Knowledge (XXG)

Geometrically (algebraic geometry)

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413: 149: 74: 230: 180: 299: 454: 260: 447: 387: 473: 440: 379: 144:{\displaystyle X\times _{k}{\overline {k}}:=X\times _{\operatorname {Spec} k}{\operatorname {Spec} {\overline {k}}}} 374: 478: 195: 158: 271: 29: 36:
of the field. In other words, a property holds geometrically if it holds after a base change to a
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The same statement also holds if "irreducible" is replaced with "
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Geometrically irreducible and geometrically reduced
293: 254: 224: 174: 143: 320:" and the separable closure is replaced by the 448: 8: 382:, vol. 52, New York: Springer-Verlag, 455: 441: 352: 340: 282: 273: 246: 240: 216: 206: 197: 162: 160: 130: 123: 111: 91: 85: 76: 333: 7: 409: 407: 71:is geometrically irreducible; i.e., 60:that is of finite type over a field 44:is a variety that is geometrically 427:. You can help Knowledge (XXG) by 14: 225:{\displaystyle X\times _{k}k_{s}} 411: 64:, the following are equivalent: 175:{\displaystyle {\overline {k}}} 1: 380:Graduate Texts in Mathematics 294:{\displaystyle X\times _{k}F} 24:, a property is said to hold 167: 135: 96: 355:, Ch II, Exercise 3.15. (b) 343:, Ch II, Exercise 3.15. (a) 495: 406: 32:if it also holds over the 474:Algebraic geometry stubs 301:is irreducible for each 423:–related article is a 295: 256: 226: 176: 145: 296: 257: 255:{\displaystyle k_{s}} 232:is irreducible for a 227: 177: 146: 272: 239: 196: 159: 75: 421:algebraic geometry 375:Algebraic Geometry 291: 252: 222: 172: 141: 18:algebraic geometry 436: 435: 389:978-0-387-90244-9 370:Hartshorne, Robin 234:separable closure 184:algebraic closure 170: 138: 99: 40:. For example, a 34:algebraic closure 486: 457: 450: 443: 415: 408: 400: 356: 350: 344: 338: 300: 298: 297: 292: 287: 286: 261: 259: 258: 253: 251: 250: 231: 229: 228: 223: 221: 220: 211: 210: 181: 179: 178: 173: 171: 163: 150: 148: 147: 142: 140: 139: 131: 122: 121: 100: 92: 90: 89: 20:, especially in 494: 493: 489: 488: 487: 485: 484: 483: 464: 463: 462: 461: 404: 390: 368: 365: 360: 359: 353:Hartshorne 1977 351: 347: 341:Hartshorne 1977 339: 335: 330: 322:perfect closure 303:field extension 278: 270: 269: 242: 237: 236: 212: 202: 194: 193: 157: 156: 107: 81: 73: 72: 56:Given a scheme 54: 38:geometric point 12: 11: 5: 492: 490: 482: 481: 476: 466: 465: 460: 459: 452: 445: 437: 434: 433: 416: 402: 401: 388: 364: 361: 358: 357: 345: 332: 331: 329: 326: 314: 313: 290: 285: 281: 277: 267: 249: 245: 219: 215: 209: 205: 201: 191: 169: 166: 137: 134: 129: 126: 120: 117: 114: 110: 106: 103: 98: 95: 88: 84: 80: 53: 50: 42:smooth variety 13: 10: 9: 6: 4: 3: 2: 491: 480: 479:Scheme theory 477: 475: 472: 471: 469: 458: 453: 451: 446: 444: 439: 438: 432: 430: 426: 422: 417: 414: 410: 405: 399: 395: 391: 385: 381: 377: 376: 371: 367: 366: 362: 354: 349: 346: 342: 337: 334: 327: 325: 323: 319: 311: 307: 304: 288: 283: 279: 275: 268: 265: 247: 243: 235: 217: 213: 207: 203: 199: 192: 189: 185: 164: 154: 132: 127: 124: 118: 115: 112: 108: 104: 101: 93: 86: 82: 78: 70: 67: 66: 65: 63: 59: 51: 49: 47: 43: 39: 35: 31: 27: 26:geometrically 23: 22:scheme theory 19: 429:expanding it 418: 403: 373: 348: 336: 315: 309: 305: 263: 187: 68: 61: 57: 55: 25: 15: 182:denotes an 153:irreducible 468:Categories 328:References 280:× 204:× 168:¯ 136:¯ 128:⁡ 116:⁡ 109:× 97:¯ 83:× 372:(1977), 155:, where 398:0463157 363:Sources 318:reduced 46:regular 28:over a 396:  386:  419:This 30:field 425:stub 384:ISBN 125:Spec 113:Spec 308:of 262:of 186:of 151:is 16:In 470:: 394:MR 392:, 378:, 324:. 102::= 48:. 456:e 449:t 442:v 431:. 312:. 310:k 306:F 289:F 284:k 276:X 266:. 264:k 248:s 244:k 218:s 214:k 208:k 200:X 190:. 188:k 165:k 133:k 119:k 105:X 94:k 87:k 79:X 69:X 62:k 58:X

Index

algebraic geometry
scheme theory
field
algebraic closure
geometric point
smooth variety
regular
irreducible
algebraic closure
separable closure
field extension
reduced
perfect closure
Hartshorne 1977
Hartshorne 1977
Hartshorne, Robin
Algebraic Geometry
Graduate Texts in Mathematics
ISBN
978-0-387-90244-9
MR
0463157
Stub icon
algebraic geometry
stub
expanding it
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