x 2 +x 3 . Now we have en
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578 comes from faithful flatness. B
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580 Spec k[x, y]/(y 2 − x 3 ) sho
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582 Applying (·) ⊗B M, and using
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584 effective Cartier divisor on al
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586 25.7.2. Corollary. — Assume t
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588 Higher pushforwards are easy to
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590 (ii) Furthermore, φ p−1 y is
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592 Proof. We build this complex in
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594 general comments on dealing wit
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596 25.9.F. EXERCISE. Prove Grauert
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598 25.10.B. EXERCISE. Show that fo
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626 structure (see Remark 29.3.5).
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628 Thus, over P 19 \ ∆, π is a
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630 Choose a point y1 ∈ Y ′′
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632 Thus the only chance we have of
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634 30.1.3. Proposition. — If a d
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636 In both cases, the subscript X
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638 30.3.3. The local-to-global spe
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640 Note that for i > 0, Ext i (L ,
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642 Because the argument relies on
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644 30.5.1. Proposition (the adjunc
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646 Foundations of Algebraic Geomet
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648 Foundations of Algebraic Geomet
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650 Foundations of Algebraic Geomet
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652 Foundations of Algebraic Geomet
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654 Foundations of Algebraic Geomet
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656 Foundations of Algebraic Geomet