2555:
129:
The syzygy theorem first appeared in
Hilbert's seminal paper "Ăber die Theorie der algebraischen Formen" (1890). The paper is split into five parts: part I proves Hilbert's basis theorem over a field, while part II proves it over the integers. Part III contains the syzygy theorem (Theorem III), which
1157:
1459:
2353:
3549:
degree necessarily occurs. However such examples are extremely rare, and this sets the question of an algorithm that is efficient when the output is not too large. At the present time, the best algorithms for computing syzygies are
2665:
1076:
1381:
2022:
2883:
2121:
957:
3207:
3311:
2186:
94:. As the relations form a module, one may consider the relations between the relations; the theorem asserts that, if one continues in this way, starting with a module over a polynomial ring in
3004:
3110:
381:
777:
601:
470:
2550:{\displaystyle G_{i_{1}}\wedge \cdots \wedge G_{i_{t}}\mapsto \sum _{j=1}^{t}(-1)^{j+1}g_{i_{j}}G_{i_{1}}\wedge \cdots \wedge {\widehat {G}}_{i_{j}}\wedge \cdots \wedge G_{i_{t}},}
529:
300:
3370:
whose unknowns are the coefficients of these monomials. Therefore, any algorithm for linear systems implies an algorithm for syzygies, as soon as a bound of the degrees is known.
3362:
of the generators of the module of syzygies. In fact, the coefficients of the syzygies are unknown polynomials. If the degree of these polynomials is bounded, the number of their
1912:
2799:
1765:
713:
222:
2345:
3528:
670:
2560:
where the hat means that the factor is omitted. A straightforward computation shows that the composition of two consecutive such maps is zero, and thus that one has a
1578:
2283:
2219:
980:
3344:
3137:
3035:
2916:
2250:
2052:
1943:
1859:
1241:
1214:
1187:
888:
861:
834:
807:
634:
411:
3621:
2740:
1832:
3455:
3743:
3723:
3699:
3679:
3655:
1676:
1542:
1317:
2815:
2569:
1152:{\displaystyle 0\longrightarrow R_{n}\longrightarrow L_{n-1}\longrightarrow \cdots \longrightarrow L_{0}\longrightarrow M\longrightarrow 0,}
1454:{\displaystyle 0\longrightarrow L_{k}\longrightarrow L_{k-1}\longrightarrow \cdots \longrightarrow L_{0}\longrightarrow M\longrightarrow 0}
993:
module is the module of the relations between generators of the first syzygy module. By continuing in this way, one may define the
1954:
3554:
algorithms. They allow the computation of the first syzygy module, and also, with almost no extra cost, all syzygies modules.
2819:
2063:
896:
3763:
3146:
3810:
Die Frage der endlich vielen
Schritte in der Theorie der Polynomideale. Unter Benutzung nachgelassener SĂ€tze von K. Hentzelt
3221:
2129:
2935:
3040:
314:
3874:
3866:
1702:
In the case of a single indeterminate, Hilbert's syzygy theorem is an instance of the theorem asserting that over a
1621:
of a ring is the supremum of the projective dimensions of all modules, Hilbert's syzygy theorem may be restated as:
724:
537:
3928:
1718:, also called "complex of exterior algebra", allows, in some cases, an explicit description of all syzygy modules.
1477:
This upper bound on the projective dimension is sharp, that is, there are modules of projective dimension exactly
3746:
3896:
420:
3918:
3913:
482:
253:
60:
1868:
3891:
3546:
2758:
1724:
675:
181:
3758:
3367:
1021:, then by taking a basis as a generating set, the next syzygy module (and every subsequent one) is the
176:
83:
48:
2304:
3923:
3886:
3796:
3634:
139:
3484:
3374:
3354:
At
Hilbert's time, there was no method available for computing syzygies. It was only known that an
1325:
64:
56:
1025:. If one does not take a basis as a generating set, then all subsequent syzygy modules are free.
607:
642:
2748:
1696:
1547:
3783:
3481:
such that the degrees occurring in a generating system of the first syzygy module is at most
2255:
2191:
1946:
983:
965:
3880:
3322:
3115:
3013:
2894:
2228:
2030:
1921:
1837:
1703:
1691:
In the case of zero indeterminates, Hilbert's syzygy theorem is simply the fact that every
1357:
1219:
1192:
1165:
1057:
866:
839:
812:
785:
612:
389:
165:
151:
91:
79:
8:
3565:
2684:
1776:
1329:
1050:. The above property of invariance, up to the sum direct with free modules, implies that
836:
is the module that is obtained with another generating set, there exist two free modules
157:
118:
114:
87:
32:
3402:
3728:
3708:
3684:
3664:
3640:
1626:
1492:
1267:
169:
131:
44:
2804:
3870:
3862:
3007:
1047:
3861:. Graduate Texts in Mathematics, 150. Springer-Verlag, New York, 1995. xvi+785 pp.
3813:
2744:
2055:
1915:
1618:
135:
40:
3551:
1216:
is projective. It can be shown that one may always choose the generating sets for
3877:
3702:
3658:
1373:
1262:
1244:
161:
52:
28:
3854:
2675:
1715:
716:
3907:
3378:
2561:
2222:
1617:
The theorem is also true for modules that are not finitely generated. As the
36:
3828:
3624:
3359:
2891:
th syzygy module is free of dimension one (generated by the product of all
1692:
2660:{\displaystyle 0\to L_{t}\to L_{t-1}\to \cdots \to L_{1}\to L_{0}\to R/I.}
16:
Theorem about linear relations in ideals and modules over polynomial rings
1862:
1350:
1022:
414:
101:
68:
20:
3821:
3817:
637:
3623:
causes the
Hilbert syzygy theorem to hold. It turns out that this is
2801:
is regular, and the Koszul complex is thus a projective resolution of
39:
in 1890, that were introduced for solving important open questions in
3355:
3218:
The same proof applies for proving that the projective dimension of
2926:
th syzygy module is thus the quotient of a free module of dimension
113:
Hilbert's syzygy theorem is now considered to be an early result of
3363:
1070:
if not. This is equivalent with the existence of an exact sequence
3745:
is equal to the Krull dimension. This result may be proven using
117:. It is the starting point of the use of homological methods in
3826:
The question of finitely many steps in polynomial ideal theory
3366:
is also bounded. Expressing that one has a syzygy provides a
477:
134:. The last part, part V, proves finite generation of certain
3627:, which is an algebraic formulation of the fact that affine
138:. Incidentally part III also contains a special case of the
2680:
it is an exact sequence if one works with a polynomial ring
3859:
Commutative algebra. With a view toward algebraic geometry
2017:{\displaystyle \Lambda (L_{1})=\bigoplus _{t=0}^{k}L_{t},}
1243:
being free, that is for the above exact sequence to be a
2878:{\displaystyle k=R/\langle x_{1},\ldots ,x_{n}\rangle .}
2116:{\displaystyle G_{i_{1}}\wedge \cdots \wedge G_{i_{t}},}
952:{\displaystyle R_{1}\oplus F_{1}\cong S_{1}\oplus F_{2}}
3812:, Mathematische Annalen, Volume 95, Number 1, 736-788,
3782:
D. Hilbert, Ăber die
Theorie der algebraischen Formen,
3202:{\displaystyle k=R/\langle x_{1},\ldots ,x_{n}\rangle }
2807:
63:, which establishes a bijective correspondence between
1056:
does not depend on the choice of generating sets. The
3731:
3711:
3687:
3667:
3643:
3568:
3530:
The same bound applies for testing the membership to
3487:
3405:
3325:
3306:{\displaystyle k/\langle g_{1},\ldots ,g_{t}\rangle }
3224:
3149:
3118:
3043:
3016:
2938:
2897:
2822:
2761:
2687:
2572:
2356:
2307:
2258:
2231:
2194:
2132:
2066:
2033:
1957:
1924:
1871:
1840:
1779:
1727:
1629:
1550:
1495:
1384:
1270:
1222:
1195:
1168:
1079:
968:
899:
869:
842:
815:
788:
727:
678:
645:
615:
540:
485:
423:
392:
317:
256:
184:
3346:
form a regular sequence of homogeneous polynomials.
100:
indeterminates over a field, one eventually finds a
2181:{\displaystyle i_{1}<i_{2}<\cdots <i_{t}.}
1706:, every submodule of a free module is itself free.
809:depends on the choice of a generating set, but, if
3737:
3717:
3693:
3673:
3649:
3637:. In fact the following generalization holds: Let
3615:
3562:One might wonder which ring-theoretic property of
3522:
3449:
3338:
3305:
3201:
3131:
3104:
3029:
2998:
2910:
2877:
2809:
2793:
2734:
2659:
2549:
2339:
2277:
2244:
2213:
2180:
2115:
2046:
2016:
1937:
1906:
1853:
1826:
1759:
1670:
1572:
1536:
1453:
1311:
1235:
1208:
1181:
1151:
974:
951:
882:
855:
828:
801:
771:
707:
664:
628:
595:
523:
464:
405:
375:
294:
216:
164:, but the concept generalizes trivially to (left)
3373:The first bound for syzygies (as well as for the
2999:{\displaystyle (x_{1},-x_{2},\ldots ,\pm x_{n}).}
3905:
3725:is finite; in that case the global dimension of
3143:). This proves that the projective dimension of
3105:{\displaystyle p_{1}x_{1}+\cdots +p_{n}x_{n}=1,}
376:{\displaystyle a_{1}g_{1}+\cdots +a_{k}g_{k}=0.}
2054:is the free module, which has, as a basis, the
1034:be the smallest integer, if any, such that the
27:is one of the three fundamental theorems about
3545:On the other hand, there are examples where a
3010:, as otherwise, there would exist polynomials
772:{\displaystyle 0\to R_{1}\to L_{0}\to M\to 0.}
596:{\displaystyle a_{1}G_{1}+\cdots +a_{k}G_{k},}
3112:which is impossible (substituting 0 for the
3300:
3268:
3196:
3164:
2869:
2837:
3681:has finite global dimension if and only if
3358:may be deduced from any upper bound of the
3799:, but extends easily to arbitrary modules.
3557:
2674:. In general the Koszul complex is not an
1356:In modern language, this implies that the
3829:(review and English-language translation)
1255:Hilbert's syzygy theorem states that, if
156:Originally, Hilbert defined syzygies for
145:
3906:
3747:Serre's theorem on regular local rings
1602:th syzygy module is free, but not the
1261:is a finitely generated module over a
465:{\displaystyle (G_{1},\ldots ,G_{k}).}
82:in Hilbert's terminology, between the
74:Hilbert's syzygy theorem concerns the
3764:Hilbert series and Hilbert polynomial
524:{\displaystyle (a_{1},\ldots ,a_{k})}
295:{\displaystyle (a_{1},\ldots ,a_{k})}
55:of polynomial rings over a field are
3457:if the coefficients over a basis of
1907:{\displaystyle G_{1},\ldots ,G_{k}.}
1483:. The standard example is the field
531:may be identified with the element
2794:{\displaystyle x_{1},\ldots ,x_{n}}
1767:be a generating system of an ideal
1760:{\displaystyle g_{1},\ldots ,g_{k}}
1611:
708:{\displaystyle G_{i}\mapsto g_{i}.}
217:{\displaystyle g_{1},\ldots ,g_{k}}
13:
2221:(because of the definition of the
1958:
1066:is this integer, if it exists, or
1015:th syzygy module is free for some
130:is used in part IV to discuss the
14:
3940:
1709:
43:, and are at the basis of modern
3139:in the latter equality provides
2340:{\displaystyle L_{t}\to L_{t-1}}
1686:
1489:, which may be considered as a
1372:, and thus that there exists a
3833:
3802:
3789:
3776:
3610:
3578:
3523:{\displaystyle (td)^{2^{cn}}.}
3498:
3488:
3441:
3409:
3349:
3260:
3228:
2990:
2939:
2932:by the submodule generated by
2729:
2697:
2640:
2627:
2614:
2608:
2589:
2576:
2434:
2424:
2400:
2318:
2301:, one may define a linear map
1974:
1961:
1821:
1789:
1665:
1633:
1531:
1499:
1445:
1439:
1426:
1420:
1401:
1388:
1306:
1274:
1140:
1134:
1121:
1115:
1096:
1083:
763:
757:
744:
731:
689:
656:
518:
486:
456:
424:
289:
257:
1:
3769:
3387:a submodule of a free module
1040:th syzygy module of a module
47:. The two other theorems are
3795:The theory is presented for
3633:-space is a variety without
3469:have a total degree at most
2755:In particular, the sequence
2743:and an ideal generated by a
1250:
244:between the generators is a
104:of relations, after at most
7:
3892:Encyclopedia of Mathematics
3752:
3475:, then there is a constant
3006:This quotient may not be a
1000:for every positive integer
715:In other words, one has an
606:and the relations form the
10:
3945:
3797:finitely generated modules
3463:of a generating system of
3368:system of linear equations
2225:), the two definitions of
665:{\displaystyle L_{0}\to M}
149:
124:
65:affine algebraic varieties
3846:, but did not prove this.
1610:th one (for a proof, see
782:This first syzygy module
61:Hilbert's Nullstellensatz
51:, which asserts that all
3375:ideal membership problem
1623:the global dimension of
1573:{\displaystyle x_{i}c=0}
121:and algebraic geometry.
90:, or, more generally, a
25:Hilbert's syzygy theorem
3929:Theorems in ring theory
3558:Syzygies and regularity
3377:) was given in 1926 by
2749:homogeneous polynomials
2278:{\displaystyle L_{t}=0}
2214:{\displaystyle L_{0}=R}
2188:In particular, one has
1596:. For this module, the
975:{\displaystyle \oplus }
49:Hilbert's basis theorem
3824:in German language) â
3759:QuillenâSuslin theorem
3739:
3719:
3695:
3675:
3651:
3617:
3524:
3451:
3340:
3307:
3203:
3133:
3106:
3031:
3000:
2912:
2879:
2811:
2795:
2736:
2661:
2551:
2423:
2341:
2279:
2246:
2215:
2182:
2117:
2048:
2018:
2000:
1939:
1908:
1855:
1828:
1761:
1672:
1574:
1538:
1455:
1313:
1237:
1210:
1183:
1153:
976:
953:
884:
857:
830:
803:
773:
709:
666:
630:
597:
525:
466:
407:
377:
296:
218:
3784:Mathematische Annalen
3740:
3720:
3696:
3676:
3652:
3618:
3525:
3452:
3341:
3339:{\displaystyle g_{i}}
3308:
3204:
3134:
3132:{\displaystyle x_{i}}
3107:
3032:
3030:{\displaystyle p_{i}}
3001:
2913:
2911:{\displaystyle G_{i}}
2880:
2812:
2796:
2737:
2662:
2552:
2403:
2342:
2295:. For every positive
2280:
2247:
2245:{\displaystyle L_{1}}
2216:
2183:
2118:
2049:
2047:{\displaystyle L_{t}}
2019:
1980:
1940:
1938:{\displaystyle L_{1}}
1909:
1856:
1854:{\displaystyle L_{1}}
1829:
1773:in a polynomial ring
1762:
1673:
1612:§ Koszul complex
1575:
1539:
1456:
1314:
1238:
1236:{\displaystyle R_{n}}
1211:
1209:{\displaystyle R_{n}}
1184:
1182:{\displaystyle L_{i}}
1154:
984:direct sum of modules
977:
954:
885:
883:{\displaystyle F_{2}}
858:
856:{\displaystyle F_{1}}
831:
829:{\displaystyle S_{1}}
804:
802:{\displaystyle R_{1}}
774:
710:
667:
631:
629:{\displaystyle R_{1}}
598:
526:
467:
408:
406:{\displaystyle L_{0}}
378:
297:
219:
140:HilbertâBurch theorem
71:of polynomial rings.
3729:
3709:
3685:
3665:
3641:
3566:
3485:
3403:
3323:
3222:
3147:
3116:
3041:
3014:
2936:
2895:
2820:
2805:
2759:
2685:
2570:
2354:
2305:
2256:
2229:
2192:
2130:
2064:
2031:
1955:
1922:
1869:
1838:
1777:
1725:
1704:principal ideal ring
1627:
1548:
1493:
1382:
1358:projective dimension
1343:th syzygy module of
1268:
1220:
1193:
1166:
1077:
1058:projective dimension
966:
897:
867:
840:
813:
786:
725:
676:
643:
613:
538:
483:
421:
390:
315:
254:
182:
152:syzygy (mathematics)
146:Syzygies (relations)
3919:Homological algebra
3914:Commutative algebra
3839:G. Hermann claimed
3701:is regular and the
3616:{\displaystyle A=k}
2735:{\displaystyle R=k}
1827:{\displaystyle R=k}
1544:-module by setting
136:rings of invariants
119:commutative algebra
115:homological algebra
3818:10.1007/BF01206635
3735:
3715:
3691:
3671:
3647:
3613:
3547:double exponential
3520:
3450:{\displaystyle k;}
3447:
3336:
3303:
3199:
3129:
3102:
3027:
2996:
2908:
2885:In this case, the
2875:
2791:
2732:
2657:
2547:
2337:
2275:
2242:
2211:
2178:
2113:
2044:
2014:
1935:
1904:
1851:
1824:
1757:
1668:
1570:
1534:
1451:
1309:
1233:
1206:
1179:
1162:where the modules
1149:
972:
949:
880:
853:
826:
799:
769:
705:
662:
626:
593:
521:
462:
403:
373:
292:
214:
132:Hilbert polynomial
57:finitely generated
45:algebraic geometry
35:, first proved by
3887:"Hilbert theorem"
3738:{\displaystyle A}
3718:{\displaystyle A}
3694:{\displaystyle A}
3674:{\displaystyle A}
3650:{\displaystyle A}
3536:of an element of
3008:projective module
2502:
2056:exterior products
1671:{\displaystyle k}
1537:{\displaystyle k}
1312:{\displaystyle k}
3936:
3924:Invariant theory
3900:
3847:
3845:
3837:
3831:
3806:
3800:
3793:
3787:
3780:
3744:
3742:
3741:
3736:
3724:
3722:
3721:
3716:
3700:
3698:
3697:
3692:
3680:
3678:
3677:
3672:
3656:
3654:
3653:
3648:
3632:
3622:
3620:
3619:
3614:
3609:
3608:
3590:
3589:
3541:
3535:
3529:
3527:
3526:
3521:
3516:
3515:
3514:
3513:
3480:
3474:
3468:
3462:
3456:
3454:
3453:
3448:
3440:
3439:
3421:
3420:
3398:
3392:
3386:
3345:
3343:
3342:
3337:
3335:
3334:
3318:
3312:
3310:
3309:
3304:
3299:
3298:
3280:
3279:
3267:
3259:
3258:
3240:
3239:
3214:
3208:
3206:
3205:
3200:
3195:
3194:
3176:
3175:
3163:
3142:
3138:
3136:
3135:
3130:
3128:
3127:
3111:
3109:
3108:
3103:
3092:
3091:
3082:
3081:
3063:
3062:
3053:
3052:
3036:
3034:
3033:
3028:
3026:
3025:
3005:
3003:
3002:
2997:
2989:
2988:
2967:
2966:
2951:
2950:
2931:
2925:
2917:
2915:
2914:
2909:
2907:
2906:
2890:
2884:
2882:
2881:
2876:
2868:
2867:
2849:
2848:
2836:
2816:
2814:
2813:
2810:{\displaystyle }
2808:
2800:
2798:
2797:
2792:
2790:
2789:
2771:
2770:
2745:regular sequence
2741:
2739:
2738:
2733:
2728:
2727:
2709:
2708:
2666:
2664:
2663:
2658:
2650:
2639:
2638:
2626:
2625:
2607:
2606:
2588:
2587:
2556:
2554:
2553:
2548:
2543:
2542:
2541:
2540:
2517:
2516:
2515:
2514:
2504:
2503:
2495:
2482:
2481:
2480:
2479:
2465:
2464:
2463:
2462:
2448:
2447:
2422:
2417:
2399:
2398:
2397:
2396:
2373:
2372:
2371:
2370:
2346:
2344:
2343:
2338:
2336:
2335:
2317:
2316:
2300:
2294:
2284:
2282:
2281:
2276:
2268:
2267:
2251:
2249:
2248:
2243:
2241:
2240:
2220:
2218:
2217:
2212:
2204:
2203:
2187:
2185:
2184:
2179:
2174:
2173:
2155:
2154:
2142:
2141:
2122:
2120:
2119:
2114:
2109:
2108:
2107:
2106:
2083:
2082:
2081:
2080:
2053:
2051:
2050:
2045:
2043:
2042:
2023:
2021:
2020:
2015:
2010:
2009:
1999:
1994:
1973:
1972:
1944:
1942:
1941:
1936:
1934:
1933:
1916:exterior algebra
1913:
1911:
1910:
1905:
1900:
1899:
1881:
1880:
1860:
1858:
1857:
1852:
1850:
1849:
1833:
1831:
1830:
1825:
1820:
1819:
1801:
1800:
1772:
1766:
1764:
1763:
1758:
1756:
1755:
1737:
1736:
1681:
1677:
1675:
1674:
1669:
1664:
1663:
1645:
1644:
1619:global dimension
1609:
1601:
1595:
1585:
1579:
1577:
1576:
1571:
1560:
1559:
1543:
1541:
1540:
1535:
1530:
1529:
1511:
1510:
1488:
1482:
1473:
1460:
1458:
1457:
1452:
1438:
1437:
1419:
1418:
1400:
1399:
1371:
1365:
1348:
1342:
1336:
1324:
1318:
1316:
1315:
1310:
1305:
1304:
1286:
1285:
1260:
1242:
1240:
1239:
1234:
1232:
1231:
1215:
1213:
1212:
1207:
1205:
1204:
1188:
1186:
1185:
1180:
1178:
1177:
1158:
1156:
1155:
1150:
1133:
1132:
1114:
1113:
1095:
1094:
1069:
1065:
1055:
1045:
1039:
1033:
1020:
1014:
1005:
998:th syzygy module
997:
981:
979:
978:
973:
958:
956:
955:
950:
948:
947:
935:
934:
922:
921:
909:
908:
889:
887:
886:
881:
879:
878:
862:
860:
859:
854:
852:
851:
835:
833:
832:
827:
825:
824:
808:
806:
805:
800:
798:
797:
778:
776:
775:
770:
756:
755:
743:
742:
714:
712:
711:
706:
701:
700:
688:
687:
671:
669:
668:
663:
655:
654:
635:
633:
632:
627:
625:
624:
602:
600:
599:
594:
589:
588:
579:
578:
560:
559:
550:
549:
530:
528:
527:
522:
517:
516:
498:
497:
475:
471:
469:
468:
463:
455:
454:
436:
435:
412:
410:
409:
404:
402:
401:
382:
380:
379:
374:
366:
365:
356:
355:
337:
336:
327:
326:
307:
301:
299:
298:
293:
288:
287:
269:
268:
249:
235:
229:
223:
221:
220:
215:
213:
212:
194:
193:
162:polynomial rings
109:
99:
41:invariant theory
29:polynomial rings
3944:
3943:
3939:
3938:
3937:
3935:
3934:
3933:
3904:
3903:
3885:
3851:
3850:
3840:
3838:
3834:
3808:Grete Hermann:
3807:
3803:
3794:
3790:
3781:
3777:
3772:
3755:
3730:
3727:
3726:
3710:
3707:
3706:
3703:Krull dimension
3686:
3683:
3682:
3666:
3663:
3662:
3659:Noetherian ring
3642:
3639:
3638:
3628:
3604:
3600:
3585:
3581:
3567:
3564:
3563:
3560:
3537:
3531:
3506:
3502:
3501:
3497:
3486:
3483:
3482:
3476:
3470:
3464:
3458:
3435:
3431:
3416:
3412:
3404:
3401:
3400:
3394:
3388:
3382:
3352:
3330:
3326:
3324:
3321:
3320:
3314:
3294:
3290:
3275:
3271:
3263:
3254:
3250:
3235:
3231:
3223:
3220:
3219:
3210:
3190:
3186:
3171:
3167:
3159:
3148:
3145:
3144:
3140:
3123:
3119:
3117:
3114:
3113:
3087:
3083:
3077:
3073:
3058:
3054:
3048:
3044:
3042:
3039:
3038:
3021:
3017:
3015:
3012:
3011:
2984:
2980:
2962:
2958:
2946:
2942:
2937:
2934:
2933:
2927:
2919:
2902:
2898:
2896:
2893:
2892:
2886:
2863:
2859:
2844:
2840:
2832:
2821:
2818:
2817:
2806:
2803:
2802:
2785:
2781:
2766:
2762:
2760:
2757:
2756:
2723:
2719:
2704:
2700:
2686:
2683:
2682:
2646:
2634:
2630:
2621:
2617:
2596:
2592:
2583:
2579:
2571:
2568:
2567:
2536:
2532:
2531:
2527:
2510:
2506:
2505:
2494:
2493:
2492:
2475:
2471:
2470:
2466:
2458:
2454:
2453:
2449:
2437:
2433:
2418:
2407:
2392:
2388:
2387:
2383:
2366:
2362:
2361:
2357:
2355:
2352:
2351:
2325:
2321:
2312:
2308:
2306:
2303:
2302:
2296:
2286:
2263:
2259:
2257:
2254:
2253:
2236:
2232:
2230:
2227:
2226:
2199:
2195:
2193:
2190:
2189:
2169:
2165:
2150:
2146:
2137:
2133:
2131:
2128:
2127:
2102:
2098:
2097:
2093:
2076:
2072:
2071:
2067:
2065:
2062:
2061:
2038:
2034:
2032:
2029:
2028:
2005:
2001:
1995:
1984:
1968:
1964:
1956:
1953:
1952:
1929:
1925:
1923:
1920:
1919:
1895:
1891:
1876:
1872:
1870:
1867:
1866:
1845:
1841:
1839:
1836:
1835:
1815:
1811:
1796:
1792:
1778:
1775:
1774:
1768:
1751:
1747:
1732:
1728:
1726:
1723:
1722:
1712:
1689:
1679:
1659:
1655:
1640:
1636:
1628:
1625:
1624:
1603:
1597:
1587:
1581:
1555:
1551:
1549:
1546:
1545:
1525:
1521:
1506:
1502:
1494:
1491:
1490:
1484:
1478:
1465:
1433:
1429:
1408:
1404:
1395:
1391:
1383:
1380:
1379:
1374:free resolution
1367:
1361:
1344:
1338:
1332:
1320:
1300:
1296:
1281:
1277:
1269:
1266:
1265:
1263:polynomial ring
1256:
1253:
1245:free resolution
1227:
1223:
1221:
1218:
1217:
1200:
1196:
1194:
1191:
1190:
1173:
1169:
1167:
1164:
1163:
1128:
1124:
1103:
1099:
1090:
1086:
1078:
1075:
1074:
1067:
1061:
1051:
1041:
1035:
1029:
1016:
1010:
1001:
995:
967:
964:
963:
943:
939:
930:
926:
917:
913:
904:
900:
898:
895:
894:
874:
870:
868:
865:
864:
847:
843:
841:
838:
837:
820:
816:
814:
811:
810:
793:
789:
787:
784:
783:
751:
747:
738:
734:
726:
723:
722:
696:
692:
683:
679:
677:
674:
673:
650:
646:
644:
641:
640:
620:
616:
614:
611:
610:
584:
580:
574:
570:
555:
551:
545:
541:
539:
536:
535:
512:
508:
493:
489:
484:
481:
480:
473:
450:
446:
431:
427:
422:
419:
418:
397:
393:
391:
388:
387:
361:
357:
351:
347:
332:
328:
322:
318:
316:
313:
312:
303:
302:of elements of
283:
279:
264:
260:
255:
252:
251:
245:
231:
225:
208:
204:
189:
185:
183:
180:
179:
154:
148:
127:
105:
95:
17:
12:
11:
5:
3942:
3932:
3931:
3926:
3921:
3916:
3902:
3901:
3883:
3855:David Eisenbud
3849:
3848:
3832:
3801:
3788:
3774:
3773:
3771:
3768:
3767:
3766:
3761:
3754:
3751:
3734:
3714:
3690:
3670:
3646:
3612:
3607:
3603:
3599:
3596:
3593:
3588:
3584:
3580:
3577:
3574:
3571:
3559:
3556:
3519:
3512:
3509:
3505:
3500:
3496:
3493:
3490:
3446:
3443:
3438:
3434:
3430:
3427:
3424:
3419:
3415:
3411:
3408:
3351:
3348:
3333:
3329:
3302:
3297:
3293:
3289:
3286:
3283:
3278:
3274:
3270:
3266:
3262:
3257:
3253:
3249:
3246:
3243:
3238:
3234:
3230:
3227:
3198:
3193:
3189:
3185:
3182:
3179:
3174:
3170:
3166:
3162:
3158:
3155:
3152:
3126:
3122:
3101:
3098:
3095:
3090:
3086:
3080:
3076:
3072:
3069:
3066:
3061:
3057:
3051:
3047:
3024:
3020:
2995:
2992:
2987:
2983:
2979:
2976:
2973:
2970:
2965:
2961:
2957:
2954:
2949:
2945:
2941:
2905:
2901:
2874:
2871:
2866:
2862:
2858:
2855:
2852:
2847:
2843:
2839:
2835:
2831:
2828:
2825:
2788:
2784:
2780:
2777:
2774:
2769:
2765:
2731:
2726:
2722:
2718:
2715:
2712:
2707:
2703:
2699:
2696:
2693:
2690:
2676:exact sequence
2672:Koszul complex
2668:
2667:
2656:
2653:
2649:
2645:
2642:
2637:
2633:
2629:
2624:
2620:
2616:
2613:
2610:
2605:
2602:
2599:
2595:
2591:
2586:
2582:
2578:
2575:
2558:
2557:
2546:
2539:
2535:
2530:
2526:
2523:
2520:
2513:
2509:
2501:
2498:
2491:
2488:
2485:
2478:
2474:
2469:
2461:
2457:
2452:
2446:
2443:
2440:
2436:
2432:
2429:
2426:
2421:
2416:
2413:
2410:
2406:
2402:
2395:
2391:
2386:
2382:
2379:
2376:
2369:
2365:
2360:
2334:
2331:
2328:
2324:
2320:
2315:
2311:
2274:
2271:
2266:
2262:
2252:coincide, and
2239:
2235:
2210:
2207:
2202:
2198:
2177:
2172:
2168:
2164:
2161:
2158:
2153:
2149:
2145:
2140:
2136:
2124:
2123:
2112:
2105:
2101:
2096:
2092:
2089:
2086:
2079:
2075:
2070:
2041:
2037:
2025:
2024:
2013:
2008:
2004:
1998:
1993:
1990:
1987:
1983:
1979:
1976:
1971:
1967:
1963:
1960:
1932:
1928:
1903:
1898:
1894:
1890:
1887:
1884:
1879:
1875:
1848:
1844:
1823:
1818:
1814:
1810:
1807:
1804:
1799:
1795:
1791:
1788:
1785:
1782:
1754:
1750:
1746:
1743:
1740:
1735:
1731:
1716:Koszul complex
1711:
1710:Koszul complex
1708:
1688:
1685:
1667:
1662:
1658:
1654:
1651:
1648:
1643:
1639:
1635:
1632:
1569:
1566:
1563:
1558:
1554:
1533:
1528:
1524:
1520:
1517:
1514:
1509:
1505:
1501:
1498:
1462:
1461:
1450:
1447:
1444:
1441:
1436:
1432:
1428:
1425:
1422:
1417:
1414:
1411:
1407:
1403:
1398:
1394:
1390:
1387:
1326:indeterminates
1308:
1303:
1299:
1295:
1292:
1289:
1284:
1280:
1276:
1273:
1252:
1249:
1230:
1226:
1203:
1199:
1176:
1172:
1160:
1159:
1148:
1145:
1142:
1139:
1136:
1131:
1127:
1123:
1120:
1117:
1112:
1109:
1106:
1102:
1098:
1093:
1089:
1085:
1082:
971:
960:
959:
946:
942:
938:
933:
929:
925:
920:
916:
912:
907:
903:
877:
873:
850:
846:
823:
819:
796:
792:
780:
779:
768:
765:
762:
759:
754:
750:
746:
741:
737:
733:
730:
717:exact sequence
704:
699:
695:
691:
686:
682:
661:
658:
653:
649:
623:
619:
604:
603:
592:
587:
583:
577:
573:
569:
566:
563:
558:
554:
548:
544:
520:
515:
511:
507:
504:
501:
496:
492:
488:
461:
458:
453:
449:
445:
442:
439:
434:
430:
426:
400:
396:
384:
383:
372:
369:
364:
360:
354:
350:
346:
343:
340:
335:
331:
325:
321:
291:
286:
282:
278:
275:
272:
267:
263:
259:
211:
207:
203:
200:
197:
192:
188:
177:generating set
150:Main article:
147:
144:
126:
123:
15:
9:
6:
4:
3:
2:
3941:
3930:
3927:
3925:
3922:
3920:
3917:
3915:
3912:
3911:
3909:
3898:
3894:
3893:
3888:
3884:
3882:
3879:
3876:
3875:0-387-94269-6
3872:
3868:
3867:0-387-94268-8
3864:
3860:
3856:
3853:
3852:
3843:
3836:
3830:
3827:
3823:
3819:
3815:
3811:
3805:
3798:
3792:
3785:
3779:
3775:
3765:
3762:
3760:
3757:
3756:
3750:
3748:
3732:
3712:
3704:
3688:
3668:
3660:
3644:
3636:
3635:singularities
3631:
3626:
3605:
3601:
3597:
3594:
3591:
3586:
3582:
3575:
3572:
3569:
3555:
3553:
3552:Gröbner basis
3548:
3543:
3540:
3534:
3517:
3510:
3507:
3503:
3494:
3491:
3479:
3473:
3467:
3461:
3444:
3436:
3432:
3428:
3425:
3422:
3417:
3413:
3406:
3397:
3393:of dimension
3391:
3385:
3380:
3379:Grete Hermann
3376:
3371:
3369:
3365:
3361:
3357:
3347:
3331:
3327:
3317:
3295:
3291:
3287:
3284:
3281:
3276:
3272:
3264:
3255:
3251:
3247:
3244:
3241:
3236:
3232:
3225:
3216:
3213:
3191:
3187:
3183:
3180:
3177:
3172:
3168:
3160:
3156:
3153:
3150:
3124:
3120:
3099:
3096:
3093:
3088:
3084:
3078:
3074:
3070:
3067:
3064:
3059:
3055:
3049:
3045:
3022:
3018:
3009:
2993:
2985:
2981:
2977:
2974:
2971:
2968:
2963:
2959:
2955:
2952:
2947:
2943:
2930:
2923:
2903:
2899:
2889:
2872:
2864:
2860:
2856:
2853:
2850:
2845:
2841:
2833:
2829:
2826:
2823:
2786:
2782:
2778:
2775:
2772:
2767:
2763:
2753:
2752:
2750:
2746:
2724:
2720:
2716:
2713:
2710:
2705:
2701:
2694:
2691:
2688:
2681:
2677:
2673:
2654:
2651:
2647:
2643:
2635:
2631:
2622:
2618:
2611:
2603:
2600:
2597:
2593:
2584:
2580:
2573:
2566:
2565:
2564:
2563:
2544:
2537:
2533:
2528:
2524:
2521:
2518:
2511:
2507:
2499:
2496:
2489:
2486:
2483:
2476:
2472:
2467:
2459:
2455:
2450:
2444:
2441:
2438:
2430:
2427:
2419:
2414:
2411:
2408:
2404:
2393:
2389:
2384:
2380:
2377:
2374:
2367:
2363:
2358:
2350:
2349:
2348:
2332:
2329:
2326:
2322:
2313:
2309:
2299:
2293:
2289:
2272:
2269:
2264:
2260:
2237:
2233:
2224:
2223:empty product
2208:
2205:
2200:
2196:
2175:
2170:
2166:
2162:
2159:
2156:
2151:
2147:
2143:
2138:
2134:
2110:
2103:
2099:
2094:
2090:
2087:
2084:
2077:
2073:
2068:
2060:
2059:
2058:
2057:
2039:
2035:
2011:
2006:
2002:
1996:
1991:
1988:
1985:
1981:
1977:
1969:
1965:
1951:
1950:
1949:
1948:
1930:
1926:
1917:
1901:
1896:
1892:
1888:
1885:
1882:
1877:
1873:
1864:
1846:
1842:
1816:
1812:
1808:
1805:
1802:
1797:
1793:
1786:
1783:
1780:
1771:
1752:
1748:
1744:
1741:
1738:
1733:
1729:
1719:
1717:
1707:
1705:
1700:
1698:
1694:
1687:Low dimension
1684:
1682:
1660:
1656:
1652:
1649:
1646:
1641:
1637:
1630:
1620:
1615:
1613:
1607:
1600:
1594:
1590:
1584:
1567:
1564:
1561:
1556:
1552:
1526:
1522:
1518:
1515:
1512:
1507:
1503:
1496:
1487:
1481:
1475:
1472:
1468:
1448:
1442:
1434:
1430:
1423:
1415:
1412:
1409:
1405:
1396:
1392:
1385:
1378:
1377:
1376:
1375:
1370:
1364:
1359:
1354:
1352:
1347:
1341:
1335:
1331:
1327:
1323:
1301:
1297:
1293:
1290:
1287:
1282:
1278:
1271:
1264:
1259:
1248:
1246:
1228:
1224:
1201:
1197:
1189:are free and
1174:
1170:
1146:
1143:
1137:
1129:
1125:
1118:
1110:
1107:
1104:
1100:
1091:
1087:
1080:
1073:
1072:
1071:
1064:
1059:
1054:
1049:
1044:
1038:
1032:
1026:
1024:
1019:
1013:
1007:
1004:
999:
992:
991:second syzygy
987:
985:
969:
944:
940:
936:
931:
927:
923:
918:
914:
910:
905:
901:
893:
892:
891:
875:
871:
848:
844:
821:
817:
794:
790:
766:
760:
752:
748:
739:
735:
728:
721:
720:
719:
718:
702:
697:
693:
684:
680:
659:
651:
647:
639:
621:
617:
609:
590:
585:
581:
575:
571:
567:
564:
561:
556:
552:
546:
542:
534:
533:
532:
513:
509:
505:
502:
499:
494:
490:
479:
459:
451:
447:
443:
440:
437:
432:
428:
416:
398:
394:
370:
367:
362:
358:
352:
348:
344:
341:
338:
333:
329:
323:
319:
311:
310:
309:
306:
284:
280:
276:
273:
270:
265:
261:
248:
243:
239:
234:
228:
209:
205:
201:
198:
195:
190:
186:
178:
173:
171:
167:
163:
159:
153:
143:
141:
137:
133:
122:
120:
116:
111:
108:
103:
98:
93:
89:
85:
81:
77:
72:
70:
66:
62:
58:
54:
50:
46:
42:
38:
37:David Hilbert
34:
30:
26:
22:
3890:
3858:
3841:
3835:
3825:
3809:
3804:
3791:
3786:36, 473â530.
3778:
3629:
3561:
3544:
3538:
3532:
3477:
3471:
3465:
3459:
3395:
3389:
3383:
3372:
3353:
3315:
3217:
3211:
2928:
2921:
2887:
2754:
2742:
2679:
2671:
2670:This is the
2669:
2559:
2297:
2291:
2287:
2125:
2026:
1769:
1720:
1713:
1701:
1693:vector space
1690:
1622:
1616:
1605:
1598:
1592:
1588:
1582:
1485:
1479:
1476:
1470:
1466:
1463:
1368:
1362:
1355:
1349:is always a
1345:
1339:
1333:
1321:
1257:
1254:
1161:
1062:
1052:
1042:
1036:
1030:
1027:
1017:
1011:
1008:
1002:
994:
990:
988:
961:
781:
605:
385:
304:
246:
241:
237:
232:
230:over a ring
226:
224:of a module
174:
155:
128:
112:
106:
96:
75:
73:
69:prime ideals
24:
18:
3350:Computation
3313:is exactly
3209:is exactly
1863:free module
1366:is at most
1351:free module
1337:, then the
1046:is free or
1023:zero module
982:denote the
890:such that
672:defined by
417:with basis
415:free module
102:zero module
21:mathematics
3908:Categories
3770:References
3625:regularity
3037:such that
2126:such that
1947:direct sum
1834:, and let
1614:, below).
1586:and every
1580:for every
1464:of length
1048:projective
638:linear map
308:such that
84:generators
3897:EMS Press
3595:…
3426:…
3364:monomials
3356:algorithm
3301:⟩
3285:…
3269:⟨
3245:…
3197:⟩
3181:…
3165:⟨
3068:⋯
2978:±
2972:…
2956:−
2870:⟩
2854:…
2838:⟨
2776:…
2714:…
2641:→
2628:→
2615:→
2612:⋯
2609:→
2601:−
2590:→
2577:→
2525:∧
2522:⋯
2519:∧
2500:^
2490:∧
2487:⋯
2484:∧
2428:−
2405:∑
2401:↦
2381:∧
2378:⋯
2375:∧
2330:−
2319:→
2160:⋯
2091:∧
2088:⋯
2085:∧
1982:⨁
1959:Λ
1886:…
1865:of basis
1806:…
1742:…
1650:…
1516:…
1446:⟶
1440:⟶
1427:⟶
1424:⋯
1421:⟶
1413:−
1402:⟶
1389:⟶
1291:…
1251:Statement
1141:⟶
1135:⟶
1122:⟶
1119:⋯
1116:⟶
1108:−
1097:⟶
1084:⟶
970:⊕
937:⊕
924:≅
911:⊕
764:→
758:→
745:→
732:→
690:↦
657:→
565:⋯
503:…
441:…
342:⋯
274:…
240:or first
199:…
168:over any
76:relations
3822:abstract
3753:See also
238:relation
175:Given a
80:syzygies
3899:, 2001
3881:1322960
3661:. Then
3319:if the
2918:); the
2562:complex
1945:is the
1328:over a
1009:If the
636:of the
250:-tuple
166:modules
125:History
110:steps.
3873:
3865:
3381:: Let
3360:degree
2678:, but
2027:where
1695:has a
962:where
608:kernel
242:syzygy
158:ideals
92:module
86:of an
78:, or
59:, and
53:ideals
33:fields
3657:be a
3399:over
3141:1 = 0
2290:>
1861:be a
1697:basis
1330:field
478:tuple
413:be a
88:ideal
31:over
3871:ISBN
3863:ISBN
2924:â 1)
2285:for
2163:<
2157:<
2144:<
1914:The
1721:Let
1714:The
1608:â 1)
1028:Let
989:The
863:and
472:The
386:Let
236:, a
170:ring
67:and
3844:= 1
3814:doi
3705:of
2747:of
2347:by
1918:of
1678:is
1360:of
1319:in
1060:of
160:in
19:In
3910::
3895:,
3889:,
3878:MR
3869:;
3857:,
3749:.
3542:.
3215:.
1699:.
1683:.
1591:â
1474:.
1469:â€
1353:.
1247:.
1006:.
986:.
767:0.
371:0.
172:.
142:.
23:,
3842:c
3820:(
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3606:n
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