Knowledge (XXG)

Six operations

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were formal consequences of a small number of axioms. These axioms hold in many cases completely unrelated to the original context, and therefore the formal consequences also hold. The six operations formalism has since been shown to apply to contexts such as
1172: 1285: 1777: 532:. Finally, when working with abelian sheaves, there is a tensor product functor ⊗ and an internal Hom functor, and these are adjoint. The six operations are the corresponding functors on the derived category: 1056: 840: 924: 1417: 2168: 2080: 1992: 1904: 1511: 1681: 1062: 1178: 657: 1552: 611: 591: 530: 492: 457: 430: 374: 347: 316: 289: 248: 216: 184: 152: 1687: 2339: 955: 1312: 2370: 221: 189: 93: 756: 846: 2086: 1998: 1910: 1822: 2418: 2189: 2184: 2423: 2283:
Laszlo, Yves; Olsson, Martin (2005). "The six operations for sheaves on Artin stacks I: Finite coefficients".
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Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique
1167:{\displaystyle \operatorname {RHom} _{Y}(Rf_{!}M,N)\to Rf_{*}\operatorname {RHom} _{X}(M,f^{!}N),} 17: 2345: 2317: 2284: 2249: 1280:{\displaystyle f^{!}\operatorname {RHom} _{Y}(M,N)\to \operatorname {RHom} _{X}(Lf^{*}M,f^{!}N).} 113: 2366: 2335: 2399:
What (if anything) unifies stable homotopy theory and Grothendieck's six functors formalism?
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are the units of the tensor product operations (which vary depending on which category of
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where the first two maps are the counit and unit, respectively of the adjunctions. If
253: 2407: 2349: 258: 71:. The basic insight was that many of the elementary facts relating cohomology on 31: 2386: 2331: 683: 1772:{\displaystyle D_{X}(M\otimes D_{X}(M'))\to \operatorname {RHom} (M,M'),} 1051:{\displaystyle (Rf_{!}M)\otimes _{Y}N\to Rf_{!}(M\otimes _{X}Lf^{*}N),} 376:. Similarly, internal tensor product is left adjoint to internal Hom. 109: 2289: 2254: 2398: 2322: 1306:, then there is a distinguished triangle in the derived category: 2390: 1412:{\displaystyle Rj_{!}j^{!}\to 1\to Ri_{*}i^{*}\to Rj_{!}j^{!},} 2359:
Le formalisme des six opérations de Grothendieck pour les D
835:{\displaystyle Lg^{*}\circ Rf_{!}\to Rf'_{!}\circ Lg'^{*},} 919:{\displaystyle Rg'_{*}\circ f'^{!}\to f^{!}\circ Rg_{*}.} 750:, respectively, then there exist natural isomorphisms: 621:. In SGA 4 III, Grothendieck and Artin proved that if 2089: 2001: 1913: 1825: 1690: 1628: 1540: 1439: 1315: 1181: 1065: 958: 849: 759: 635: 599: 579: 511: 473: 438: 411: 355: 328: 297: 270: 229: 197: 165: 133: 573:
Suppose that we restrict ourselves to a category of
2312:Cisinski, Denis-Charles; DĂ©glise, FrĂ©dĂ©ric (2019). 2163:{\displaystyle D_{Y}(f_{*}M)\cong f_{!}(D_{X}(M)).} 2075:{\displaystyle D_{Y}(f_{!}M)\cong f_{*}(D_{X}(M)),} 1987:{\displaystyle D_{X}(f^{!}N)\cong f^{*}(D_{Y}(N)),} 1899:{\displaystyle D_{X}(f^{*}N)\cong f^{!}(D_{Y}(N)),} 1579:is an invertible object in the derived category on 2365:. Travaux en Cours. Vol. 35. Paris: Hermann. 2162: 2074: 1986: 1898: 1771: 1675: 1546: 1505: 1411: 1279: 1166: 1050: 918: 834: 651: 605: 585: 524: 486: 451: 424: 402:induces several functors. Specifically, it gives 368: 341: 310: 283: 242: 210: 178: 146: 933:is separated and of finite type, for any objects 1554:-adic torsion sheaves is under consideration). 2239:"Isomorphisms between left and right adjoints" 494:of direct image with proper support. In the 50:. It originally sprang from the relations in 8: 1430:are regular, then there is an isomorphism: 2221:"An introduction to six-functor formalism" 2321: 2288: 2253: 2139: 2126: 2107: 2094: 2088: 2051: 2038: 2019: 2006: 2000: 1963: 1950: 1931: 1918: 1912: 1875: 1862: 1843: 1830: 1824: 1808:are objects in the derived categories of 1714: 1695: 1689: 1652: 1639: 1627: 1539: 1494: 1484: 1444: 1438: 1391: 1381: 1365: 1355: 1333: 1323: 1314: 1262: 1246: 1227: 1196: 1186: 1180: 1149: 1127: 1117: 1089: 1070: 1064: 1033: 1020: 1004: 985: 969: 957: 907: 891: 877: 857: 848: 822: 799: 783: 767: 758: 643: 634: 598: 578: 516: 510: 478: 472: 443: 437: 416: 410: 360: 354: 333: 327: 302: 296: 275: 269: 234: 228: 202: 196: 170: 164: 138: 132: 2314:Triangulated categories of mixed motives 1506:{\displaystyle 1_{Z}(-c)\to i^{!}1_{S},} 398:be a morphism of schemes. The morphism 2211: 1816:, then there are natural isomorphisms: 874: 819: 222:proper (or extraordinary) inverse image 2316:. Springer Monographs in Mathematics. 190:proper (or extraordinary) direct image 112:. Usually these are functors between 1676:{\displaystyle M\to D_{X}(D_{X}(M)),} 697:is separated and of finite type. If 459:between the categories of sheaves on 7: 2237:Fausk, H.; P. Hu; J. P. May (2003). 949:, there exist natural isomorphisms: 613:is coprime to the characteristic of 1615:′ in the derived category on 734:′ denote the base changes of 711:is another morphism of schemes, if 116:and so are actually left and right 1302:with complementary open immersion 686:and denotes a shift in degree by 380:Six operations in Ă©tale cohomology 94:locally compact topological spaces 25: 18:Grothendieck's six operations 625:is smooth of relative dimension 27:Formalism in homological algebra 1782:are isomorphisms. Finally, if 2154: 2151: 2145: 2132: 2116: 2100: 2066: 2063: 2057: 2044: 2028: 2012: 1978: 1975: 1969: 1956: 1940: 1924: 1890: 1887: 1881: 1868: 1852: 1836: 1763: 1746: 1737: 1734: 1731: 1720: 1701: 1667: 1664: 1658: 1645: 1632: 1477: 1474: 1462: 1459: 1450: 1403: 1397: 1371: 1345: 1339: 1271: 1236: 1220: 1217: 1205: 1158: 1136: 1107: 1104: 1079: 1042: 1010: 994: 978: 959: 884: 789: 54:that arise from a morphism of 1: 693:. Furthermore, suppose that 593:-adic torsion sheaves, where 36:Grothendieck's six operations 945:in the derived category of 937:in the derived category of 718:denotes the base change of 467:, and it gives the functor 2440: 2357:Mebkhout, Zoghman (1989). 2190:Image functors for sheaves 2185:Grothendieck local duality 2332:10.1007/978-3-030-33242-6 2219:Gallauer, Martin (2021). 1294:is a closed immersion of 505:admits a right adjoint 108:The operations are six 2164: 2076: 1988: 1900: 1773: 1677: 1619:, the canonical maps: 1548: 1507: 1413: 1281: 1168: 1052: 920: 836: 653: 652:{\displaystyle Lf^{*}} 607: 587: 526: 488: 453: 426: 370: 343: 312: 285: 244: 212: 180: 148: 40:Alexander Grothendieck 2165: 2077: 1989: 1901: 1774: 1678: 1607:. Then, for objects 1549: 1547:{\displaystyle \ell } 1508: 1414: 1282: 1169: 1053: 921: 837: 654: 608: 606:{\displaystyle \ell } 588: 586:{\displaystyle \ell } 527: 525:{\displaystyle f^{!}} 489: 487:{\displaystyle f_{!}} 454: 452:{\displaystyle f_{*}} 427: 425:{\displaystyle f^{*}} 371: 369:{\displaystyle f^{!}} 344: 342:{\displaystyle f_{!}} 313: 311:{\displaystyle f_{*}} 286: 284:{\displaystyle f^{*}} 245: 243:{\displaystyle f^{!}} 213: 211:{\displaystyle f_{!}} 181: 179:{\displaystyle f^{*}} 149: 147:{\displaystyle f_{*}} 48:six-functor formalism 2087: 1999: 1911: 1823: 1688: 1626: 1538: 1437: 1313: 1179: 1063: 956: 929:Again assuming that 847: 757: 633: 597: 577: 509: 471: 436: 409: 353: 326: 295: 268: 227: 195: 163: 131: 46:, also known as the 42:, is a formalism in 2419:Homological algebra 2264:2002math......6079F 2246:Theory Appl. Categ. 865: 807: 86:algebraic varieties 44:homological algebra 2363:-modules cohĂ©rents 2160: 2072: 1984: 1896: 1769: 1673: 1596:to be the functor 1544: 1503: 1409: 1277: 1164: 1048: 916: 853: 832: 795: 649: 603: 583: 522: 484: 449: 422: 366: 339: 308: 281: 240: 208: 176: 144: 114:derived categories 2341:978-3-030-33241-9 1800:-schemes, and if 1796:is a morphism of 659:is isomorphic to 16:(Redirected from 2431: 2424:Duality theories 2376: 2353: 2325: 2308: 2306: 2294: 2292: 2275: 2274: 2272: 2270: 2257: 2243: 2234: 2228: 2227: 2225: 2216: 2180:Coherent duality 2169: 2167: 2166: 2161: 2144: 2143: 2131: 2130: 2112: 2111: 2099: 2098: 2081: 2079: 2078: 2073: 2056: 2055: 2043: 2042: 2024: 2023: 2011: 2010: 1993: 1991: 1990: 1985: 1968: 1967: 1955: 1954: 1936: 1935: 1923: 1922: 1905: 1903: 1902: 1897: 1880: 1879: 1867: 1866: 1848: 1847: 1835: 1834: 1795: 1778: 1776: 1775: 1770: 1762: 1730: 1719: 1718: 1700: 1699: 1682: 1680: 1679: 1674: 1657: 1656: 1644: 1643: 1606: 1586: 1583:with respect to 1574: 1553: 1551: 1550: 1545: 1533: 1524: 1512: 1510: 1509: 1504: 1499: 1498: 1489: 1488: 1449: 1448: 1418: 1416: 1415: 1410: 1396: 1395: 1386: 1385: 1370: 1369: 1360: 1359: 1338: 1337: 1328: 1327: 1286: 1284: 1283: 1278: 1267: 1266: 1251: 1250: 1232: 1231: 1201: 1200: 1191: 1190: 1173: 1171: 1170: 1165: 1154: 1153: 1132: 1131: 1122: 1121: 1094: 1093: 1075: 1074: 1057: 1055: 1054: 1049: 1038: 1037: 1025: 1024: 1009: 1008: 990: 989: 974: 973: 925: 923: 922: 917: 912: 911: 896: 895: 883: 882: 881: 861: 841: 839: 838: 833: 828: 827: 826: 803: 788: 787: 772: 771: 717: 710: 706:′ → 692: 677: 669: 658: 656: 655: 650: 648: 647: 612: 610: 609: 604: 592: 590: 589: 584: 569: 565: 561: 555: 546: 537: 531: 529: 528: 523: 521: 520: 496:derived category 493: 491: 490: 485: 483: 482: 458: 456: 455: 450: 448: 447: 431: 429: 428: 423: 421: 420: 404:adjoint functors 397: 375: 373: 372: 367: 365: 364: 348: 346: 345: 340: 338: 337: 317: 315: 314: 309: 307: 306: 290: 288: 287: 282: 280: 279: 249: 247: 246: 241: 239: 238: 217: 215: 214: 209: 207: 206: 185: 183: 182: 177: 175: 174: 153: 151: 150: 145: 143: 142: 118:derived functors 70: 52:Ă©tale cohomology 21: 2439: 2438: 2434: 2433: 2432: 2430: 2429: 2428: 2404: 2403: 2383: 2373: 2362: 2356: 2342: 2311: 2304: 2298:Ayoub, Joseph. 2297: 2282: 2279: 2278: 2268: 2266: 2241: 2236: 2235: 2231: 2223: 2218: 2217: 2213: 2208: 2200:Change of rings 2195:Verdier duality 2176: 2135: 2122: 2103: 2090: 2085: 2084: 2047: 2034: 2015: 2002: 1997: 1996: 1959: 1946: 1927: 1914: 1909: 1908: 1871: 1858: 1839: 1826: 1821: 1820: 1783: 1755: 1723: 1710: 1691: 1686: 1685: 1648: 1635: 1624: 1623: 1597: 1595: 1584: 1562: 1561:is regular and 1536: 1535: 1532: 1526: 1523: 1517: 1490: 1480: 1440: 1435: 1434: 1387: 1377: 1361: 1351: 1329: 1319: 1311: 1310: 1258: 1242: 1223: 1192: 1182: 1177: 1176: 1145: 1123: 1113: 1085: 1066: 1061: 1060: 1029: 1016: 1000: 981: 965: 954: 953: 903: 887: 873: 869: 845: 844: 818: 814: 779: 763: 755: 754: 712: 698: 687: 671: 660: 639: 631: 630: 595: 594: 575: 574: 567: 563: 557: 554: 548: 545: 539: 533: 512: 507: 506: 504: 474: 469: 468: 439: 434: 433: 412: 407: 406: 385: 382: 356: 351: 350: 329: 324: 323: 320:adjoint functor 298: 293: 292: 271: 266: 265: 230: 225: 224: 198: 193: 192: 166: 161: 160: 134: 129: 128: 106: 58: 28: 23: 22: 15: 12: 11: 5: 2437: 2435: 2427: 2426: 2421: 2416: 2406: 2405: 2402: 2401: 2396: 2387:six operations 2382: 2381:External links 2379: 2378: 2377: 2371: 2360: 2354: 2340: 2309: 2295: 2277: 2276: 2229: 2210: 2209: 2207: 2204: 2203: 2202: 2197: 2192: 2187: 2182: 2175: 2172: 2171: 2170: 2159: 2156: 2153: 2150: 2147: 2142: 2138: 2134: 2129: 2125: 2121: 2118: 2115: 2110: 2106: 2102: 2097: 2093: 2082: 2071: 2068: 2065: 2062: 2059: 2054: 2050: 2046: 2041: 2037: 2033: 2030: 2027: 2022: 2018: 2014: 2009: 2005: 1994: 1983: 1980: 1977: 1974: 1971: 1966: 1962: 1958: 1953: 1949: 1945: 1942: 1939: 1934: 1930: 1926: 1921: 1917: 1906: 1895: 1892: 1889: 1886: 1883: 1878: 1874: 1870: 1865: 1861: 1857: 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1098: 1095: 1090: 1086: 1082: 1076: 1071: 1067: 1059: 1045: 1039: 1034: 1030: 1026: 1021: 1017: 1013: 1005: 1001: 997: 991: 986: 982: 975: 970: 966: 962: 952: 951: 950: 948: 944: 940: 936: 932: 913: 908: 904: 900: 897: 892: 888: 878: 870: 866: 862: 858: 854: 850: 843: 829: 823: 815: 811: 808: 804: 800: 796: 792: 784: 780: 776: 773: 768: 764: 760: 753: 752: 751: 749: 745: 741: 737: 733: 729: 725: 721: 715: 709: 705: 701: 696: 691: 685: 681: 675: 667: 663: 644: 640: 636: 628: 624: 620: 616: 600: 580: 571: 560: 551: 542: 536: 517: 513: 501: 497: 479: 475: 466: 462: 444: 440: 417: 413: 405: 401: 396: 392: 388: 379: 377: 361: 357: 334: 330: 321: 303: 299: 276: 272: 264:The functors 260: 257: 255: 251: 235: 231: 223: 219: 203: 199: 191: 187: 171: 167: 159: 158:inverse image 155: 139: 135: 127: 123: 122: 121: 119: 115: 111: 103: 101: 99: 95: 91: 87: 83: 78: 74: 69: 65: 61: 57: 53: 49: 45: 41: 37: 33: 19: 2414:Sheaf theory 2391: 2358: 2313: 2300: 2290:math/0512097 2267:. 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Index

Grothendieck's six operations
mathematics
Alexander Grothendieck
homological algebra
Ă©tale cohomology
schemes
algebraic varieties
sheaves
locally compact topological spaces
motives
functors
derived categories
derived functors
direct image
inverse image
proper (or extraordinary) direct image
proper (or extraordinary) inverse image
tensor product
internal Hom
adjoint functor
adjoint functors
derived category
Tate twist
Coherent duality
Grothendieck local duality
Image functors for sheaves
Verdier duality
Change of rings
"An introduction to six-functor formalism"
"Isomorphisms between left and right adjoints"

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