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Standard complex

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665: 867: 43: 447: 263: 274: 650: 578: 202: 792: 178: 514: 442:{\displaystyle d(a_{0}\otimes \cdots \otimes a_{n+1})=\sum _{i=0}^{n}(-1)^{i}a_{0}\otimes \cdots \otimes a_{i}a_{i+1}\otimes \cdots \otimes a_{n+1}\,.} 64: 583: 539: 908: 735: 17: 88: 31: 927: 258:{\displaystyle \cdots \rightarrow A\otimes A\otimes A\rightarrow A\otimes A\rightarrow A\rightarrow 0\,,} 901: 727: 675: 51: 932: 762: 69: 163: 894: 796: 833: 745: 122: 882: 8: 126: 844: 821: 463: 813: 756: 731: 138: 805: 752: 717: 134: 130: 840: 829: 741: 721: 878: 700: 57: 921: 817: 713: 146: 30:"Standard resolution" redirects here. For the television monitor size, see 849: 664: 825: 866: 809: 645:{\displaystyle A\otimes (A/K)\otimes \cdots \otimes (A/K)\otimes A} 129:. It was first introduced for the special case of algebras over a 874: 160:
used a vertical bar | as a shortened form of the tensor product
27:
Technique for constructing resolutions in homological algebra
573:{\displaystyle A\otimes A\otimes \cdots \otimes A\otimes A} 153:, IX.6) and has since been generalized in many ways. 536:
The normalized (or reduced) standard complex replaces
765: 586: 542: 466: 277: 205: 166: 460:-algebra, the standard complex is exact. Moreover, 786: 644: 572: 508: 441: 257: 172: 751: 157: 142: 919: 156:The name "bar complex" comes from the fact that 843:(2005). "Lectures on Noncommutative Geometry". 726:, Princeton Mathematical Series, vol. 19, 712: 150: 531: 902: 909: 895: 848: 435: 251: 89:Learn how and when to remove this message 839: 192:is an associative algebra over a field 14: 920: 861: 659: 36: 180:in their notation for the complex. 24: 772: 25: 944: 865: 663: 41: 268:with the differential given by 158:Eilenberg & Mac Lane (1953) 781: 769: 633: 619: 607: 593: 503: 491: 473: 467: 356: 346: 319: 281: 245: 239: 227: 209: 32:Standard-definition television 13: 1: 706: 183: 149: and Eilenberg ( 881:. You can help Knowledge by 520:-bimodule resolution of the 7: 694: 532:Normalized standard complex 121:, is a way of constructing 10: 949: 860: 759:(1953), "On the groups of 728:Princeton University Press 196:, the standard complex is 29: 787:{\displaystyle H(\Pi ,n)} 655: 173:{\displaystyle \otimes } 877:-related article is a 788: 672:This section is empty. 646: 574: 510: 443: 345: 259: 174: 50:This article includes 797:Annals of Mathematics 789: 647: 575: 511: 444: 325: 260: 175: 763: 584: 540: 464: 275: 203: 164: 135:Samuel Eilenberg 101:In mathematics, the 928:Homological algebra 723:Homological algebra 127:homological algebra 107:standard resolution 18:Standard resolution 784: 757:Mac Lane, Saunders 642: 570: 506: 439: 255: 170: 58:properly formatted 890: 889: 800:, Second Series, 753:Eilenberg, Samuel 737:978-0-691-04991-5 718:Eilenberg, Samuel 692: 691: 139:Saunders Mac Lane 99: 98: 91: 16:(Redirected from 940: 911: 904: 897: 869: 862: 854: 852: 841:Ginzburg, Victor 836: 793: 791: 790: 785: 748: 687: 684: 674:You can help by 667: 660: 651: 649: 648: 643: 629: 603: 579: 577: 576: 571: 515: 513: 512: 509:{\displaystyle } 507: 448: 446: 445: 440: 434: 433: 409: 408: 393: 392: 374: 373: 364: 363: 344: 339: 318: 317: 293: 292: 264: 262: 261: 256: 179: 177: 176: 171: 147:Henri Cartan 131:commutative ring 119:bar construction 103:standard complex 94: 87: 83: 80: 74: 72: 67:this article by 52:inline citations 45: 44: 37: 21: 948: 947: 943: 942: 941: 939: 938: 937: 918: 917: 916: 915: 858: 850:math.AG/0506603 810:10.2307/1969820 764: 761: 760: 738: 709: 697: 688: 682: 679: 658: 625: 599: 585: 582: 581: 541: 538: 537: 534: 465: 462: 461: 423: 419: 398: 394: 388: 384: 369: 365: 359: 355: 340: 329: 307: 303: 288: 284: 276: 273: 272: 204: 201: 200: 186: 165: 162: 161: 95: 84: 78: 75: 70:correcting them 68: 62: 46: 42: 35: 28: 23: 22: 15: 12: 11: 5: 946: 936: 935: 930: 914: 913: 906: 899: 891: 888: 887: 870: 856: 855: 837: 783: 780: 777: 774: 771: 768: 749: 736: 708: 705: 704: 703: 701:Koszul complex 696: 693: 690: 689: 670: 668: 657: 654: 641: 638: 635: 632: 628: 624: 621: 618: 615: 612: 609: 606: 602: 598: 595: 592: 589: 569: 566: 563: 560: 557: 554: 551: 548: 545: 533: 530: 505: 502: 499: 496: 493: 490: 487: 484: 481: 478: 475: 472: 469: 450: 449: 438: 432: 429: 426: 422: 418: 415: 412: 407: 404: 401: 397: 391: 387: 383: 380: 377: 372: 368: 362: 358: 354: 351: 348: 343: 338: 335: 332: 328: 324: 321: 316: 313: 310: 306: 302: 299: 296: 291: 287: 283: 280: 266: 265: 254: 250: 247: 244: 241: 238: 235: 232: 229: 226: 223: 220: 217: 214: 211: 208: 185: 182: 169: 111:bar resolution 105:, also called 97: 96: 49: 47: 40: 26: 9: 6: 4: 3: 2: 945: 934: 933:Algebra stubs 931: 929: 926: 925: 923: 912: 907: 905: 900: 898: 893: 892: 886: 884: 880: 876: 871: 868: 864: 863: 859: 851: 846: 842: 838: 835: 831: 827: 823: 819: 815: 811: 807: 803: 799: 798: 778: 775: 766: 758: 754: 750: 747: 743: 739: 733: 729: 725: 724: 719: 715: 714:Cartan, Henri 711: 710: 702: 699: 698: 686: 677: 673: 669: 666: 662: 661: 653: 639: 636: 630: 626: 622: 616: 613: 610: 604: 600: 596: 590: 587: 567: 564: 561: 558: 555: 552: 549: 546: 543: 529: 527: 523: 519: 500: 497: 494: 488: 485: 482: 479: 476: 470: 459: 455: 436: 430: 427: 424: 420: 416: 413: 410: 405: 402: 399: 395: 389: 385: 381: 378: 375: 370: 366: 360: 352: 349: 341: 336: 333: 330: 326: 322: 314: 311: 308: 304: 300: 297: 294: 289: 285: 278: 271: 270: 269: 252: 248: 242: 236: 233: 230: 224: 221: 218: 215: 212: 206: 199: 198: 197: 195: 191: 181: 167: 159: 154: 152: 148: 144: 140: 137: and 136: 132: 128: 124: 120: 116: 112: 108: 104: 93: 90: 82: 71: 66: 61: 59: 56:they are not 53: 48: 39: 38: 33: 19: 883:expanding it 872: 857: 801: 795: 722: 680: 676:adding to it 671: 535: 525: 521: 517: 457: 456:is a unital 453: 451: 267: 193: 189: 187: 155: 118: 114: 110: 106: 102: 100: 85: 76: 55: 123:resolutions 115:bar complex 922:Categories 804:: 55–106, 707:References 524:-bimodule 516:is a free 184:Definition 818:0003-486X 773:Π 683:June 2011 637:⊗ 617:⊗ 614:⋯ 611:⊗ 591:⊗ 565:⊗ 559:⊗ 556:⋯ 553:⊗ 547:⊗ 498:⊗ 492:→ 486:⊗ 480:⊗ 474:→ 471:⋯ 417:⊗ 414:⋯ 411:⊗ 382:⊗ 379:⋯ 376:⊗ 350:− 327:∑ 301:⊗ 298:⋯ 295:⊗ 246:→ 240:→ 234:⊗ 228:→ 222:⊗ 216:⊗ 210:→ 207:⋯ 168:⊗ 720:(1956), 695:See also 79:May 2024 875:algebra 834:0056295 826:1969820 746:0077480 141: ( 65:improve 63:Please 832:  824:  816:  794:. 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Index

Standard resolution
Standard-definition television
inline citations
properly formatted
improve
correcting them
Learn how and when to remove this message
resolutions
homological algebra
commutative ring
Samuel Eilenberg
Saunders Mac Lane
1953
Henri Cartan
1956
Eilenberg & Mac Lane (1953)

adding to it
Koszul complex
Cartan, Henri
Eilenberg, Samuel
Homological algebra
Princeton University Press
ISBN
978-0-691-04991-5
MR
0077480
Eilenberg, Samuel
Mac Lane, Saunders
Annals of Mathematics

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