Knowledge (XXG)

Morphism of algebraic stacks

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One particular important example is a presentation of a stack, which is widely used in the study of stacks.
449: 374: 95: 514: 86: 495: 190: 152: 518: 256: 456: 445: 90: 491: 452: 561: 506: 476: 17: 223: 184: 364:{\displaystyle \operatorname {Spec} (k)\to \operatorname {Vect} _{n}} 29: 222:
More generally, one can also consider a morphism between
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may be too technical for most readers to understand
429: 397: 363: 310: 271: 211: 173: 135: 542: 8: 549: 535: 226:; (a stackification would be an example.) 410: 389: 384: 380: 379: 376: 355: 331: 311:{\displaystyle \operatorname {Vect} _{n}} 302: 296: 258: 192: 154: 117: 97: 73:Learn how and when to remove this message 57:, without removing the technical details. 430:{\displaystyle \operatorname {Spec} (k)} 468: 55:make it understandable to non-experts 7: 503: 501: 398:{\displaystyle \mathbb {A} _{k}^{n}} 448:is a morphism that factorizes as a 136:{\displaystyle p:X\to C,\,q:Y\to C} 521:. You can help Knowledge (XXG) by 253:if there is a smooth presentation 25: 505: 34: 326:, then there is a presentation 424: 418: 348: 345: 339: 263: 165: 127: 108: 1: 496:Morphisms of algebraic stacks 371:given by the trivial bundle 589: 500: 212:{\displaystyle q\circ f=p} 477:Notes on algebraic stacks 568:Algebraic geometry stubs 174:{\displaystyle f:X\to Y} 283:for some smooth scheme 517:–related article is a 431: 399: 365: 312: 279:of relative dimension 273: 272:{\displaystyle U\to X} 213: 175: 137: 442:quasi-affine morphism 432: 400: 366: 320:moduli stack of rank- 313: 274: 214: 176: 143:over a base category 138: 409: 375: 330: 295: 257: 191: 153: 96: 475:§ 8.6 of F. Meyer, 394: 237:An algebraic stack 181:of algebraic stacks 573:Algebraic geometry 515:algebraic geometry 427: 395: 378: 361: 308: 291:. For example, if 269: 209: 171: 133: 87:algebraic geometry 530: 529: 83: 82: 75: 16:(Redirected from 580: 551: 544: 537: 509: 502: 479: 473: 446:algebraic stacks 436: 434: 433: 428: 404: 402: 401: 396: 393: 388: 383: 370: 368: 367: 362: 360: 359: 317: 315: 314: 309: 307: 306: 278: 276: 275: 270: 218: 216: 215: 210: 180: 178: 177: 172: 142: 140: 139: 134: 91:algebraic stacks 78: 71: 67: 64: 58: 38: 37: 30: 21: 588: 587: 583: 582: 581: 579: 578: 577: 558: 557: 556: 555: 488: 483: 482: 474: 470: 465: 457:affine morphism 455:followed by an 407: 406: 373: 372: 351: 328: 327: 298: 293: 292: 255: 254: 232: 189: 188: 151: 150: 94: 93: 79: 68: 62: 59: 51:help improve it 48: 39: 35: 28: 27:Type of functor 23: 22: 15: 12: 11: 5: 586: 584: 576: 575: 570: 560: 559: 554: 553: 546: 539: 531: 528: 527: 510: 499: 498: 492:Stacks Project 487: 484: 481: 480: 467: 466: 464: 461: 453:open immersion 426: 423: 420: 417: 414: 392: 387: 382: 358: 354: 350: 347: 344: 341: 338: 335: 324:vector bundles 305: 301: 268: 265: 262: 241:is said to be 231: 228: 208: 205: 202: 199: 196: 170: 167: 164: 161: 158: 132: 129: 126: 123: 120: 116: 113: 110: 107: 104: 101: 81: 80: 42: 40: 33: 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 585: 574: 571: 569: 566: 565: 563: 552: 547: 545: 540: 538: 533: 532: 526: 524: 520: 516: 511: 508: 504: 497: 493: 490: 489: 485: 478: 472: 469: 462: 460: 458: 454: 451: 450:quasi-compact 447: 443: 438: 421: 415: 412: 390: 385: 356: 352: 342: 336: 333: 325: 323: 303: 299: 290: 287:of dimension 286: 282: 266: 260: 252: 248: 245:of dimension 244: 240: 235: 229: 227: 225: 220: 206: 203: 200: 197: 194: 186: 182: 168: 162: 159: 156: 146: 130: 124: 121: 118: 114: 111: 105: 102: 99: 92: 88: 77: 74: 66: 63:November 2023 56: 52: 46: 43:This article 41: 32: 31: 19: 523:expanding it 512: 471: 441: 439: 321: 318:denotes the 288: 284: 280: 250: 246: 242: 238: 236: 233: 221: 148: 144: 84: 69: 60: 44: 18:Smooth stack 562:Categories 494:, Ch, 83, 486:References 187:such that 416:⁡ 349:→ 337:⁡ 264:→ 224:prestacks 198:∘ 166:→ 149:morphism 128:→ 109:→ 444:between 89:, given 185:functor 49:Please 243:smooth 513:This 463:Notes 405:over 230:Types 183:is a 519:stub 413:Spec 353:Vect 334:Spec 300:Vect 147:, a 85:In 53:to 564:: 459:. 440:A 437:. 249:- 219:. 550:e 543:t 536:v 525:. 425:) 422:k 419:( 391:n 386:k 381:A 357:n 346:) 343:k 340:( 322:n 304:n 289:n 285:U 281:j 267:X 261:U 251:j 247:n 239:X 207:p 204:= 201:f 195:q 169:Y 163:X 160:: 157:f 145:C 131:C 125:Y 122:: 119:q 115:, 112:C 106:X 103:: 100:p 76:) 70:( 65:) 61:( 47:. 20:)

Index

Smooth stack
help improve it
make it understandable to non-experts
Learn how and when to remove this message
algebraic geometry
algebraic stacks
functor
prestacks
moduli stack of rank-n vector bundles
algebraic stacks
quasi-compact
open immersion
affine morphism
Notes on algebraic stacks
Stacks Project
Morphisms of algebraic stacks
Stub icon
algebraic geometry
stub
expanding it
v
t
e
Categories
Algebraic geometry stubs
Algebraic geometry

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