4977:
7862:
predicts that these groups are finitely generated. Moreover, the rank of the group of cycles modulo homological equivalence, and also of the group of cycles homologically equivalent to zero, should be equal to the order of vanishing of an L-function of the given variety at certain integer points.
4633:
7184:
7803:
1864:
6590:
1584:
Rationally equivalent cycles defined by hypersurfaces are easy to construct on projective space because they can all be constructed as the vanishing loci of the same vector bundle. For example, given two homogeneous polynomials of degree
5129:
4785:
6122:
7432:
4712:
7550:), these homomorphisms form a ring homomorphism from the Chow ring to the cohomology ring. Intuitively, this is because the products in both the Chow ring and the cohomology ring describe the intersection of cycles.
7537:
6744:
3654:
2367:
6857:
4459:
4385:
2902:
1679:
7696:
617:
1719:
7098:
2265:
5925:
1915:
6325:
5473:
8128:
gave the current standard foundation for Chow groups, working with singular varieties wherever possible. In their theory, the intersection product for smooth varieties is constructed by
2473:
798:
719:
7019:
5583:
6471:
5675:
6389:
2758:
7225:
6903:
4431:
3941:
2188:
7090:
6455:
6426:
4741:
4020:
3839:
3536:
4272:
2415:
2115:
2007:
6628:
3129:
2153:
1477:
6780:
2797:
2604:
1141:
6228:
6183:
5797:
5373:
4777:
1221:
961:
303:
6923:
4086:
3436:
3155:
7319:
7056:
6151:
4198:
3770:
3971:
2317:
1711:
1019:
5278:
5181:
4972:{\displaystyle CH^{\bullet }(F_{a})\cong {\frac {CH^{\bullet }(\mathbb {P} ^{1})}{(\zeta ^{2}+aH\zeta )}}\cong {\frac {\mathbf {Z} }{(H^{2},\zeta ^{2}+aH\zeta )}}}
5060:
3181:
987:
460:
4156:
4109:
7245:
6248:
6012:
5992:
5972:
5952:
5837:
5817:
5759:
5739:
5715:
5695:
5643:
5623:
5603:
5536:
5516:
5493:
5393:
5322:
5302:
5245:
5225:
5201:
5152:
5049:
5029:
5005:
4451:
4312:
4292:
4238:
4218:
4129:
4060:
4040:
3991:
3879:
3859:
3810:
3790:
3740:
3720:
3697:
3677:
3556:
3456:
3402:
3382:
3362:
3342:
3283:
3263:
3243:
3045:
3025:
3005:
2985:
2965:
2945:
2925:
2688:
2668:
2648:
2628:
2565:
2545:
2521:
2501:
2285:
2208:
2071:
2027:
1603:
1569:
1541:
1521:
1501:
1432:
1412:
1385:
1365:
1345:
1307:
1287:
1267:
1181:
1161:
1099:
1079:
1059:
1039:
921:
901:
881:
858:
838:
818:
759:
739:
680:
660:
640:
547:
527:
503:
483:
434:
411:
391:
367:
343:
323:
267:
244:
220:
196:
172:
152:
132:
101:
3491:
3318:
3223:
3084:
1961:
1247:
8116:
gave an influential proof that the intersection product is well-defined on cycles modulo rational equivalence for a smooth quasi-projective variety, using
6023:
8666:
7350:
7659:), with the same formal properties as in topology. The Chern classes give a close connection between vector bundles and Chow groups. Namely, let
7926:(tensored with the rationals) with strong properties. The conjecture would imply a tight connection between the singular or etale cohomology of
4650:
7463:
8076:. The key advantage of this extension is that it is easier to form quotients in the latter category and thus it is more natural to consider
6636:
4628:{\displaystyle CH^{\bullet }(\mathbb {P} (E))\cong {\frac {CH^{\bullet }(X)}{\zeta ^{r}+c_{1}\zeta ^{r-1}+c_{2}\zeta ^{r-2}+\cdots +c_{r}}}}
8218:
Bloch, Algebraic cycles and higher K-groups; Voevodsky, Triangulated categories of motives over a field, section 2.2 and
Proposition 4.2.9.
8019:" and more generally a bivariant theory associated to any morphism of schemes. A bivariant theory is a pair of covariant and contravariant
7886:) of the cycle map from the Chow groups to singular cohomology. For a smooth projective variety over a finitely generated field (such as a
3564:
2322:
6785:
4317:
2809:
1608:
8671:
8146:
7850:
over a number field. It is an open problem whether all Chow groups are finitely generated for every variety over a number field. The
7679:
111:
7798:{\displaystyle K_{0}(X)\otimes _{\mathbf {Z} }\mathbf {Q} \cong \prod _{i}{\mathit {CH}}^{i}(X)\otimes _{\mathbf {Z} }\mathbf {Q} .}
1859:{\displaystyle X={\text{Proj}}\left({\frac {\mathbb {C} }{(sf+tg)}}\right)\hookrightarrow \mathbb {P} ^{1}\times \mathbb {P} ^{n}}
555:
8097:
463:
71:. The Chow groups carry rich information about an algebraic variety, and they are correspondingly hard to compute in general.
7179:{\displaystyle {\begin{matrix}S'\times _{S}X&\to &X\\\downarrow &&\downarrow \\S'&\to &S\end{matrix}}}
8630:
8533:
8500:
7820:
Some of the deepest conjectures in algebraic geometry and number theory are attempts to understand Chow groups. For example:
104:
59:) in a similar way to how simplicial or cellular homology groups are formed out of subcomplexes. When the variety is
8597:
8315:
8125:
8012:
3409:
2221:
5847:
1872:
8553:(1932), "La serie canonica e la teoria delle serie principali di gruppi di punti sopra una superficie algebrica",
6253:
5401:
7986:
should be finite-dimensional; more precisely, it should map isomorphically to the group of complex points of the
6585:{\displaystyle f:\operatorname {Spec} \left({\frac {\mathbb {C} }{(f(x)-g(x,y))}}\right)\to \mathbb {A} _{x}^{1}}
764:
685:
8042:
This is in a sense the most elementary extension of the Chow ring to singular varieties; other theories such as
8517:
8121:
8008:
6931:
5541:
3700:
3405:
8039:. The name "bivariant" refers to the fact that the theory contains both covariant and contravariant functors.
5648:
8307:
7859:
7809:
6399:
Note that non-examples can be constructed using blowups; for example, if we take the blowup of the origin in
6185:. The localization sequence can be extended to the left using a generalization of Chow groups, (Borel–Moore)
8084:, which has been constructed only in some special case and which is needed in particular to make sense of a
6330:
2287:, then the vanishing loci of a generic section of both line bundles defines non-equivalent cycle classes in
8661:
4241:
3048:
2693:
506:
8656:
8156:
8108:
in the theory of algebraic curves. For higher-codimension cycles, rational equivalence was introduced by
6862:
5718:
4390:
3887:
2158:
8622:
8589:
7616:, this homomorphism can be identified with a ring homomorphism from the Chow ring to etale cohomology.
4132:
3679:
is the class of a hyperplane (the zero locus of a single linear function). Furthermore, any subvariety
2420:
6431:
6402:
4717:
3996:
3815:
3512:
8085:
4246:
2372:
7192:
8331:
7825:
7341:
3265:
are subvarieties of complementary dimension (meaning that their dimensions sum to the dimension of
2076:
1966:
6598:
8379:
7064:
3089:
2120:
1437:
6752:
5304:, and some of the deepest problems in number theory are attempts to understand this group. When
8117:
8077:
7683:
5281:
2763:
2570:
1107:
48:
8236:
Voisin, Hodge Theory and
Complex Algebraic Geometry, v. 1, section 12.3.3; v. 2, Theorem 9.24.
6201:
6156:
5770:
5346:
4746:
4647:
can be readily computed using the projective bundle formula. Recall that it is constructed as
4159:
1543:(or more generally, a locally Noetherian normal factorial scheme ), this is isomorphic to the
1190:
930:
272:
8448:
8129:
8081:
8054:
8016:
7566:
6908:
4065:
3415:
3134:
7970:
is "infinite-dimensional" (not the image of any finite-dimensional family of zero-cycles on
7253:
7035:
6130:
5124:{\displaystyle 0\rightarrow X(k)\rightarrow CH_{0}(X)\rightarrow \mathbf {Z} \rightarrow 0.}
4177:
3745:
8640:
8607:
8543:
8510:
8477:
8410:
8072:
The theory of Chow groups of schemes of finite type over a field extends easily to that of
7570:
7557:, the cycle map from the Chow ring to ordinary cohomology factors through a richer theory,
4163:
3949:
2290:
1684:
992:
107:
68:
8574:
5254:
5157:
8:
8141:
8024:
7919:
7562:
3321:
3187:
3160:
1480:
966:
439:
44:
7454:
4138:
4091:
64:
8581:
8043:
8028:
7671:
7558:
7554:
7230:
6233:
6191:
6186:
5997:
5977:
5957:
5937:
5841:
5822:
5802:
5744:
5724:
5700:
5680:
5628:
5608:
5588:
5521:
5501:
5478:
5378:
5307:
5287:
5230:
5210:
5186:
5137:
5034:
5014:
4990:
4644:
4436:
4297:
4277:
4223:
4203:
4114:
4045:
4025:
3976:
3864:
3844:
3795:
3775:
3725:
3705:
3682:
3662:
3541:
3441:
3387:
3367:
3347:
3327:
3268:
3248:
3228:
3030:
3010:
2990:
2970:
2950:
2930:
2910:
2673:
2653:
2633:
2613:
2550:
2530:
2524:
2506:
2486:
2270:
2193:
2032:
2012:
1588:
1554:
1526:
1506:
1486:
1417:
1397:
1370:
1350:
1312:
1292:
1272:
1252:
1166:
1146:
1084:
1064:
1044:
1024:
906:
886:
866:
843:
823:
803:
744:
724:
665:
645:
625:
532:
512:
488:
468:
419:
396:
376:
370:
352:
328:
308:
252:
229:
205:
199:
181:
157:
137:
117:
86:
20:
4987:
For other algebraic varieties, Chow groups can have richer behavior. For example, let
3461:
3288:
3193:
3054:
1920:
1226:
8626:
8593:
8529:
8496:
8465:
8398:
8393:
8311:
8101:
8066:
8032:
7904:
7808:
This isomorphism shows the importance of rational equivalence, compared to any other
52:
40:
8570:
8562:
8550:
8457:
8388:
8151:
8109:
8105:
7987:
7875:
7605:
6117:{\displaystyle CH_{i}(Z)\rightarrow CH_{i}(X)\rightarrow CH_{i}(X-Z)\rightarrow 0,}
5496:
3507:
1548:
56:
32:
8431:
8418:
6127:
where the first homomorphism is the pushforward associated to the proper morphism
8636:
8603:
8539:
8525:
8506:
8492:
8473:
8406:
8301:
8073:
8062:
8036:
7952:
7891:
7864:
7687:
5341:
2009:. This can be used to show that the cycle class of every hypersurface of degree
8484:
8443:
8113:
7879:
6015:
5325:
5204:
5052:
5008:
3320:
is equal to the sum of the points of the intersection with coefficients called
1681:, we can construct a family of hypersurfaces defined as the vanishing locus of
1184:
247:
28:
1223:
by the subgroup of cycles rationally equivalent to zero. Sometimes one writes
8650:
8614:
8469:
8402:
8370:
7963:
7915:
7851:
7625:
7427:{\displaystyle {\mathit {CH}}_{i}(X)\rightarrow H_{2i}^{BM}(X,\mathbf {Z} ).}
5765:
60:
8263:
Voisin, Hodge Theory and
Complex Algebraic Geometry, v. 2, Conjecture 11.21.
8189:
8035:, which is a contravariant functor that assigns to a space a ring, namely a
6153:, and the second homomorphism is pullback with respect to the flat morphism
3412:'s intersection theory constructs a canonical element of the Chow groups of
226:
coefficients. (Here and below, subvarieties are understood to be closed in
7887:
5248:
1544:
55:. The elements of the Chow group are formed out of subvarieties (so-called
7453:
is smooth over the complex numbers, this cycle map can be rewritten using
4707:{\displaystyle F_{a}=\mathbb {P} ({\mathcal {O}}\oplus {\mathcal {O}}(a))}
2117:
can be used to establish a rational equivalence. Notice that the locus of
7943:
be a smooth complex projective surface. The Chow group of zero-cycles on
7855:
7636:
5329:
2800:
2607:
8272:
Voisin, Hodge Theory and
Complex Algebraic Geometry, v. 2, Theorem 10.1.
7532:{\displaystyle {\mathit {CH}}^{j}(X)\rightarrow H^{2j}(X,\mathbf {Z} ).}
8566:
8100:) was studied in various forms during the 19th century, leading to the
2907:
The product arises from intersecting algebraic cycles. For example, if
7340:
over the complex numbers, there is a homomorphism from Chow groups to
6739:{\displaystyle g(\alpha ,y)=(y-a_{1})^{e_{1}}\cdots (y-a_{k})^{e_{k}}}
7974:). The Bloch–Beilinson conjecture would imply a satisfying converse,
3649:{\displaystyle CH^{*}(\mathbb {P} ^{n})\cong \mathbf {Z} /(H^{n+1}),}
8461:
8446:(1956), "On equivalence classes of cycles in an algebraic variety",
7604:, there is an analogous cycle map from Chow groups to (Borel–Moore)
2362:{\displaystyle \operatorname {Div} (C)\cong \operatorname {Pic} (C)}
6852:{\displaystyle \{\alpha _{1},\ldots ,\alpha _{k}\}=f^{-1}(\alpha )}
840:.) The definition of the order of vanishing requires some care for
63:, the Chow groups can be interpreted as cohomology groups (compare
8281:
Voisin, Hodge Theory and
Complex Algebraic Geometry, v. 2, Ch. 11.
4743:. Then, the only non-trivial Chern class of this vector bundle is
8080:
of algebraic spaces. A much more formidable extension is that of
8020:
5328:, the example of an elliptic curve shows that Chow groups can be
4380:{\displaystyle \zeta =c_{1}({\mathcal {O}}_{\mathbb {P} (E)}(1))}
2897:{\displaystyle CH^{i}(X)\times CH^{j}(X)\rightarrow CH^{i+j}(X).}
223:
7581:) maps isomorphically to Deligne cohomology, but that fails for
1674:{\displaystyle f,g\in H^{0}(\mathbb {P} ^{n},{\mathcal {O}}(d))}
8015:
extended the Chow ring to singular varieties by defining the "
3131:
is the sum of the irreducible components of the intersection
2213:
8584:(2000), "Triangulated categories of motives over a field",
8254:
Fulton, Intersection Theory, section 3.2 and
Example 8.3.3.
3404:, with no assumption on the dimension of the intersection,
1579:
8179:
Fulton. Intersection Theory, section 1.2 and
Appendix A.3.
612:{\displaystyle (f)=\sum _{Z}\operatorname {ord} _{Z}(f)Z,}
7755:
7473:
7360:
3881:, respectively, their product in the Chow ring is simply
3190:
constructs an explicit cycle that represents the product
8357:
3264 and All That: A Second Course in
Algebraic Geometry
7947:
maps onto the integers by the degree homomorphism; let
7922:
conjecture predicts a filtration on the Chow groups of
74:
8303:
Categorical
Framework for the Study of Singular Spaces
7103:
8619:
Hodge Theory and
Complex Algebraic Geometry (2 vols.)
8057:
are an amalgamation of Chow groups of varieties over
7863:
Finiteness of these ranks would also follow from the
7699:
7466:
7353:
7256:
7233:
7195:
7101:
7067:
7038:
6934:
6911:
6865:
6788:
6755:
6639:
6601:
6474:
6434:
6405:
6333:
6256:
6236:
6204:
6159:
6133:
6026:
6000:
5980:
5960:
5940:
5850:
5825:
5805:
5773:
5747:
5727:
5703:
5683:
5651:
5631:
5611:
5591:
5544:
5524:
5504:
5481:
5404:
5381:
5349:
5310:
5290:
5257:
5233:
5213:
5189:
5160:
5140:
5063:
5037:
5017:
4993:
4788:
4749:
4720:
4653:
4462:
4439:
4393:
4320:
4300:
4280:
4249:
4226:
4206:
4180:
4141:
4117:
4094:
4068:
4048:
4028:
3999:
3979:
3952:
3890:
3867:
3847:
3818:
3798:
3778:
3748:
3728:
3708:
3685:
3665:
3567:
3544:
3515:
3464:
3444:
3418:
3390:
3370:
3350:
3330:
3291:
3271:
3251:
3231:
3196:
3163:
3137:
3092:
3057:
3033:
3013:
2993:
2973:
2953:
2933:
2913:
2812:
2766:
2696:
2676:
2656:
2636:
2616:
2573:
2553:
2533:
2509:
2489:
2423:
2375:
2325:
2293:
2273:
2224:
2196:
2161:
2123:
2079:
2035:
2015:
1969:
1923:
1875:
1722:
1687:
1611:
1591:
1557:
1529:
1509:
1489:
1440:
1420:
1400:
1373:
1353:
1315:
1295:
1275:
1255:
1229:
1193:
1169:
1149:
1110:
1087:
1067:
1047:
1027:
995:
969:
933:
909:
889:
869:
846:
826:
806:
767:
747:
727:
688:
668:
648:
628:
558:
535:
515:
491:
471:
442:
422:
399:
379:
355:
331:
311:
275:
255:
232:
208:
184:
160:
140:
120:
89:
7565:
from cycles homologically equivalent to zero to the
4779:. This implies that the Chow ring is isomorphic to
1574:
7797:
7531:
7426:
7313:
7239:
7219:
7178:
7084:
7050:
7013:
6917:
6897:
6851:
6774:
6738:
6622:
6584:
6449:
6420:
6383:
6319:
6242:
6222:
6177:
6145:
6116:
6006:
5986:
5966:
5946:
5919:
5831:
5811:
5791:
5753:
5733:
5709:
5689:
5669:
5637:
5617:
5597:
5577:
5530:
5510:
5487:
5467:
5387:
5367:
5316:
5296:
5272:
5239:
5219:
5195:
5175:
5146:
5123:
5043:
5023:
4999:
4971:
4771:
4735:
4706:
4627:
4445:
4425:
4379:
4306:
4286:
4266:
4232:
4212:
4192:
4150:
4123:
4103:
4080:
4054:
4034:
4014:
3985:
3965:
3935:
3873:
3853:
3833:
3804:
3784:
3764:
3734:
3714:
3691:
3671:
3648:
3550:
3530:
3485:
3450:
3430:
3396:
3376:
3356:
3336:
3312:
3277:
3257:
3237:
3217:
3175:
3149:
3123:
3078:
3039:
3019:
2999:
2979:
2959:
2939:
2919:
2896:
2791:
2752:
2682:
2662:
2642:
2622:
2598:
2559:
2539:
2515:
2495:
2467:
2409:
2361:
2311:
2279:
2259:
2202:
2182:
2147:
2109:
2065:
2021:
2001:
1955:
1909:
1858:
1705:
1673:
1597:
1563:
1535:
1515:
1495:
1471:
1426:
1406:
1379:
1359:
1339:
1301:
1281:
1261:
1241:
1215:
1175:
1155:
1135:
1093:
1073:
1053:
1033:
1013:
981:
955:
915:
895:
875:
852:
832:
812:
792:
753:
733:
713:
674:
654:
634:
611:
541:
521:
497:
477:
454:
428:
405:
385:
361:
337:
317:
297:
261:
238:
214:
190:
166:
146:
126:
95:
16:Analogs of homology groups for algebraic varieties
8332:Chow groups, Chow cohomology and linear varieties
8245:Deligne, Cohomologie Etale (SGA 4 1/2), Expose 4.
8648:
8586:Cycles, Transfers, and Motivic Homology Theories
8299:
7333:) from Chow groups to more computable theories.
6428:then the fiber over the origin is isomorphic to
5930:A key computational tool for Chow groups is the
3742:in projective space is rationally equivalent to
2527:, not just a graded abelian group. Namely, when
2369:for smooth varieties, so the divisor classes of
6460:
2260:{\displaystyle L,L'\in \operatorname {Pic} (C)}
2210:, which is the coefficient of its cycle class.
8342:Fulton, Intersection Theory, Chapters 5, 6, 8.
5920:{\displaystyle f^{*}:CH_{i}(Y)\to CH_{i+r}(X)}
3285:) whose intersection has dimension zero, then
1910:{\displaystyle \pi _{1}:X\to \mathbb {P} ^{1}}
8209:Fulton, Intersection Theory, Proposition 1.8.
6905:respectively. The flat pullback of the point
6394:
4158:points of intersection; this is a version of
8300:Fulton, William; MacPherson, Robert (1981).
6821:
6789:
6320:{\displaystyle f^{*}:CH^{i}(Y)\to CH^{i}(X)}
5468:{\displaystyle f_{*}:CH_{i}(X)\to CH_{i}(Y)}
4169:
1713:. Schematically, this can be constructed as
8432:"Les classes d'Ă©quivalence rationnelle, II"
8096:Rational equivalence of divisors (known as
7024:
3772:. It follows that for any two subvarieties
1269:in the Chow group, and if two subvarieties
793:{\displaystyle \operatorname {ord} _{Z}(f)}
714:{\displaystyle \operatorname {ord} _{Z}(f)}
8419:"Les classes d'Ă©quivalence rationnelle, I"
8354:
7978:: for a smooth complex projective surface
7329:There are several homomorphisms (known as
1963:is the projective hypersurface defined by
8667:Topological methods of algebraic geometry
8580:
8438:, SĂ©minaire Claude Chevalley, vol. 3
8425:, SĂ©minaire Claude Chevalley, vol. 3
8392:
8227:Fulton, Intersection Theory, section 19.1
8200:Fulton, Intersection Theory, section 8.1.
8190:https://stacks.math.columbia.edu/tag/0BE9
7014:{\displaystyle f^{*}=e_{1}+\cdots +e_{k}}
6567:
6495:
6465:Consider the branched covering of curves
6437:
6408:
5578:{\displaystyle CH_{0}(X)\to \mathbf {Z} }
5134:Thus the Chow group of an elliptic curve
4842:
4723:
4668:
4480:
4350:
4251:
4002:
3922:
3918:
3821:
3586:
3518:
2214:Rational Equivalence of Cycles on a Curve
2164:
2044:
1897:
1846:
1831:
1742:
1639:
36:
8290:Fulton, Intersection Theory, Chapter 17.
7870:For a smooth complex projective variety
7842:) is finitely generated for any variety
5670:{\displaystyle \operatorname {Spec} (E)}
5031:. Then the Chow group of zero-cycles on
4453:, then there is an isomorphism of rings
4062:intersect transversely, it follows that
1580:Rational Equivalence on Projective Space
8483:
7903:) of the cycle map from Chow groups to
7619:
6327:, which is in fact a ring homomorphism
4274:can be computed using the Chow ring of
67:) and have a multiplication called the
8649:
8613:
8549:
8516:
8429:
8416:
6384:{\displaystyle CH^{*}(Y)\to CH^{*}(X)}
4638:
8373:(1986), "Algebraic cycles and higher
8369:
7828:implies that the divisor class group
6595:Since the morphism ramifies whenever
2753:{\displaystyle CH^{i}(X)=CH_{n-i}(X)}
2218:If we take two distinct line bundles
47:are algebro-geometric analogs of the
8442:
8069:on the associated complex manifold.
7029:Consider a flat family of varieties
4135:, this means that there are exactly
75:Rational equivalence and Chow groups
8061:together with a component encoding
8003:
7437:The factor of 2 appears because an
7092:. Then, using the cartesian square
6898:{\displaystyle e_{1},\ldots ,e_{k}}
6250:, there is a pullback homomorphism
4426:{\displaystyle c_{1},\ldots ,c_{r}}
4240:over a field, the Chow ring of the
3501:
1061:and all nonzero rational functions
505:which is not identically zero, the
13:
8046:map to the operational Chow ring.
7894:predicts the image (tensored with
7752:
7470:
7357:
4687:
4677:
4343:
3936:{\displaystyle \cdot =a\,b\,H^{n}}
3438:whose image in the Chow groups of
3225:in the Chow ring. For example, if
3186:More generally, in various cases,
2183:{\displaystyle \mathbb {P} ^{n-1}}
1917:we can see the fiber over a point
1654:
721:denotes the order of vanishing of
246:, unless stated otherwise.) For a
14:
8683:
8555:Commentarii Mathematici Helvetici
8147:Grothendieck–Riemann–Roch theorem
8049:
7976:Bloch's conjecture on zero-cycles
7680:Grothendieck–Riemann–Roch theorem
2468:{\displaystyle s'\in H^{0}(C,L')}
8672:Chinese mathematical discoveries
7910:For a smooth projective variety
7788:
7781:
7732:
7725:
7519:
7414:
6450:{\displaystyle \mathbb {P} ^{1}}
6421:{\displaystyle \mathbb {A} ^{2}}
5585:, which takes a closed point in
5571:
5335:
5154:is closely related to the group
5111:
4904:
4736:{\displaystyle \mathbb {P} ^{1}}
4643:For example, the Chow ring of a
4015:{\displaystyle \mathbb {P} ^{n}}
3834:{\displaystyle \mathbb {P} ^{n}}
3603:
3531:{\displaystyle \mathbb {P} ^{n}}
2478:
1575:Examples of Rational Equivalence
8436:Anneaux de Chow et applications
8423:Anneaux de Chow et applications
8348:
8336:
8324:
8293:
8284:
8275:
8266:
8257:
8031:respectively. It generalizes a
6782:. This implies that the points
4267:{\displaystyle \mathbb {P} (E)}
2410:{\displaystyle s\in H^{0}(C,L)}
8355:Eisenbud, David; Harris, Joe,
8248:
8239:
8230:
8221:
8212:
8203:
8194:
8182:
8173:
8130:deformation to the normal cone
7815:
7772:
7766:
7716:
7710:
7523:
7509:
7493:
7490:
7484:
7418:
7404:
7380:
7377:
7371:
7308:
7284:
7278:
7267:
7220:{\displaystyle S'\times _{S}X}
7164:
7147:
7141:
7129:
7042:
7008:
6995:
6973:
6967:
6951:
6945:
6846:
6840:
6720:
6700:
6681:
6661:
6655:
6643:
6611:
6605:
6562:
6552:
6549:
6537:
6528:
6522:
6516:
6511:
6499:
6378:
6372:
6356:
6353:
6347:
6314:
6308:
6292:
6289:
6283:
6214:
6169:
6137:
6105:
6102:
6090:
6074:
6071:
6065:
6049:
6046:
6040:
5914:
5908:
5886:
5883:
5877:
5839:(possibly empty), there is a
5783:
5664:
5658:
5567:
5564:
5558:
5462:
5456:
5440:
5437:
5431:
5359:
5267:
5261:
5170:
5164:
5115:
5107:
5104:
5098:
5082:
5079:
5073:
5067:
4963:
4925:
4920:
4908:
4891:
4866:
4861:
4855:
4852:
4837:
4815:
4802:
4701:
4698:
4692:
4672:
4530:
4524:
4521:
4515:
4493:
4490:
4484:
4476:
4374:
4371:
4365:
4360:
4354:
4337:
4261:
4255:
4184:
3909:
3903:
3897:
3891:
3812:of complementary dimension in
3640:
3621:
3613:
3607:
3596:
3581:
3480:
3474:
3471:
3465:
3307:
3301:
3298:
3292:
3212:
3206:
3203:
3197:
3118:
3112:
3073:
3067:
3064:
3058:
2888:
2882:
2860:
2857:
2851:
2832:
2826:
2786:
2780:
2747:
2741:
2716:
2710:
2593:
2587:
2462:
2445:
2404:
2392:
2356:
2350:
2338:
2332:
2306:
2300:
2254:
2248:
2060:
2039:
1950:
1924:
1892:
1826:
1816:
1798:
1793:
1761:
1758:
1746:
1668:
1665:
1659:
1634:
1466:
1460:
1334:
1328:
1322:
1316:
1249:for the class of a subvariety
1236:
1230:
1210:
1204:
1130:
1124:
1008:
996:
976:
970:
950:
944:
787:
781:
708:
702:
600:
594:
565:
559:
292:
286:
1:
8489:Cohomologie Etale (SGA 4 1/2)
8308:American Mathematical Society
8162:
7810:adequate equivalence relation
7324:
5677:for a finite extension field
3157:, which all have codimension
2475:define inequivalent classes.
2267:of a smooth projective curve
2110:{\displaystyle sf+tx_{0}^{d}}
2002:{\displaystyle s_{0}f+t_{0}g}
925:rationally equivalent to zero
393:-dimensional subvarieties of
8394:10.1016/0001-8708(86)90081-2
8167:
6623:{\displaystyle f(\alpha )=0}
6461:Branched coverings of curves
5538:, this gives a homomorphism
4242:associated projective bundle
4220:over a smooth proper scheme
2029:is rationally equivalent to
622:where the sum runs over all
7:
8363:
8157:Motive (algebraic geometry)
8135:
7998:
7982:with geometric genus zero,
7441:-dimensional subvariety of
7085:{\displaystyle S'\subset S}
5934:, as follows. For a scheme
5717:, and its degree means the
3496:
3124:{\displaystyle CH^{i+j}(X)}
2947:are smooth subvarieties of
2148:{\displaystyle x_{0}^{d}=0}
1472:{\displaystyle CH_{n-1}(X)}
79:For what follows, define a
10:
8688:
8623:Cambridge University Press
8590:Princeton University Press
8430:Claude, Chevalley (1958),
8417:Claude, Chevalley (1958),
8091:
6775:{\displaystyle e_{i}>1}
6395:Examples of flat pullbacks
4982:
4088:is a zero-cycle of degree
2670:is a variety of dimension
1414:is a variety of dimension
1021:-dimensional subvarieties
642:-dimensional subvarieties
8120:. Starting in the 1970s,
8104:in number theory and the
8086:virtual fundamental class
7189:we see that the image of
5819:with fibers of dimension
4294:and the Chern classes of
4170:Projective bundle formula
2792:{\displaystyle CH^{*}(X)}
2599:{\displaystyle CH^{i}(X)}
2523:, the Chow groups form a
1136:{\displaystyle CH_{i}(X)}
7600:over an arbitrary field
7561:. This incorporates the
7025:Flat family of varieties
6223:{\displaystyle f:X\to Y}
6178:{\displaystyle X-Z\to X}
5792:{\displaystyle f:X\to Y}
5397:pushforward homomorphism
5368:{\displaystyle f:X\to Y}
4772:{\displaystyle c_{1}=aH}
2606:to be the Chow group of
2190:and it has multiplicity
1216:{\displaystyle Z_{i}(X)}
963:generated by the cycles
956:{\displaystyle Z_{i}(X)}
325:-dimensional cycles (or
298:{\displaystyle Z_{i}(X)}
8380:Advances in Mathematics
8112:in the 1930s. In 1956,
8078:equivariant Chow groups
8023:that assign to a map a
7930:and the Chow groups of
6918:{\displaystyle \alpha }
6630:we get a factorization
6230:of smooth schemes over
5974:and a closed subscheme
4081:{\displaystyle Y\cap Z}
3431:{\displaystyle Y\cap Z}
3324:. For any subvarieties
3150:{\displaystyle Y\cap Z}
2690:, this just means that
2503:is smooth over a field
1163:-dimensional cycles on
8065:information, that is,
8055:Arithmetic Chow groups
7951:be the kernel. If the
7890:or number field), the
7867:in algebraic K-theory.
7799:
7533:
7428:
7315:
7314:{\displaystyle f^{*}=}
7241:
7221:
7180:
7086:
7052:
7051:{\displaystyle X\to S}
7015:
6919:
6899:
6853:
6776:
6740:
6624:
6586:
6451:
6422:
6385:
6321:
6244:
6224:
6189:groups, also known as
6179:
6147:
6146:{\displaystyle Z\to X}
6118:
6008:
5988:
5968:
5948:
5921:
5833:
5813:
5793:
5755:
5735:
5711:
5691:
5671:
5639:
5619:
5599:
5579:
5532:
5512:
5489:
5469:
5389:
5369:
5318:
5298:
5274:
5241:
5221:
5197:
5177:
5148:
5125:
5045:
5025:
5001:
4973:
4773:
4737:
4708:
4629:
4447:
4427:
4381:
4308:
4288:
4268:
4234:
4214:
4194:
4193:{\displaystyle E\to X}
4174:Given a vector bundle
4162:, a classic result of
4152:
4125:
4105:
4082:
4056:
4036:
4016:
3987:
3967:
3937:
3875:
3855:
3835:
3806:
3786:
3766:
3765:{\displaystyle dH^{a}}
3736:
3716:
3693:
3673:
3650:
3552:
3532:
3487:
3452:
3432:
3398:
3378:
3358:
3338:
3314:
3279:
3259:
3239:
3219:
3177:
3151:
3125:
3080:
3041:
3021:
3001:
2981:
2961:
2941:
2921:
2898:
2793:
2754:
2684:
2664:
2644:
2624:
2600:
2561:
2541:
2517:
2497:
2469:
2411:
2363:
2313:
2281:
2261:
2204:
2184:
2149:
2111:
2067:
2023:
2003:
1957:
1911:
1860:
1707:
1675:
1599:
1565:
1537:
1517:
1497:
1473:
1428:
1408:
1381:
1361:
1341:
1303:
1283:
1263:
1243:
1217:
1177:
1157:
1137:
1095:
1075:
1055:
1035:
1015:
983:
957:
917:
897:
877:
854:
834:
814:
794:
755:
735:
715:
676:
656:
636:
613:
543:
523:
499:
479:
456:
430:
407:
387:
363:
339:
319:
299:
263:
240:
216:
192:
168:
148:
128:
97:
8449:Annals of Mathematics
8082:Chow group of a stack
8017:operational Chow ring
7860:values of L-functions
7812:on algebraic cycles.
7800:
7690:gives an isomorphism
7674:of vector bundles on
7567:intermediate Jacobian
7553:For a smooth complex
7534:
7429:
7316:
7247:. Therefore, we have
7242:
7222:
7181:
7087:
7053:
7016:
6920:
6900:
6854:
6777:
6741:
6625:
6587:
6452:
6423:
6386:
6322:
6245:
6225:
6180:
6148:
6119:
6009:
5989:
5969:
5949:
5932:localization sequence
5922:
5834:
5814:
5794:
5756:
5736:
5712:
5692:
5672:
5640:
5625:. (A closed point in
5620:
5600:
5580:
5533:
5513:
5495:. For example, for a
5490:
5470:
5390:
5370:
5319:
5299:
5275:
5242:
5222:
5198:
5178:
5149:
5126:
5046:
5026:
5002:
4974:
4774:
4738:
4709:
4630:
4448:
4433:the Chern classes of
4428:
4382:
4309:
4289:
4269:
4235:
4215:
4195:
4153:
4126:
4106:
4083:
4057:
4037:
4017:
3988:
3968:
3966:{\displaystyle H^{n}}
3938:
3876:
3856:
3836:
3807:
3787:
3767:
3737:
3717:
3694:
3674:
3651:
3553:
3533:
3488:
3453:
3433:
3399:
3379:
3359:
3339:
3315:
3280:
3260:
3240:
3220:
3178:
3152:
3126:
3081:
3042:
3022:
3007:respectively, and if
3002:
2982:
2962:
2942:
2922:
2899:
2794:
2755:
2685:
2665:
2645:
2625:
2601:
2562:
2542:
2518:
2498:
2470:
2412:
2364:
2314:
2312:{\displaystyle CH(C)}
2282:
2262:
2205:
2185:
2150:
2112:
2068:
2024:
2004:
1958:
1912:
1869:using the projection
1861:
1708:
1706:{\displaystyle sf+tg}
1676:
1600:
1566:
1538:
1518:
1498:
1474:
1429:
1409:
1389:rationally equivalent
1382:
1362:
1342:
1304:
1284:
1264:
1244:
1218:
1178:
1158:
1138:
1096:
1076:
1056:
1036:
1016:
1014:{\displaystyle (i+1)}
984:
958:
918:
898:
878:
855:
835:
815:
795:
756:
736:
716:
677:
657:
637:
614:
544:
524:
500:
480:
457:
431:
408:
388:
364:
340:
320:
300:
264:
241:
217:
193:
169:
149:
129:
98:
8592:, pp. 188–238,
8524:, Berlin, New York:
8063:Arakelov-theoretical
7914:over any field, the
7878:predicts the image (
7826:Mordell–Weil theorem
7697:
7620:Relation to K-theory
7571:exponential sequence
7464:
7445:has real dimension 2
7351:
7342:Borel–Moore homology
7336:First, for a scheme
7254:
7231:
7193:
7099:
7065:
7036:
6932:
6909:
6863:
6859:have multiplicities
6786:
6753:
6637:
6599:
6472:
6432:
6403:
6331:
6254:
6234:
6202:
6157:
6131:
6024:
5998:
5978:
5958:
5938:
5848:
5823:
5803:
5771:
5745:
5725:
5701:
5681:
5649:
5629:
5609:
5589:
5542:
5522:
5502:
5479:
5402:
5379:
5347:
5308:
5288:
5273:{\displaystyle X(k)}
5255:
5231:
5211:
5187:
5176:{\displaystyle X(k)}
5158:
5138:
5061:
5035:
5015:
4991:
4786:
4747:
4718:
4651:
4460:
4437:
4391:
4318:
4298:
4278:
4247:
4224:
4204:
4178:
4164:enumerative geometry
4139:
4133:algebraically closed
4115:
4111:. If the base field
4092:
4066:
4046:
4026:
3997:
3977:
3950:
3888:
3865:
3845:
3816:
3796:
3776:
3746:
3726:
3706:
3683:
3663:
3565:
3542:
3513:
3462:
3442:
3416:
3388:
3368:
3348:
3328:
3322:intersection numbers
3289:
3269:
3249:
3229:
3194:
3161:
3135:
3090:
3055:
3031:
3011:
2991:
2971:
2951:
2931:
2911:
2810:
2764:
2694:
2674:
2654:
2634:
2614:
2571:
2551:
2531:
2507:
2487:
2421:
2373:
2323:
2291:
2271:
2222:
2194:
2159:
2121:
2077:
2033:
2013:
1967:
1921:
1873:
1720:
1685:
1609:
1589:
1555:
1527:
1507:
1487:
1438:
1418:
1398:
1371:
1351:
1313:
1293:
1273:
1253:
1227:
1191:
1167:
1147:
1108:
1085:
1065:
1045:
1025:
993:
967:
931:
907:
887:
883:of finite type over
867:
844:
824:
804:
765:
745:
725:
686:
666:
646:
626:
556:
533:
513:
489:
469:
440:
420:
397:
377:
353:
329:
309:
273:
253:
230:
206:
182:
158:
154:of finite type over
138:
118:
87:
69:intersection product
33:Claude Chevalley
8662:Intersection theory
8582:Voevodsky, Vladimir
8522:Intersection Theory
8142:Intersection theory
8118:Chow's moving lemma
7882:with the rationals
7631:on a smooth scheme
7403:
7227:is a subvariety of
6581:
5605:to its degree over
4639:Hirzebruch surfaces
3993:-rational point in
3364:of a smooth scheme
3188:intersection theory
3176:{\displaystyle i+j}
3051:, then the product
2799:form a commutative
2760:.) Then the groups
2138:
2106:
1481:divisor class group
982:{\displaystyle (f)}
927:is the subgroup of
455:{\displaystyle i+1}
202:of subvarieties of
8657:Algebraic geometry
8567:10.1007/bf01202721
8098:linear equivalence
8067:differential forms
8044:motivic cohomology
7962:, Ω) is not zero,
7795:
7748:
7672:Grothendieck group
7559:Deligne cohomology
7555:projective variety
7529:
7457:as a homomorphism
7424:
7383:
7311:
7237:
7217:
7176:
7174:
7082:
7048:
7011:
6915:
6895:
6849:
6772:
6736:
6620:
6582:
6565:
6447:
6418:
6381:
6317:
6240:
6220:
6192:higher Chow groups
6175:
6143:
6114:
6004:
5984:
5964:
5944:
5917:
5829:
5809:
5789:
5751:
5731:
5707:
5687:
5667:
5635:
5615:
5595:
5575:
5528:
5508:
5485:
5465:
5385:
5365:
5314:
5294:
5282:Mordell–Weil group
5270:
5237:
5217:
5193:
5173:
5144:
5121:
5041:
5021:
4997:
4969:
4769:
4733:
4704:
4645:Hirzebruch surface
4625:
4443:
4423:
4377:
4304:
4284:
4264:
4230:
4210:
4190:
4151:{\displaystyle ab}
4148:
4121:
4104:{\displaystyle ab}
4101:
4078:
4052:
4032:
4022:. For example, if
4012:
3983:
3973:is the class of a
3963:
3933:
3871:
3851:
3831:
3802:
3782:
3762:
3732:
3712:
3689:
3669:
3646:
3548:
3528:
3483:
3448:
3428:
3394:
3374:
3354:
3334:
3310:
3275:
3255:
3235:
3215:
3173:
3147:
3121:
3076:
3037:
3017:
2997:
2977:
2957:
2937:
2917:
2894:
2803:with the product:
2789:
2750:
2680:
2660:
2640:
2620:
2596:
2557:
2537:
2513:
2493:
2465:
2407:
2359:
2319:. This is because
2309:
2277:
2257:
2200:
2180:
2145:
2124:
2107:
2092:
2063:
2019:
1999:
1953:
1907:
1856:
1703:
1671:
1595:
1561:
1533:
1513:
1493:
1469:
1424:
1404:
1394:For example, when
1377:
1357:
1337:
1299:
1279:
1259:
1239:
1213:
1173:
1153:
1133:
1091:
1071:
1051:
1031:
1011:
979:
953:
913:
893:
873:
850:
830:
810:
790:
751:
731:
711:
672:
652:
632:
609:
580:
539:
519:
495:
475:
452:
426:
403:
383:
371:free abelian group
359:
335:
315:
295:
259:
236:
212:
200:linear combination
188:
164:
144:
124:
93:
21:algebraic geometry
8632:978-0-521-71801-1
8551:Severi, Francesco
8535:978-0-387-98549-7
8502:978-3-540-08066-4
8102:ideal class group
8033:cohomology theory
7939:For example, let
7905:l-adic cohomology
7739:
7678:. As part of the
7635:over a field has
7240:{\displaystyle X}
7061:and a subvariety
6749:where one of the
6556:
6243:{\displaystyle k}
6198:For any morphism
6007:{\displaystyle X}
5987:{\displaystyle Z}
5967:{\displaystyle k}
5947:{\displaystyle X}
5832:{\displaystyle r}
5812:{\displaystyle k}
5754:{\displaystyle k}
5734:{\displaystyle E}
5710:{\displaystyle k}
5690:{\displaystyle E}
5638:{\displaystyle X}
5618:{\displaystyle k}
5598:{\displaystyle X}
5531:{\displaystyle k}
5511:{\displaystyle X}
5488:{\displaystyle i}
5475:for each integer
5388:{\displaystyle k}
5317:{\displaystyle k}
5297:{\displaystyle X}
5240:{\displaystyle k}
5220:{\displaystyle X}
5196:{\displaystyle k}
5147:{\displaystyle X}
5044:{\displaystyle X}
5024:{\displaystyle k}
5000:{\displaystyle X}
4967:
4895:
4623:
4446:{\displaystyle E}
4307:{\displaystyle E}
4287:{\displaystyle X}
4233:{\displaystyle X}
4213:{\displaystyle r}
4124:{\displaystyle k}
4055:{\displaystyle Z}
4035:{\displaystyle Y}
3986:{\displaystyle k}
3874:{\displaystyle b}
3854:{\displaystyle a}
3805:{\displaystyle Z}
3785:{\displaystyle Y}
3735:{\displaystyle a}
3715:{\displaystyle d}
3692:{\displaystyle Y}
3672:{\displaystyle H}
3551:{\displaystyle k}
3506:The Chow ring of
3451:{\displaystyle X}
3410:Robert MacPherson
3397:{\displaystyle k}
3377:{\displaystyle X}
3357:{\displaystyle Z}
3337:{\displaystyle Y}
3278:{\displaystyle X}
3258:{\displaystyle Z}
3238:{\displaystyle Y}
3040:{\displaystyle Z}
3020:{\displaystyle Y}
3000:{\displaystyle j}
2980:{\displaystyle i}
2960:{\displaystyle X}
2940:{\displaystyle Z}
2920:{\displaystyle Y}
2683:{\displaystyle n}
2663:{\displaystyle X}
2643:{\displaystyle X}
2623:{\displaystyle i}
2560:{\displaystyle k}
2540:{\displaystyle X}
2516:{\displaystyle k}
2496:{\displaystyle X}
2280:{\displaystyle C}
2203:{\displaystyle d}
2066:{\displaystyle d}
2022:{\displaystyle d}
1820:
1732:
1598:{\displaystyle d}
1564:{\displaystyle X}
1536:{\displaystyle k}
1516:{\displaystyle X}
1496:{\displaystyle X}
1434:, the Chow group
1427:{\displaystyle n}
1407:{\displaystyle X}
1380:{\displaystyle W}
1360:{\displaystyle Z}
1340:{\displaystyle =}
1302:{\displaystyle W}
1282:{\displaystyle Z}
1262:{\displaystyle Z}
1176:{\displaystyle X}
1156:{\displaystyle i}
1094:{\displaystyle W}
1074:{\displaystyle f}
1054:{\displaystyle X}
1034:{\displaystyle W}
916:{\displaystyle i}
896:{\displaystyle k}
876:{\displaystyle X}
853:{\displaystyle W}
833:{\displaystyle Z}
820:has a pole along
813:{\displaystyle f}
754:{\displaystyle Z}
734:{\displaystyle f}
675:{\displaystyle W}
655:{\displaystyle Z}
635:{\displaystyle i}
571:
542:{\displaystyle i}
522:{\displaystyle f}
498:{\displaystyle W}
478:{\displaystyle f}
464:rational function
429:{\displaystyle W}
406:{\displaystyle X}
386:{\displaystyle i}
362:{\displaystyle X}
338:{\displaystyle i}
318:{\displaystyle i}
262:{\displaystyle i}
239:{\displaystyle X}
215:{\displaystyle X}
191:{\displaystyle X}
167:{\displaystyle k}
147:{\displaystyle X}
134:. For any scheme
127:{\displaystyle k}
96:{\displaystyle k}
53:topological space
41:algebraic variety
8679:
8643:
8610:
8577:
8546:
8513:
8480:
8439:
8426:
8413:
8396:
8359:
8343:
8340:
8334:
8328:
8322:
8321:
8297:
8291:
8288:
8282:
8279:
8273:
8270:
8264:
8261:
8255:
8252:
8246:
8243:
8237:
8234:
8228:
8225:
8219:
8216:
8210:
8207:
8201:
8198:
8192:
8188:Stacks Project,
8186:
8180:
8177:
8152:Hodge conjecture
8110:Francesco Severi
8106:Jacobian variety
8074:algebraic spaces
8004:Bivariant theory
7988:Albanese variety
7876:Hodge conjecture
7804:
7802:
7801:
7796:
7791:
7786:
7785:
7784:
7765:
7764:
7759:
7758:
7747:
7735:
7730:
7729:
7728:
7709:
7708:
7686:showed that the
7538:
7536:
7535:
7530:
7522:
7508:
7507:
7483:
7482:
7477:
7476:
7455:Poincaré duality
7433:
7431:
7430:
7425:
7417:
7402:
7394:
7370:
7369:
7364:
7363:
7320:
7318:
7317:
7312:
7304:
7303:
7294:
7277:
7266:
7265:
7246:
7244:
7243:
7238:
7226:
7224:
7223:
7218:
7213:
7212:
7203:
7185:
7183:
7182:
7177:
7175:
7161:
7145:
7123:
7122:
7113:
7091:
7089:
7088:
7083:
7075:
7057:
7055:
7054:
7049:
7020:
7018:
7017:
7012:
7007:
7006:
6994:
6993:
6966:
6965:
6944:
6943:
6924:
6922:
6921:
6916:
6904:
6902:
6901:
6896:
6894:
6893:
6875:
6874:
6858:
6856:
6855:
6850:
6839:
6838:
6820:
6819:
6801:
6800:
6781:
6779:
6778:
6773:
6765:
6764:
6745:
6743:
6742:
6737:
6735:
6734:
6733:
6732:
6718:
6717:
6696:
6695:
6694:
6693:
6679:
6678:
6629:
6627:
6626:
6621:
6591:
6589:
6588:
6583:
6580:
6575:
6570:
6561:
6557:
6555:
6514:
6498:
6492:
6456:
6454:
6453:
6448:
6446:
6445:
6440:
6427:
6425:
6424:
6419:
6417:
6416:
6411:
6390:
6388:
6387:
6382:
6371:
6370:
6346:
6345:
6326:
6324:
6323:
6318:
6307:
6306:
6282:
6281:
6266:
6265:
6249:
6247:
6246:
6241:
6229:
6227:
6226:
6221:
6187:motivic homology
6184:
6182:
6181:
6176:
6152:
6150:
6149:
6144:
6123:
6121:
6120:
6115:
6089:
6088:
6064:
6063:
6039:
6038:
6013:
6011:
6010:
6005:
5993:
5991:
5990:
5985:
5973:
5971:
5970:
5965:
5953:
5951:
5950:
5945:
5926:
5924:
5923:
5918:
5907:
5906:
5876:
5875:
5860:
5859:
5838:
5836:
5835:
5830:
5818:
5816:
5815:
5810:
5799:of schemes over
5798:
5796:
5795:
5790:
5760:
5758:
5757:
5752:
5740:
5738:
5737:
5732:
5716:
5714:
5713:
5708:
5696:
5694:
5693:
5688:
5676:
5674:
5673:
5668:
5644:
5642:
5641:
5636:
5624:
5622:
5621:
5616:
5604:
5602:
5601:
5596:
5584:
5582:
5581:
5576:
5574:
5557:
5556:
5537:
5535:
5534:
5529:
5517:
5515:
5514:
5509:
5494:
5492:
5491:
5486:
5474:
5472:
5471:
5466:
5455:
5454:
5430:
5429:
5414:
5413:
5394:
5392:
5391:
5386:
5375:of schemes over
5374:
5372:
5371:
5366:
5332:abelian groups.
5323:
5321:
5320:
5315:
5303:
5301:
5300:
5295:
5279:
5277:
5276:
5271:
5246:
5244:
5243:
5238:
5226:
5224:
5223:
5218:
5202:
5200:
5199:
5194:
5182:
5180:
5179:
5174:
5153:
5151:
5150:
5145:
5130:
5128:
5127:
5122:
5114:
5097:
5096:
5050:
5048:
5047:
5042:
5030:
5028:
5027:
5022:
5006:
5004:
5003:
4998:
4978:
4976:
4975:
4970:
4968:
4966:
4950:
4949:
4937:
4936:
4923:
4907:
4901:
4896:
4894:
4878:
4877:
4864:
4851:
4850:
4845:
4836:
4835:
4822:
4814:
4813:
4801:
4800:
4778:
4776:
4775:
4770:
4759:
4758:
4742:
4740:
4739:
4734:
4732:
4731:
4726:
4713:
4711:
4710:
4705:
4691:
4690:
4681:
4680:
4671:
4663:
4662:
4634:
4632:
4631:
4626:
4624:
4622:
4621:
4620:
4602:
4601:
4586:
4585:
4573:
4572:
4557:
4556:
4544:
4543:
4533:
4514:
4513:
4500:
4483:
4475:
4474:
4452:
4450:
4449:
4444:
4432:
4430:
4429:
4424:
4422:
4421:
4403:
4402:
4386:
4384:
4383:
4378:
4364:
4363:
4353:
4347:
4346:
4336:
4335:
4313:
4311:
4310:
4305:
4293:
4291:
4290:
4285:
4273:
4271:
4270:
4265:
4254:
4239:
4237:
4236:
4231:
4219:
4217:
4216:
4211:
4199:
4197:
4196:
4191:
4160:BĂ©zout's theorem
4157:
4155:
4154:
4149:
4130:
4128:
4127:
4122:
4110:
4108:
4107:
4102:
4087:
4085:
4084:
4079:
4061:
4059:
4058:
4053:
4041:
4039:
4038:
4033:
4021:
4019:
4018:
4013:
4011:
4010:
4005:
3992:
3990:
3989:
3984:
3972:
3970:
3969:
3964:
3962:
3961:
3942:
3940:
3939:
3934:
3932:
3931:
3880:
3878:
3877:
3872:
3860:
3858:
3857:
3852:
3840:
3838:
3837:
3832:
3830:
3829:
3824:
3811:
3809:
3808:
3803:
3791:
3789:
3788:
3783:
3771:
3769:
3768:
3763:
3761:
3760:
3741:
3739:
3738:
3733:
3722:and codimension
3721:
3719:
3718:
3713:
3698:
3696:
3695:
3690:
3678:
3676:
3675:
3670:
3655:
3653:
3652:
3647:
3639:
3638:
3620:
3606:
3595:
3594:
3589:
3580:
3579:
3557:
3555:
3554:
3549:
3537:
3535:
3534:
3529:
3527:
3526:
3521:
3508:projective space
3502:Projective space
3492:
3490:
3489:
3486:{\displaystyle }
3484:
3457:
3455:
3454:
3449:
3437:
3435:
3434:
3429:
3403:
3401:
3400:
3395:
3383:
3381:
3380:
3375:
3363:
3361:
3360:
3355:
3343:
3341:
3340:
3335:
3319:
3317:
3316:
3313:{\displaystyle }
3311:
3284:
3282:
3281:
3276:
3264:
3262:
3261:
3256:
3244:
3242:
3241:
3236:
3224:
3222:
3221:
3218:{\displaystyle }
3216:
3182:
3180:
3179:
3174:
3156:
3154:
3153:
3148:
3130:
3128:
3127:
3122:
3111:
3110:
3085:
3083:
3082:
3079:{\displaystyle }
3077:
3046:
3044:
3043:
3038:
3026:
3024:
3023:
3018:
3006:
3004:
3003:
2998:
2986:
2984:
2983:
2978:
2966:
2964:
2963:
2958:
2946:
2944:
2943:
2938:
2926:
2924:
2923:
2918:
2903:
2901:
2900:
2895:
2881:
2880:
2850:
2849:
2825:
2824:
2798:
2796:
2795:
2790:
2779:
2778:
2759:
2757:
2756:
2751:
2740:
2739:
2709:
2708:
2689:
2687:
2686:
2681:
2669:
2667:
2666:
2661:
2649:
2647:
2646:
2641:
2629:
2627:
2626:
2621:
2605:
2603:
2602:
2597:
2586:
2585:
2566:
2564:
2563:
2558:
2546:
2544:
2543:
2538:
2522:
2520:
2519:
2514:
2502:
2500:
2499:
2494:
2483:When the scheme
2474:
2472:
2471:
2466:
2461:
2444:
2443:
2431:
2416:
2414:
2413:
2408:
2391:
2390:
2368:
2366:
2365:
2360:
2318:
2316:
2315:
2310:
2286:
2284:
2283:
2278:
2266:
2264:
2263:
2258:
2238:
2209:
2207:
2206:
2201:
2189:
2187:
2186:
2181:
2179:
2178:
2167:
2154:
2152:
2151:
2146:
2137:
2132:
2116:
2114:
2113:
2108:
2105:
2100:
2072:
2070:
2069:
2064:
2059:
2058:
2047:
2028:
2026:
2025:
2020:
2008:
2006:
2005:
2000:
1995:
1994:
1979:
1978:
1962:
1960:
1959:
1956:{\displaystyle }
1954:
1949:
1948:
1936:
1935:
1916:
1914:
1913:
1908:
1906:
1905:
1900:
1885:
1884:
1865:
1863:
1862:
1857:
1855:
1854:
1849:
1840:
1839:
1834:
1825:
1821:
1819:
1796:
1792:
1791:
1773:
1772:
1745:
1739:
1733:
1730:
1712:
1710:
1709:
1704:
1680:
1678:
1677:
1672:
1658:
1657:
1648:
1647:
1642:
1633:
1632:
1604:
1602:
1601:
1596:
1570:
1568:
1567:
1562:
1542:
1540:
1539:
1534:
1522:
1520:
1519:
1514:
1502:
1500:
1499:
1494:
1478:
1476:
1475:
1470:
1459:
1458:
1433:
1431:
1430:
1425:
1413:
1411:
1410:
1405:
1386:
1384:
1383:
1378:
1366:
1364:
1363:
1358:
1346:
1344:
1343:
1338:
1308:
1306:
1305:
1300:
1288:
1286:
1285:
1280:
1268:
1266:
1265:
1260:
1248:
1246:
1245:
1242:{\displaystyle }
1240:
1222:
1220:
1219:
1214:
1203:
1202:
1182:
1180:
1179:
1174:
1162:
1160:
1159:
1154:
1142:
1140:
1139:
1134:
1123:
1122:
1100:
1098:
1097:
1092:
1080:
1078:
1077:
1072:
1060:
1058:
1057:
1052:
1040:
1038:
1037:
1032:
1020:
1018:
1017:
1012:
988:
986:
985:
980:
962:
960:
959:
954:
943:
942:
922:
920:
919:
914:
902:
900:
899:
894:
882:
880:
879:
874:
859:
857:
856:
851:
839:
837:
836:
831:
819:
817:
816:
811:
799:
797:
796:
791:
777:
776:
760:
758:
757:
752:
740:
738:
737:
732:
720:
718:
717:
712:
698:
697:
682:and the integer
681:
679:
678:
673:
661:
659:
658:
653:
641:
639:
638:
633:
618:
616:
615:
610:
590:
589:
579:
548:
546:
545:
540:
528:
526:
525:
520:
504:
502:
501:
496:
484:
482:
481:
476:
461:
459:
458:
453:
435:
433:
432:
427:
412:
410:
409:
404:
392:
390:
389:
384:
368:
366:
365:
360:
349:, for short) on
344:
342:
341:
336:
324:
322:
321:
316:
304:
302:
301:
296:
285:
284:
268:
266:
265:
260:
245:
243:
242:
237:
221:
219:
218:
213:
197:
195:
194:
189:
173:
171:
170:
165:
153:
151:
150:
145:
133:
131:
130:
125:
102:
100:
99:
94:
65:Poincaré duality
57:algebraic cycles
8687:
8686:
8682:
8681:
8680:
8678:
8677:
8676:
8647:
8646:
8633:
8600:
8536:
8526:Springer-Verlag
8518:Fulton, William
8503:
8493:Springer-Verlag
8485:Deligne, Pierre
8462:10.2307/1969596
8444:Chow, Wei-Liang
8366:
8351:
8346:
8341:
8337:
8329:
8325:
8318:
8298:
8294:
8289:
8285:
8280:
8276:
8271:
8267:
8262:
8258:
8253:
8249:
8244:
8240:
8235:
8231:
8226:
8222:
8217:
8213:
8208:
8204:
8199:
8195:
8187:
8183:
8178:
8174:
8170:
8165:
8138:
8094:
8052:
8037:cohomology ring
8006:
8001:
7953:geometric genus
7902:
7892:Tate conjecture
7865:Bass conjecture
7837:
7818:
7787:
7780:
7779:
7775:
7760:
7751:
7750:
7749:
7743:
7731:
7724:
7723:
7719:
7704:
7700:
7698:
7695:
7694:
7688:Chern character
7665:
7646:
7624:An (algebraic)
7622:
7612:is smooth over
7563:Abel–Jacobi map
7518:
7500:
7496:
7478:
7469:
7468:
7467:
7465:
7462:
7461:
7413:
7395:
7387:
7365:
7356:
7355:
7354:
7352:
7349:
7348:
7327:
7299:
7295:
7287:
7270:
7261:
7257:
7255:
7252:
7251:
7232:
7229:
7228:
7208:
7204:
7196:
7194:
7191:
7190:
7173:
7172:
7167:
7162:
7154:
7151:
7150:
7144:
7138:
7137:
7132:
7127:
7118:
7114:
7106:
7102:
7100:
7097:
7096:
7068:
7066:
7063:
7062:
7037:
7034:
7033:
7027:
7002:
6998:
6989:
6985:
6961:
6957:
6939:
6935:
6933:
6930:
6929:
6910:
6907:
6906:
6889:
6885:
6870:
6866:
6864:
6861:
6860:
6831:
6827:
6815:
6811:
6796:
6792:
6787:
6784:
6783:
6760:
6756:
6754:
6751:
6750:
6728:
6724:
6723:
6719:
6713:
6709:
6689:
6685:
6684:
6680:
6674:
6670:
6638:
6635:
6634:
6600:
6597:
6596:
6576:
6571:
6566:
6515:
6494:
6493:
6491:
6487:
6473:
6470:
6469:
6463:
6441:
6436:
6435:
6433:
6430:
6429:
6412:
6407:
6406:
6404:
6401:
6400:
6397:
6366:
6362:
6341:
6337:
6332:
6329:
6328:
6302:
6298:
6277:
6273:
6261:
6257:
6255:
6252:
6251:
6235:
6232:
6231:
6203:
6200:
6199:
6158:
6155:
6154:
6132:
6129:
6128:
6084:
6080:
6059:
6055:
6034:
6030:
6025:
6022:
6021:
5999:
5996:
5995:
5979:
5976:
5975:
5959:
5956:
5955:
5939:
5936:
5935:
5896:
5892:
5871:
5867:
5855:
5851:
5849:
5846:
5845:
5824:
5821:
5820:
5804:
5801:
5800:
5772:
5769:
5768:
5746:
5743:
5742:
5726:
5723:
5722:
5702:
5699:
5698:
5682:
5679:
5678:
5650:
5647:
5646:
5630:
5627:
5626:
5610:
5607:
5606:
5590:
5587:
5586:
5570:
5552:
5548:
5543:
5540:
5539:
5523:
5520:
5519:
5503:
5500:
5499:
5480:
5477:
5476:
5450:
5446:
5425:
5421:
5409:
5405:
5403:
5400:
5399:
5380:
5377:
5376:
5348:
5345:
5344:
5342:proper morphism
5338:
5326:complex numbers
5309:
5306:
5305:
5289:
5286:
5285:
5256:
5253:
5252:
5232:
5229:
5228:
5212:
5209:
5208:
5205:rational points
5188:
5185:
5184:
5159:
5156:
5155:
5139:
5136:
5135:
5110:
5092:
5088:
5062:
5059:
5058:
5036:
5033:
5032:
5016:
5013:
5012:
4992:
4989:
4988:
4985:
4945:
4941:
4932:
4928:
4924:
4903:
4902:
4900:
4873:
4869:
4865:
4846:
4841:
4840:
4831:
4827:
4823:
4821:
4809:
4805:
4796:
4792:
4787:
4784:
4783:
4754:
4750:
4748:
4745:
4744:
4727:
4722:
4721:
4719:
4716:
4715:
4686:
4685:
4676:
4675:
4667:
4658:
4654:
4652:
4649:
4648:
4641:
4616:
4612:
4591:
4587:
4581:
4577:
4562:
4558:
4552:
4548:
4539:
4535:
4534:
4509:
4505:
4501:
4499:
4479:
4470:
4466:
4461:
4458:
4457:
4438:
4435:
4434:
4417:
4413:
4398:
4394:
4392:
4389:
4388:
4349:
4348:
4342:
4341:
4340:
4331:
4327:
4319:
4316:
4315:
4299:
4296:
4295:
4279:
4276:
4275:
4250:
4248:
4245:
4244:
4225:
4222:
4221:
4205:
4202:
4201:
4179:
4176:
4175:
4172:
4140:
4137:
4136:
4116:
4113:
4112:
4093:
4090:
4089:
4067:
4064:
4063:
4047:
4044:
4043:
4027:
4024:
4023:
4006:
4001:
4000:
3998:
3995:
3994:
3978:
3975:
3974:
3957:
3953:
3951:
3948:
3947:
3927:
3923:
3889:
3886:
3885:
3866:
3863:
3862:
3846:
3843:
3842:
3825:
3820:
3819:
3817:
3814:
3813:
3797:
3794:
3793:
3777:
3774:
3773:
3756:
3752:
3747:
3744:
3743:
3727:
3724:
3723:
3707:
3704:
3703:
3684:
3681:
3680:
3664:
3661:
3660:
3628:
3624:
3616:
3602:
3590:
3585:
3584:
3575:
3571:
3566:
3563:
3562:
3543:
3540:
3539:
3538:over any field
3522:
3517:
3516:
3514:
3511:
3510:
3504:
3499:
3463:
3460:
3459:
3458:is the product
3443:
3440:
3439:
3417:
3414:
3413:
3389:
3386:
3385:
3369:
3366:
3365:
3349:
3346:
3345:
3329:
3326:
3325:
3290:
3287:
3286:
3270:
3267:
3266:
3250:
3247:
3246:
3230:
3227:
3226:
3195:
3192:
3191:
3162:
3159:
3158:
3136:
3133:
3132:
3100:
3096:
3091:
3088:
3087:
3056:
3053:
3052:
3032:
3029:
3028:
3012:
3009:
3008:
2992:
2989:
2988:
2972:
2969:
2968:
2967:of codimension
2952:
2949:
2948:
2932:
2929:
2928:
2912:
2909:
2908:
2870:
2866:
2845:
2841:
2820:
2816:
2811:
2808:
2807:
2774:
2770:
2765:
2762:
2761:
2729:
2725:
2704:
2700:
2695:
2692:
2691:
2675:
2672:
2671:
2655:
2652:
2651:
2635:
2632:
2631:
2615:
2612:
2611:
2581:
2577:
2572:
2569:
2568:
2552:
2549:
2548:
2547:is smooth over
2532:
2529:
2528:
2508:
2505:
2504:
2488:
2485:
2484:
2481:
2454:
2439:
2435:
2424:
2422:
2419:
2418:
2386:
2382:
2374:
2371:
2370:
2324:
2321:
2320:
2292:
2289:
2288:
2272:
2269:
2268:
2231:
2223:
2220:
2219:
2216:
2195:
2192:
2191:
2168:
2163:
2162:
2160:
2157:
2156:
2133:
2128:
2122:
2119:
2118:
2101:
2096:
2078:
2075:
2074:
2048:
2043:
2042:
2034:
2031:
2030:
2014:
2011:
2010:
1990:
1986:
1974:
1970:
1968:
1965:
1964:
1944:
1940:
1931:
1927:
1922:
1919:
1918:
1901:
1896:
1895:
1880:
1876:
1874:
1871:
1870:
1850:
1845:
1844:
1835:
1830:
1829:
1797:
1787:
1783:
1768:
1764:
1741:
1740:
1738:
1734:
1729:
1721:
1718:
1717:
1686:
1683:
1682:
1653:
1652:
1643:
1638:
1637:
1628:
1624:
1610:
1607:
1606:
1590:
1587:
1586:
1582:
1577:
1556:
1553:
1552:
1528:
1525:
1524:
1523:is smooth over
1508:
1505:
1504:
1488:
1485:
1484:
1448:
1444:
1439:
1436:
1435:
1419:
1416:
1415:
1399:
1396:
1395:
1387:are said to be
1372:
1369:
1368:
1352:
1349:
1348:
1314:
1311:
1310:
1294:
1291:
1290:
1274:
1271:
1270:
1254:
1251:
1250:
1228:
1225:
1224:
1198:
1194:
1192:
1189:
1188:
1168:
1165:
1164:
1148:
1145:
1144:
1118:
1114:
1109:
1106:
1105:
1086:
1083:
1082:
1066:
1063:
1062:
1046:
1043:
1042:
1026:
1023:
1022:
994:
991:
990:
968:
965:
964:
938:
934:
932:
929:
928:
908:
905:
904:
903:, the group of
888:
885:
884:
868:
865:
864:
845:
842:
841:
825:
822:
821:
805:
802:
801:
800:is negative if
772:
768:
766:
763:
762:
746:
743:
742:
726:
723:
722:
693:
689:
687:
684:
683:
667:
664:
663:
647:
644:
643:
627:
624:
623:
585:
581:
575:
557:
554:
553:
534:
531:
530:
514:
511:
510:
490:
487:
486:
470:
467:
466:
441:
438:
437:
421:
418:
417:
398:
395:
394:
378:
375:
374:
354:
351:
350:
330:
327:
326:
310:
307:
306:
280:
276:
274:
271:
270:
254:
251:
250:
231:
228:
227:
207:
204:
203:
198:means a finite
183:
180:
179:
176:algebraic cycle
159:
156:
155:
139:
136:
135:
119:
116:
115:
88:
85:
84:
77:
17:
12:
11:
5:
8685:
8675:
8674:
8669:
8664:
8659:
8645:
8644:
8631:
8615:Voisin, Claire
8611:
8598:
8578:
8547:
8534:
8514:
8501:
8481:
8440:
8427:
8414:
8387:(3): 267–304,
8371:Bloch, Spencer
8365:
8362:
8361:
8360:
8350:
8347:
8345:
8344:
8335:
8323:
8316:
8292:
8283:
8274:
8265:
8256:
8247:
8238:
8229:
8220:
8211:
8202:
8193:
8181:
8171:
8169:
8166:
8164:
8161:
8160:
8159:
8154:
8149:
8144:
8137:
8134:
8114:Wei-Liang Chow
8093:
8090:
8051:
8050:Other variants
8048:
8005:
8002:
8000:
7997:
7996:
7995:
7936:
7935:
7908:
7898:
7868:
7858:conjecture on
7832:
7817:
7814:
7806:
7805:
7794:
7790:
7783:
7778:
7774:
7771:
7768:
7763:
7757:
7754:
7746:
7742:
7738:
7734:
7727:
7722:
7718:
7715:
7712:
7707:
7703:
7663:
7642:
7621:
7618:
7606:etale homology
7542:In this case (
7540:
7539:
7528:
7525:
7521:
7517:
7514:
7511:
7506:
7503:
7499:
7495:
7492:
7489:
7486:
7481:
7475:
7472:
7435:
7434:
7423:
7420:
7416:
7412:
7409:
7406:
7401:
7398:
7393:
7390:
7386:
7382:
7379:
7376:
7373:
7368:
7362:
7359:
7326:
7323:
7322:
7321:
7310:
7307:
7302:
7298:
7293:
7290:
7286:
7283:
7280:
7276:
7273:
7269:
7264:
7260:
7236:
7216:
7211:
7207:
7202:
7199:
7187:
7186:
7171:
7168:
7166:
7163:
7160:
7157:
7153:
7152:
7149:
7146:
7143:
7140:
7139:
7136:
7133:
7131:
7128:
7126:
7121:
7117:
7112:
7109:
7105:
7104:
7081:
7078:
7074:
7071:
7059:
7058:
7047:
7044:
7041:
7026:
7023:
7022:
7021:
7010:
7005:
7001:
6997:
6992:
6988:
6984:
6981:
6978:
6975:
6972:
6969:
6964:
6960:
6956:
6953:
6950:
6947:
6942:
6938:
6914:
6892:
6888:
6884:
6881:
6878:
6873:
6869:
6848:
6845:
6842:
6837:
6834:
6830:
6826:
6823:
6818:
6814:
6810:
6807:
6804:
6799:
6795:
6791:
6771:
6768:
6763:
6759:
6747:
6746:
6731:
6727:
6722:
6716:
6712:
6708:
6705:
6702:
6699:
6692:
6688:
6683:
6677:
6673:
6669:
6666:
6663:
6660:
6657:
6654:
6651:
6648:
6645:
6642:
6619:
6616:
6613:
6610:
6607:
6604:
6593:
6592:
6579:
6574:
6569:
6564:
6560:
6554:
6551:
6548:
6545:
6542:
6539:
6536:
6533:
6530:
6527:
6524:
6521:
6518:
6513:
6510:
6507:
6504:
6501:
6497:
6490:
6486:
6483:
6480:
6477:
6462:
6459:
6444:
6439:
6415:
6410:
6396:
6393:
6380:
6377:
6374:
6369:
6365:
6361:
6358:
6355:
6352:
6349:
6344:
6340:
6336:
6316:
6313:
6310:
6305:
6301:
6297:
6294:
6291:
6288:
6285:
6280:
6276:
6272:
6269:
6264:
6260:
6239:
6219:
6216:
6213:
6210:
6207:
6174:
6171:
6168:
6165:
6162:
6142:
6139:
6136:
6125:
6124:
6113:
6110:
6107:
6104:
6101:
6098:
6095:
6092:
6087:
6083:
6079:
6076:
6073:
6070:
6067:
6062:
6058:
6054:
6051:
6048:
6045:
6042:
6037:
6033:
6029:
6016:exact sequence
6014:, there is an
6003:
5983:
5963:
5943:
5916:
5913:
5910:
5905:
5902:
5899:
5895:
5891:
5888:
5885:
5882:
5879:
5874:
5870:
5866:
5863:
5858:
5854:
5828:
5808:
5788:
5785:
5782:
5779:
5776:
5750:
5730:
5706:
5686:
5666:
5663:
5660:
5657:
5654:
5634:
5614:
5594:
5573:
5569:
5566:
5563:
5560:
5555:
5551:
5547:
5527:
5507:
5484:
5464:
5461:
5458:
5453:
5449:
5445:
5442:
5439:
5436:
5433:
5428:
5424:
5420:
5417:
5412:
5408:
5384:
5364:
5361:
5358:
5355:
5352:
5337:
5334:
5313:
5293:
5280:is called the
5269:
5266:
5263:
5260:
5236:
5216:
5192:
5172:
5169:
5166:
5163:
5143:
5132:
5131:
5120:
5117:
5113:
5109:
5106:
5103:
5100:
5095:
5091:
5087:
5084:
5081:
5078:
5075:
5072:
5069:
5066:
5053:exact sequence
5040:
5020:
5009:elliptic curve
4996:
4984:
4981:
4980:
4979:
4965:
4962:
4959:
4956:
4953:
4948:
4944:
4940:
4935:
4931:
4927:
4922:
4919:
4916:
4913:
4910:
4906:
4899:
4893:
4890:
4887:
4884:
4881:
4876:
4872:
4868:
4863:
4860:
4857:
4854:
4849:
4844:
4839:
4834:
4830:
4826:
4820:
4817:
4812:
4808:
4804:
4799:
4795:
4791:
4768:
4765:
4762:
4757:
4753:
4730:
4725:
4703:
4700:
4697:
4694:
4689:
4684:
4679:
4674:
4670:
4666:
4661:
4657:
4640:
4637:
4636:
4635:
4619:
4615:
4611:
4608:
4605:
4600:
4597:
4594:
4590:
4584:
4580:
4576:
4571:
4568:
4565:
4561:
4555:
4551:
4547:
4542:
4538:
4532:
4529:
4526:
4523:
4520:
4517:
4512:
4508:
4504:
4498:
4495:
4492:
4489:
4486:
4482:
4478:
4473:
4469:
4465:
4442:
4420:
4416:
4412:
4409:
4406:
4401:
4397:
4376:
4373:
4370:
4367:
4362:
4359:
4356:
4352:
4345:
4339:
4334:
4330:
4326:
4323:
4303:
4283:
4263:
4260:
4257:
4253:
4229:
4209:
4189:
4186:
4183:
4171:
4168:
4147:
4144:
4120:
4100:
4097:
4077:
4074:
4071:
4051:
4031:
4009:
4004:
3982:
3960:
3956:
3944:
3943:
3930:
3926:
3921:
3917:
3914:
3911:
3908:
3905:
3902:
3899:
3896:
3893:
3870:
3850:
3828:
3823:
3801:
3781:
3759:
3755:
3751:
3731:
3711:
3688:
3668:
3657:
3656:
3645:
3642:
3637:
3634:
3631:
3627:
3623:
3619:
3615:
3612:
3609:
3605:
3601:
3598:
3593:
3588:
3583:
3578:
3574:
3570:
3547:
3525:
3520:
3503:
3500:
3498:
3495:
3482:
3479:
3476:
3473:
3470:
3467:
3447:
3427:
3424:
3421:
3406:William Fulton
3393:
3373:
3353:
3333:
3309:
3306:
3303:
3300:
3297:
3294:
3274:
3254:
3234:
3214:
3211:
3208:
3205:
3202:
3199:
3172:
3169:
3166:
3146:
3143:
3140:
3120:
3117:
3114:
3109:
3106:
3103:
3099:
3095:
3075:
3072:
3069:
3066:
3063:
3060:
3036:
3016:
2996:
2976:
2956:
2936:
2916:
2905:
2904:
2893:
2890:
2887:
2884:
2879:
2876:
2873:
2869:
2865:
2862:
2859:
2856:
2853:
2848:
2844:
2840:
2837:
2834:
2831:
2828:
2823:
2819:
2815:
2788:
2785:
2782:
2777:
2773:
2769:
2749:
2746:
2743:
2738:
2735:
2732:
2728:
2724:
2721:
2718:
2715:
2712:
2707:
2703:
2699:
2679:
2659:
2639:
2619:
2595:
2592:
2589:
2584:
2580:
2576:
2556:
2536:
2512:
2492:
2480:
2477:
2464:
2460:
2457:
2453:
2450:
2447:
2442:
2438:
2434:
2430:
2427:
2406:
2403:
2400:
2397:
2394:
2389:
2385:
2381:
2378:
2358:
2355:
2352:
2349:
2346:
2343:
2340:
2337:
2334:
2331:
2328:
2308:
2305:
2302:
2299:
2296:
2276:
2256:
2253:
2250:
2247:
2244:
2241:
2237:
2234:
2230:
2227:
2215:
2212:
2199:
2177:
2174:
2171:
2166:
2144:
2141:
2136:
2131:
2127:
2104:
2099:
2095:
2091:
2088:
2085:
2082:
2062:
2057:
2054:
2051:
2046:
2041:
2038:
2018:
1998:
1993:
1989:
1985:
1982:
1977:
1973:
1952:
1947:
1943:
1939:
1934:
1930:
1926:
1904:
1899:
1894:
1891:
1888:
1883:
1879:
1867:
1866:
1853:
1848:
1843:
1838:
1833:
1828:
1824:
1818:
1815:
1812:
1809:
1806:
1803:
1800:
1795:
1790:
1786:
1782:
1779:
1776:
1771:
1767:
1763:
1760:
1757:
1754:
1751:
1748:
1744:
1737:
1728:
1725:
1702:
1699:
1696:
1693:
1690:
1670:
1667:
1664:
1661:
1656:
1651:
1646:
1641:
1636:
1631:
1627:
1623:
1620:
1617:
1614:
1594:
1581:
1578:
1576:
1573:
1560:
1532:
1512:
1492:
1468:
1465:
1462:
1457:
1454:
1451:
1447:
1443:
1423:
1403:
1376:
1356:
1336:
1333:
1330:
1327:
1324:
1321:
1318:
1298:
1278:
1258:
1238:
1235:
1232:
1212:
1209:
1206:
1201:
1197:
1185:quotient group
1172:
1152:
1132:
1129:
1126:
1121:
1117:
1113:
1090:
1070:
1050:
1030:
1010:
1007:
1004:
1001:
998:
978:
975:
972:
952:
949:
946:
941:
937:
912:
892:
872:
849:
829:
809:
789:
786:
783:
780:
775:
771:
750:
730:
710:
707:
704:
701:
696:
692:
671:
651:
631:
620:
619:
608:
605:
602:
599:
596:
593:
588:
584:
578:
574:
570:
567:
564:
561:
538:
518:
494:
474:
451:
448:
445:
425:
416:For a variety
402:
382:
373:on the set of
358:
334:
314:
294:
291:
288:
283:
279:
258:
248:natural number
235:
211:
187:
163:
143:
123:
92:
76:
73:
29:Wei-Liang Chow
15:
9:
6:
4:
3:
2:
8684:
8673:
8670:
8668:
8665:
8663:
8660:
8658:
8655:
8654:
8652:
8642:
8638:
8634:
8628:
8624:
8620:
8616:
8612:
8609:
8605:
8601:
8599:9781400837120
8595:
8591:
8587:
8583:
8579:
8576:
8572:
8568:
8564:
8560:
8556:
8552:
8548:
8545:
8541:
8537:
8531:
8527:
8523:
8519:
8515:
8512:
8508:
8504:
8498:
8494:
8490:
8486:
8482:
8479:
8475:
8471:
8467:
8463:
8459:
8455:
8451:
8450:
8445:
8441:
8437:
8433:
8428:
8424:
8420:
8415:
8412:
8408:
8404:
8400:
8395:
8390:
8386:
8382:
8381:
8376:
8372:
8368:
8367:
8358:
8353:
8352:
8339:
8333:
8327:
8319:
8317:9780821822432
8313:
8309:
8305:
8304:
8296:
8287:
8278:
8269:
8260:
8251:
8242:
8233:
8224:
8215:
8206:
8197:
8191:
8185:
8176:
8172:
8158:
8155:
8153:
8150:
8148:
8145:
8143:
8140:
8139:
8133:
8131:
8127:
8123:
8119:
8115:
8111:
8107:
8103:
8099:
8089:
8087:
8083:
8079:
8075:
8070:
8068:
8064:
8060:
8056:
8047:
8045:
8040:
8038:
8034:
8030:
8026:
8022:
8018:
8014:
8010:
7993:
7989:
7985:
7981:
7977:
7973:
7969:
7965:
7961:
7957:
7954:
7950:
7946:
7942:
7938:
7937:
7933:
7929:
7925:
7921:
7917:
7913:
7909:
7906:
7901:
7897:
7893:
7889:
7885:
7881:
7877:
7873:
7869:
7866:
7861:
7857:
7853:
7849:
7846:of dimension
7845:
7841:
7835:
7831:
7827:
7823:
7822:
7821:
7813:
7811:
7792:
7776:
7769:
7761:
7744:
7740:
7736:
7720:
7713:
7705:
7701:
7693:
7692:
7691:
7689:
7685:
7681:
7677:
7673:
7669:
7662:
7658:
7654:
7650:
7645:
7641:
7638:
7637:Chern classes
7634:
7630:
7627:
7626:vector bundle
7617:
7615:
7611:
7607:
7603:
7599:
7596:For a scheme
7594:
7592:
7588:
7584:
7580:
7576:
7572:
7568:
7564:
7560:
7556:
7551:
7549:
7545:
7526:
7515:
7512:
7504:
7501:
7497:
7487:
7479:
7460:
7459:
7458:
7456:
7452:
7448:
7444:
7440:
7421:
7410:
7407:
7399:
7396:
7391:
7388:
7384:
7374:
7366:
7347:
7346:
7345:
7343:
7339:
7334:
7332:
7305:
7300:
7296:
7291:
7288:
7281:
7274:
7271:
7262:
7258:
7250:
7249:
7248:
7234:
7214:
7209:
7205:
7200:
7197:
7169:
7158:
7155:
7134:
7124:
7119:
7115:
7110:
7107:
7095:
7094:
7093:
7079:
7076:
7072:
7069:
7045:
7039:
7032:
7031:
7030:
7003:
6999:
6990:
6986:
6982:
6979:
6976:
6970:
6962:
6958:
6954:
6948:
6940:
6936:
6928:
6927:
6926:
6912:
6890:
6886:
6882:
6879:
6876:
6871:
6867:
6843:
6835:
6832:
6828:
6824:
6816:
6812:
6808:
6805:
6802:
6797:
6793:
6769:
6766:
6761:
6757:
6729:
6725:
6714:
6710:
6706:
6703:
6697:
6690:
6686:
6675:
6671:
6667:
6664:
6658:
6652:
6649:
6646:
6640:
6633:
6632:
6631:
6617:
6614:
6608:
6602:
6577:
6572:
6558:
6546:
6543:
6540:
6534:
6531:
6525:
6519:
6508:
6505:
6502:
6488:
6484:
6481:
6478:
6475:
6468:
6467:
6466:
6458:
6442:
6413:
6392:
6375:
6367:
6363:
6359:
6350:
6342:
6338:
6334:
6311:
6303:
6299:
6295:
6286:
6278:
6274:
6270:
6267:
6262:
6258:
6237:
6217:
6211:
6208:
6205:
6196:
6194:
6193:
6188:
6172:
6166:
6163:
6160:
6140:
6134:
6111:
6108:
6099:
6096:
6093:
6085:
6081:
6077:
6068:
6060:
6056:
6052:
6043:
6035:
6031:
6027:
6020:
6019:
6018:
6017:
6001:
5981:
5961:
5954:over a field
5941:
5933:
5928:
5911:
5903:
5900:
5897:
5893:
5889:
5880:
5872:
5868:
5864:
5861:
5856:
5852:
5844:
5843:
5826:
5806:
5786:
5780:
5777:
5774:
5767:
5766:flat morphism
5762:
5748:
5728:
5721:of the field
5720:
5704:
5684:
5661:
5655:
5652:
5645:has the form
5632:
5612:
5592:
5561:
5553:
5549:
5545:
5525:
5505:
5498:
5497:proper scheme
5482:
5459:
5451:
5447:
5443:
5434:
5426:
5422:
5418:
5415:
5410:
5406:
5398:
5395:, there is a
5382:
5362:
5356:
5353:
5350:
5343:
5336:Functoriality
5333:
5331:
5327:
5311:
5291:
5283:
5264:
5258:
5250:
5234:
5214:
5206:
5190:
5167:
5161:
5141:
5118:
5101:
5093:
5089:
5085:
5076:
5070:
5064:
5057:
5056:
5055:
5054:
5051:fits into an
5038:
5018:
5011:over a field
5010:
4994:
4960:
4957:
4954:
4951:
4946:
4942:
4938:
4933:
4929:
4917:
4914:
4911:
4897:
4888:
4885:
4882:
4879:
4874:
4870:
4858:
4847:
4832:
4828:
4824:
4818:
4810:
4806:
4797:
4793:
4789:
4782:
4781:
4780:
4766:
4763:
4760:
4755:
4751:
4728:
4695:
4682:
4664:
4659:
4655:
4646:
4617:
4613:
4609:
4606:
4603:
4598:
4595:
4592:
4588:
4582:
4578:
4574:
4569:
4566:
4563:
4559:
4553:
4549:
4545:
4540:
4536:
4527:
4518:
4510:
4506:
4502:
4496:
4487:
4471:
4467:
4463:
4456:
4455:
4454:
4440:
4418:
4414:
4410:
4407:
4404:
4399:
4395:
4368:
4357:
4332:
4328:
4324:
4321:
4301:
4281:
4258:
4243:
4227:
4207:
4187:
4181:
4167:
4165:
4161:
4145:
4142:
4134:
4118:
4098:
4095:
4075:
4072:
4069:
4049:
4029:
4007:
3980:
3958:
3954:
3928:
3924:
3919:
3915:
3912:
3906:
3900:
3894:
3884:
3883:
3882:
3868:
3848:
3826:
3799:
3779:
3757:
3753:
3749:
3729:
3709:
3702:
3686:
3666:
3643:
3635:
3632:
3629:
3625:
3617:
3610:
3599:
3591:
3576:
3572:
3568:
3561:
3560:
3559:
3545:
3523:
3509:
3494:
3477:
3468:
3445:
3425:
3422:
3419:
3411:
3407:
3391:
3371:
3351:
3331:
3323:
3304:
3295:
3272:
3252:
3232:
3209:
3200:
3189:
3184:
3170:
3167:
3164:
3144:
3141:
3138:
3115:
3107:
3104:
3101:
3097:
3093:
3070:
3061:
3050:
3034:
3014:
2994:
2974:
2954:
2934:
2914:
2891:
2885:
2877:
2874:
2871:
2867:
2863:
2854:
2846:
2842:
2838:
2835:
2829:
2821:
2817:
2813:
2806:
2805:
2804:
2802:
2783:
2775:
2771:
2767:
2744:
2736:
2733:
2730:
2726:
2722:
2719:
2713:
2705:
2701:
2697:
2677:
2657:
2637:
2617:
2609:
2590:
2582:
2578:
2574:
2554:
2534:
2526:
2510:
2490:
2479:The Chow ring
2476:
2458:
2455:
2451:
2448:
2440:
2436:
2432:
2428:
2425:
2401:
2398:
2395:
2387:
2383:
2379:
2376:
2353:
2347:
2344:
2341:
2335:
2329:
2326:
2303:
2297:
2294:
2274:
2251:
2245:
2242:
2239:
2235:
2232:
2228:
2225:
2211:
2197:
2175:
2172:
2169:
2142:
2139:
2134:
2129:
2125:
2102:
2097:
2093:
2089:
2086:
2083:
2080:
2055:
2052:
2049:
2036:
2016:
1996:
1991:
1987:
1983:
1980:
1975:
1971:
1945:
1941:
1937:
1932:
1928:
1902:
1889:
1886:
1881:
1877:
1851:
1841:
1836:
1822:
1813:
1810:
1807:
1804:
1801:
1788:
1784:
1780:
1777:
1774:
1769:
1765:
1755:
1752:
1749:
1735:
1726:
1723:
1716:
1715:
1714:
1700:
1697:
1694:
1691:
1688:
1662:
1649:
1644:
1629:
1625:
1621:
1618:
1615:
1612:
1592:
1572:
1558:
1550:
1546:
1530:
1510:
1490:
1482:
1463:
1455:
1452:
1449:
1445:
1441:
1421:
1401:
1392:
1390:
1374:
1354:
1331:
1325:
1319:
1296:
1276:
1256:
1233:
1207:
1199:
1195:
1186:
1170:
1150:
1127:
1119:
1115:
1111:
1104:
1088:
1068:
1048:
1028:
1005:
1002:
999:
973:
947:
939:
935:
926:
910:
890:
870:
863:For a scheme
861:
847:
827:
807:
784:
778:
773:
769:
748:
728:
705:
699:
694:
690:
669:
649:
629:
606:
603:
597:
591:
586:
582:
576:
572:
568:
562:
552:
551:
550:
536:
516:
508:
492:
472:
465:
449:
446:
443:
436:of dimension
423:
414:
400:
380:
372:
356:
348:
332:
312:
289:
281:
277:
256:
249:
233:
225:
209:
201:
185:
177:
161:
141:
121:
113:
109:
106:
90:
83:over a field
82:
72:
70:
66:
62:
58:
54:
50:
46:
42:
38:
34:
30:
27:(named after
26:
22:
8618:
8585:
8558:
8554:
8521:
8488:
8453:
8447:
8435:
8422:
8384:
8378:
8374:
8356:
8349:Introductory
8338:
8326:
8302:
8295:
8286:
8277:
8268:
8259:
8250:
8241:
8232:
8223:
8214:
8205:
8196:
8184:
8175:
8095:
8071:
8058:
8053:
8041:
8007:
7991:
7983:
7979:
7975:
7971:
7967:
7966:showed that
7959:
7955:
7948:
7944:
7940:
7931:
7927:
7923:
7911:
7899:
7895:
7888:finite field
7883:
7871:
7847:
7843:
7839:
7833:
7829:
7819:
7807:
7684:Grothendieck
7675:
7667:
7660:
7656:
7652:
7648:
7643:
7639:
7632:
7628:
7623:
7613:
7609:
7601:
7597:
7595:
7590:
7586:
7582:
7578:
7574:
7552:
7547:
7546:smooth over
7543:
7541:
7450:
7446:
7442:
7438:
7436:
7337:
7335:
7330:
7328:
7188:
7060:
7028:
6748:
6594:
6464:
6398:
6197:
6190:
6126:
5931:
5929:
5842:homomorphism
5840:
5763:
5396:
5339:
5249:number field
5133:
4986:
4642:
4314:. If we let
4173:
3945:
3841:and degrees
3658:
3558:is the ring
3505:
3185:
3049:transversely
2906:
2482:
2217:
1868:
1583:
1549:line bundles
1545:Picard group
1393:
1388:
1102:
924:
862:
621:
415:
346:
269:, the group
175:
80:
78:
24:
18:
8561:: 268–326,
8456:: 450–479,
8330:B. Totaro,
7816:Conjectures
7573:shows that
5330:uncountable
2801:graded ring
2608:codimension
112:finite type
25:Chow groups
8651:Categories
8575:58.1229.01
8377:-theory",
8163:References
8126:MacPherson
8013:MacPherson
7331:cycle maps
7325:Cycle maps
3047:intersect
2630:cycles on
1103:Chow group
860:singular.
8470:0003-486X
8403:0001-8708
8168:Citations
7920:Beilinson
7777:⊗
7741:∏
7737:≅
7721:⊗
7670:) be the
7494:→
7381:→
7297:×
7263:∗
7206:×
7165:→
7148:↓
7142:↓
7130:→
7116:×
7077:⊂
7043:→
7000:α
6980:⋯
6971:α
6949:α
6941:∗
6913:α
6880:…
6844:α
6833:−
6813:α
6806:…
6794:α
6707:−
6698:⋯
6668:−
6647:α
6609:α
6563:→
6532:−
6485:
6368:∗
6357:→
6343:∗
6293:→
6263:∗
6215:→
6170:→
6164:−
6138:→
6106:→
6097:−
6075:→
6050:→
5887:→
5857:∗
5784:→
5656:
5568:→
5441:→
5411:∗
5360:→
5116:→
5108:→
5083:→
5068:→
4961:ζ
4943:ζ
4918:ζ
4898:≅
4889:ζ
4871:ζ
4859:ζ
4833:∙
4819:≅
4798:∙
4683:⊕
4607:⋯
4596:−
4589:ζ
4567:−
4560:ζ
4537:ζ
4528:ζ
4511:∙
4497:≅
4472:∙
4408:…
4322:ζ
4185:→
4073:∩
3901:⋅
3600:≅
3577:∗
3423:∩
3142:∩
2861:→
2836:×
2776:∗
2734:−
2567:, define
2433:∈
2380:∈
2348:
2342:≅
2330:
2246:
2240:∈
2173:−
2053:−
1893:→
1878:π
1842:×
1827:↪
1778:…
1622:∈
1453:−
779:
700:
592:
573:∑
103:to be an
43:over any
39:)) of an
8617:(2002),
8520:(1998),
8487:(1977),
8364:Advanced
8136:See also
8021:functors
7999:Variants
7880:tensored
7593:> 1.
7292:′
7275:′
7201:′
7159:′
7111:′
7073:′
6925:is then
4200:of rank
3497:Examples
2650:. (When
2459:′
2429:′
2236:′
2073:, since
989:for all
923:-cycles
761:. (Thus
462:and any
105:integral
49:homology
8641:1997577
8608:1764202
8544:1644323
8511:0463174
8478:0082173
8411:0852815
8092:History
7964:Mumford
7608:. When
7589:) with
7449:. When
5324:is the
5227:. When
4983:Remarks
1503:. When
1479:is the
1347:, then
1183:is the
549:-cycle
529:is the
507:divisor
369:is the
224:integer
81:variety
35: (
8639:
8629:
8606:
8596:
8573:
8542:
8532:
8509:
8499:
8476:
8468:
8409:
8401:
8314:
8122:Fulton
8027:and a
8009:Fulton
7874:, the
7569:. The
5764:For a
5719:degree
5340:For a
5007:be an
3946:where
3701:degree
3659:where
1101:. The
741:along
347:cycles
108:scheme
61:smooth
23:, the
8025:group
7916:Bloch
7852:Bloch
7651:) in
5741:over
5518:over
5247:is a
4714:over
3384:over
1605:, so
1309:have
222:with
174:, an
114:over
51:of a
45:field
8627:ISBN
8594:ISBN
8530:ISBN
8497:ISBN
8466:ISSN
8399:ISSN
8312:ISBN
8124:and
8029:ring
8011:and
7856:Kato
7824:The
6767:>
6482:Spec
5653:Spec
4387:and
4042:and
3792:and
3408:and
3344:and
3245:and
3027:and
2987:and
2927:and
2525:ring
2417:and
1731:Proj
1367:and
1289:and
37:1958
8571:JFM
8563:doi
8458:doi
8389:doi
7990:of
5994:of
5761:.)
5697:of
5284:of
5207:of
5183:of
4131:is
3699:of
3086:in
2345:Pic
2327:Div
2243:Pic
2155:is
1551:on
1547:of
1483:of
1187:of
1143:of
1081:on
1041:of
770:ord
691:ord
662:of
583:ord
509:of
485:on
305:of
178:on
110:of
31:by
19:In
8653::
8637:MR
8635:,
8625:,
8621:,
8604:MR
8602:,
8588:,
8569:,
8557:,
8540:MR
8538:,
8528:,
8507:MR
8505:,
8495:,
8491:,
8474:MR
8472:,
8464:,
8454:64
8452:,
8434:,
8421:,
8407:MR
8405:,
8397:,
8385:61
8383:,
8310:.
8306:.
8132:.
8088:.
7836:-1
7830:CH
7682:,
7653:CH
7583:CH
7575:CH
7344::
6457:.
6391:.
6195:.
5927:.
5251:,
5119:0.
4166:.
3861:,
3493:.
3183:.
1571:.
1391:.
413:.
8565::
8559:4
8460::
8391::
8375:K
8320:.
8059:Q
7994:.
7992:X
7984:K
7980:X
7972:X
7968:K
7960:X
7958:(
7956:h
7949:K
7945:X
7941:X
7934:.
7932:X
7928:X
7924:X
7918:–
7912:X
7907:.
7900:l
7896:Q
7884:Q
7872:X
7854:–
7848:n
7844:X
7840:X
7838:(
7834:n
7793:.
7789:Q
7782:Z
7773:)
7770:X
7767:(
7762:i
7756:H
7753:C
7745:i
7733:Q
7726:Z
7717:)
7714:X
7711:(
7706:0
7702:K
7676:X
7668:X
7666:(
7664:0
7661:K
7657:X
7655:(
7649:E
7647:(
7644:i
7640:c
7633:X
7629:E
7614:k
7610:X
7602:k
7598:X
7591:j
7587:X
7585:(
7579:X
7577:(
7548:C
7544:X
7527:.
7524:)
7520:Z
7516:,
7513:X
7510:(
7505:j
7502:2
7498:H
7491:)
7488:X
7485:(
7480:j
7474:H
7471:C
7451:X
7447:i
7443:X
7439:i
7422:.
7419:)
7415:Z
7411:,
7408:X
7405:(
7400:M
7397:B
7392:i
7389:2
7385:H
7378:)
7375:X
7372:(
7367:i
7361:H
7358:C
7338:X
7309:]
7306:X
7301:S
7289:S
7285:[
7282:=
7279:]
7272:S
7268:[
7259:f
7235:X
7215:X
7210:S
7198:S
7170:S
7156:S
7135:X
7125:X
7120:S
7108:S
7080:S
7070:S
7046:S
7040:X
7009:]
7004:k
6996:[
6991:k
6987:e
6983:+
6977:+
6974:]
6968:[
6963:1
6959:e
6955:=
6952:]
6946:[
6937:f
6891:k
6887:e
6883:,
6877:,
6872:1
6868:e
6847:)
6841:(
6836:1
6829:f
6825:=
6822:}
6817:k
6809:,
6803:,
6798:1
6790:{
6770:1
6762:i
6758:e
6730:k
6726:e
6721:)
6715:k
6711:a
6704:y
6701:(
6691:1
6687:e
6682:)
6676:1
6672:a
6665:y
6662:(
6659:=
6656:)
6653:y
6650:,
6644:(
6641:g
6618:0
6615:=
6612:)
6606:(
6603:f
6578:1
6573:x
6568:A
6559:)
6553:)
6550:)
6547:y
6544:,
6541:x
6538:(
6535:g
6529:)
6526:x
6523:(
6520:f
6517:(
6512:]
6509:y
6506:,
6503:x
6500:[
6496:C
6489:(
6479::
6476:f
6443:1
6438:P
6414:2
6409:A
6379:)
6376:X
6373:(
6364:H
6360:C
6354:)
6351:Y
6348:(
6339:H
6335:C
6315:)
6312:X
6309:(
6304:i
6300:H
6296:C
6290:)
6287:Y
6284:(
6279:i
6275:H
6271:C
6268::
6259:f
6238:k
6218:Y
6212:X
6209::
6206:f
6173:X
6167:Z
6161:X
6141:X
6135:Z
6112:,
6109:0
6103:)
6100:Z
6094:X
6091:(
6086:i
6082:H
6078:C
6072:)
6069:X
6066:(
6061:i
6057:H
6053:C
6047:)
6044:Z
6041:(
6036:i
6032:H
6028:C
6002:X
5982:Z
5962:k
5942:X
5915:)
5912:X
5909:(
5904:r
5901:+
5898:i
5894:H
5890:C
5884:)
5881:Y
5878:(
5873:i
5869:H
5865:C
5862::
5853:f
5827:r
5807:k
5787:Y
5781:X
5778::
5775:f
5749:k
5729:E
5705:k
5685:E
5665:)
5662:E
5659:(
5633:X
5613:k
5593:X
5572:Z
5565:)
5562:X
5559:(
5554:0
5550:H
5546:C
5526:k
5506:X
5483:i
5463:)
5460:Y
5457:(
5452:i
5448:H
5444:C
5438:)
5435:X
5432:(
5427:i
5423:H
5419:C
5416::
5407:f
5383:k
5363:Y
5357:X
5354::
5351:f
5312:k
5292:X
5268:)
5265:k
5262:(
5259:X
5235:k
5215:X
5203:-
5191:k
5171:)
5168:k
5165:(
5162:X
5142:X
5112:Z
5105:)
5102:X
5099:(
5094:0
5090:H
5086:C
5080:)
5077:k
5074:(
5071:X
5065:0
5039:X
5019:k
4995:X
4964:)
4958:H
4955:a
4952:+
4947:2
4939:,
4934:2
4930:H
4926:(
4921:]
4915:,
4912:H
4909:[
4905:Z
4892:)
4886:H
4883:a
4880:+
4875:2
4867:(
4862:]
4856:[
4853:)
4848:1
4843:P
4838:(
4829:H
4825:C
4816:)
4811:a
4807:F
4803:(
4794:H
4790:C
4767:H
4764:a
4761:=
4756:1
4752:c
4729:1
4724:P
4702:)
4699:)
4696:a
4693:(
4688:O
4678:O
4673:(
4669:P
4665:=
4660:a
4656:F
4618:r
4614:c
4610:+
4604:+
4599:2
4593:r
4583:2
4579:c
4575:+
4570:1
4564:r
4554:1
4550:c
4546:+
4541:r
4531:]
4525:[
4522:)
4519:X
4516:(
4507:H
4503:C
4494:)
4491:)
4488:E
4485:(
4481:P
4477:(
4468:H
4464:C
4441:E
4419:r
4415:c
4411:,
4405:,
4400:1
4396:c
4375:)
4372:)
4369:1
4366:(
4361:)
4358:E
4355:(
4351:P
4344:O
4338:(
4333:1
4329:c
4325:=
4302:E
4282:X
4262:)
4259:E
4256:(
4252:P
4228:X
4208:r
4188:X
4182:E
4146:b
4143:a
4119:k
4099:b
4096:a
4076:Z
4070:Y
4050:Z
4030:Y
4008:n
4003:P
3981:k
3959:n
3955:H
3929:n
3925:H
3920:b
3916:a
3913:=
3910:]
3907:Z
3904:[
3898:]
3895:Y
3892:[
3869:b
3849:a
3827:n
3822:P
3800:Z
3780:Y
3758:a
3754:H
3750:d
3730:a
3710:d
3687:Y
3667:H
3644:,
3641:)
3636:1
3633:+
3630:n
3626:H
3622:(
3618:/
3614:]
3611:H
3608:[
3604:Z
3597:)
3592:n
3587:P
3582:(
3573:H
3569:C
3546:k
3524:n
3519:P
3481:]
3478:Z
3475:[
3472:]
3469:Y
3466:[
3446:X
3426:Z
3420:Y
3392:k
3372:X
3352:Z
3332:Y
3308:]
3305:Z
3302:[
3299:]
3296:Y
3293:[
3273:X
3253:Z
3233:Y
3213:]
3210:Z
3207:[
3204:]
3201:Y
3198:[
3171:j
3168:+
3165:i
3145:Z
3139:Y
3119:)
3116:X
3113:(
3108:j
3105:+
3102:i
3098:H
3094:C
3074:]
3071:Z
3068:[
3065:]
3062:Y
3059:[
3035:Z
3015:Y
2995:j
2975:i
2955:X
2935:Z
2915:Y
2892:.
2889:)
2886:X
2883:(
2878:j
2875:+
2872:i
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2855:X
2852:(
2847:j
2843:H
2839:C
2833:)
2830:X
2827:(
2822:i
2818:H
2814:C
2787:)
2784:X
2781:(
2772:H
2768:C
2748:)
2745:X
2742:(
2737:i
2731:n
2727:H
2723:C
2720:=
2717:)
2714:X
2711:(
2706:i
2702:H
2698:C
2678:n
2658:X
2638:X
2618:i
2610:-
2594:)
2591:X
2588:(
2583:i
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2351:(
2339:)
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2333:(
2307:)
2304:C
2301:(
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2275:C
2255:)
2252:C
2249:(
2233:L
2229:,
2226:L
2198:d
2176:1
2170:n
2165:P
2143:0
2140:=
2135:d
2130:0
2126:x
2103:d
2098:0
2094:x
2090:t
2087:+
2084:f
2081:s
2061:]
2056:1
2050:n
2045:P
2040:[
2037:d
2017:d
1997:g
1992:0
1988:t
1984:+
1981:f
1976:0
1972:s
1951:]
1946:0
1942:t
1938::
1933:0
1929:s
1925:[
1903:1
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1317:[
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