math 216: foundations of algebraic geometry - Stanford University

More documents

Recommendations

Info

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

$Math 115 HW 4 Part 1$

$MATH 220: Problem Set 8 Solutions - Stanford University$

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

$Math 205b Homework 2 Solutions$

5.5. Projective schemes, and the Proj construction 132 Chapter 6. Some properties of schemes 141 6.1. Topological properties 141 6.2. Reducedness and integrality 143 6.3. Properties of schemes that can be checked “affine-locally” 144 6.4. Normality and factoriality 149 6.5. Where functions are supported: Associated points of schemes 153 Part III. Morphisms of schemes 161 Chapter 7. Morphisms of schemes 163 7.1. Introduction 163 7.2. Morphisms of ringed spaces 164 7.3. From locally ringed spaces to morphisms of schemes 166 7.4. Maps of graded rings and maps of projective schemes 172 7.5. Rational maps from reduced schemes 174 7.6. ⋆ Representable functors and group schemes 179 7.7. ⋆⋆ The Grassmannian (initial construction) 183 Chapter 8. Useful classes of morphisms of schemes 185 8.1. An example of a reasonable class of morphisms: Open embeddings 185 8.2. Algebraic interlude: Lying Over and Nakayama 186 8.3. A gazillion finiteness conditions on morphisms 190 8.4. Images of morphisms: Chevalley’s theorem and elimination theory 199 Chapter 9. Closed embeddings and related notions 207 9.1. Closed embeddings and closed subschemes 207 9.2. More projective geometry 212 9.3. “Smallest closed subschemes such that ...” 217 Chapter 10. Fibered products of schemes 223 10.1. They exist 223 10.2. Computing fibered products in practice 229 10.3. Interpretations: Pulling back families and fibers of morphisms 232 10.4. Properties preserved by base change 237 10.5. ⋆ Properties not preserved by base change, and how to fix (some of) them 239 10.6. Products of projective schemes: The Segre embedding 245 10.7. Normalization 247 Chapter 11. Separated and proper morphisms, and (finally!) varieties 253 11.1. Separated morphisms (and quasiseparatedness done properly) 253 11.2. Rational maps to separated schemes 263 11.3. Proper morphisms 266 Part IV. Harder properties of schemes 271 Chapter 12. Dimension 273 12.1. Dimension and codimension 273 12.2. Dimension, transcendence degree, and Noether normalization 277 12.3. Codimension one miracles: Krull and Hartogs 284
12.4. Dimensions of fibers of morphisms of varieties 290 12.5. ⋆⋆ Proof of Krull’s Principal Ideal Theorem 12.3.3 294 Chapter 13. Nonsingularity (“smoothness”) of Noetherian schemes 297 13.1. The Zariski tangent space 297 13.2. Nonsingularity 302 13.3. Two pleasant facts about regular local rings 306 13.4. Discrete valuation rings: Dimension 1 Noetherian regular local rings 310 13.5. Valuative criteria for separatedness and properness 316 13.6. ⋆ Filtered rings and modules, and the Artin-Rees Lemma 319 13.7. ⋆ Completions 321 Part V. Quasicoherent sheaves 325 Chapter 14. Quasicoherent and coherent sheaves 327 14.1. Vector bundles and locally free sheaves 327 14.2. Quasicoherent sheaves 332 14.3. Characterizing quasicoherence using the distinguished affine base 334 14.4. Quasicoherent sheaves form an abelian category 339 14.5. Module-like constructions 340 14.6. Finite type and coherent sheaves 344 14.7. Pleasant properties of finite type and coherent sheaves 346 14.8. ⋆⋆ Coherent modules over non-Noetherian rings 350 Chapter 15. Line bundles: Invertible sheaves and divisors 353 15.1. Some line bundles on projective space 353 15.2. Line bundles and Weil divisors 355 15.3. ⋆ Effective Cartier divisors “=” invertible ideal sheaves 363 Chapter 16. Quasicoherent sheaves on projective A-schemes 365 16.1. The quasicoherent sheaf corresponding to a graded module 365 16.2. Invertible sheaves (line bundles) on projective A-schemes 366 16.3. Globally generated and base-point-free line bundles 367 16.4. Quasicoherent sheaves and graded modules 370 Chapter 17. Pushforwards and pullbacks of quasicoherent sheaves 375 17.1. Introduction 375 17.2. Pushforwards of quasicoherent sheaves 375 17.3. Pullbacks of quasicoherent sheaves 376 17.4. Invertible sheaves and maps to projective schemes 382 17.5. The Curve-to-Projective Extension Theorem 387 17.6. Ample and very ample line bundles 389 17.7. ⋆ The Grassmannian as a moduli space 394 Chapter 18. Relative versions of Spec and Proj, and projective morphisms 397 18.1. Relative Spec of a (quasicoherent) sheaf of algebras 397 18.2. Relative Proj of a sheaf of graded algebras 400 18.3. Projective morphisms 403 18.4. Applications to curves 409 Chapter 19. ⋆ Blowing up a scheme along a closed subscheme 415
Page 1: MATH 216: FOUNDATIONS OF ALGEBRAIC
Page 6 and 7: 19.1. Motivating example: blowing u
Page 9 and 10: CHAPTER 1 Introduction I can illust
Page 11: April 4, 2012 draft 11 into thinkin
Page 15 and 16: CHAPTER 2 Some category theory That
Page 17 and 18: April 4, 2012 draft 17 the universa
Page 19 and 20: April 4, 2012 draft 19 2.2.7. Examp
Page 21 and 22: April 4, 2012 draft 21 It is wise t
Page 23 and 24: April 4, 2012 draft 23 made using t
Page 25 and 26: April 4, 2012 draft 25 2.3.E. EXERC
Page 27 and 28: April 4, 2012 draft 27 2.3.6. Essen
Page 29 and 30: diagram April 4, 2012 draft 29 ≤1
Page 31 and 32: April 4, 2012 draft 31 2.4 Limits a
Page 33 and 34: April 4, 2012 draft 33 Even though
Page 35 and 36: we require (2.5.0.2) MorB(F(A ′ )
Page 37 and 38: April 4, 2012 draft 37 and S is a m
Page 39 and 40: April 4, 2012 draft 39 2.6.2. Remar
Page 41 and 42: April 4, 2012 draft 41 is exact if
Page 43 and 44: April 4, 2012 draft 43 2.6.6. Theor
Page 45 and 46: April 4, 2012 draft 45 obvious to y
Page 47 and 48: April 4, 2012 draft 47 is an exact
Page 49 and 50: April 4, 2012 draft 49 The order of
Page 51 and 52: April 4, 2012 draft 51 arrowheads a
Page 53 and 54: April 4, 2012 draft 53 Lemma! (Noti
Page 55 and 56:
April 4, 2012 draft 55 2.7.8. Goals
Page 57 and 58:
2.7.10. Defining E p,q r . Define X
Page 59:
April 4, 2012 draft 59 This complet
Page 62 and 63:
62 Math 216: Foundations of Algebra
Page 64 and 65:
Page 66 and 67:
Page 68 and 69:
Page 70 and 71:
Page 72 and 73:
Page 74 and 75:
Page 76 and 77:
Page 78 and 79:
Page 80 and 81:
Page 82 and 83:
Page 84 and 85:
Page 87:
Part II Schemes
Page 90 and 91:
Page 92 and 93:
Page 94 and 95:
Page 96 and 97:
Page 98 and 99:
Page 100 and 101:
100 Math 216: Foundations of Algebr
Page 102 and 103:
Page 104 and 105:
Page 106 and 107:
Page 108 and 109:
Page 110 and 111:
Page 112 and 113:
Page 114 and 115:
Page 116 and 117:
Page 118 and 119:
Page 120 and 121:
Page 122 and 123:
Page 124 and 125:
Page 126 and 127:
Page 128 and 129:
Page 130 and 131:
Page 132 and 133:
Page 134 and 135:
Page 136 and 137:
Page 138 and 139:
Page 141 and 142:
CHAPTER 6 Some properties of scheme
Page 143 and 144:
April 4, 2012 draft 143 6.1.2. Dime
Page 145 and 146:
April 4, 2012 draft 145 6.3.1. Prop
Page 147 and 148:
April 4, 2012 draft 147 6.3.5. Unim
Page 149 and 150:
April 4, 2012 draft 149 irreducible
Page 151 and 152:
April 4, 2012 draft 151 6.4.5. Thus
Page 153 and 154:
April 4, 2012 draft 153 that Z[ √
Page 155 and 156:
The next property makes (A) more st
Page 157 and 158:
April 4, 2012 draft 157 6.5.G. EXER
Page 159 and 160:
We prove Theorem 6.5.6 in a series
Page 161:
Part III Morphisms of schemes
Page 164 and 165:
Page 166 and 167:
Page 168 and 169:
Page 170 and 171:
Page 172 and 173:
Page 174 and 175:
Page 176 and 177:
Page 178 and 179:
Page 180 and 181:
Page 182 and 183:
Page 184 and 185:
Page 186 and 187:
Page 188 and 189:
Page 190 and 191:
Page 192 and 193:
Page 194 and 195:
Page 196 and 197:
Page 198 and 199:
Page 200 and 201:
Page 202 and 203:
Page 204 and 205:
Page 207 and 208:
CHAPTER 9 Closed embeddings and rel
Page 209 and 210:
April 4, 2012 draft 209 closed. (ii
Page 211 and 212:
April 4, 2012 draft 211 FIGURE 9.1.
Page 213 and 214:
April 4, 2012 draft 213 P3 k . In f
Page 215 and 216:
April 4, 2012 draft 215 (perhaps no
Page 217 and 218:
April 4, 2012 draft 217 FIGURE 9.3.
Page 219 and 220:
April 4, 2012 draft 219 Example 4.
Page 221:
April 4, 2012 draft 221 “p to p
Page 224 and 225:
Page 226 and 227:
Page 228 and 229:
Page 230 and 231:
Page 232 and 233:
Page 234 and 235:
Page 236 and 237:
Page 238 and 239:
Page 240 and 241:
Page 242 and 243:
Page 244 and 245:
Page 246 and 247:
Page 248 and 249:
Page 250 and 251:
Page 252 and 253:
Page 254 and 255:
Page 256 and 257:
Page 258 and 259:
Page 260 and 261:
Page 262 and 263:
Page 264 and 265:
Page 266 and 267:
Page 268 and 269:
Page 271:
Part IV Harder properties of scheme
Page 274 and 275:
Page 276 and 277:
Page 278 and 279:
Page 280 and 281:
Page 282 and 283:
Page 284 and 285:
Page 286 and 287:
Page 288 and 289:
Page 290 and 291:
Page 292 and 293:
Page 294 and 295:
Page 296 and 297:
Page 298 and 299:
Page 300 and 301:
Page 302 and 303:
Page 304 and 305:
Page 306 and 307:
Page 308 and 309:
Page 310 and 311:
Page 312 and 313:
Page 314 and 315:
Page 316 and 317:
Page 318 and 319:
Page 320 and 321:
Page 322 and 323:
Page 325:
Part V Quasicoherent sheaves
Page 328 and 329:
Page 330 and 331:
Page 332 and 333:
Page 334 and 335:
Page 336 and 337:
Page 338 and 339:
Page 340 and 341:
Page 342 and 343:
Page 344 and 345:
Page 346 and 347:
Page 348 and 349:
Page 350 and 351:
Page 352 and 353:
Page 354 and 355:
Page 356 and 357:
Page 358 and 359:
Page 360 and 361:
Page 362 and 363:
Page 364 and 365:
Page 366 and 367:
Page 368 and 369:
Page 370 and 371:
Page 372 and 373:
Page 375 and 376:
CHAPTER 17 Pushforwards and pullbac
Page 377 and 378:
April 4, 2012 draft 377 The similar
Page 379 and 380:
April 4, 2012 draft 379 (π ∗ G )
Page 381 and 382:
April 4, 2012 draft 381 pullback of
Page 383 and 384:
April 4, 2012 draft 383 Every time
Page 385 and 386:
April 4, 2012 draft 385 of π : P 1
Page 387 and 388:
April 4, 2012 draft 387 We next wa
Page 389 and 390:
April 4, 2012 draft 389 We next red
Page 391 and 392:
April 4, 2012 draft 391 [EGA, III1.
Page 393 and 394:
April 4, 2012 draft 393 cover of (q
Page 395 and 396:
April 4, 2012 draft 395 see a key e
Page 397 and 398:
CHAPTER 18 Relative versions of Spe
Page 399 and 400:
April 4, 2012 draft 399 18.1.D. EXE
Page 401 and 402:
April 4, 2012 draft 401 Show that t
Page 403 and 404:
April 4, 2012 draft 403 as Proj S
Page 405 and 406:
finite type quasicoherent sheaves.
Page 407 and 408:
April 4, 2012 draft 407 quasicohere
Page 409 and 410:
April 4, 2012 draft 409 namely X).
Page 411 and 412:
April 4, 2012 draft 411 18.4.A. EXE
Page 413 and 414:
April 4, 2012 draft 413 (b) Suppose
Page 415 and 416:
CHAPTER 19 ⋆ Blowing up a scheme
Page 417 and 418:
April 4, 2012 draft 417 which lets
Page 419 and 420:
April 4, 2012 draft 419 Spec A. (A
Page 421 and 422:
April 4, 2012 draft 421 given by xi
Page 423 and 424:
April 4, 2012 draft 423 to you (in
Page 425 and 426:
April 4, 2012 draft 425 family is
Page 427 and 428:
April 4, 2012 draft 427 that the no
Page 429 and 430:
April 4, 2012 draft 429 rational ma
Page 431:
April 4, 2012 draft 431 (cf. §19.3
Page 434 and 435:
Page 436 and 437:
Page 438 and 439:
Page 440 and 441:
Page 442 and 443:
Page 444 and 445:
Page 446 and 447:
Page 448 and 449:
Page 450 and 451:
Page 452 and 453:
Page 454 and 455:
Page 456 and 457:
Page 458 and 459:
Page 460 and 461:
Page 462 and 463:
Page 464 and 465:
Page 466 and 467:
Page 468 and 469:
Page 470 and 471:
Page 472 and 473:
Page 474 and 475:
Page 476 and 477:
Page 478 and 479:
Page 480 and 481:
Page 482 and 483:
Page 484 and 485:
Page 486 and 487:
Page 488 and 489:
Page 490 and 491:
Page 492 and 493:
Page 494 and 495:
Page 496 and 497:
Page 498 and 499:
Page 500 and 501:
Page 502 and 503:
Page 504 and 505:
Page 506 and 507:
Page 508 and 509:
Page 510 and 511:
Page 512 and 513:
Page 514 and 515:
Page 516 and 517:
Page 518 and 519:
Page 520 and 521:
Page 522 and 523:
Page 524 and 525:
Page 526 and 527:
Page 528 and 529:
Page 530 and 531:
Page 532 and 533:
Page 534 and 535:
Page 536 and 537:
Page 538 and 539:
Page 541:
Part VI More
Page 544 and 545:
544 we hope M ⊗A N ′ → M ⊗A
Page 546 and 547:
546 Proof. Given a short exact sequ
Page 548 and 549:
548 products to products (a consequ
Page 550 and 551:
550 24.3.C. EXERCISE. Show that you
Page 552 and 553:
552 Suppose · · · → C p−1
Page 554 and 555:
554 Exercise 24.3.D describes what
Page 556 and 557:
556 — Z U (V) = Z if V ⊂ U, and
Page 558 and 559:
558 Then we take cohomology in the
Page 560 and 561:
560 Prove this by endowing X with t
Page 563 and 564:
CHAPTER 25 Flatness The concept of
Page 565 and 566:
• In §25.2, we discuss some of t
Page 567 and 568:
Motivated by Proposition 25.2.3, th
Page 569 and 570:
the same idea as the proof of Theor
Page 571 and 572:
(b) If M ′ and M ′′ are both
Page 573 and 574:
is everything. As M is finitely pre
Page 575 and 576:
25.4.K. EXERCISE. Suppose (with the
Page 577 and 578:
y 2 . Then show that t is not a zer
Page 579 and 580:
(c) Recall the Going-Up Theorem, de
Page 581 and 582:
25.6 Local criteria for flatness (T
Page 583 and 584:
t is a non-zerodivisor on B. Then M
Page 585 and 586:
Proof. The question is local on the
Page 587 and 588:
In §17.4.8, we produced a proper n
Page 589 and 590:
25.8.1. Semicontinuity theorem. —
Page 591 and 592:
pullback: given a fibered square (2
Page 593 and 594:
we will give a new proof of Theorem
Page 595 and 596:
25.9.C. EXERCISE. (a) Describe a ma
Page 597 and 598:
25.9.K. EXERCISE. Put all the piece
Page 599 and 600:
CHAPTER 29 Twenty-seven lines 29.1
Page 601 and 602:
(29.2.0.1) if and only if (29.2.0.2
Page 603 and 604:
involving precisely one x or y surv
Page 605 and 606:
29.4.B. EXERCISE. Show that the bir
Page 607 and 608:
CHAPTER 30 ⋆ Proof of Serre duali
Page 609 and 610:
30.2.A. EXERCISE. Suppose F = O(m).
Page 611 and 612:
and consider the spectral sequence
Page 613 and 614:
where in the middle term, the “a
Page 615 and 616:
By Exercise 30.3.H again, then stro
Page 617 and 618:
30.4.B. EXERCISE (STRONG SERRE DUAL
Page 619 and 620:
Bibliography [ACGH] E. Arbarello, M
Page 621 and 622:
0-ring, 11 A-scheme, 147 A[[x]], 11
Page 623 and 624:
Cohen-Seidenberg Lying Over Theorem
Page 625 and 626:
generic fiber, 575 generic point, 1
Page 627 and 628:
Noetherian module, 114 Noetherian r
Page 629 and 630:
Serre duality (strong form), 634 Se
show all

math 216: foundations of algebraic geometry - Stanford University

Create successful ePaper yourself

Delete template?

Save as template?