7762:
7461:
4149:
7757:{\displaystyle 0\to {\mathcal {O}}_{\mathbb {P} ^{n}}(-d_{1}-d_{2}){\xrightarrow {\begin{bmatrix}f_{2}\\-f_{1}\end{bmatrix}}}{\mathcal {O}}_{\mathbb {P} ^{n}}(-d_{1})\oplus {\mathcal {O}}_{\mathbb {P} ^{n}}(-d_{2}){\xrightarrow {\begin{bmatrix}f_{1}&f_{2}\end{bmatrix}}}{\mathcal {O}}_{\mathbb {P} ^{n}}\to {\mathcal {O}}_{X}\to 0}
6205:
3932:
5186:
4861:
5808:
2195:
Thus, the graded algebras generated by elements of degree 1 are exactly, up to an isomorphism, the quotients of polynomial rings by homogeneous ideals. Therefore, the remainder of this article will be restricted to the quotients of polynomial rings by ideals.
5942:
1136:
980:
4364:
1785:
121:, as they are the easiest known way for computing the dimension and the degree of an algebraic variety defined by explicit polynomial equations. In addition, they provide useful invariants for families of algebraic varieties because a flat family
2643:
7040:
of the whole computation depends mainly on the regularity, which is the degree of the numerator of the
Hilbert series. In fact the Gröbner basis may be computed by linear algebra over the polynomials of degree bounded by the regularity.
3154:
4144:{\displaystyle 0\longrightarrow \left(R/\langle h_{0},\ldots ,h_{k-1}\rangle \right)^{}{\stackrel {h_{k}}{\longrightarrow }}R/\langle h_{1},\ldots ,h_{k-1}\rangle \longrightarrow R/\langle h_{1},\ldots ,h_{k}\rangle \longrightarrow 0,}
7035:
Thus the computation of the
Hilbert series is reduced, through the computation of a Gröbner basis, to the same problem for an ideal generated by monomials, which is usually much easier than the computation of the Gröbner basis. The
6875:
2266:
7388:
is equivalent to the category of graded-modules modulo a finite number of graded-pieces, we can use the results in the previous section to construct
Hilbert polynomials of coherent sheaves. For example, a complete intersection
684:
1455:
6880:
These formulas may be viewed as a way for computing
Hilbert series. This is rarely the case, as, with the known algorithms, the computation of the Hilbert series and the computation of a free resolution start from the same
5299:
5931:
6485:
7210:
501:
4482:
5682:
792:
6321:
2983:
4978:
2552:
5104:
4701:
3540:
367:
6754:
5606:
4772:
2801:
1257:
3430:
6200:{\displaystyle HS_{R_{k}}(t)={\frac {(1-t^{\delta _{1}})\cdots (1-t^{\delta _{k}})}{(1-t)^{n}}}={\frac {(1+t+\cdots +t^{\delta _{1}})\cdots (1+t+\cdots +t^{\delta _{k}})}{(1-t)^{n-k}}}\,.}
5690:
263:
7321:
3811:
In this section, one does not need irreducibility of algebraic sets nor primality of ideals. Also, as
Hilbert series are not changed by extending the field of coefficients, the field
991:
6693:
2405:
2334:
5490:
3917:
1861:
1586:
840:
562:
3625:
1499:
7251:
4201:
1916:
1680:
7138:
7106:
4193:
3233:
1175:
7453:
7358:
151:
3729:
6618:
1664:
1625:
1533:
1326:
5388:
4569:
2563:
5333:
5092:
4734:
2876:
2184:
2133:
821:
299:
6579:
4764:
4603:
177:
6351:
5015:
4630:
4536:
4509:
3802:
3184:
2994:
1287:
4396:
2691:
6967:
6549:
5557:
7407:
7386:
5361:
5072:
3860:
2896:
6762:
3159:
The proof that the
Hilbert series has this simple form is obtained by applying recursively the previous formula for the quotient by a non zero divisor (here
2219:
6901:). This section describes how the Hilbert series may be computed in the case of a quotient of a polynomial ring, filtered or graded by the total degree.
570:
1334:
5197:
7928:
7850:
5819:
7773:
6370:
7880:
7143:
408:
4408:
7842:
5611:
3319:
of degree one which is not a zero divisor (in fact almost all elements of degree one have this property). The Krull dimension of
710:
7037:
6259:
396:
107:
3636:
2904:
5181:{\displaystyle 0\longrightarrow R^{}{\stackrel {f}{\longrightarrow }}R\longrightarrow R/\langle f\rangle \longrightarrow 0,}
4872:
2452:
43:
are three strongly related notions which measure the growth of the dimension of the homogeneous components of the algebra.
5033:
4856:{\displaystyle 0\longrightarrow R_{1}{\stackrel {h_{d}-1}{\longrightarrow }}R_{1}\longrightarrow R_{0}\longrightarrow 0,}
4639:
3461:
7987:
310:
95:
7992:
6710:
5562:
2714:
1992:
6250:
5803:{\displaystyle 0\;\rightarrow \;R_{i-1}^{}\;{\xrightarrow {f_{i}}}\;R_{i-1}\;\rightarrow \;R_{i}\;\rightarrow \;0\,.}
3926:
homogeneous polynomials of degree one. The definition of a regular sequence implies the existence of exact sequences
1186:
3341:
7910:
7832:
7324:
7258:
219:
7271:
1131:{\displaystyle {\binom {n+\delta -1}{\delta -1}}={\frac {(n+\delta -1)(n+\delta -2)\cdots (n+1)}{(\delta -1)!}}}
6886:
3647:
3252:
zero if the maximal homogeneous ideal, that is the ideal generated by the homogeneous elements of degree 1, is
7950:
7045:
6623:
5047:. In fact, the multiplicity of a point is the number of occurrences of the corresponding maximal ideal in a
975:{\displaystyle HS_{S}(t)=P(t)\left(1+\delta t+\cdots +{\binom {n+\delta -1}{\delta -1}}t^{n}+\cdots \right)}
74:
4987:, and the fact that the Hilbert series of a graded algebra is also its Hilbert series as filtered algebra.
2345:
7914:
2277:
5449:
4359:{\displaystyle HS_{R/\langle h_{0},\ldots ,h_{d-1}\rangle }(t)=(1-t)^{d}\,HS_{R}(t)={\frac {P(t)}{1-t}},}
3876:
3640:
1802:
1780:{\displaystyle HP_{S}(n)={\frac {P(1)}{(\delta -1)!}}n^{\delta -1}+{\text{ terms of lower degree in }}n.}
1538:
521:
5436:
5405:
3587:
1463:
7223:
5420:
The usual BĂ©zout's theorem is easily deduced by starting from a hypersurface, and intersecting it with
2009:
Polynomial rings and their quotients by homogeneous ideals are typical graded algebras. Conversely, if
3639:
as the value at 1 of the numerator of the
Hilbert series. This provides also a rather simple proof of
1883:
7217:
4633:
1922:
1142:
51:
7119:
7087:
4157:
3189:
1148:
7919:
7412:
205:
111:
70:
40:
20:
7334:
7945:
7073:
6981:
be the homogeneous ideal generated by the homogeneous parts of highest degree of the elements of
5048:
2638:{\displaystyle 0\;\rightarrow \;A^{}\;{\xrightarrow {f}}\;A\;\rightarrow \;A/f\rightarrow \;0\,,}
1977:
124:
118:
7807:
3660:
6584:
1925:, since the dimensions are integers, but the polynomial almost never has integer coefficients (
1630:
1591:
1512:
1292:
7924:
7876:
7846:
7081:
7077:
7010:
6358:
5370:
4541:
3655:
3149:{\displaystyle HP_{R_{n}}(k)={{k+n-1} \choose {n-1}}={\frac {(k+1)\cdots (k+n-1)}{(n-1)!}}\,.}
59:
5318:
5077:
4706:
2813:
2154:
2070:
806:
271:
7955:
7868:
7053:
7049:
7002:
6882:
6554:
5440:
4739:
4578:
4484:
has Krull dimension one, and is the ring of regular functions of a projective algebraic set
3871:
3863:
3805:
2142:
1970:
704:
is generated by elements of degree 1 then the sum of the
Hilbert series may be rewritten as
156:
55:
47:
7969:
7938:
7890:
6329:
4993:
4608:
4514:
4487:
3738:
3162:
1265:
7965:
7934:
7886:
6246:
4372:
3827:
3253:
3249:
2663:
824:
83:
The quotient by an ideal of a multivariate polynomial ring, filtered by the total degree.
77:
7030:
the filtered algebra R/I and the graded algebras R/H and R/G have the same
Hilbert series
6911:
6493:
5501:
4538:
belongs to a regular sequence, none of these points belong to the hyperplane of equation
4511:
of dimension 0 consisting of a finite number of points, which may be multiple points. As
3578:
This formula for the
Hilbert series implies that the degree of the Hilbert polynomial is
6889:
which is not higher than that the complexity of the computation of the free resolution.
6222:, and that its degree is the product of the degrees of the polynomials in the sequence.
5304:
Looking on the numerators this proves the following generalization of BĂ©zout's theorem:
7860:
7778:
7392:
7371:
7109:
6870:{\displaystyle HS_{L}(t)={\frac {t^{\delta _{1}}+\cdots +t^{\delta _{h}}}{(1-t)^{n}}}.}
5346:
5057:
3845:
2881:
2210:
196:
180:
36:
7981:
7960:
5018:
2261:{\displaystyle 0\;\rightarrow \;A\;\rightarrow \;B\;\rightarrow \;C\;\rightarrow \;0}
1933:
400:
103:
91:
2410:
This follows immediately from the same property for the dimension of vector spaces.
7014:
6237:
4572:
3438:, we get eventually an algebra of dimension 0 whose Hilbert series is a polynomial
2443:
679:{\displaystyle HS_{S}(t)={\frac {Q(t)}{\prod _{i=1}^{h}\left(1-t^{d_{i}}\right)}},}
1450:{\displaystyle HF_{S}(n)=\sum _{i=0}^{d}a_{i}{\binom {n-i+\delta -1}{\delta -1}}.}
7836:
7898:
7783:
7254:
7044:
The computation of Hilbert series and Hilbert polynomials are available in most
6551:
is a polynomial ring, and if one knows the degrees of the basis elements of the
6354:
5294:{\displaystyle HS_{R/\langle f\rangle }(t)=\left(1-t^{\delta }\right)HS_{R}(t).}
184:
7360:
the associated coherent sheaf the two definitions of Hilbert polynomial agree.
213:, which is finitely generated by elements of positive degree. This means that
7872:
7803:
7021:
the (homogeneous) ideal generated by the leading monomials of the elements of
1954:
is the minimal degree of the generators of the module, which may be negative.
87:
7906:
7867:, Graduate Texts in Mathematics, vol. 150, New York: Springer-Verlag,
7084:
while finitely generated graded modules correspond to coherent sheaves. If
6210:
This shows that the complete intersection defined by a regular sequence of
5443:. In this case, there is a simple explicit formula for the Hilbert series.
3838:
is the number of points of intersection, counted with multiplicities, of
117:
The Hilbert polynomial and Hilbert series are important in computational
6897:
The Hilbert polynomial is easily deducible from the Hilbert series (see
65:
The typical situations where these notions are used are the following:
3815:
is supposed, without loss of generality, to be algebraically closed.
7668:
7531:
7257:. The Euler characteristic in this case is a well-defined number by
5742:
2597:
5926:{\displaystyle HS_{R_{i}}(t)=(1-t^{\delta _{i}})HS_{R_{i-1}}(t)\,.}
3432:. Iterating this a number of times equal to the Krull dimension of
2271:
is an exact sequence of graded or filtered modules, then we have
6480:{\displaystyle HS_{M}(t)=\sum _{i=1}^{k}(-1)^{i-1}HS_{L_{i}}(t).}
2209:
Hilbert series and Hilbert polynomial are additive relatively to
7028:
The computation of the Hilbert series is based on the fact that
6885:, from which the Hilbert series may be directly computed with a
7368:
Since the category of coherent sheaves on a projective variety
834:
In this case the series expansion of this rational fraction is
7205:{\displaystyle p_{\mathcal {F}}(m)=\chi (X,{\mathcal {F}}(m))}
496:{\displaystyle HS_{S}(t)=\sum _{n=0}^{\infty }HF_{S}(n)t^{n}.}
4477:{\displaystyle R_{1}=R/\langle h_{0},\ldots ,h_{d-1}\rangle }
1932:
All these definitions may be extended to finitely generated
7737:
7711:
7624:
7579:
7474:
7296:
7229:
7185:
7153:
7125:
7093:
6695:
In fact, these formulas imply that, if a graded free module
5677:{\displaystyle R_{i}=R/\langle f_{1},\ldots ,f_{i}\rangle ,}
5054:
For proving BĂ©zout's theorem, one may proceed similarly. If
3650:
and the Hilbert series, consider a projective algebraic set
564:, then the sum of the Hilbert series is a rational fraction
399:
in the more general setting of graded vector spaces, is the
6581:
then the formulas of the preceding sections allow deducing
7865:
Commutative algebra. With a view toward algebraic geometry
2806:
Hilbert series and Hilbert polynomial of a polynomial ring
2151:
and this defines an isomorphism of graded algebra between
787:{\displaystyle HS_{S}(t)={\frac {P(t)}{(1-t)^{\delta }}},}
106:
series of an algebra or a module is a special case of the
7076:, graded rings generated by elements of degree 1 produce
6316:{\displaystyle 0\to L_{k}\to \cdots \to L_{1}\to M\to 0,}
4703:
as its ring of regular functions. The linear polynomial
2978:{\displaystyle HS_{R_{n}}(t)={\frac {1}{(1-t)^{n}}}\,.}
7673:
7536:
4973:{\displaystyle HS_{R_{0}}(t)=(1-t)HS_{R_{1}}(t)=P(t).}
2557:
It follows from the additivity on the exact sequence
153:
has the same Hilbert polynomial over any closed point
7464:
7415:
7395:
7374:
7337:
7274:
7226:
7146:
7122:
7090:
6914:
6765:
6713:
6626:
6587:
6557:
6496:
6373:
6332:
6262:
5945:
5822:
5693:
5614:
5565:
5504:
5452:
5373:
5349:
5321:
5200:
5107:
5080:
5060:
4996:
4875:
4775:
4742:
4709:
4642:
4611:
4581:
4544:
4517:
4490:
4411:
4375:
4204:
4160:
3935:
3879:
3848:
3741:
3663:
3646:
For showing the relationship between the degree of a
3590:
3464:
3344:
3192:
3165:
2997:
2907:
2884:
2816:
2717:
2666:
2566:
2547:{\displaystyle HS_{A/(f)}(t)=(1-t^{d})\,HS_{A}(t)\,.}
2455:
2348:
2280:
2222:
2157:
2073:
1886:
1805:
1683:
1633:
1594:
1541:
1515:
1466:
1337:
1295:
1268:
1189:
1151:
994:
843:
809:
713:
573:
524:
411:
313:
274:
222:
159:
127:
6898:
6893:
Computation of Hilbert series and Hilbert polynomial
94:. In this case the Hilbert polynomial is called the
4696:{\displaystyle R_{0}=R_{1}/\langle h_{d}-1\rangle }
3631:
Degree of a projective variety and BĂ©zout's theorem
3535:{\displaystyle HS_{A}(t)={\frac {P(t)}{(1-t)^{d}}}}
7756:
7447:
7401:
7380:
7352:
7315:
7245:
7204:
7132:
7100:
6961:
6869:
6748:
6687:
6612:
6573:
6543:
6479:
6345:
6315:
6199:
5925:
5802:
5676:
5600:
5551:
5484:
5393:In a more geometrical form, this may restated as:
5382:
5355:
5327:
5293:
5180:
5086:
5066:
5009:
4972:
4855:
4758:
4728:
4695:
4624:
4597:
4563:
4530:
4503:
4476:
4390:
4358:
4187:
4143:
3911:
3854:
3796:
3723:
3619:
3534:
3424:
3268:-vector space is finite and the Hilbert series of
3248:generated by homogeneous elements of degree 1 has
3227:
3178:
3148:
2977:
2890:
2870:
2795:
2685:
2637:
2546:
2399:
2328:
2260:
2178:
2127:
1910:
1855:
1779:
1658:
1619:
1580:
1527:
1493:
1449:
1320:
1281:
1251:
1169:
1130:
974:
815:
786:
678:
556:
495:
362:{\displaystyle HF_{S}:n\longmapsto \dim _{K}S_{n}}
361:
293:
257:
171:
145:
3068:
3033:
1588:This shows that there exists a unique polynomial
1438:
1397:
1033:
998:
945:
910:
7948:(1978), "Hilbert functions of graded algebras",
7264:This function is indeed a polynomial. For large
6749:{\displaystyle \delta _{1},\ldots ,\delta _{h},}
5601:{\displaystyle \delta _{1},\ldots ,\delta _{k}.}
2796:{\displaystyle HS_{A^{}}(t)=t^{d}\,HS_{A}(t)\,.}
5813:The additivity of Hilbert series implies thus
1252:{\displaystyle P(t)=\sum _{i=0}^{d}a_{i}t^{i},}
803:is a polynomial with integer coefficients, and
6364:The additivity of Hilbert series implies that
3425:{\displaystyle HS_{A/(f)}(t)=(1-t)\,HS_{A}(t)}
2067:defines an homomorphism of graded rings from
2015:is a graded algebra generated over the field
1627:with rational coefficients which is equal to
8:
7954:, vol. 28, no. 1, pp. 57â83,
5668:
5636:
5223:
5217:
5166:
5160:
4690:
4671:
4471:
4433:
4259:
4221:
4129:
4097:
4083:
4045:
3994:
3956:
3635:The Hilbert series allows us to compute the
3338:The additivity of Hilbert series shows that
3313:is positive, there is a homogeneous element
2693:is the graded module which is obtained from
1991:is defined as the Hilbert polynomial of the
1973:are those of the associated graded algebra.
5412:, then the degree of their intersection is
2988:It follows that the Hilbert polynomial is
695:is a polynomial with integer coefficients.
258:{\displaystyle S=\bigoplus _{i\geq 0}S_{i}}
7331:is a finitely generated graded module and
7316:{\displaystyle H^{0}(X,{\mathcal {F}}(m))}
5792:
5788:
5777:
5773:
5756:
5736:
5701:
5697:
5427:other hypersurfaces, one after the other.
4398:is the numerator of the Hilbert series of
2810:The Hilbert series of the polynomial ring
2627:
2612:
2608:
2604:
2591:
2574:
2570:
2254:
2250:
2246:
2242:
2238:
2234:
2230:
2226:
179:. This is used in the construction of the
7959:
7918:
7742:
7736:
7735:
7723:
7719:
7718:
7716:
7710:
7709:
7692:
7680:
7663:
7654:
7636:
7632:
7631:
7629:
7623:
7622:
7609:
7591:
7587:
7586:
7584:
7578:
7577:
7560:
7543:
7526:
7517:
7504:
7486:
7482:
7481:
7479:
7473:
7472:
7463:
7436:
7423:
7414:
7394:
7373:
7339:
7338:
7336:
7295:
7294:
7279:
7273:
7228:
7227:
7225:
7184:
7183:
7152:
7151:
7145:
7124:
7123:
7121:
7092:
7091:
7089:
6950:
6931:
6913:
6855:
6829:
6824:
6803:
6798:
6791:
6773:
6764:
6737:
6718:
6712:
6676:
6655:
6634:
6625:
6595:
6586:
6562:
6556:
6532:
6513:
6495:
6457:
6452:
6433:
6414:
6403:
6381:
6372:
6337:
6331:
6292:
6273:
6261:
6253:, meaning there exists an exact sequence
6193:
6178:
6149:
6144:
6105:
6100:
6072:
6060:
6031:
6026:
5999:
5994:
5978:
5958:
5953:
5944:
5919:
5896:
5891:
5873:
5868:
5835:
5830:
5821:
5796:
5782:
5761:
5747:
5737:
5725:
5717:
5706:
5692:
5662:
5643:
5631:
5619:
5613:
5589:
5570:
5564:
5540:
5521:
5503:
5476:
5457:
5451:
5400:- If a projective hypersurface of degree
5372:
5348:
5339:, then the degree of the intersection of
5320:
5273:
5255:
5212:
5208:
5199:
5155:
5138:
5133:
5131:
5130:
5118:
5106:
5079:
5059:
5001:
4995:
4935:
4930:
4888:
4883:
4874:
4838:
4825:
4805:
4800:
4795:
4793:
4792:
4786:
4774:
4747:
4741:
4714:
4708:
4678:
4666:
4660:
4647:
4641:
4616:
4610:
4586:
4580:
4549:
4543:
4522:
4516:
4495:
4489:
4459:
4440:
4428:
4416:
4410:
4374:
4324:
4306:
4298:
4292:
4247:
4228:
4216:
4212:
4203:
4159:
4123:
4104:
4092:
4071:
4052:
4040:
4027:
4022:
4017:
4015:
4014:
4002:
3982:
3963:
3951:
3934:
3903:
3884:
3878:
3847:
3786:
3777:
3758:
3740:
3712:
3693:
3680:
3662:
3591:
3589:
3523:
3490:
3472:
3463:
3407:
3399:
3356:
3352:
3343:
3238:Shape of the Hilbert series and dimension
3221:
3200:
3191:
3170:
3164:
3142:
3077:
3067:
3055:
3038:
3032:
3030:
3010:
3005:
2996:
2971:
2962:
2940:
2920:
2915:
2906:
2883:
2859:
2840:
2821:
2815:
2789:
2774:
2766:
2760:
2730:
2725:
2716:
2671:
2665:
2631:
2616:
2592:
2579:
2565:
2540:
2525:
2517:
2508:
2467:
2463:
2454:
2388:
2372:
2356:
2347:
2320:
2304:
2288:
2279:
2221:
2168:
2162:
2156:
2116:
2097:
2078:
2072:
1942:, with the only difference that a factor
1885:
1838:
1813:
1804:
1766:
1751:
1709:
1691:
1682:
1641:
1632:
1602:
1593:
1552:
1546:
1540:
1514:
1465:
1437:
1396:
1394:
1388:
1378:
1367:
1345:
1336:
1303:
1294:
1273:
1267:
1240:
1230:
1220:
1209:
1188:
1150:
1042:
1032:
997:
995:
993:
955:
944:
909:
907:
851:
842:
808:
772:
739:
721:
712:
657:
652:
631:
620:
599:
581:
572:
548:
529:
523:
518:homogeneous elements of positive degrees
484:
465:
452:
441:
419:
410:
353:
340:
321:
312:
279:
273:
249:
233:
221:
158:
126:
5439:if its defining ideal is generated by a
4571:The complement of this hyperplane is an
7795:
5684:one has the following exact sequences
3654:, defined as the set of the zeros of a
3449:. This show that the Hilbert series of
1926:
7116:, we define the Hilbert polynomial of
6688:{\displaystyle HS_{R}(t)=1/(1-t)^{n}.}
5315:is a homogeneous polynomial of degree
5074:is a homogeneous polynomial of degree
3584:, and that its leading coefficient is
2705:, in order that the multiplication by
2045:of degree 1, then the map which sends
395:. The Hilbert series, which is called
3256:. This implies that the dimension of
2400:{\displaystyle HP_{B}=HP_{A}+HP_{C}.}
1948:appears in the Hilbert series, where
1670:large enough. This polynomial is the
16:Tool in mathematical dimension theory
7:
2329:{\displaystyle HS_{B}=HS_{A}+HS_{C}}
1768: terms of lower degree in
46:These notions have been extended to
5485:{\displaystyle f_{1},\ldots ,f_{k}}
5043:is the degree of the algebraic set
4985:Hilbert series of filtered algebras
4766:and one has thus an exact sequence
3912:{\displaystyle h_{0},\ldots ,h_{d}}
2005:Graded algebra and polynomial rings
1856:{\displaystyle HP_{S}(n)=HF_{S}(n)}
1581:{\displaystyle a_{i}/(\delta -1)!.}
557:{\displaystyle d_{1},\ldots ,d_{h}}
54:over these algebras, as well as to
7838:Algebraic Geometry, A First Course
7068:Generalization to coherent sheaves
3620:{\displaystyle {\frac {P(1)}{d!}}}
3037:
1494:{\displaystyle n\geq i-\delta +1,}
1401:
1002:
914:
453:
14:
7809:Foundations of Algebraic Geometry
7259:Grothendieck's finiteness theorem
7246:{\displaystyle {\mathcal {F}}(m)}
6216:polynomials has a codimension of
5343:with the hypersurface defined by
5335:, which is not a zero divisor in
5094:, which is not a zero divisor in
3866:. This implies the existence, in
7903:Computational Algebraic Geometry
6707:homogeneous elements of degrees
5435:A projective algebraic set is a
5363:is the product of the degree of
2711:has degree 0. This implies that
2430:a homogeneous element of degree
1911:{\displaystyle \deg P-\delta +1}
1505:in this sum is a polynomial in
191:Definitions and main properties
7774:CastelnuovoâMumford regularity
7748:
7731:
7660:
7644:
7615:
7599:
7523:
7494:
7468:
7442:
7416:
7344:
7310:
7307:
7301:
7285:
7240:
7234:
7199:
7196:
7190:
7174:
7165:
7159:
7133:{\displaystyle {\mathcal {F}}}
7101:{\displaystyle {\mathcal {F}}}
6956:
6924:
6852:
6839:
6785:
6779:
6673:
6660:
6646:
6640:
6607:
6601:
6538:
6506:
6471:
6465:
6430:
6420:
6393:
6387:
6304:
6298:
6285:
6279:
6266:
6226:Relation with free resolutions
6175:
6162:
6157:
6119:
6113:
6075:
6057:
6044:
6039:
6013:
6007:
5981:
5972:
5966:
5916:
5910:
5881:
5855:
5849:
5843:
5789:
5774:
5731:
5718:
5698:
5546:
5514:
5408:of an algebraic set of degree
5285:
5279:
5234:
5228:
5169:
5149:
5134:
5125:
5119:
5111:
4964:
4958:
4949:
4943:
4920:
4908:
4902:
4896:
4844:
4831:
4796:
4779:
4385:
4379:
4336:
4330:
4318:
4312:
4289:
4276:
4270:
4264:
4188:{\displaystyle k=0,\ldots ,d.}
4132:
4086:
4018:
4009:
4003:
3939:
3783:
3751:
3718:
3673:
3637:degree of an algebraic variety
3603:
3597:
3520:
3507:
3502:
3496:
3484:
3478:
3419:
3413:
3396:
3384:
3378:
3372:
3367:
3361:
3228:{\displaystyle HS_{K}(t)=1\,.}
3212:
3206:
3133:
3121:
3116:
3098:
3092:
3080:
3024:
3018:
2959:
2946:
2934:
2928:
2865:
2833:
2786:
2780:
2750:
2744:
2737:
2731:
2678:
2672:
2624:
2609:
2586:
2580:
2571:
2537:
2531:
2514:
2495:
2489:
2483:
2478:
2472:
2414:Quotient by a non-zero divisor
2251:
2243:
2235:
2227:
2122:
2090:
1850:
1844:
1825:
1819:
1738:
1726:
1721:
1715:
1703:
1697:
1653:
1647:
1614:
1608:
1569:
1557:
1357:
1351:
1315:
1309:
1199:
1193:
1170:{\displaystyle n>-\delta ,}
1119:
1107:
1102:
1090:
1084:
1066:
1063:
1045:
878:
872:
863:
857:
769:
756:
751:
745:
733:
727:
611:
605:
593:
587:
477:
471:
431:
425:
333:
195:Consider a finitely generated
137:
69:The quotient by a homogeneous
1:
7448:{\displaystyle (d_{1},d_{2})}
5036:may be used for proving that
4984:
3292:is equal to the dimension of
80:, graded by the total degree.
7961:10.1016/0001-8708(78)90045-2
7353:{\displaystyle {\tilde {M}}}
6756:then its Hilbert series is
2200:Properties of Hilbert series
1976:The Hilbert polynomial of a
1921:The Hilbert polynomial is a
5498:homogeneous polynomials in
5025:-vector space of dimension
2699:by shifting the degrees by
1993:homogeneous coordinate ring
146:{\displaystyle \pi :X\to S}
8009:
7911:Cambridge University Press
7056:these functions are named
5936:A simple recursion gives
3724:{\displaystyle I\subset k}
3569:is the Krull dimension of
3329:is the Krull dimension of
3307:If the Krull dimension of
197:graded commutative algebra
39:finitely generated over a
37:graded commutative algebra
7873:10.1007/978-1-4612-5350-1
7325:Serre's vanishing theorem
7112:over a projective scheme
6969:be a polynomial ring and
6613:{\displaystyle HS_{M}(t)}
4736:is not a zero divisor in
3842:with the intersection of
2654:is the multiplication by
1659:{\displaystyle HF_{S}(n)}
1620:{\displaystyle HP_{S}(n)}
1535:with leading coefficient
1528:{\displaystyle \delta -1}
1321:{\displaystyle HS_{S}(t)}
96:HilbertâSamuel polynomial
50:, and graded or filtered
7046:computer algebra systems
7038:computational complexity
6887:computational complexity
5559:, of respective degrees
5383:{\displaystyle \delta .}
4564:{\displaystyle h_{d}=0.}
3648:projective algebraic set
2648:where the arrow labeled
2424:be a graded algebra and
378:to the dimension of the
7951:Advances in Mathematics
7364:Graded free resolutions
7220:of coherent sheaf, and
5328:{\displaystyle \delta }
5087:{\displaystyle \delta }
4729:{\displaystyle h_{d}-1}
3630:
2871:{\displaystyle R_{n}=K}
2179:{\displaystyle R_{n}/I}
2145:is a homogeneous ideal
2128:{\displaystyle R_{n}=K}
1880:. It may be lower than
816:{\displaystyle \delta }
397:HilbertâPoincarĂ© series
294:{\displaystyle S_{0}=K}
108:HilbertâPoincarĂ© series
75:multivariate polynomial
7758:
7449:
7403:
7382:
7354:
7317:
7247:
7206:
7134:
7102:
7048:. For example in both
6963:
6871:
6750:
6689:
6614:
6575:
6574:{\displaystyle L_{i},}
6545:
6481:
6419:
6347:
6317:
6251:Hilbert syzygy theorem
6201:
5927:
5804:
5678:
5602:
5553:
5486:
5384:
5357:
5329:
5295:
5182:
5098:, the exact sequence
5088:
5068:
5011:
4974:
4857:
4760:
4759:{\displaystyle R_{1},}
4730:
4697:
4626:
4599:
4598:{\displaystyle V_{0}.}
4565:
4532:
4505:
4478:
4392:
4360:
4189:
4145:
3913:
3856:
3808:on the algebraic set.
3798:
3725:
3621:
3536:
3426:
3229:
3180:
3150:
2979:
2892:
2872:
2797:
2687:
2639:
2548:
2401:
2330:
2262:
2213:. More precisely, if
2180:
2129:
1912:
1857:
1781:
1660:
1621:
1582:
1529:
1495:
1451:
1383:
1322:
1283:
1253:
1225:
1171:
1132:
976:
817:
788:
680:
636:
558:
497:
457:
363:
295:
259:
173:
172:{\displaystyle s\in S}
147:
7759:
7450:
7404:
7383:
7355:
7318:
7248:
7207:
7135:
7103:
7017:partial ordering and
6989:is homogeneous, then
6964:
6872:
6751:
6690:
6615:
6576:
6546:
6482:
6399:
6357:, and the arrows are
6348:
6346:{\displaystyle L_{i}}
6318:
6202:
5928:
5805:
5679:
5603:
5554:
5487:
5437:complete intersection
5431:Complete intersection
5406:irreducible component
5404:does not contain any
5385:
5358:
5330:
5296:
5183:
5089:
5069:
5034:JordanâHölder theorem
5012:
5010:{\displaystyle R_{0}}
4975:
4858:
4761:
4731:
4698:
4627:
4625:{\displaystyle V_{0}}
4600:
4566:
4533:
4531:{\displaystyle h_{d}}
4506:
4504:{\displaystyle V_{0}}
4479:
4393:
4361:
4190:
4146:
3914:
3857:
3799:
3797:{\displaystyle R=k/I}
3726:
3622:
3545:where the polynomial
3537:
3427:
3230:
3186:) and remarking that
3181:
3179:{\displaystyle x_{n}}
3151:
2980:
2893:
2873:
2798:
2688:
2640:
2549:
2402:
2331:
2263:
2181:
2130:
2027:homogeneous elements
1913:
1858:
1782:
1661:
1622:
1583:
1530:
1496:
1452:
1363:
1323:
1284:
1282:{\displaystyle t^{n}}
1254:
1205:
1172:
1133:
977:
818:
789:
681:
616:
559:
498:
437:
364:
304:The Hilbert function
296:
260:
174:
148:
90:by the powers of its
7462:
7413:
7393:
7372:
7335:
7272:
7224:
7218:Euler characteristic
7144:
7120:
7088:
6912:
6763:
6711:
6624:
6585:
6555:
6494:
6371:
6330:
6260:
6230:Every graded module
5943:
5820:
5691:
5612:
5563:
5502:
5450:
5371:
5347:
5319:
5198:
5105:
5078:
5058:
4994:
4873:
4866:which implies that
4773:
4740:
4707:
4640:
4634:affine algebraic set
4609:
4579:
4542:
4515:
4488:
4409:
4391:{\displaystyle P(t)}
4373:
4202:
4158:
3933:
3877:
3846:
3834:, and the degree of
3739:
3735:is a field, and let
3661:
3588:
3462:
3342:
3190:
3163:
2995:
2905:
2882:
2814:
2715:
2686:{\displaystyle A^{}}
2664:
2564:
2453:
2346:
2278:
2220:
2155:
2071:
1923:numerical polynomial
1884:
1803:
1681:
1631:
1592:
1539:
1513:
1464:
1335:
1293:
1266:
1187:
1177:and is 0 otherwise.
1149:
1143:binomial coefficient
992:
841:
807:
711:
571:
522:
409:
311:
272:
220:
157:
125:
86:The filtration of a
7988:Commutative algebra
7705:
7573:
7455:has the resolution
7268:it agrees with dim
6962:{\displaystyle R=K}
6544:{\displaystyle R=k}
5753:
5735:
5552:{\displaystyle R=K}
4195:This implies that
3804:be the ring of the
2898:indeterminates is
2601:
1674:, and has the form
1262:the coefficient of
112:graded vector space
21:commutative algebra
7993:Algebraic geometry
7754:
7700:
7568:
7445:
7399:
7378:
7350:
7313:
7243:
7202:
7130:
7098:
7078:projective schemes
7074:algebraic geometry
6959:
6867:
6746:
6685:
6610:
6571:
6541:
6477:
6359:graded linear maps
6343:
6313:
6197:
5923:
5800:
5702:
5674:
5598:
5549:
5482:
5380:
5353:
5325:
5291:
5178:
5084:
5064:
5049:composition series
5007:
4983:Here we are using
4970:
4853:
4756:
4726:
4693:
4622:
4595:
4561:
4528:
4501:
4474:
4388:
4356:
4185:
4141:
3909:
3852:
3794:
3721:
3617:
3532:
3422:
3225:
3176:
3146:
2975:
2888:
2868:
2793:
2683:
2635:
2544:
2397:
2326:
2258:
2176:
2125:
1978:projective variety
1967:Hilbert polynomial
1908:
1878:Hilbert regularity
1853:
1777:
1672:Hilbert polynomial
1656:
1617:
1578:
1525:
1501:the term of index
1491:
1447:
1318:
1279:
1249:
1167:
1128:
972:
813:
784:
676:
554:
493:
359:
291:
255:
244:
169:
143:
119:algebraic geometry
60:projective schemes
29:Hilbert polynomial
7930:978-0-521-53650-9
7852:978-0-387-97716-4
7706:
7574:
7402:{\displaystyle X}
7381:{\displaystyle X}
7347:
7082:Proj construction
7062:HilbertPolynomial
7011:monomial ordering
6862:
6191:
6067:
5754:
5356:{\displaystyle f}
5143:
5067:{\displaystyle f}
4818:
4351:
4034:
3855:{\displaystyle d}
3806:regular functions
3656:homogeneous ideal
3615:
3530:
3242:A graded algebra
3140:
3066:
2969:
2891:{\displaystyle n}
2602:
1769:
1745:
1436:
1126:
1031:
943:
779:
671:
372:maps the integer
229:
48:filtered algebras
8000:
7972:
7963:
7946:Stanley, Richard
7941:
7922:
7893:
7856:
7843:Springer Science
7818:
7817:, Theorem 18.6.1
7816:
7814:
7800:
7763:
7761:
7760:
7755:
7747:
7746:
7741:
7740:
7730:
7729:
7728:
7727:
7722:
7715:
7714:
7707:
7704:
7697:
7696:
7685:
7684:
7664:
7659:
7658:
7643:
7642:
7641:
7640:
7635:
7628:
7627:
7614:
7613:
7598:
7597:
7596:
7595:
7590:
7583:
7582:
7575:
7572:
7565:
7564:
7548:
7547:
7527:
7522:
7521:
7509:
7508:
7493:
7492:
7491:
7490:
7485:
7478:
7477:
7454:
7452:
7451:
7446:
7441:
7440:
7428:
7427:
7409:of multi-degree
7408:
7406:
7405:
7400:
7387:
7385:
7384:
7379:
7359:
7357:
7356:
7351:
7349:
7348:
7340:
7322:
7320:
7319:
7314:
7300:
7299:
7284:
7283:
7252:
7250:
7249:
7244:
7233:
7232:
7211:
7209:
7208:
7203:
7189:
7188:
7158:
7157:
7156:
7139:
7137:
7136:
7131:
7129:
7128:
7107:
7105:
7104:
7099:
7097:
7096:
6968:
6966:
6965:
6960:
6955:
6954:
6936:
6935:
6876:
6874:
6873:
6868:
6863:
6861:
6860:
6859:
6837:
6836:
6835:
6834:
6833:
6810:
6809:
6808:
6807:
6792:
6778:
6777:
6755:
6753:
6752:
6747:
6742:
6741:
6723:
6722:
6706:
6700:
6694:
6692:
6691:
6686:
6681:
6680:
6659:
6639:
6638:
6619:
6617:
6616:
6611:
6600:
6599:
6580:
6578:
6577:
6572:
6567:
6566:
6550:
6548:
6547:
6542:
6537:
6536:
6518:
6517:
6486:
6484:
6483:
6478:
6464:
6463:
6462:
6461:
6444:
6443:
6418:
6413:
6386:
6385:
6361:of degree zero.
6352:
6350:
6349:
6344:
6342:
6341:
6322:
6320:
6319:
6314:
6297:
6296:
6278:
6277:
6244:
6235:
6221:
6215:
6206:
6204:
6203:
6198:
6192:
6190:
6189:
6188:
6160:
6156:
6155:
6154:
6153:
6112:
6111:
6110:
6109:
6073:
6068:
6066:
6065:
6064:
6042:
6038:
6037:
6036:
6035:
6006:
6005:
6004:
6003:
5979:
5965:
5964:
5963:
5962:
5932:
5930:
5929:
5924:
5909:
5908:
5907:
5906:
5880:
5879:
5878:
5877:
5842:
5841:
5840:
5839:
5809:
5807:
5806:
5801:
5787:
5786:
5772:
5771:
5755:
5752:
5751:
5738:
5734:
5730:
5729:
5716:
5683:
5681:
5680:
5675:
5667:
5666:
5648:
5647:
5635:
5624:
5623:
5607:
5605:
5604:
5599:
5594:
5593:
5575:
5574:
5558:
5556:
5555:
5550:
5545:
5544:
5526:
5525:
5497:
5491:
5489:
5488:
5483:
5481:
5480:
5462:
5461:
5441:regular sequence
5426:
5415:
5411:
5403:
5389:
5387:
5386:
5381:
5366:
5362:
5360:
5359:
5354:
5342:
5338:
5334:
5332:
5331:
5326:
5314:
5300:
5298:
5297:
5292:
5278:
5277:
5265:
5261:
5260:
5259:
5227:
5226:
5216:
5187:
5185:
5184:
5179:
5159:
5145:
5144:
5142:
5137:
5132:
5129:
5128:
5097:
5093:
5091:
5090:
5085:
5073:
5071:
5070:
5065:
5046:
5042:
5031:
5024:
5016:
5014:
5013:
5008:
5006:
5005:
4979:
4977:
4976:
4971:
4942:
4941:
4940:
4939:
4895:
4894:
4893:
4892:
4862:
4860:
4859:
4854:
4843:
4842:
4830:
4829:
4820:
4819:
4817:
4810:
4809:
4799:
4794:
4791:
4790:
4765:
4763:
4762:
4757:
4752:
4751:
4735:
4733:
4732:
4727:
4719:
4718:
4702:
4700:
4699:
4694:
4683:
4682:
4670:
4665:
4664:
4652:
4651:
4631:
4629:
4628:
4623:
4621:
4620:
4604:
4602:
4601:
4596:
4591:
4590:
4570:
4568:
4567:
4562:
4554:
4553:
4537:
4535:
4534:
4529:
4527:
4526:
4510:
4508:
4507:
4502:
4500:
4499:
4483:
4481:
4480:
4475:
4470:
4469:
4445:
4444:
4432:
4421:
4420:
4401:
4397:
4395:
4394:
4389:
4365:
4363:
4362:
4357:
4352:
4350:
4339:
4325:
4311:
4310:
4297:
4296:
4263:
4262:
4258:
4257:
4233:
4232:
4220:
4194:
4192:
4191:
4186:
4150:
4148:
4147:
4142:
4128:
4127:
4109:
4108:
4096:
4082:
4081:
4057:
4056:
4044:
4036:
4035:
4033:
4032:
4031:
4021:
4016:
4013:
4012:
4001:
3997:
3993:
3992:
3968:
3967:
3955:
3925:
3918:
3916:
3915:
3910:
3908:
3907:
3889:
3888:
3872:regular sequence
3869:
3864:general position
3861:
3859:
3858:
3853:
3841:
3837:
3833:
3826:is equal to the
3825:
3821:
3814:
3803:
3801:
3800:
3795:
3790:
3782:
3781:
3763:
3762:
3734:
3730:
3728:
3727:
3722:
3717:
3716:
3698:
3697:
3685:
3684:
3653:
3641:BĂ©zout's theorem
3626:
3624:
3623:
3618:
3616:
3614:
3606:
3592:
3583:
3574:
3568:
3562:
3555:
3541:
3539:
3538:
3533:
3531:
3529:
3528:
3527:
3505:
3491:
3477:
3476:
3454:
3448:
3437:
3431:
3429:
3428:
3423:
3412:
3411:
3371:
3370:
3360:
3334:
3328:
3318:
3312:
3303:
3297:
3291:
3284:
3274:is a polynomial
3273:
3267:
3261:
3247:
3234:
3232:
3231:
3226:
3205:
3204:
3185:
3183:
3182:
3177:
3175:
3174:
3155:
3153:
3152:
3147:
3141:
3139:
3119:
3078:
3073:
3072:
3071:
3065:
3054:
3036:
3017:
3016:
3015:
3014:
2984:
2982:
2981:
2976:
2970:
2968:
2967:
2966:
2941:
2927:
2926:
2925:
2924:
2897:
2895:
2894:
2889:
2877:
2875:
2874:
2869:
2864:
2863:
2845:
2844:
2826:
2825:
2802:
2800:
2799:
2794:
2779:
2778:
2765:
2764:
2743:
2742:
2741:
2740:
2710:
2704:
2698:
2692:
2690:
2689:
2684:
2682:
2681:
2659:
2653:
2644:
2642:
2641:
2636:
2620:
2603:
2593:
2590:
2589:
2553:
2551:
2550:
2545:
2530:
2529:
2513:
2512:
2482:
2481:
2471:
2446:. Then we have
2441:
2435:
2429:
2423:
2406:
2404:
2403:
2398:
2393:
2392:
2377:
2376:
2361:
2360:
2335:
2333:
2332:
2327:
2325:
2324:
2309:
2308:
2293:
2292:
2267:
2265:
2264:
2259:
2191:
2185:
2183:
2182:
2177:
2172:
2167:
2166:
2150:
2140:
2134:
2132:
2131:
2126:
2121:
2120:
2102:
2101:
2083:
2082:
2066:
2055:
2044:
2026:
2020:
2014:
2000:
1990:
1984:
1971:filtered algebra
1959:Hilbert function
1953:
1947:
1941:
1929:, pp. 41).
1917:
1915:
1914:
1909:
1875:
1862:
1860:
1859:
1854:
1843:
1842:
1818:
1817:
1798:
1786:
1784:
1783:
1778:
1770:
1767:
1762:
1761:
1746:
1744:
1724:
1710:
1696:
1695:
1669:
1665:
1663:
1662:
1657:
1646:
1645:
1626:
1624:
1623:
1618:
1607:
1606:
1587:
1585:
1584:
1579:
1556:
1551:
1550:
1534:
1532:
1531:
1526:
1508:
1504:
1500:
1498:
1497:
1492:
1456:
1454:
1453:
1448:
1443:
1442:
1441:
1435:
1424:
1400:
1393:
1392:
1382:
1377:
1350:
1349:
1327:
1325:
1324:
1319:
1308:
1307:
1288:
1286:
1285:
1280:
1278:
1277:
1258:
1256:
1255:
1250:
1245:
1244:
1235:
1234:
1224:
1219:
1176:
1174:
1173:
1168:
1137:
1135:
1134:
1129:
1127:
1125:
1105:
1043:
1038:
1037:
1036:
1030:
1019:
1001:
981:
979:
978:
973:
971:
967:
960:
959:
950:
949:
948:
942:
931:
913:
856:
855:
830:
822:
820:
819:
814:
802:
793:
791:
790:
785:
780:
778:
777:
776:
754:
740:
726:
725:
703:
694:
685:
683:
682:
677:
672:
670:
669:
665:
664:
663:
662:
661:
635:
630:
614:
600:
586:
585:
563:
561:
560:
555:
553:
552:
534:
533:
517:
512:is generated by
511:
502:
500:
499:
494:
489:
488:
470:
469:
456:
451:
424:
423:
394:
383:
377:
368:
366:
365:
360:
358:
357:
345:
344:
326:
325:
300:
298:
297:
292:
284:
283:
264:
262:
261:
256:
254:
253:
243:
212:
203:
178:
176:
175:
170:
152:
150:
149:
144:
56:coherent sheaves
25:Hilbert function
8008:
8007:
8003:
8002:
8001:
7999:
7998:
7997:
7978:
7977:
7976:
7944:
7931:
7897:
7883:
7861:Eisenbud, David
7859:
7853:
7831:
7827:
7822:
7821:
7812:
7802:
7801:
7797:
7792:
7770:
7734:
7717:
7708:
7699:
7698:
7688:
7686:
7676:
7669:
7650:
7630:
7621:
7605:
7585:
7576:
7567:
7566:
7556:
7550:
7549:
7539:
7532:
7513:
7500:
7480:
7471:
7460:
7459:
7432:
7419:
7411:
7410:
7391:
7390:
7370:
7369:
7366:
7333:
7332:
7275:
7270:
7269:
7222:
7221:
7147:
7142:
7141:
7118:
7117:
7086:
7085:
7070:
6973:be an ideal in
6946:
6927:
6910:
6909:
6895:
6851:
6838:
6825:
6820:
6799:
6794:
6793:
6769:
6761:
6760:
6733:
6714:
6709:
6708:
6702:
6701:has a basis of
6696:
6672:
6630:
6622:
6621:
6591:
6583:
6582:
6558:
6553:
6552:
6528:
6509:
6492:
6491:
6453:
6448:
6429:
6377:
6369:
6368:
6333:
6328:
6327:
6288:
6269:
6258:
6257:
6249:because of the
6247:free resolution
6240:
6231:
6228:
6217:
6211:
6174:
6161:
6145:
6140:
6101:
6096:
6074:
6056:
6043:
6027:
6022:
5995:
5990:
5980:
5954:
5949:
5941:
5940:
5892:
5887:
5869:
5864:
5831:
5826:
5818:
5817:
5778:
5757:
5743:
5721:
5689:
5688:
5658:
5639:
5615:
5610:
5609:
5585:
5566:
5561:
5560:
5536:
5517:
5500:
5499:
5493:
5472:
5453:
5448:
5447:
5433:
5421:
5413:
5409:
5401:
5369:
5368:
5364:
5345:
5344:
5340:
5336:
5317:
5316:
5312:
5269:
5251:
5244:
5240:
5204:
5196:
5195:
5114:
5103:
5102:
5095:
5076:
5075:
5056:
5055:
5044:
5037:
5026:
5022:
4997:
4992:
4991:
4931:
4926:
4884:
4879:
4871:
4870:
4834:
4821:
4801:
4782:
4771:
4770:
4743:
4738:
4737:
4710:
4705:
4704:
4674:
4656:
4643:
4638:
4637:
4612:
4607:
4606:
4582:
4577:
4576:
4545:
4540:
4539:
4518:
4513:
4512:
4491:
4486:
4485:
4455:
4436:
4412:
4407:
4406:
4399:
4371:
4370:
4340:
4326:
4302:
4288:
4243:
4224:
4208:
4200:
4199:
4156:
4155:
4119:
4100:
4067:
4048:
4023:
3978:
3959:
3947:
3943:
3942:
3931:
3930:
3920:
3899:
3880:
3875:
3874:
3867:
3862:hyperplanes in
3844:
3843:
3839:
3835:
3831:
3828:Krull dimension
3823:
3819:
3812:
3773:
3754:
3737:
3736:
3732:
3708:
3689:
3676:
3659:
3658:
3651:
3633:
3607:
3593:
3586:
3585:
3579:
3570:
3564:
3557:
3546:
3519:
3506:
3492:
3468:
3460:
3459:
3450:
3439:
3433:
3403:
3348:
3340:
3339:
3330:
3320:
3314:
3308:
3304:-vector space.
3299:
3293:
3286:
3275:
3269:
3263:
3257:
3250:Krull dimension
3243:
3240:
3196:
3188:
3187:
3166:
3161:
3160:
3120:
3079:
3031:
3006:
3001:
2993:
2992:
2958:
2945:
2916:
2911:
2903:
2902:
2880:
2879:
2855:
2836:
2817:
2812:
2811:
2808:
2770:
2756:
2726:
2721:
2713:
2712:
2706:
2700:
2694:
2667:
2662:
2661:
2655:
2649:
2575:
2562:
2561:
2521:
2504:
2459:
2451:
2450:
2442:which is not a
2437:
2431:
2425:
2419:
2416:
2384:
2368:
2352:
2344:
2343:
2316:
2300:
2284:
2276:
2275:
2218:
2217:
2211:exact sequences
2207:
2202:
2187:
2158:
2153:
2152:
2146:
2136:
2112:
2093:
2074:
2069:
2068:
2065:
2057:
2054:
2046:
2043:
2034:
2028:
2022:
2016:
2010:
2007:
1996:
1986:
1980:
1949:
1943:
1937:
1882:
1881:
1874:
1864:
1834:
1809:
1801:
1800:
1797:
1791:
1747:
1725:
1711:
1687:
1679:
1678:
1667:
1637:
1629:
1628:
1598:
1590:
1589:
1542:
1537:
1536:
1511:
1510:
1506:
1502:
1462:
1461:
1425:
1402:
1395:
1384:
1341:
1333:
1332:
1299:
1291:
1290:
1269:
1264:
1263:
1236:
1226:
1185:
1184:
1147:
1146:
1106:
1044:
1020:
1003:
996:
990:
989:
951:
932:
915:
908:
885:
881:
847:
839:
838:
828:
825:Krull dimension
805:
804:
798:
768:
755:
741:
717:
709:
708:
699:
690:
653:
648:
641:
637:
615:
601:
577:
569:
568:
544:
525:
520:
519:
513:
507:
480:
461:
415:
407:
406:
393:
385:
379:
373:
349:
336:
317:
309:
308:
275:
270:
269:
245:
218:
217:
208:
199:
193:
155:
154:
123:
122:
17:
12:
11:
5:
8006:
8004:
7996:
7995:
7990:
7980:
7979:
7975:
7974:
7942:
7929:
7920:10.1.1.57.7472
7895:
7881:
7857:
7851:
7828:
7826:
7823:
7820:
7819:
7794:
7793:
7791:
7788:
7787:
7786:
7781:
7779:Hilbert scheme
7776:
7769:
7766:
7765:
7764:
7753:
7750:
7745:
7739:
7733:
7726:
7721:
7713:
7703:
7695:
7691:
7687:
7683:
7679:
7675:
7674:
7672:
7667:
7662:
7657:
7653:
7649:
7646:
7639:
7634:
7626:
7620:
7617:
7612:
7608:
7604:
7601:
7594:
7589:
7581:
7571:
7563:
7559:
7555:
7552:
7551:
7546:
7542:
7538:
7537:
7535:
7530:
7525:
7520:
7516:
7512:
7507:
7503:
7499:
7496:
7489:
7484:
7476:
7470:
7467:
7444:
7439:
7435:
7431:
7426:
7422:
7418:
7398:
7377:
7365:
7362:
7346:
7343:
7312:
7309:
7306:
7303:
7298:
7293:
7290:
7287:
7282:
7278:
7242:
7239:
7236:
7231:
7201:
7198:
7195:
7192:
7187:
7182:
7179:
7176:
7173:
7170:
7167:
7164:
7161:
7155:
7150:
7140:as a function
7127:
7110:coherent sheaf
7095:
7069:
7066:
6997:. Finally let
6958:
6953:
6949:
6945:
6942:
6939:
6934:
6930:
6926:
6923:
6920:
6917:
6894:
6891:
6878:
6877:
6866:
6858:
6854:
6850:
6847:
6844:
6841:
6832:
6828:
6823:
6819:
6816:
6813:
6806:
6802:
6797:
6790:
6787:
6784:
6781:
6776:
6772:
6768:
6745:
6740:
6736:
6732:
6729:
6726:
6721:
6717:
6684:
6679:
6675:
6671:
6668:
6665:
6662:
6658:
6654:
6651:
6648:
6645:
6642:
6637:
6633:
6629:
6609:
6606:
6603:
6598:
6594:
6590:
6570:
6565:
6561:
6540:
6535:
6531:
6527:
6524:
6521:
6516:
6512:
6508:
6505:
6502:
6499:
6488:
6487:
6476:
6473:
6470:
6467:
6460:
6456:
6451:
6447:
6442:
6439:
6436:
6432:
6428:
6425:
6422:
6417:
6412:
6409:
6406:
6402:
6398:
6395:
6392:
6389:
6384:
6380:
6376:
6340:
6336:
6324:
6323:
6312:
6309:
6306:
6303:
6300:
6295:
6291:
6287:
6284:
6281:
6276:
6272:
6268:
6265:
6236:over a graded
6227:
6224:
6208:
6207:
6196:
6187:
6184:
6181:
6177:
6173:
6170:
6167:
6164:
6159:
6152:
6148:
6143:
6139:
6136:
6133:
6130:
6127:
6124:
6121:
6118:
6115:
6108:
6104:
6099:
6095:
6092:
6089:
6086:
6083:
6080:
6077:
6071:
6063:
6059:
6055:
6052:
6049:
6046:
6041:
6034:
6030:
6025:
6021:
6018:
6015:
6012:
6009:
6002:
5998:
5993:
5989:
5986:
5983:
5977:
5974:
5971:
5968:
5961:
5957:
5952:
5948:
5934:
5933:
5922:
5918:
5915:
5912:
5905:
5902:
5899:
5895:
5890:
5886:
5883:
5876:
5872:
5867:
5863:
5860:
5857:
5854:
5851:
5848:
5845:
5838:
5834:
5829:
5825:
5811:
5810:
5799:
5795:
5791:
5785:
5781:
5776:
5770:
5767:
5764:
5760:
5750:
5746:
5741:
5733:
5728:
5724:
5720:
5715:
5712:
5709:
5705:
5700:
5696:
5673:
5670:
5665:
5661:
5657:
5654:
5651:
5646:
5642:
5638:
5634:
5630:
5627:
5622:
5618:
5597:
5592:
5588:
5584:
5581:
5578:
5573:
5569:
5548:
5543:
5539:
5535:
5532:
5529:
5524:
5520:
5516:
5513:
5510:
5507:
5479:
5475:
5471:
5468:
5465:
5460:
5456:
5432:
5429:
5418:
5417:
5391:
5390:
5379:
5376:
5352:
5324:
5302:
5301:
5290:
5287:
5284:
5281:
5276:
5272:
5268:
5264:
5258:
5254:
5250:
5247:
5243:
5239:
5236:
5233:
5230:
5225:
5222:
5219:
5215:
5211:
5207:
5203:
5189:
5188:
5177:
5174:
5171:
5168:
5165:
5162:
5158:
5154:
5151:
5148:
5141:
5136:
5127:
5124:
5121:
5117:
5113:
5110:
5083:
5063:
5004:
5000:
4981:
4980:
4969:
4966:
4963:
4960:
4957:
4954:
4951:
4948:
4945:
4938:
4934:
4929:
4925:
4922:
4919:
4916:
4913:
4910:
4907:
4904:
4901:
4898:
4891:
4887:
4882:
4878:
4864:
4863:
4852:
4849:
4846:
4841:
4837:
4833:
4828:
4824:
4816:
4813:
4808:
4804:
4798:
4789:
4785:
4781:
4778:
4755:
4750:
4746:
4725:
4722:
4717:
4713:
4692:
4689:
4686:
4681:
4677:
4673:
4669:
4663:
4659:
4655:
4650:
4646:
4619:
4615:
4594:
4589:
4585:
4575:that contains
4560:
4557:
4552:
4548:
4525:
4521:
4498:
4494:
4473:
4468:
4465:
4462:
4458:
4454:
4451:
4448:
4443:
4439:
4435:
4431:
4427:
4424:
4419:
4415:
4387:
4384:
4381:
4378:
4367:
4366:
4355:
4349:
4346:
4343:
4338:
4335:
4332:
4329:
4323:
4320:
4317:
4314:
4309:
4305:
4301:
4295:
4291:
4287:
4284:
4281:
4278:
4275:
4272:
4269:
4266:
4261:
4256:
4253:
4250:
4246:
4242:
4239:
4236:
4231:
4227:
4223:
4219:
4215:
4211:
4207:
4184:
4181:
4178:
4175:
4172:
4169:
4166:
4163:
4152:
4151:
4140:
4137:
4134:
4131:
4126:
4122:
4118:
4115:
4112:
4107:
4103:
4099:
4095:
4091:
4088:
4085:
4080:
4077:
4074:
4070:
4066:
4063:
4060:
4055:
4051:
4047:
4043:
4039:
4030:
4026:
4020:
4011:
4008:
4005:
4000:
3996:
3991:
3988:
3985:
3981:
3977:
3974:
3971:
3966:
3962:
3958:
3954:
3950:
3946:
3941:
3938:
3906:
3902:
3898:
3895:
3892:
3887:
3883:
3851:
3818:The dimension
3793:
3789:
3785:
3780:
3776:
3772:
3769:
3766:
3761:
3757:
3753:
3750:
3747:
3744:
3720:
3715:
3711:
3707:
3704:
3701:
3696:
3692:
3688:
3683:
3679:
3675:
3672:
3669:
3666:
3632:
3629:
3613:
3610:
3605:
3602:
3599:
3596:
3543:
3542:
3526:
3522:
3518:
3515:
3512:
3509:
3504:
3501:
3498:
3495:
3489:
3486:
3483:
3480:
3475:
3471:
3467:
3421:
3418:
3415:
3410:
3406:
3402:
3398:
3395:
3392:
3389:
3386:
3383:
3380:
3377:
3374:
3369:
3366:
3363:
3359:
3355:
3351:
3347:
3239:
3236:
3224:
3220:
3217:
3214:
3211:
3208:
3203:
3199:
3195:
3173:
3169:
3157:
3156:
3145:
3138:
3135:
3132:
3129:
3126:
3123:
3118:
3115:
3112:
3109:
3106:
3103:
3100:
3097:
3094:
3091:
3088:
3085:
3082:
3076:
3070:
3064:
3061:
3058:
3053:
3050:
3047:
3044:
3041:
3035:
3029:
3026:
3023:
3020:
3013:
3009:
3004:
3000:
2986:
2985:
2974:
2965:
2961:
2957:
2954:
2951:
2948:
2944:
2939:
2936:
2933:
2930:
2923:
2919:
2914:
2910:
2887:
2867:
2862:
2858:
2854:
2851:
2848:
2843:
2839:
2835:
2832:
2829:
2824:
2820:
2807:
2804:
2792:
2788:
2785:
2782:
2777:
2773:
2769:
2763:
2759:
2755:
2752:
2749:
2746:
2739:
2736:
2733:
2729:
2724:
2720:
2680:
2677:
2674:
2670:
2646:
2645:
2634:
2630:
2626:
2623:
2619:
2615:
2611:
2607:
2600:
2596:
2588:
2585:
2582:
2578:
2573:
2569:
2555:
2554:
2543:
2539:
2536:
2533:
2528:
2524:
2520:
2516:
2511:
2507:
2503:
2500:
2497:
2494:
2491:
2488:
2485:
2480:
2477:
2474:
2470:
2466:
2462:
2458:
2415:
2412:
2408:
2407:
2396:
2391:
2387:
2383:
2380:
2375:
2371:
2367:
2364:
2359:
2355:
2351:
2337:
2336:
2323:
2319:
2315:
2312:
2307:
2303:
2299:
2296:
2291:
2287:
2283:
2269:
2268:
2257:
2253:
2249:
2245:
2241:
2237:
2233:
2229:
2225:
2206:
2203:
2201:
2198:
2175:
2171:
2165:
2161:
2124:
2119:
2115:
2111:
2108:
2105:
2100:
2096:
2092:
2089:
2086:
2081:
2077:
2061:
2050:
2039:
2032:
2006:
2003:
1963:Hilbert series
1934:graded modules
1907:
1904:
1901:
1898:
1895:
1892:
1889:
1876:is called the
1872:
1852:
1849:
1846:
1841:
1837:
1833:
1830:
1827:
1824:
1821:
1816:
1812:
1808:
1795:
1788:
1787:
1776:
1773:
1765:
1760:
1757:
1754:
1750:
1743:
1740:
1737:
1734:
1731:
1728:
1723:
1720:
1717:
1714:
1708:
1705:
1702:
1699:
1694:
1690:
1686:
1655:
1652:
1649:
1644:
1640:
1636:
1616:
1613:
1610:
1605:
1601:
1597:
1577:
1574:
1571:
1568:
1565:
1562:
1559:
1555:
1549:
1545:
1524:
1521:
1518:
1490:
1487:
1484:
1481:
1478:
1475:
1472:
1469:
1458:
1457:
1446:
1440:
1434:
1431:
1428:
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1399:
1391:
1387:
1381:
1376:
1373:
1370:
1366:
1362:
1359:
1356:
1353:
1348:
1344:
1340:
1317:
1314:
1311:
1306:
1302:
1298:
1276:
1272:
1260:
1259:
1248:
1243:
1239:
1233:
1229:
1223:
1218:
1215:
1212:
1208:
1204:
1201:
1198:
1195:
1192:
1166:
1163:
1160:
1157:
1154:
1139:
1138:
1124:
1121:
1118:
1115:
1112:
1109:
1104:
1101:
1098:
1095:
1092:
1089:
1086:
1083:
1080:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1041:
1035:
1029:
1026:
1023:
1018:
1015:
1012:
1009:
1006:
1000:
983:
982:
970:
966:
963:
958:
954:
947:
941:
938:
935:
930:
927:
924:
921:
918:
912:
906:
903:
900:
897:
894:
891:
888:
884:
880:
877:
874:
871:
868:
865:
862:
859:
854:
850:
846:
812:
795:
794:
783:
775:
771:
767:
764:
761:
758:
753:
750:
747:
744:
738:
735:
732:
729:
724:
720:
716:
687:
686:
675:
668:
660:
656:
651:
647:
644:
640:
634:
629:
626:
623:
619:
613:
610:
607:
604:
598:
595:
592:
589:
584:
580:
576:
551:
547:
543:
540:
537:
532:
528:
504:
503:
492:
487:
483:
479:
476:
473:
468:
464:
460:
455:
450:
447:
444:
440:
436:
433:
430:
427:
422:
418:
414:
389:
384:-vector space
370:
369:
356:
352:
348:
343:
339:
335:
332:
329:
324:
320:
316:
290:
287:
282:
278:
266:
265:
252:
248:
242:
239:
236:
232:
228:
225:
192:
189:
181:Hilbert scheme
168:
165:
162:
142:
139:
136:
133:
130:
100:
99:
84:
81:
33:Hilbert series
15:
13:
10:
9:
6:
4:
3:
2:
8005:
7994:
7991:
7989:
7986:
7985:
7983:
7971:
7967:
7962:
7957:
7953:
7952:
7947:
7943:
7940:
7936:
7932:
7926:
7921:
7916:
7912:
7908:
7904:
7900:
7896:
7892:
7888:
7884:
7882:0-387-94268-8
7878:
7874:
7870:
7866:
7862:
7858:
7854:
7848:
7844:
7840:
7839:
7834:
7830:
7829:
7824:
7811:
7810:
7805:
7799:
7796:
7789:
7785:
7782:
7780:
7777:
7775:
7772:
7771:
7767:
7751:
7743:
7724:
7701:
7693:
7689:
7681:
7677:
7670:
7665:
7655:
7651:
7647:
7637:
7618:
7610:
7606:
7602:
7592:
7569:
7561:
7557:
7553:
7544:
7540:
7533:
7528:
7518:
7514:
7510:
7505:
7501:
7497:
7487:
7465:
7458:
7457:
7456:
7437:
7433:
7429:
7424:
7420:
7396:
7375:
7363:
7361:
7341:
7330:
7326:
7304:
7291:
7288:
7280:
7276:
7267:
7262:
7260:
7256:
7237:
7219:
7215:
7193:
7180:
7177:
7171:
7168:
7162:
7148:
7115:
7111:
7083:
7079:
7075:
7067:
7065:
7063:
7059:
7058:HilbertSeries
7055:
7051:
7047:
7042:
7039:
7033:
7031:
7026:
7024:
7020:
7016:
7013:refining the
7012:
7008:
7004:
7003:Gröbner basis
7000:
6996:
6992:
6988:
6984:
6980:
6976:
6972:
6951:
6947:
6943:
6940:
6937:
6932:
6928:
6921:
6918:
6915:
6907:
6902:
6900:
6892:
6890:
6888:
6884:
6883:Gröbner basis
6864:
6856:
6848:
6845:
6842:
6830:
6826:
6821:
6817:
6814:
6811:
6804:
6800:
6795:
6788:
6782:
6774:
6770:
6766:
6759:
6758:
6757:
6743:
6738:
6734:
6730:
6727:
6724:
6719:
6715:
6705:
6699:
6682:
6677:
6669:
6666:
6663:
6656:
6652:
6649:
6643:
6635:
6631:
6627:
6604:
6596:
6592:
6588:
6568:
6563:
6559:
6533:
6529:
6525:
6522:
6519:
6514:
6510:
6503:
6500:
6497:
6474:
6468:
6458:
6454:
6449:
6445:
6440:
6437:
6434:
6426:
6423:
6415:
6410:
6407:
6404:
6400:
6396:
6390:
6382:
6378:
6374:
6367:
6366:
6365:
6362:
6360:
6356:
6338:
6334:
6310:
6307:
6301:
6293:
6289:
6282:
6274:
6270:
6263:
6256:
6255:
6254:
6252:
6248:
6245:has a graded
6243:
6239:
6234:
6225:
6223:
6220:
6214:
6194:
6185:
6182:
6179:
6171:
6168:
6165:
6150:
6146:
6141:
6137:
6134:
6131:
6128:
6125:
6122:
6116:
6106:
6102:
6097:
6093:
6090:
6087:
6084:
6081:
6078:
6069:
6061:
6053:
6050:
6047:
6032:
6028:
6023:
6019:
6016:
6010:
6000:
5996:
5991:
5987:
5984:
5975:
5969:
5959:
5955:
5950:
5946:
5939:
5938:
5937:
5920:
5913:
5903:
5900:
5897:
5893:
5888:
5884:
5874:
5870:
5865:
5861:
5858:
5852:
5846:
5836:
5832:
5827:
5823:
5816:
5815:
5814:
5797:
5793:
5783:
5779:
5768:
5765:
5762:
5758:
5748:
5744:
5739:
5726:
5722:
5713:
5710:
5707:
5703:
5694:
5687:
5686:
5685:
5671:
5663:
5659:
5655:
5652:
5649:
5644:
5640:
5632:
5628:
5625:
5620:
5616:
5595:
5590:
5586:
5582:
5579:
5576:
5571:
5567:
5541:
5537:
5533:
5530:
5527:
5522:
5518:
5511:
5508:
5505:
5496:
5477:
5473:
5469:
5466:
5463:
5458:
5454:
5444:
5442:
5438:
5430:
5428:
5424:
5407:
5399:
5396:
5395:
5394:
5377:
5374:
5350:
5322:
5310:
5307:
5306:
5305:
5288:
5282:
5274:
5270:
5266:
5262:
5256:
5252:
5248:
5245:
5241:
5237:
5231:
5220:
5213:
5209:
5205:
5201:
5194:
5193:
5192:
5175:
5172:
5163:
5156:
5152:
5146:
5139:
5122:
5115:
5108:
5101:
5100:
5099:
5081:
5061:
5052:
5050:
5040:
5035:
5029:
5021:, which is a
5020:
5019:Artinian ring
5002:
4998:
4988:
4986:
4967:
4961:
4955:
4952:
4946:
4936:
4932:
4927:
4923:
4917:
4914:
4911:
4905:
4899:
4889:
4885:
4880:
4876:
4869:
4868:
4867:
4850:
4847:
4839:
4835:
4826:
4822:
4814:
4811:
4806:
4802:
4787:
4783:
4776:
4769:
4768:
4767:
4753:
4748:
4744:
4723:
4720:
4715:
4711:
4687:
4684:
4679:
4675:
4667:
4661:
4657:
4653:
4648:
4644:
4635:
4617:
4613:
4592:
4587:
4583:
4574:
4558:
4555:
4550:
4546:
4523:
4519:
4496:
4492:
4466:
4463:
4460:
4456:
4452:
4449:
4446:
4441:
4437:
4429:
4425:
4422:
4417:
4413:
4403:
4382:
4376:
4353:
4347:
4344:
4341:
4333:
4327:
4321:
4315:
4307:
4303:
4299:
4293:
4285:
4282:
4279:
4273:
4267:
4254:
4251:
4248:
4244:
4240:
4237:
4234:
4229:
4225:
4217:
4213:
4209:
4205:
4198:
4197:
4196:
4182:
4179:
4176:
4173:
4170:
4167:
4164:
4161:
4138:
4135:
4124:
4120:
4116:
4113:
4110:
4105:
4101:
4093:
4089:
4078:
4075:
4072:
4068:
4064:
4061:
4058:
4053:
4049:
4041:
4037:
4028:
4024:
4006:
3998:
3989:
3986:
3983:
3979:
3975:
3972:
3969:
3964:
3960:
3952:
3948:
3944:
3936:
3929:
3928:
3927:
3923:
3904:
3900:
3896:
3893:
3890:
3885:
3881:
3873:
3865:
3849:
3830:minus one of
3829:
3816:
3809:
3807:
3791:
3787:
3778:
3774:
3770:
3767:
3764:
3759:
3755:
3748:
3745:
3742:
3713:
3709:
3705:
3702:
3699:
3694:
3690:
3686:
3681:
3677:
3670:
3667:
3664:
3657:
3649:
3644:
3642:
3638:
3628:
3611:
3608:
3600:
3594:
3582:
3576:
3573:
3567:
3560:
3556:is such that
3553:
3549:
3524:
3516:
3513:
3510:
3499:
3493:
3487:
3481:
3473:
3469:
3465:
3458:
3457:
3456:
3453:
3446:
3442:
3436:
3416:
3408:
3404:
3400:
3393:
3390:
3387:
3381:
3375:
3364:
3357:
3353:
3349:
3345:
3336:
3333:
3327:
3323:
3317:
3311:
3305:
3302:
3296:
3289:
3282:
3278:
3272:
3266:
3260:
3255:
3251:
3246:
3237:
3235:
3222:
3218:
3215:
3209:
3201:
3197:
3193:
3171:
3167:
3143:
3136:
3130:
3127:
3124:
3113:
3110:
3107:
3104:
3101:
3095:
3089:
3086:
3083:
3074:
3062:
3059:
3056:
3051:
3048:
3045:
3042:
3039:
3027:
3021:
3011:
3007:
3002:
2998:
2991:
2990:
2989:
2972:
2963:
2955:
2952:
2949:
2942:
2937:
2931:
2921:
2917:
2912:
2908:
2901:
2900:
2899:
2885:
2860:
2856:
2852:
2849:
2846:
2841:
2837:
2830:
2827:
2822:
2818:
2805:
2803:
2790:
2783:
2775:
2771:
2767:
2761:
2757:
2753:
2747:
2734:
2727:
2722:
2718:
2709:
2703:
2697:
2675:
2668:
2658:
2652:
2632:
2628:
2621:
2617:
2613:
2605:
2598:
2594:
2583:
2576:
2567:
2560:
2559:
2558:
2541:
2534:
2526:
2522:
2518:
2509:
2505:
2501:
2498:
2492:
2486:
2475:
2468:
2464:
2460:
2456:
2449:
2448:
2447:
2445:
2440:
2434:
2428:
2422:
2413:
2411:
2394:
2389:
2385:
2381:
2378:
2373:
2369:
2365:
2362:
2357:
2353:
2349:
2342:
2341:
2340:
2321:
2317:
2313:
2310:
2305:
2301:
2297:
2294:
2289:
2285:
2281:
2274:
2273:
2272:
2255:
2247:
2239:
2231:
2223:
2216:
2215:
2214:
2212:
2204:
2199:
2197:
2193:
2190:
2173:
2169:
2163:
2159:
2149:
2144:
2139:
2117:
2113:
2109:
2106:
2103:
2098:
2094:
2087:
2084:
2079:
2075:
2064:
2060:
2053:
2049:
2042:
2038:
2031:
2025:
2019:
2013:
2004:
2002:
1999:
1994:
1989:
1983:
1979:
1974:
1972:
1968:
1964:
1960:
1955:
1952:
1946:
1940:
1935:
1930:
1928:
1924:
1919:
1905:
1902:
1899:
1896:
1893:
1890:
1887:
1879:
1871:
1867:
1847:
1839:
1835:
1831:
1828:
1822:
1814:
1810:
1806:
1794:
1774:
1771:
1763:
1758:
1755:
1752:
1748:
1741:
1735:
1732:
1729:
1718:
1712:
1706:
1700:
1692:
1688:
1684:
1677:
1676:
1675:
1673:
1650:
1642:
1638:
1634:
1611:
1603:
1599:
1595:
1575:
1572:
1566:
1563:
1560:
1553:
1547:
1543:
1522:
1519:
1516:
1488:
1485:
1482:
1479:
1476:
1473:
1470:
1467:
1444:
1432:
1429:
1426:
1421:
1418:
1415:
1412:
1409:
1406:
1403:
1389:
1385:
1379:
1374:
1371:
1368:
1364:
1360:
1354:
1346:
1342:
1338:
1331:
1330:
1329:
1312:
1304:
1300:
1296:
1274:
1270:
1246:
1241:
1237:
1231:
1227:
1221:
1216:
1213:
1210:
1206:
1202:
1196:
1190:
1183:
1182:
1181:
1178:
1164:
1161:
1158:
1155:
1152:
1144:
1122:
1116:
1113:
1110:
1099:
1096:
1093:
1087:
1081:
1078:
1075:
1072:
1069:
1060:
1057:
1054:
1051:
1048:
1039:
1027:
1024:
1021:
1016:
1013:
1010:
1007:
1004:
988:
987:
986:
968:
964:
961:
956:
952:
939:
936:
933:
928:
925:
922:
919:
916:
904:
901:
898:
895:
892:
889:
886:
882:
875:
869:
866:
860:
852:
848:
844:
837:
836:
835:
832:
826:
810:
801:
781:
773:
765:
762:
759:
748:
742:
736:
730:
722:
718:
714:
707:
706:
705:
702:
696:
693:
673:
666:
658:
654:
649:
645:
642:
638:
632:
627:
624:
621:
617:
608:
602:
596:
590:
582:
578:
574:
567:
566:
565:
549:
545:
541:
538:
535:
530:
526:
516:
510:
490:
485:
481:
474:
466:
462:
458:
448:
445:
442:
438:
434:
428:
420:
416:
412:
405:
404:
403:
402:
401:formal series
398:
392:
388:
382:
376:
354:
350:
346:
341:
337:
330:
327:
322:
318:
314:
307:
306:
305:
302:
288:
285:
280:
276:
250:
246:
240:
237:
234:
230:
226:
223:
216:
215:
214:
211:
207:
202:
198:
190:
188:
186:
182:
166:
163:
160:
140:
134:
131:
128:
120:
115:
113:
109:
105:
97:
93:
92:maximal ideal
89:
85:
82:
79:
76:
72:
68:
67:
66:
63:
61:
57:
53:
49:
44:
42:
38:
34:
30:
26:
22:
7949:
7902:
7899:Schenck, Hal
7864:
7837:
7808:
7798:
7367:
7328:
7265:
7263:
7213:
7113:
7071:
7061:
7057:
7043:
7034:
7029:
7027:
7022:
7018:
7015:total degree
7006:
6998:
6994:
6990:
6986:
6982:
6978:
6974:
6970:
6905:
6903:
6896:
6879:
6703:
6697:
6489:
6363:
6355:free modules
6325:
6241:
6238:regular ring
6232:
6229:
6218:
6212:
6209:
5935:
5812:
5494:
5445:
5434:
5422:
5419:
5397:
5392:
5308:
5303:
5190:
5053:
5038:
5027:
4989:
4982:
4865:
4636:, which has
4573:affine space
4404:
4368:
4153:
3921:
3817:
3810:
3645:
3634:
3580:
3577:
3571:
3565:
3558:
3551:
3547:
3544:
3451:
3444:
3440:
3434:
3337:
3331:
3325:
3321:
3315:
3309:
3306:
3300:
3294:
3287:
3280:
3276:
3270:
3264:
3258:
3244:
3241:
3158:
2987:
2809:
2707:
2701:
2695:
2656:
2650:
2647:
2556:
2444:zero divisor
2438:
2432:
2426:
2420:
2417:
2409:
2338:
2270:
2208:
2194:
2188:
2147:
2137:
2062:
2058:
2051:
2047:
2040:
2036:
2029:
2023:
2017:
2011:
2008:
1997:
1987:
1981:
1975:
1966:
1962:
1958:
1956:
1950:
1944:
1938:
1931:
1927:Schenck 2003
1920:
1877:
1869:
1865:
1792:
1789:
1671:
1459:
1261:
1179:
1140:
984:
833:
799:
796:
700:
697:
691:
688:
514:
508:
505:
390:
386:
380:
374:
371:
303:
267:
209:
200:
194:
116:
101:
64:
45:
32:
28:
24:
18:
7833:Harris, Joe
7784:Quot scheme
7255:Serre twist
6353:are graded
5191:shows that
4605:This makes
3335:minus one.
185:Quot scheme
7982:Categories
7825:References
7804:Ravi Vakil
6326:where the
3285:such that
2205:Additivity
1799:such that
1790:The least
1509:of degree
88:local ring
31:, and the
7915:CiteSeerX
7907:Cambridge
7790:Citations
7749:→
7732:→
7648:−
7619:⊕
7603:−
7554:−
7511:−
7498:−
7469:→
7345:~
7172:χ
6941:…
6908:a field,
6904:Thus let
6846:−
6827:δ
6815:⋯
6801:δ
6735:δ
6728:…
6716:δ
6667:−
6523:…
6438:−
6424:−
6401:∑
6305:→
6299:→
6286:→
6283:⋯
6280:→
6267:→
6183:−
6169:−
6147:δ
6135:⋯
6117:⋯
6103:δ
6091:⋯
6051:−
6029:δ
6020:−
6011:⋯
5997:δ
5988:−
5901:−
5871:δ
5862:−
5790:→
5775:→
5766:−
5723:δ
5711:−
5699:→
5669:⟩
5653:…
5637:⟨
5587:δ
5580:…
5568:δ
5531:…
5467:…
5375:δ
5323:δ
5257:δ
5249:−
5224:⟩
5218:⟨
5170:⟶
5167:⟩
5161:⟨
5150:⟶
5135:⟶
5123:δ
5112:⟶
5082:δ
4915:−
4845:⟶
4832:⟶
4812:−
4797:⟶
4780:⟶
4721:−
4691:⟩
4685:−
4672:⟨
4472:⟩
4464:−
4450:…
4434:⟨
4405:The ring
4345:−
4283:−
4260:⟩
4252:−
4238:…
4222:⟨
4174:…
4133:⟶
4130:⟩
4114:…
4098:⟨
4087:⟶
4084:⟩
4076:−
4062:…
4046:⟨
4019:⟶
3995:⟩
3987:−
3973:…
3957:⟨
3940:⟶
3894:…
3768:…
3703:…
3668:⊂
3514:−
3391:−
3254:nilpotent
3128:−
3111:−
3096:⋯
3060:−
3049:−
2953:−
2850:…
2625:→
2610:→
2572:→
2502:−
2252:→
2244:→
2236:→
2228:→
2107:…
1900:δ
1897:−
1891:
1756:−
1753:δ
1733:−
1730:δ
1564:−
1561:δ
1520:−
1517:δ
1480:δ
1477:−
1471:≥
1430:−
1427:δ
1419:−
1416:δ
1407:−
1365:∑
1207:∑
1162:δ
1159:−
1114:−
1111:δ
1088:⋯
1079:−
1076:δ
1058:−
1055:δ
1025:−
1022:δ
1014:−
1011:δ
965:⋯
937:−
934:δ
926:−
923:δ
902:⋯
893:δ
811:δ
774:δ
763:−
646:−
618:∏
539:…
454:∞
439:∑
347:
334:⟼
268:and that
238:≥
231:⨁
164:∈
138:→
129:π
7901:(2003),
7863:(1995),
7835:(1992).
7806:(2015).
7768:See also
7666:→
7529:→
7212:, where
5740:→
5608:Setting
3731:, where
2595:→
1965:and the
1328:is thus
7970:0485835
7939:0011360
7891:1322960
7216:is the
5398:Theorem
5309:Theorem
3870:, of a
3561:(1) â 0
2035:, ...,
1141:is the
823:is the
204:over a
104:Hilbert
52:modules
7968:
7937:
7927:
7917:
7889:
7879:
7849:
7009:for a
6977:. Let
5032:, and
5017:is an
4369:where
2660:, and
2143:kernel
2141:. Its
1961:, the
985:where
797:where
689:where
73:of a
27:, the
23:, the
7813:(PDF)
7327:. If
7108:is a
7054:Magma
7050:Maple
7001:be a
6985:. If
6899:above
6620:from
5311:- If
4990:Thus
3298:as a
3262:as a
2339:and
2135:onto
2056:onto
1969:of a
1936:over
206:field
110:of a
71:ideal
58:over
41:field
35:of a
7925:ISBN
7877:ISBN
7847:ISBN
7060:and
7052:and
5446:Let
4154:for
3563:and
2418:Let
2186:and
1957:The
1863:for
1666:for
1460:For
1180:If
1156:>
1145:for
183:and
102:The
78:ring
7956:doi
7869:doi
7323:by
7080:by
7072:In
7005:of
6490:If
5492:be
5425:â 1
5367:by
5041:(1)
5030:(1)
4632:an
3924:+ 1
3919:of
3822:of
3455:is
3326:(f)
3290:(1)
2878:in
2436:in
2021:by
1995:of
1985:in
1888:deg
1289:in
827:of
698:If
506:If
338:dim
19:In
7984::
7966:MR
7964:,
7935:MR
7933:,
7923:,
7913:,
7909::
7905:,
7887:MR
7885:,
7875:,
7845:.
7841:.
7261:.
7253:a
7064:.
7032:.
7025:.
5414:dÎŽ
5051:.
4559:0.
4402:.
3643:.
3627:.
3575:.
2192:.
2001:.
1918:.
1868:â„
831:.
301:.
187:.
114:.
62:.
7973:.
7958::
7894:.
7871::
7855:.
7815:.
7752:0
7744:X
7738:O
7725:n
7720:P
7712:O
7702:]
7694:2
7690:f
7682:1
7678:f
7671:[
7661:)
7656:2
7652:d
7645:(
7638:n
7633:P
7625:O
7616:)
7611:1
7607:d
7600:(
7593:n
7588:P
7580:O
7570:]
7562:1
7558:f
7545:2
7541:f
7534:[
7524:)
7519:2
7515:d
7506:1
7502:d
7495:(
7488:n
7483:P
7475:O
7466:0
7443:)
7438:2
7434:d
7430:,
7425:1
7421:d
7417:(
7397:X
7376:X
7342:M
7329:M
7311:)
7308:)
7305:m
7302:(
7297:F
7292:,
7289:X
7286:(
7281:0
7277:H
7266:m
7241:)
7238:m
7235:(
7230:F
7214:Ï
7200:)
7197:)
7194:m
7191:(
7186:F
7181:,
7178:X
7175:(
7169:=
7166:)
7163:m
7160:(
7154:F
7149:p
7126:F
7114:X
7094:F
7023:B
7019:G
7007:I
6999:B
6995:I
6993:=
6991:H
6987:I
6983:I
6979:H
6975:R
6971:I
6957:]
6952:n
6948:x
6944:,
6938:,
6933:1
6929:x
6925:[
6922:K
6919:=
6916:R
6906:K
6865:.
6857:n
6853:)
6849:t
6843:1
6840:(
6831:h
6822:t
6818:+
6812:+
6805:1
6796:t
6789:=
6786:)
6783:t
6780:(
6775:L
6771:S
6767:H
6744:,
6739:h
6731:,
6725:,
6720:1
6704:h
6698:L
6683:.
6678:n
6674:)
6670:t
6664:1
6661:(
6657:/
6653:1
6650:=
6647:)
6644:t
6641:(
6636:R
6632:S
6628:H
6608:)
6605:t
6602:(
6597:M
6593:S
6589:H
6569:,
6564:i
6560:L
6539:]
6534:n
6530:x
6526:,
6520:,
6515:1
6511:x
6507:[
6504:k
6501:=
6498:R
6475:.
6472:)
6469:t
6466:(
6459:i
6455:L
6450:S
6446:H
6441:1
6435:i
6431:)
6427:1
6421:(
6416:k
6411:1
6408:=
6405:i
6397:=
6394:)
6391:t
6388:(
6383:M
6379:S
6375:H
6339:i
6335:L
6311:,
6308:0
6302:M
6294:1
6290:L
6275:k
6271:L
6264:0
6242:R
6233:M
6219:k
6213:k
6195:.
6186:k
6180:n
6176:)
6172:t
6166:1
6163:(
6158:)
6151:k
6142:t
6138:+
6132:+
6129:t
6126:+
6123:1
6120:(
6114:)
6107:1
6098:t
6094:+
6088:+
6085:t
6082:+
6079:1
6076:(
6070:=
6062:n
6058:)
6054:t
6048:1
6045:(
6040:)
6033:k
6024:t
6017:1
6014:(
6008:)
6001:1
5992:t
5985:1
5982:(
5976:=
5973:)
5970:t
5967:(
5960:k
5956:R
5951:S
5947:H
5921:.
5917:)
5914:t
5911:(
5904:1
5898:i
5894:R
5889:S
5885:H
5882:)
5875:i
5866:t
5859:1
5856:(
5853:=
5850:)
5847:t
5844:(
5837:i
5833:R
5828:S
5824:H
5798:.
5794:0
5784:i
5780:R
5769:1
5763:i
5759:R
5749:i
5745:f
5732:]
5727:i
5719:[
5714:1
5708:i
5704:R
5695:0
5672:,
5664:i
5660:f
5656:,
5650:,
5645:1
5641:f
5633:/
5629:R
5626:=
5621:i
5617:R
5596:.
5591:k
5583:,
5577:,
5572:1
5547:]
5542:n
5538:x
5534:,
5528:,
5523:1
5519:x
5515:[
5512:K
5509:=
5506:R
5495:k
5478:k
5474:f
5470:,
5464:,
5459:1
5455:f
5423:n
5416:.
5410:ÎŽ
5402:d
5378:.
5365:V
5351:f
5341:V
5337:R
5313:f
5289:.
5286:)
5283:t
5280:(
5275:R
5271:S
5267:H
5263:)
5253:t
5246:1
5242:(
5238:=
5235:)
5232:t
5229:(
5221:f
5214:/
5210:R
5206:S
5202:H
5176:,
5173:0
5164:f
5157:/
5153:R
5147:R
5140:f
5126:]
5120:[
5116:R
5109:0
5096:R
5062:f
5045:V
5039:P
5028:P
5023:k
5003:0
4999:R
4968:.
4965:)
4962:t
4959:(
4956:P
4953:=
4950:)
4947:t
4944:(
4937:1
4933:R
4928:S
4924:H
4921:)
4918:t
4912:1
4909:(
4906:=
4903:)
4900:t
4897:(
4890:0
4886:R
4881:S
4877:H
4851:,
4848:0
4840:0
4836:R
4827:1
4823:R
4815:1
4807:d
4803:h
4788:1
4784:R
4777:0
4754:,
4749:1
4745:R
4724:1
4716:d
4712:h
4688:1
4680:d
4676:h
4668:/
4662:1
4658:R
4654:=
4649:0
4645:R
4618:0
4614:V
4593:.
4588:0
4584:V
4556:=
4551:d
4547:h
4524:d
4520:h
4497:0
4493:V
4467:1
4461:d
4457:h
4453:,
4447:,
4442:0
4438:h
4430:/
4426:R
4423:=
4418:1
4414:R
4400:R
4386:)
4383:t
4380:(
4377:P
4354:,
4348:t
4342:1
4337:)
4334:t
4331:(
4328:P
4322:=
4319:)
4316:t
4313:(
4308:R
4304:S
4300:H
4294:d
4290:)
4286:t
4280:1
4277:(
4274:=
4271:)
4268:t
4265:(
4255:1
4249:d
4245:h
4241:,
4235:,
4230:0
4226:h
4218:/
4214:R
4210:S
4206:H
4183:.
4180:d
4177:,
4171:,
4168:0
4165:=
4162:k
4139:,
4136:0
4125:k
4121:h
4117:,
4111:,
4106:1
4102:h
4094:/
4090:R
4079:1
4073:k
4069:h
4065:,
4059:,
4054:1
4050:h
4042:/
4038:R
4029:k
4025:h
4010:]
4007:1
4004:[
3999:)
3990:1
3984:k
3980:h
3976:,
3970:,
3965:0
3961:h
3953:/
3949:R
3945:(
3937:0
3922:d
3905:d
3901:h
3897:,
3891:,
3886:0
3882:h
3868:R
3850:d
3840:V
3836:V
3832:R
3824:V
3820:d
3813:k
3792:I
3788:/
3784:]
3779:n
3775:x
3771:,
3765:,
3760:0
3756:x
3752:[
3749:k
3746:=
3743:R
3733:k
3719:]
3714:n
3710:x
3706:,
3700:,
3695:1
3691:x
3687:,
3682:0
3678:x
3674:[
3671:k
3665:I
3652:V
3612:!
3609:d
3604:)
3601:1
3598:(
3595:P
3581:d
3572:A
3566:d
3559:P
3554:)
3552:t
3550:(
3548:P
3525:d
3521:)
3517:t
3511:1
3508:(
3503:)
3500:t
3497:(
3494:P
3488:=
3485:)
3482:t
3479:(
3474:A
3470:S
3466:H
3452:A
3447:)
3445:t
3443:(
3441:P
3435:A
3420:)
3417:t
3414:(
3409:A
3405:S
3401:H
3397:)
3394:t
3388:1
3385:(
3382:=
3379:)
3376:t
3373:(
3368:)
3365:f
3362:(
3358:/
3354:A
3350:S
3346:H
3332:A
3324:/
3322:A
3316:f
3310:A
3301:K
3295:A
3288:P
3283:)
3281:t
3279:(
3277:P
3271:A
3265:K
3259:A
3245:A
3223:.
3219:1
3216:=
3213:)
3210:t
3207:(
3202:K
3198:S
3194:H
3172:n
3168:x
3144:.
3137:!
3134:)
3131:1
3125:n
3122:(
3117:)
3114:1
3108:n
3105:+
3102:k
3099:(
3093:)
3090:1
3087:+
3084:k
3081:(
3075:=
3069:)
3063:1
3057:n
3052:1
3046:n
3043:+
3040:k
3034:(
3028:=
3025:)
3022:k
3019:(
3012:n
3008:R
3003:P
2999:H
2973:.
2964:n
2960:)
2956:t
2950:1
2947:(
2943:1
2938:=
2935:)
2932:t
2929:(
2922:n
2918:R
2913:S
2909:H
2886:n
2866:]
2861:n
2857:x
2853:,
2847:,
2842:1
2838:x
2834:[
2831:K
2828:=
2823:n
2819:R
2791:.
2787:)
2784:t
2781:(
2776:A
2772:S
2768:H
2762:d
2758:t
2754:=
2751:)
2748:t
2745:(
2738:]
2735:d
2732:[
2728:A
2723:S
2719:H
2708:f
2702:d
2696:A
2679:]
2676:d
2673:[
2669:A
2657:f
2651:f
2633:,
2629:0
2622:f
2618:/
2614:A
2606:A
2599:f
2587:]
2584:d
2581:[
2577:A
2568:0
2542:.
2538:)
2535:t
2532:(
2527:A
2523:S
2519:H
2515:)
2510:d
2506:t
2499:1
2496:(
2493:=
2490:)
2487:t
2484:(
2479:)
2476:f
2473:(
2469:/
2465:A
2461:S
2457:H
2439:A
2433:d
2427:f
2421:A
2395:.
2390:C
2386:P
2382:H
2379:+
2374:A
2370:P
2366:H
2363:=
2358:B
2354:P
2350:H
2322:C
2318:S
2314:H
2311:+
2306:A
2302:S
2298:H
2295:=
2290:B
2286:S
2282:H
2256:0
2248:C
2240:B
2232:A
2224:0
2189:S
2174:I
2170:/
2164:n
2160:R
2148:I
2138:S
2123:]
2118:n
2114:X
2110:,
2104:,
2099:1
2095:X
2091:[
2088:K
2085:=
2080:n
2076:R
2063:i
2059:g
2052:i
2048:X
2041:n
2037:g
2033:1
2030:g
2024:n
2018:K
2012:S
1998:V
1988:P
1982:V
1951:m
1945:t
1939:S
1906:1
1903:+
1894:P
1873:0
1870:n
1866:n
1851:)
1848:n
1845:(
1840:S
1836:F
1832:H
1829:=
1826:)
1823:n
1820:(
1815:S
1811:P
1807:H
1796:0
1793:n
1775:.
1772:n
1764:+
1759:1
1749:n
1742:!
1739:)
1736:1
1727:(
1722:)
1719:1
1716:(
1713:P
1707:=
1704:)
1701:n
1698:(
1693:S
1689:P
1685:H
1668:n
1654:)
1651:n
1648:(
1643:S
1639:F
1635:H
1615:)
1612:n
1609:(
1604:S
1600:P
1596:H
1576:.
1573:!
1570:)
1567:1
1558:(
1554:/
1548:i
1544:a
1523:1
1507:n
1503:i
1489:,
1486:1
1483:+
1474:i
1468:n
1445:.
1439:)
1433:1
1422:1
1413:+
1410:i
1404:n
1398:(
1390:i
1386:a
1380:d
1375:0
1372:=
1369:i
1361:=
1358:)
1355:n
1352:(
1347:S
1343:F
1339:H
1316:)
1313:t
1310:(
1305:S
1301:S
1297:H
1275:n
1271:t
1247:,
1242:i
1238:t
1232:i
1228:a
1222:d
1217:0
1214:=
1211:i
1203:=
1200:)
1197:t
1194:(
1191:P
1165:,
1153:n
1123:!
1120:)
1117:1
1108:(
1103:)
1100:1
1097:+
1094:n
1091:(
1085:)
1082:2
1073:+
1070:n
1067:(
1064:)
1061:1
1052:+
1049:n
1046:(
1040:=
1034:)
1028:1
1017:1
1008:+
1005:n
999:(
969:)
962:+
957:n
953:t
946:)
940:1
929:1
920:+
917:n
911:(
905:+
899:+
896:t
890:+
887:1
883:(
879:)
876:t
873:(
870:P
867:=
864:)
861:t
858:(
853:S
849:S
845:H
829:S
800:P
782:,
770:)
766:t
760:1
757:(
752:)
749:t
746:(
743:P
737:=
734:)
731:t
728:(
723:S
719:S
715:H
701:S
692:Q
674:,
667:)
659:i
655:d
650:t
643:1
639:(
633:h
628:1
625:=
622:i
612:)
609:t
606:(
603:Q
597:=
594:)
591:t
588:(
583:S
579:S
575:H
550:h
546:d
542:,
536:,
531:1
527:d
515:h
509:S
491:.
486:n
482:t
478:)
475:n
472:(
467:S
463:F
459:H
449:0
446:=
443:n
435:=
432:)
429:t
426:(
421:S
417:S
413:H
391:n
387:S
381:K
375:n
355:n
351:S
342:K
331:n
328::
323:S
319:F
315:H
289:K
286:=
281:0
277:S
251:i
247:S
241:0
235:i
227:=
224:S
210:K
201:S
167:S
161:s
141:S
135:X
132::
98:.
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