2566:
140:
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
1168:
1470:
2364:
3560:
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
2676:
1087:
1392:
2033:
2894:
2132:
968:
3218:
3322:
2197:
105:. 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
3015:
3121:
392:
788:
612:
481:
2561:{\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}},}
540:
311:
3381:
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.
3373:
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
1923:
2810:
1776:
724:
233:
2356:
3539:
681:
2571:
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
1589:
2294:
2230:
991:
3355:
3148:
3046:
2927:
2261:
2063:
1954:
1870:
1252:
1225:
1198:
899:
872:
845:
818:
645:
422:
3632:
2751:
1843:
3466:
3754:
3734:
3710:
3690:
3666:
1687:
1553:
1328:
2826:
2580:
1163:{\displaystyle 0\longrightarrow R_{n}\longrightarrow L_{n-1}\longrightarrow \cdots \longrightarrow L_{0}\longrightarrow M\longrightarrow 0,}
1465:{\displaystyle 0\longrightarrow L_{k}\longrightarrow L_{k-1}\longrightarrow \cdots \longrightarrow L_{0}\longrightarrow M\longrightarrow 0}
1004:
module is the module of the relations between generators of the first syzygy module. By continuing in this way, one may define the
1965:
3565:
algorithms. They allow the computation of the first syzygy module, and also, with almost no extra cost, all syzygies modules.
2830:
2074:
907:
3774:
3157:
3821:
Die Frage der endlich vielen
Schritte in der Theorie der Polynomideale. Unter Benutzung nachgelassener SĂ€tze von K. Hentzelt
3232:
2140:
2946:
3051:
325:
3885:
3877:
1713:
In the case of a single indeterminate, Hilbert's syzygy theorem is an instance of the theorem asserting that over a
1632:
of a ring is the supremum of the projective dimensions of all modules, Hilbert's syzygy theorem may be restated as:
735:
548:
3939:
1729:, also called "complex of exterior algebra", allows, in some cases, an explicit description of all syzygy modules.
1488:
This upper bound on the projective dimension is sharp, that is, there are modules of projective dimension exactly
3757:
17:
3907:
431:
3929:
3924:
493:
264:
71:
1879:
3902:
3557:
2769:
1735:
686:
192:
3769:
3378:
1032:, then by taking a basis as a generating set, the next syzygy module (and every subsequent one) is the
187:
94:
59:
2315:
3934:
3897:
3807:
3645:
150:
3495:
3385:
3365:
At
Hilbert's time, there was no method available for computing syzygies. It was only known that an
1336:
75:
67:
1036:. If one does not take a basis as a generating set, then all subsequent syzygy modules are free.
618:
653:
2759:
1707:
1558:
3794:
3492:
such that the degrees occurring in a generating system of the first syzygy module is at most
2266:
2202:
1957:
994:
976:
3891:
3333:
3126:
3024:
2905:
2239:
2041:
1932:
1848:
1714:
1702:
In the case of zero indeterminates, Hilbert's syzygy theorem is simply the fact that every
1368:
1230:
1203:
1176:
1068:
877:
850:
823:
796:
623:
400:
176:
162:
102:
90:
8:
3576:
2695:
1787:
1340:
1061:. The above property of invariance, up to the sum direct with free modules, implies that
847:
is the module that is obtained with another generating set, there exist two free modules
168:
129:
125:
98:
43:
3413:
3739:
3719:
3695:
3675:
3651:
1637:
1503:
1278:
180:
142:
55:
2815:
3881:
3873:
3018:
1058:
3872:. Graduate Texts in Mathematics, 150. Springer-Verlag, New York, 1995. xvi+785 pp.
3824:
2755:
2066:
1926:
1629:
146:
51:
3562:
1227:
is projective. It can be shown that one may always choose the generating sets for
3888:
3713:
3669:
1384:
1273:
1255:
172:
63:
39:
3865:
2686:
1726:
727:
3918:
3389:
2572:
2233:
1628:
The theorem is also true for modules that are not finitely generated. As the
47:
3839:
3635:
3370:
2902:
th syzygy module is free of dimension one (generated by the product of all
1703:
2671:{\displaystyle 0\to L_{t}\to L_{t-1}\to \cdots \to L_{1}\to L_{0}\to R/I.}
27:
Theorem about linear relations in ideals and modules over polynomial rings
1873:
1361:
1033:
425:
112:
79:
31:
3832:
3828:
648:
3634:
causes the
Hilbert syzygy theorem to hold. It turns out that this is
2812:
is regular, and the Koszul complex is thus a projective resolution of
50:
in 1890, that were introduced for solving important open questions in
3366:
3229:
The same proof applies for proving that the projective dimension of
2937:
th syzygy module is thus the quotient of a free module of dimension
124:
Hilbert's syzygy theorem is now considered to be an early result of
3374:
1081:
if not. This is equivalent with the existence of an exact sequence
3756:
is equal to the Krull dimension. This result may be proven using
128:. It is the starting point of the use of homological methods in
3837:
The question of finitely many steps in polynomial ideal theory
3377:
is also bounded. Expressing that one has a syzygy provides a
488:
145:. The last part, part V, proves finite generation of certain
3638:, which is an algebraic formulation of the fact that affine
149:. Incidentally part III also contains a special case of the
2691:
it is an exact sequence if one works with a polynomial ring
3870:
Commutative algebra. With a view toward algebraic geometry
2028:{\displaystyle \Lambda (L_{1})=\bigoplus _{t=0}^{k}L_{t},}
1254:
being free, that is for the above exact sequence to be a
2889:{\displaystyle k=R/\langle x_{1},\ldots ,x_{n}\rangle .}
2127:{\displaystyle G_{i_{1}}\wedge \cdots \wedge G_{i_{t}},}
963:{\displaystyle R_{1}\oplus F_{1}\cong S_{1}\oplus F_{2}}
3823:, Mathematische Annalen, Volume 95, Number 1, 736-788,
3793:
D. Hilbert, Ăber die
Theorie der algebraischen Formen,
3213:{\displaystyle k=R/\langle x_{1},\ldots ,x_{n}\rangle }
2818:
74:, which establishes a bijective correspondence between
1067:
does not depend on the choice of generating sets. The
3742:
3722:
3698:
3678:
3654:
3579:
3541:
The same bound applies for testing the membership to
3498:
3416:
3336:
3317:{\displaystyle k/\langle g_{1},\ldots ,g_{t}\rangle }
3235:
3160:
3129:
3054:
3027:
2949:
2908:
2833:
2772:
2698:
2583:
2367:
2318:
2269:
2242:
2205:
2143:
2077:
2044:
1968:
1935:
1882:
1851:
1790:
1738:
1640:
1561:
1506:
1395:
1281:
1233:
1206:
1179:
1090:
979:
910:
880:
853:
826:
799:
738:
689:
656:
626:
551:
496:
434:
403:
328:
267:
195:
3357:
form a regular sequence of homogeneous polynomials.
111:
indeterminates over a field, one eventually finds a
2192:{\displaystyle i_{1}<i_{2}<\cdots <i_{t}.}
1717:, every submodule of a free module is itself free.
820:depends on the choice of a generating set, but, if
3748:
3728:
3704:
3684:
3660:
3648:. In fact the following generalization holds: Let
3626:
3573:One might wonder which ring-theoretic property of
3533:
3460:
3349:
3316:
3212:
3142:
3115:
3040:
3009:
2921:
2888:
2820:
2804:
2745:
2670:
2560:
2350:
2288:
2255:
2224:
2191:
2126:
2057:
2027:
1948:
1917:
1864:
1837:
1770:
1681:
1583:
1547:
1464:
1322:
1246:
1219:
1192:
1162:
985:
962:
893:
866:
839:
812:
782:
718:
675:
639:
606:
534:
475:
416:
386:
305:
227:
175:, but the concept generalizes trivially to (left)
3384:The first bound for syzygies (as well as for the
3010:{\displaystyle (x_{1},-x_{2},\ldots ,\pm x_{n}).}
3916:
3736:is finite; in that case the global dimension of
3154:). This proves that the projective dimension of
3116:{\displaystyle p_{1}x_{1}+\cdots +p_{n}x_{n}=1,}
387:{\displaystyle a_{1}g_{1}+\cdots +a_{k}g_{k}=0.}
2065:is the free module, which has, as a basis, the
1045:be the smallest integer, if any, such that the
38:is one of the three fundamental theorems about
3556:On the other hand, there are examples where a
3021:, as otherwise, there would exist polynomials
783:{\displaystyle 0\to R_{1}\to L_{0}\to M\to 0.}
607:{\displaystyle a_{1}G_{1}+\cdots +a_{k}G_{k},}
3123:which is impossible (substituting 0 for the
3311:
3279:
3207:
3175:
2880:
2848:
3692:has finite global dimension if and only if
3369:may be deduced from any upper bound of the
3810:, but extends easily to arbitrary modules.
3568:
2685:. In general the Koszul complex is not an
1367:In modern language, this implies that the
3840:(review and English-language translation)
1266:Hilbert's syzygy theorem states that, if
167:Originally, Hilbert defined syzygies for
156:
14:
3917:
3758:Serre's theorem on regular local rings
1613:th syzygy module is free, but not the
1272:is a finitely generated module over a
476:{\displaystyle (G_{1},\ldots ,G_{k}).}
93:in Hilbert's terminology, between the
85:Hilbert's syzygy theorem concerns the
3775:Hilbert series and Hilbert polynomial
535:{\displaystyle (a_{1},\ldots ,a_{k})}
306:{\displaystyle (a_{1},\ldots ,a_{k})}
66:of polynomial rings over a field are
3468:if the coefficients over a basis of
1918:{\displaystyle G_{1},\ldots ,G_{k}.}
1494:. The standard example is the field
542:may be identified with the element
2805:{\displaystyle x_{1},\ldots ,x_{n}}
1778:be a generating system of an ideal
1771:{\displaystyle g_{1},\ldots ,g_{k}}
1622:
719:{\displaystyle G_{i}\mapsto g_{i}.}
228:{\displaystyle g_{1},\ldots ,g_{k}}
24:
2232:(because of the definition of the
1969:
1077:is this integer, if it exists, or
1026:th syzygy module is free for some
141:is used in part IV to discuss the
25:
3951:
1720:
54:, and are at the basis of modern
3150:in the latter equality provides
2351:{\displaystyle L_{t}\to L_{t-1}}
1697:
1500:, which may be considered as a
1383:, and thus that there exists a
3844:
3813:
3800:
3787:
3621:
3589:
3534:{\displaystyle (td)^{2^{cn}}.}
3509:
3499:
3452:
3420:
3360:
3271:
3239:
3001:
2950:
2943:by the submodule generated by
2740:
2708:
2651:
2638:
2625:
2619:
2600:
2587:
2445:
2435:
2411:
2329:
2312:, one may define a linear map
1985:
1972:
1832:
1800:
1676:
1644:
1542:
1510:
1456:
1450:
1437:
1431:
1412:
1399:
1317:
1285:
1151:
1145:
1132:
1126:
1107:
1094:
774:
768:
755:
742:
700:
667:
529:
497:
467:
435:
300:
268:
13:
1:
3780:
3398:a submodule of a free module
1051:th syzygy module of a module
58:. The two other theorems are
3806:The theory is presented for
3644:-space is a variety without
3480:have a total degree at most
2766:In particular, the sequence
2754:and an ideal generated by a
1261:
255:between the generators is a
115:of relations, after at most
7:
3903:Encyclopedia of Mathematics
3763:
3486:, then there is a constant
3017:This quotient may not be a
1011:for every positive integer
726:In other words, one has an
617:and the relations form the
10:
3956:
3808:finitely generated modules
3474:of a generating system of
3379:system of linear equations
2236:), the two definitions of
676:{\displaystyle L_{0}\to M}
160:
135:
76:affine algebraic varieties
3857:, but did not prove this.
1621:th one (for a proof, see
793:This first syzygy module
72:Hilbert's Nullstellensatz
62:, which asserts that all
3386:ideal membership problem
1634:the global dimension of
1584:{\displaystyle x_{i}c=0}
132:and algebraic geometry.
101:, or, more generally, a
36:Hilbert's syzygy theorem
3940:Theorems in ring theory
3569:Syzygies and regularity
3388:) was given in 1926 by
2760:homogeneous polynomials
2289:{\displaystyle L_{t}=0}
2225:{\displaystyle L_{0}=R}
2199:In particular, one has
1607:. For this module, the
986:{\displaystyle \oplus }
60:Hilbert's basis theorem
3835:in German language) â
3770:QuillenâSuslin theorem
3750:
3730:
3706:
3686:
3662:
3628:
3535:
3462:
3351:
3318:
3214:
3144:
3117:
3042:
3011:
2923:
2890:
2822:
2806:
2747:
2672:
2562:
2434:
2352:
2290:
2257:
2226:
2193:
2128:
2059:
2029:
2011:
1950:
1919:
1866:
1839:
1772:
1683:
1585:
1549:
1466:
1324:
1248:
1221:
1194:
1164:
987:
964:
895:
868:
841:
814:
784:
720:
677:
641:
608:
536:
477:
418:
388:
307:
229:
3795:Mathematische Annalen
3751:
3731:
3707:
3687:
3663:
3629:
3536:
3463:
3352:
3350:{\displaystyle g_{i}}
3319:
3215:
3145:
3143:{\displaystyle x_{i}}
3118:
3043:
3041:{\displaystyle p_{i}}
3012:
2924:
2922:{\displaystyle G_{i}}
2891:
2823:
2807:
2748:
2673:
2563:
2414:
2353:
2306:. For every positive
2291:
2258:
2256:{\displaystyle L_{1}}
2227:
2194:
2129:
2060:
2058:{\displaystyle L_{t}}
2030:
1991:
1951:
1949:{\displaystyle L_{1}}
1920:
1867:
1865:{\displaystyle L_{1}}
1840:
1784:in a polynomial ring
1773:
1684:
1623:§ Koszul complex
1586:
1550:
1467:
1325:
1249:
1247:{\displaystyle R_{n}}
1222:
1220:{\displaystyle R_{n}}
1195:
1193:{\displaystyle L_{i}}
1165:
995:direct sum of modules
988:
965:
896:
894:{\displaystyle F_{2}}
869:
867:{\displaystyle F_{1}}
842:
840:{\displaystyle S_{1}}
815:
813:{\displaystyle R_{1}}
785:
721:
678:
642:
640:{\displaystyle R_{1}}
609:
537:
478:
419:
417:{\displaystyle L_{0}}
389:
308:
230:
151:HilbertâBurch theorem
82:of polynomial rings.
3740:
3720:
3696:
3676:
3652:
3577:
3496:
3414:
3334:
3233:
3158:
3127:
3052:
3025:
2947:
2906:
2831:
2816:
2770:
2696:
2581:
2365:
2316:
2267:
2240:
2203:
2141:
2075:
2042:
1966:
1933:
1880:
1849:
1788:
1736:
1715:principal ideal ring
1638:
1559:
1504:
1393:
1369:projective dimension
1354:th syzygy module of
1279:
1231:
1204:
1177:
1088:
1069:projective dimension
977:
908:
878:
851:
824:
797:
736:
687:
654:
624:
549:
494:
432:
401:
326:
265:
193:
163:syzygy (mathematics)
157:Syzygies (relations)
3930:Homological algebra
3925:Commutative algebra
3850:G. Hermann claimed
3712:is regular and the
3627:{\displaystyle A=k}
2746:{\displaystyle R=k}
1838:{\displaystyle R=k}
1555:-module by setting
147:rings of invariants
130:commutative algebra
126:homological algebra
3829:10.1007/BF01206635
3746:
3726:
3702:
3682:
3658:
3624:
3558:double exponential
3531:
3461:{\displaystyle k;}
3458:
3347:
3314:
3210:
3140:
3113:
3038:
3007:
2919:
2896:In this case, the
2886:
2802:
2743:
2668:
2558:
2348:
2286:
2253:
2222:
2189:
2124:
2055:
2025:
1946:
1915:
1862:
1835:
1768:
1679:
1581:
1545:
1462:
1320:
1244:
1217:
1190:
1173:where the modules
1160:
983:
960:
891:
864:
837:
810:
780:
716:
673:
637:
604:
532:
473:
414:
384:
303:
225:
143:Hilbert polynomial
68:finitely generated
56:algebraic geometry
46:, first proved by
3898:"Hilbert theorem"
3749:{\displaystyle A}
3729:{\displaystyle A}
3705:{\displaystyle A}
3685:{\displaystyle A}
3661:{\displaystyle A}
3547:of an element of
3019:projective module
2513:
2067:exterior products
1682:{\displaystyle k}
1548:{\displaystyle k}
1323:{\displaystyle k}
16:(Redirected from
3947:
3935:Invariant theory
3911:
3858:
3856:
3848:
3842:
3817:
3811:
3804:
3798:
3791:
3755:
3753:
3752:
3747:
3735:
3733:
3732:
3727:
3711:
3709:
3708:
3703:
3691:
3689:
3688:
3683:
3667:
3665:
3664:
3659:
3643:
3633:
3631:
3630:
3625:
3620:
3619:
3601:
3600:
3552:
3546:
3540:
3538:
3537:
3532:
3527:
3526:
3525:
3524:
3491:
3485:
3479:
3473:
3467:
3465:
3464:
3459:
3451:
3450:
3432:
3431:
3409:
3403:
3397:
3356:
3354:
3353:
3348:
3346:
3345:
3329:
3323:
3321:
3320:
3315:
3310:
3309:
3291:
3290:
3278:
3270:
3269:
3251:
3250:
3225:
3219:
3217:
3216:
3211:
3206:
3205:
3187:
3186:
3174:
3153:
3149:
3147:
3146:
3141:
3139:
3138:
3122:
3120:
3119:
3114:
3103:
3102:
3093:
3092:
3074:
3073:
3064:
3063:
3047:
3045:
3044:
3039:
3037:
3036:
3016:
3014:
3013:
3008:
3000:
2999:
2978:
2977:
2962:
2961:
2942:
2936:
2928:
2926:
2925:
2920:
2918:
2917:
2901:
2895:
2893:
2892:
2887:
2879:
2878:
2860:
2859:
2847:
2827:
2825:
2824:
2821:{\displaystyle }
2819:
2811:
2809:
2808:
2803:
2801:
2800:
2782:
2781:
2756:regular sequence
2752:
2750:
2749:
2744:
2739:
2738:
2720:
2719:
2677:
2675:
2674:
2669:
2661:
2650:
2649:
2637:
2636:
2618:
2617:
2599:
2598:
2567:
2565:
2564:
2559:
2554:
2553:
2552:
2551:
2528:
2527:
2526:
2525:
2515:
2514:
2506:
2493:
2492:
2491:
2490:
2476:
2475:
2474:
2473:
2459:
2458:
2433:
2428:
2410:
2409:
2408:
2407:
2384:
2383:
2382:
2381:
2357:
2355:
2354:
2349:
2347:
2346:
2328:
2327:
2311:
2305:
2295:
2293:
2292:
2287:
2279:
2278:
2262:
2260:
2259:
2254:
2252:
2251:
2231:
2229:
2228:
2223:
2215:
2214:
2198:
2196:
2195:
2190:
2185:
2184:
2166:
2165:
2153:
2152:
2133:
2131:
2130:
2125:
2120:
2119:
2118:
2117:
2094:
2093:
2092:
2091:
2064:
2062:
2061:
2056:
2054:
2053:
2034:
2032:
2031:
2026:
2021:
2020:
2010:
2005:
1984:
1983:
1955:
1953:
1952:
1947:
1945:
1944:
1927:exterior algebra
1924:
1922:
1921:
1916:
1911:
1910:
1892:
1891:
1871:
1869:
1868:
1863:
1861:
1860:
1844:
1842:
1841:
1836:
1831:
1830:
1812:
1811:
1783:
1777:
1775:
1774:
1769:
1767:
1766:
1748:
1747:
1692:
1688:
1686:
1685:
1680:
1675:
1674:
1656:
1655:
1630:global dimension
1620:
1612:
1606:
1596:
1590:
1588:
1587:
1582:
1571:
1570:
1554:
1552:
1551:
1546:
1541:
1540:
1522:
1521:
1499:
1493:
1484:
1471:
1469:
1468:
1463:
1449:
1448:
1430:
1429:
1411:
1410:
1382:
1376:
1359:
1353:
1347:
1335:
1329:
1327:
1326:
1321:
1316:
1315:
1297:
1296:
1271:
1253:
1251:
1250:
1245:
1243:
1242:
1226:
1224:
1223:
1218:
1216:
1215:
1199:
1197:
1196:
1191:
1189:
1188:
1169:
1167:
1166:
1161:
1144:
1143:
1125:
1124:
1106:
1105:
1080:
1076:
1066:
1056:
1050:
1044:
1031:
1025:
1016:
1009:th syzygy module
1008:
992:
990:
989:
984:
969:
967:
966:
961:
959:
958:
946:
945:
933:
932:
920:
919:
900:
898:
897:
892:
890:
889:
873:
871:
870:
865:
863:
862:
846:
844:
843:
838:
836:
835:
819:
817:
816:
811:
809:
808:
789:
787:
786:
781:
767:
766:
754:
753:
725:
723:
722:
717:
712:
711:
699:
698:
682:
680:
679:
674:
666:
665:
646:
644:
643:
638:
636:
635:
613:
611:
610:
605:
600:
599:
590:
589:
571:
570:
561:
560:
541:
539:
538:
533:
528:
527:
509:
508:
486:
482:
480:
479:
474:
466:
465:
447:
446:
423:
421:
420:
415:
413:
412:
393:
391:
390:
385:
377:
376:
367:
366:
348:
347:
338:
337:
318:
312:
310:
309:
304:
299:
298:
280:
279:
260:
246:
240:
234:
232:
231:
226:
224:
223:
205:
204:
173:polynomial rings
120:
110:
52:invariant theory
40:polynomial rings
21:
3955:
3954:
3950:
3949:
3948:
3946:
3945:
3944:
3915:
3914:
3896:
3862:
3861:
3851:
3849:
3845:
3819:Grete Hermann:
3818:
3814:
3805:
3801:
3792:
3788:
3783:
3766:
3741:
3738:
3737:
3721:
3718:
3717:
3714:Krull dimension
3697:
3694:
3693:
3677:
3674:
3673:
3670:Noetherian ring
3653:
3650:
3649:
3639:
3615:
3611:
3596:
3592:
3578:
3575:
3574:
3571:
3548:
3542:
3517:
3513:
3512:
3508:
3497:
3494:
3493:
3487:
3481:
3475:
3469:
3446:
3442:
3427:
3423:
3415:
3412:
3411:
3405:
3399:
3393:
3363:
3341:
3337:
3335:
3332:
3331:
3325:
3305:
3301:
3286:
3282:
3274:
3265:
3261:
3246:
3242:
3234:
3231:
3230:
3221:
3201:
3197:
3182:
3178:
3170:
3159:
3156:
3155:
3151:
3134:
3130:
3128:
3125:
3124:
3098:
3094:
3088:
3084:
3069:
3065:
3059:
3055:
3053:
3050:
3049:
3032:
3028:
3026:
3023:
3022:
2995:
2991:
2973:
2969:
2957:
2953:
2948:
2945:
2944:
2938:
2930:
2913:
2909:
2907:
2904:
2903:
2897:
2874:
2870:
2855:
2851:
2843:
2832:
2829:
2828:
2817:
2814:
2813:
2796:
2792:
2777:
2773:
2771:
2768:
2767:
2734:
2730:
2715:
2711:
2697:
2694:
2693:
2657:
2645:
2641:
2632:
2628:
2607:
2603:
2594:
2590:
2582:
2579:
2578:
2547:
2543:
2542:
2538:
2521:
2517:
2516:
2505:
2504:
2503:
2486:
2482:
2481:
2477:
2469:
2465:
2464:
2460:
2448:
2444:
2429:
2418:
2403:
2399:
2398:
2394:
2377:
2373:
2372:
2368:
2366:
2363:
2362:
2336:
2332:
2323:
2319:
2317:
2314:
2313:
2307:
2297:
2274:
2270:
2268:
2265:
2264:
2247:
2243:
2241:
2238:
2237:
2210:
2206:
2204:
2201:
2200:
2180:
2176:
2161:
2157:
2148:
2144:
2142:
2139:
2138:
2113:
2109:
2108:
2104:
2087:
2083:
2082:
2078:
2076:
2073:
2072:
2049:
2045:
2043:
2040:
2039:
2016:
2012:
2006:
1995:
1979:
1975:
1967:
1964:
1963:
1940:
1936:
1934:
1931:
1930:
1906:
1902:
1887:
1883:
1881:
1878:
1877:
1856:
1852:
1850:
1847:
1846:
1826:
1822:
1807:
1803:
1789:
1786:
1785:
1779:
1762:
1758:
1743:
1739:
1737:
1734:
1733:
1723:
1700:
1690:
1670:
1666:
1651:
1647:
1639:
1636:
1635:
1614:
1608:
1598:
1592:
1566:
1562:
1560:
1557:
1556:
1536:
1532:
1517:
1513:
1505:
1502:
1501:
1495:
1489:
1476:
1444:
1440:
1419:
1415:
1406:
1402:
1394:
1391:
1390:
1385:free resolution
1378:
1372:
1355:
1349:
1343:
1331:
1311:
1307:
1292:
1288:
1280:
1277:
1276:
1274:polynomial ring
1267:
1264:
1256:free resolution
1238:
1234:
1232:
1229:
1228:
1211:
1207:
1205:
1202:
1201:
1184:
1180:
1178:
1175:
1174:
1139:
1135:
1114:
1110:
1101:
1097:
1089:
1086:
1085:
1078:
1072:
1062:
1052:
1046:
1040:
1027:
1021:
1012:
1006:
978:
975:
974:
954:
950:
941:
937:
928:
924:
915:
911:
909:
906:
905:
885:
881:
879:
876:
875:
858:
854:
852:
849:
848:
831:
827:
825:
822:
821:
804:
800:
798:
795:
794:
762:
758:
749:
745:
737:
734:
733:
707:
703:
694:
690:
688:
685:
684:
661:
657:
655:
652:
651:
631:
627:
625:
622:
621:
595:
591:
585:
581:
566:
562:
556:
552:
550:
547:
546:
523:
519:
504:
500:
495:
492:
491:
484:
461:
457:
442:
438:
433:
430:
429:
408:
404:
402:
399:
398:
372:
368:
362:
358:
343:
339:
333:
329:
327:
324:
323:
314:
313:of elements of
294:
290:
275:
271:
266:
263:
262:
256:
242:
236:
219:
215:
200:
196:
194:
191:
190:
165:
159:
138:
116:
106:
28:
23:
22:
15:
12:
11:
5:
3953:
3943:
3942:
3937:
3932:
3927:
3913:
3912:
3894:
3866:David Eisenbud
3860:
3859:
3843:
3812:
3799:
3785:
3784:
3782:
3779:
3778:
3777:
3772:
3765:
3762:
3745:
3725:
3701:
3681:
3657:
3623:
3618:
3614:
3610:
3607:
3604:
3599:
3595:
3591:
3588:
3585:
3582:
3570:
3567:
3530:
3523:
3520:
3516:
3511:
3507:
3504:
3501:
3457:
3454:
3449:
3445:
3441:
3438:
3435:
3430:
3426:
3422:
3419:
3362:
3359:
3344:
3340:
3313:
3308:
3304:
3300:
3297:
3294:
3289:
3285:
3281:
3277:
3273:
3268:
3264:
3260:
3257:
3254:
3249:
3245:
3241:
3238:
3209:
3204:
3200:
3196:
3193:
3190:
3185:
3181:
3177:
3173:
3169:
3166:
3163:
3137:
3133:
3112:
3109:
3106:
3101:
3097:
3091:
3087:
3083:
3080:
3077:
3072:
3068:
3062:
3058:
3035:
3031:
3006:
3003:
2998:
2994:
2990:
2987:
2984:
2981:
2976:
2972:
2968:
2965:
2960:
2956:
2952:
2916:
2912:
2885:
2882:
2877:
2873:
2869:
2866:
2863:
2858:
2854:
2850:
2846:
2842:
2839:
2836:
2799:
2795:
2791:
2788:
2785:
2780:
2776:
2742:
2737:
2733:
2729:
2726:
2723:
2718:
2714:
2710:
2707:
2704:
2701:
2687:exact sequence
2683:Koszul complex
2679:
2678:
2667:
2664:
2660:
2656:
2653:
2648:
2644:
2640:
2635:
2631:
2627:
2624:
2621:
2616:
2613:
2610:
2606:
2602:
2597:
2593:
2589:
2586:
2569:
2568:
2557:
2550:
2546:
2541:
2537:
2534:
2531:
2524:
2520:
2512:
2509:
2502:
2499:
2496:
2489:
2485:
2480:
2472:
2468:
2463:
2457:
2454:
2451:
2447:
2443:
2440:
2437:
2432:
2427:
2424:
2421:
2417:
2413:
2406:
2402:
2397:
2393:
2390:
2387:
2380:
2376:
2371:
2345:
2342:
2339:
2335:
2331:
2326:
2322:
2285:
2282:
2277:
2273:
2263:coincide, and
2250:
2246:
2221:
2218:
2213:
2209:
2188:
2183:
2179:
2175:
2172:
2169:
2164:
2160:
2156:
2151:
2147:
2135:
2134:
2123:
2116:
2112:
2107:
2103:
2100:
2097:
2090:
2086:
2081:
2052:
2048:
2036:
2035:
2024:
2019:
2015:
2009:
2004:
2001:
1998:
1994:
1990:
1987:
1982:
1978:
1974:
1971:
1943:
1939:
1914:
1909:
1905:
1901:
1898:
1895:
1890:
1886:
1859:
1855:
1834:
1829:
1825:
1821:
1818:
1815:
1810:
1806:
1802:
1799:
1796:
1793:
1765:
1761:
1757:
1754:
1751:
1746:
1742:
1727:Koszul complex
1722:
1721:Koszul complex
1719:
1699:
1696:
1678:
1673:
1669:
1665:
1662:
1659:
1654:
1650:
1646:
1643:
1580:
1577:
1574:
1569:
1565:
1544:
1539:
1535:
1531:
1528:
1525:
1520:
1516:
1512:
1509:
1473:
1472:
1461:
1458:
1455:
1452:
1447:
1443:
1439:
1436:
1433:
1428:
1425:
1422:
1418:
1414:
1409:
1405:
1401:
1398:
1337:indeterminates
1319:
1314:
1310:
1306:
1303:
1300:
1295:
1291:
1287:
1284:
1263:
1260:
1241:
1237:
1214:
1210:
1187:
1183:
1171:
1170:
1159:
1156:
1153:
1150:
1147:
1142:
1138:
1134:
1131:
1128:
1123:
1120:
1117:
1113:
1109:
1104:
1100:
1096:
1093:
982:
971:
970:
957:
953:
949:
944:
940:
936:
931:
927:
923:
918:
914:
888:
884:
861:
857:
834:
830:
807:
803:
791:
790:
779:
776:
773:
770:
765:
761:
757:
752:
748:
744:
741:
728:exact sequence
715:
710:
706:
702:
697:
693:
672:
669:
664:
660:
634:
630:
615:
614:
603:
598:
594:
588:
584:
580:
577:
574:
569:
565:
559:
555:
531:
526:
522:
518:
515:
512:
507:
503:
499:
472:
469:
464:
460:
456:
453:
450:
445:
441:
437:
411:
407:
395:
394:
383:
380:
375:
371:
365:
361:
357:
354:
351:
346:
342:
336:
332:
302:
297:
293:
289:
286:
283:
278:
274:
270:
222:
218:
214:
211:
208:
203:
199:
188:generating set
161:Main article:
158:
155:
137:
134:
26:
18:Syzygy theorem
9:
6:
4:
3:
2:
3952:
3941:
3938:
3936:
3933:
3931:
3928:
3926:
3923:
3922:
3920:
3909:
3905:
3904:
3899:
3895:
3893:
3890:
3887:
3886:0-387-94269-6
3883:
3879:
3878:0-387-94268-8
3875:
3871:
3867:
3864:
3863:
3854:
3847:
3841:
3838:
3834:
3830:
3826:
3822:
3816:
3809:
3803:
3796:
3790:
3786:
3776:
3773:
3771:
3768:
3767:
3761:
3759:
3743:
3723:
3715:
3699:
3679:
3671:
3655:
3647:
3646:singularities
3642:
3637:
3616:
3612:
3608:
3605:
3602:
3597:
3593:
3586:
3583:
3580:
3566:
3564:
3563:Gröbner basis
3559:
3554:
3551:
3545:
3528:
3521:
3518:
3514:
3505:
3502:
3490:
3484:
3478:
3472:
3455:
3447:
3443:
3439:
3436:
3433:
3428:
3424:
3417:
3408:
3404:of dimension
3402:
3396:
3391:
3390:Grete Hermann
3387:
3382:
3380:
3376:
3372:
3368:
3358:
3342:
3338:
3328:
3306:
3302:
3298:
3295:
3292:
3287:
3283:
3275:
3266:
3262:
3258:
3255:
3252:
3247:
3243:
3236:
3227:
3224:
3202:
3198:
3194:
3191:
3188:
3183:
3179:
3171:
3167:
3164:
3161:
3135:
3131:
3110:
3107:
3104:
3099:
3095:
3089:
3085:
3081:
3078:
3075:
3070:
3066:
3060:
3056:
3033:
3029:
3020:
3004:
2996:
2992:
2988:
2985:
2982:
2979:
2974:
2970:
2966:
2963:
2958:
2954:
2941:
2934:
2914:
2910:
2900:
2883:
2875:
2871:
2867:
2864:
2861:
2856:
2852:
2844:
2840:
2837:
2834:
2797:
2793:
2789:
2786:
2783:
2778:
2774:
2764:
2763:
2761:
2757:
2735:
2731:
2727:
2724:
2721:
2716:
2712:
2705:
2702:
2699:
2692:
2688:
2684:
2665:
2662:
2658:
2654:
2646:
2642:
2633:
2629:
2622:
2614:
2611:
2608:
2604:
2595:
2591:
2584:
2577:
2576:
2575:
2574:
2555:
2548:
2544:
2539:
2535:
2532:
2529:
2522:
2518:
2510:
2507:
2500:
2497:
2494:
2487:
2483:
2478:
2470:
2466:
2461:
2455:
2452:
2449:
2441:
2438:
2430:
2425:
2422:
2419:
2415:
2404:
2400:
2395:
2391:
2388:
2385:
2378:
2374:
2369:
2361:
2360:
2359:
2343:
2340:
2337:
2333:
2324:
2320:
2310:
2304:
2300:
2283:
2280:
2275:
2271:
2248:
2244:
2235:
2234:empty product
2219:
2216:
2211:
2207:
2186:
2181:
2177:
2173:
2170:
2167:
2162:
2158:
2154:
2149:
2145:
2121:
2114:
2110:
2105:
2101:
2098:
2095:
2088:
2084:
2079:
2071:
2070:
2069:
2068:
2050:
2046:
2022:
2017:
2013:
2007:
2002:
1999:
1996:
1992:
1988:
1980:
1976:
1962:
1961:
1960:
1959:
1941:
1937:
1928:
1912:
1907:
1903:
1899:
1896:
1893:
1888:
1884:
1875:
1857:
1853:
1827:
1823:
1819:
1816:
1813:
1808:
1804:
1797:
1794:
1791:
1782:
1763:
1759:
1755:
1752:
1749:
1744:
1740:
1730:
1728:
1718:
1716:
1711:
1709:
1705:
1698:Low dimension
1695:
1693:
1671:
1667:
1663:
1660:
1657:
1652:
1648:
1641:
1631:
1626:
1624:
1618:
1611:
1605:
1601:
1595:
1578:
1575:
1572:
1567:
1563:
1537:
1533:
1529:
1526:
1523:
1518:
1514:
1507:
1498:
1492:
1486:
1483:
1479:
1459:
1453:
1445:
1441:
1434:
1426:
1423:
1420:
1416:
1407:
1403:
1396:
1389:
1388:
1387:
1386:
1381:
1375:
1370:
1365:
1363:
1358:
1352:
1346:
1342:
1338:
1334:
1312:
1308:
1304:
1301:
1298:
1293:
1289:
1282:
1275:
1270:
1259:
1257:
1239:
1235:
1212:
1208:
1200:are free and
1185:
1181:
1157:
1154:
1148:
1140:
1136:
1129:
1121:
1118:
1115:
1111:
1102:
1098:
1091:
1084:
1083:
1082:
1075:
1070:
1065:
1060:
1055:
1049:
1043:
1037:
1035:
1030:
1024:
1018:
1015:
1010:
1003:
1002:second syzygy
998:
996:
980:
955:
951:
947:
942:
938:
934:
929:
925:
921:
916:
912:
904:
903:
902:
886:
882:
859:
855:
832:
828:
805:
801:
777:
771:
763:
759:
750:
746:
739:
732:
731:
730:
729:
713:
708:
704:
695:
691:
670:
662:
658:
650:
632:
628:
620:
601:
596:
592:
586:
582:
578:
575:
572:
567:
563:
557:
553:
545:
544:
543:
524:
520:
516:
513:
510:
505:
501:
490:
470:
462:
458:
454:
451:
448:
443:
439:
427:
409:
405:
381:
378:
373:
369:
363:
359:
355:
352:
349:
344:
340:
334:
330:
322:
321:
320:
317:
295:
291:
287:
284:
281:
276:
272:
259:
254:
250:
245:
239:
220:
216:
212:
209:
206:
201:
197:
189:
184:
182:
178:
174:
170:
164:
154:
152:
148:
144:
133:
131:
127:
122:
119:
114:
109:
104:
100:
96:
92:
88:
83:
81:
77:
73:
69:
65:
61:
57:
53:
49:
48:David Hilbert
45:
41:
37:
33:
19:
3901:
3869:
3852:
3846:
3836:
3820:
3815:
3802:
3797:36, 473â530.
3789:
3640:
3572:
3555:
3549:
3543:
3488:
3482:
3476:
3470:
3406:
3400:
3394:
3383:
3364:
3326:
3228:
3222:
2939:
2932:
2898:
2765:
2753:
2690:
2682:
2681:This is the
2680:
2570:
2308:
2302:
2298:
2136:
2037:
1780:
1731:
1724:
1712:
1704:vector space
1701:
1633:
1627:
1616:
1609:
1603:
1599:
1593:
1496:
1490:
1487:
1481:
1477:
1474:
1379:
1373:
1366:
1360:is always a
1356:
1350:
1344:
1332:
1268:
1265:
1172:
1073:
1063:
1053:
1047:
1041:
1038:
1028:
1022:
1019:
1013:
1005:
1001:
999:
972:
792:
616:
396:
315:
257:
252:
248:
243:
241:over a ring
237:
235:of a module
185:
166:
139:
123:
117:
107:
86:
84:
80:prime ideals
35:
29:
3361:Computation
3324:is exactly
3220:is exactly
1874:free module
1377:is at most
1362:free module
1348:, then the
1057:is free or
1034:zero module
993:denote the
901:such that
683:defined by
428:with basis
426:free module
113:zero module
32:mathematics
3919:Categories
3781:References
3636:regularity
3048:such that
2137:such that
1958:direct sum
1845:, and let
1625:, below).
1597:and every
1591:for every
1475:of length
1059:projective
649:linear map
319:such that
95:generators
3908:EMS Press
3606:…
3437:…
3375:monomials
3367:algorithm
3312:⟩
3296:…
3280:⟨
3256:…
3208:⟩
3192:…
3176:⟨
3079:⋯
2989:±
2983:…
2967:−
2881:⟩
2865:…
2849:⟨
2787:…
2725:…
2652:→
2639:→
2626:→
2623:⋯
2620:→
2612:−
2601:→
2588:→
2536:∧
2533:⋯
2530:∧
2511:^
2501:∧
2498:⋯
2495:∧
2439:−
2416:∑
2412:↦
2392:∧
2389:⋯
2386:∧
2341:−
2330:→
2171:⋯
2102:∧
2099:⋯
2096:∧
1993:⨁
1970:Λ
1897:…
1876:of basis
1817:…
1753:…
1661:…
1527:…
1457:⟶
1451:⟶
1438:⟶
1435:⋯
1432:⟶
1424:−
1413:⟶
1400:⟶
1302:…
1262:Statement
1152:⟶
1146:⟶
1133:⟶
1130:⋯
1127:⟶
1119:−
1108:⟶
1095:⟶
981:⊕
948:⊕
935:≅
922:⊕
775:→
769:→
756:→
743:→
701:↦
668:→
576:⋯
514:…
452:…
353:⋯
285:…
251:or first
210:…
179:over any
87:relations
3833:abstract
3764:See also
249:relation
186:Given a
91:syzygies
3910:, 2001
3892:1322960
3672:. Then
3330:if the
2929:); the
2573:complex
1956:is the
1339:over a
1020:If the
647:of the
261:-tuple
177:modules
136:History
121:steps.
3884:
3876:
3392:: Let
3371:degree
2689:, but
2038:where
1706:has a
973:where
619:kernel
253:syzygy
169:ideals
103:module
97:of an
89:, or
70:, and
64:ideals
44:fields
3668:be a
3410:over
3152:1 = 0
2301:>
1872:be a
1708:basis
1341:field
489:tuple
424:be a
99:ideal
42:over
3882:ISBN
3874:ISBN
2935:â 1)
2296:for
2174:<
2168:<
2155:<
1925:The
1732:Let
1725:The
1619:â 1)
1039:Let
1000:The
874:and
483:The
397:Let
247:, a
181:ring
78:and
3855:= 1
3825:doi
3716:of
2758:of
2358:by
1929:of
1689:is
1371:of
1330:in
1071:of
171:in
30:In
3921::
3906:,
3900:,
3889:MR
3880:;
3868:,
3760:.
3553:.
3226:.
1710:.
1694:.
1602:â
1485:.
1480:â€
1364:.
1258:.
1017:.
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382:0.
183:.
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34:,
3853:c
3831:(
3827::
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3641:n
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3609:,
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1986:)
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