31:
1005:
614:
6615:
The affine scheme has "classical points", which correspond with points of the variety (and hence maximal ideals of the coordinate ring of the variety), and also a point for each closed subvariety of the variety (these points correspond to prime, non-maximal ideals of the coordinate ring). This
6528:
If an author requires the base field of an affine variety to be algebraically closed (as this article does), then irreducible affine algebraic sets over non-algebraically closed fields are a generalization of affine varieties. This generalization notably includes affine varieties over the
2425:
2430:
If the point is singular, the affine subspace defined by these equations is also called a tangent space by some authors, while other authors say that there is no tangent space at a singular point. A more intrinsic definition, which does not use coordinates is given by
3762:
3320:, is the inverse of the function assigning an algebraic set to a radical ideal, by the nullstellensatz. Hence the correspondence between affine algebraic sets and radical ideals is a bijection. The coordinate ring of an affine algebraic set is
6616:
creates a more well-defined notion of the "generic point" of an affine variety, by assigning to each closed subvariety an open point that is dense in the subvariety. More generally, an affine scheme is an affine variety if it is
5750:
388:
2158:
1548:
5111:
1871:
3422:
is not prime. Affine subvarieties are precisely those whose coordinate ring is an integral domain. This is because an ideal is prime if and only if the quotient of the ring by the ideal is an integral domain.
1343:
4500:
between the coordinate rings (going in the opposite direction), and for each such homomorphism, there is a morphism of the varieties associated to the coordinate rings. This can be shown explicitly: let
1737:
1628:
3625:
2258:
5825:
5293:
2909:
3630:
2750:
4806:
Similarly, for each homomorphism of the coordinate rings, a morphism of the affine varieties can be constructed in the opposite direction. Mirroring the paragraph above, a homomorphism
6027:.) This makes the cohomological study of an affine variety non-existent, in a sharp contrast to the projective case in which cohomology groups of line bundles are of central interest.
4891:
3811:
5552:
1972:
5918:
2030:
5364:
4750:
3207:
2670:
2605:
4784:
3298:
374:
5856:
5185:
5039:
3484:
3416:
948:
1143:
2541:
1067:
91:
1413:
910:
3513:
5984:
3858:
3251:
5600:
3066:
2504:
5620:
5465:
2814:
6010:
2247:
1199:
982:
716:
4263:
A morphism, or regular map, of affine varieties is a function between affine varieties that is polynomial in each coordinate: more precisely, for affine varieties
6546:
are obtained by gluing affine varieties. Linear structures that are attached to varieties are also (trivially) affine varieties; e.g., tangent spaces, fibers of
3152:
3038:
178:
879:
5634:
780:
is an integer associated to every variety, and even to every algebraic set, whose importance relies on the large number of its equivalent definitions (see
3515:
As maximal ideals are radical, maximal ideals correspond to minimal algebraic sets (those that contain no proper algebraic subsets), which are points in
609:{\displaystyle V(f_{1},\ldots ,f_{m})=\left\{(a_{1},\ldots ,a_{n})\in k^{n}\;|\;f_{1}(a_{1},\ldots ,a_{n})=\ldots =f_{m}(a_{1},\ldots ,a_{n})=0\right\}.}
3860:
An algebraic subset is a point if and only if the coordinate ring of the subset is a field, as the quotient of a ring by a maximal ideal is a field.
2072:
1666:-rational points; each such point is the second intersection point of the curve and a line with a rational slope passing through the rational point.
1453:
3863:
The following table summarises this correspondence, for algebraic subsets of an affine variety and ideals of the corresponding coordinate ring:
4434:
going in the opposite direction. Because of this, along with the fact that there is a one-to-one correspondence between affine varieties over
623:
is an affine algebraic set that is not the union of two proper affine algebraic subsets. Such an affine algebraic set is often said to be
5044:
1787:
1269:
2420:{\displaystyle \sum _{i=1}^{n}{\frac {\partial f_{j}}{\partial {x_{i}}}}(a_{1},\dots ,a_{n})(x_{i}-a_{i})=0,\quad j=1,\dots ,r.}
1672:
1563:
6490:
3526:
6759:
2962:
The geometric structure of an affine variety is linked in a deep way to the algebraic structure of its coordinate ring. Let
5758:
17:
4172:
of the
Zariski topologies on the two spaces. Indeed, the product topology is generated by products of the basic open sets
6725:
3757:{\displaystyle (a_{1},\ldots ,a_{n})\mapsto \langle {\overline {x_{1}-a_{1}}},\ldots ,{\overline {x_{n}-a_{n}}}\rangle ,}
5218:
6686:
This is because, over an algebraically closed field, the tensor product of integral domains is an integral domain; see
3088:. The reason that the base field is required to be algebraically closed is that affine varieties automatically satisfy
2823:
995:
of the coordinate ring of the variety. (Similarly, the normalization of a projective variety is a projective variety.)
6887:
2682:
2188:
781:
6793:
5938:
gives a cohomological characterization of an affine variety; it says an algebraic variety is affine if and only if
5935:
5929:
817:
6617:
4258:
5196:
6861:
6776:
4834:
3767:
5493:
4421:
There is a one-to-one correspondence between morphisms of affine varieties over an algebraically closed field
1887:
6751:
951:
5880:
1977:
3089:
211:
6842:
The Red Book of
Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians
6625:
5314:
4718:
3161:
2610:
2545:
6906:
6803:
4755:
3256:
333:
5830:
5159:
5013:
3445:
3362:
918:
1106:
109:
2508:
1039:
41:
6746:
6736:
The original article was written as a partial human translation of the corresponding French article.
6564:
6353:
Using the multiplicative notation, the associativity, identity and inverse laws can be rewritten as:
1382:
226:) over which the common zeros are considered (that is, the points of the affine algebraic set are in
887:
3489:
288:
5941:
3820:
3230:
6547:
5569:
5555:
5203:
3047:
2464:
6441:
6036:
4447:
4415:
3339:
Prime ideals of the coordinate ring correspond to affine subvarieties. An affine algebraic set
2928:
831:
5605:
5401:
6621:
6024:
2755:
2432:
6687:
5989:
1778:
can be drawn on a piece of paper or by graphing software. The figure on the right shows the
991:
of an irreducible affine variety is affine; the coordinate ring of the normalization is the
6769:
6420:
6013:
4992:
2225:
1172:
960:
652:
8:
6539:
6283:
3069:
2924:
2672:(in fact, a countable intersection of affine algebraic sets is an affine algebraic set).
189:
3099:
2985:
125:
6876:
6780:
6560:
6543:
1641:
836:
105:
97:
5745:{\displaystyle {\mathcal {O}}_{X,x}=\varinjlim _{f(x)\neq 0}A=A_{{\mathfrak {m}}_{x}}}
6883:
6857:
6789:
6755:
6715:
6494:
6286:
on the variety. The above morphisms are often written using ordinary group notation:
35:
984:
are exactly the hypersurfaces, that is the varieties defined by a single polynomial.
207:
an algebraic set whose defining ideal is prime (affine variety in the above sense).
6849:
6741:
5114:
4169:
2444:
992:
757:
6830:
6845:
6807:
6765:
2916:
2676:
2250:
2219:
2040:
1631:
1240:
739:
284:
120:
6468:
It can be shown that any affine algebraic group is isomorphic to a subgroup of
2935:
is finitely-generated, so every open set is a finite union of basic open sets.
2153:{\displaystyle {\frac {\partial f_{j}}{\partial {x_{i}}}}(a_{1},\dots ,a_{n}).}
1031:
218:
in which the coefficients are considered, from the algebraically closed field
4463:
is precisely the category of finitely-generated, nilpotent-free algebras over
6900:
6837:
6720:
6556:
6083:
1543:{\displaystyle \left({\frac {1-t^{2}}{1+t^{2}}},{\frac {2t}{1+t^{2}}}\right)}
988:
647:
6431:
3321:
4991:
Equipped with the structure sheaf described below, an affine variety is a
30:
6530:
1438:
1220:
950:(the affine plane with the origin removed) is not an affine variety; cf.
264:
193:
6871:
3343:
can be written as the union of two other algebraic sets if and only if
2950:
is simply the subspace topology inherited from the
Zariski topology on
116:
291:
asserts that the affine algebraic variety (it is a curve) defined by
5858:
is a sheaf; indeed, it says if a function is regular (pointwise) on
1004:
6853:
5106:{\displaystyle {\mathcal {O}}_{X}(U)=\Gamma (U,{\mathcal {O}}_{X})}
4411:
3919:
A product of affine varieties can be defined using the isomorphism
2679:, where Zariski-open sets are countable unions of sets of the form
1866:{\displaystyle V(y^{2}-x^{3}+x^{2}+16x)\subseteq \mathbf {C} ^{2}.}
4440:
and their coordinate rings, the category of affine varieties over
6567:). Each affine variety has an affine scheme associated to it: if
769:), it is the space of global sections of the structure sheaf of
6844:. Lecture Notes in Mathematics. Vol. 1358 (2nd ed.).
4563:
be a morphism. Indeed, a homomorphism between polynomial rings
4428:
and homomorphisms of coordinate rings of affine varieties over
4238:
will define algebraic sets that are in the
Zariski topology on
1428:
1654:
It can be proved that an algebraic curve of degree two with a
4450:
to the category of coordinate rings of affine varieties over
3316:, the set of all functions that also vanish on all points of
1338:{\displaystyle V=V(x^{2}+y^{2}-1)\subseteq \mathbf {C} ^{2},}
6482:. For this reason, affine algebraic groups are often called
6405:
The most prominent example of an affine algebraic group is
3308:. Furthermore, the function taking an affine algebraic set
4457:
The category of coordinate rings of affine varieties over
1732:{\displaystyle V(x^{2}+y^{2}+1)\subseteq \mathbf {C} ^{2}}
1623:{\displaystyle V(x^{2}+y^{2}-3)\subseteq \mathbf {C} ^{2}}
3934:
then embedding the product in this new affine space. Let
1640:-rational point. This can be deduced from the fact that,
324:
of a system of polynomial equations with coefficients in
318:
is the set of solutions in an algebraically closed field
3620:{\displaystyle R=k/\langle f_{1},\ldots ,f_{m}\rangle ,}
957:
The subvarieties of codimension one in the affine space
5307:
Proof: The inclusion ⊃ is clear. For the opposite, let
4226:
but cannot be obtained as a product of a polynomial in
6542:; that is to say, general algebraic varieties such as
6489:
Affine algebraic groups play an important role in the
3211:
Radical ideals (ideals that are their own radical) of
793:
The complement of a hypersurface in an affine variety
6508:-rational points of an affine algebraic group, where
5992:
5944:
5883:
5833:
5820:{\displaystyle {\mathfrak {m}}_{x}=\{f\in A|f(x)=0\}}
5761:
5637:
5608:
5572:
5496:
5404:
5317:
5221:
5162:
5047:
5016:
4837:
4758:
4721:
3823:
3770:
3633:
3627:
this correspondence becomes explicit through the map
3529:
3492:
3448:
3365:
3259:
3233:
3164:
3102:
3050:
2988:
2826:
2758:
2685:
2675:
The
Zariski topology can also be described by way of
2613:
2548:
2511:
2467:
2261:
2228:
2075:
1980:
1890:
1790:
1675:
1566:
1456:
1385:
1272:
1175:
1109:
1042:
963:
921:
890:
839:
655:
391:
336:
128:
44:
6538:
An affine variety plays a role of a local chart for
5874:); that is, "regular-ness" can be patched together.
1875:
912:(the affine line with the origin removed) is affine.
816:) is affine. Its defining equations are obtained by
6559:, a locally-ringed space that is isomorphic to the
6430:This is the group of linear transformations of the
4643:corresponds uniquely to a choice of image for each
6875:
6004:
5978:
5912:
5850:
5819:
5744:
5614:
5594:
5546:
5459:
5358:
5288:{\displaystyle \Gamma (D(f),{\mathcal {O}}_{X})=A}
5287:
5179:
5105:
5033:
4885:
4778:
4744:
3852:
3805:
3756:
3619:
3507:
3478:
3410:
3292:
3245:
3201:
3146:
3060:
3032:
2957:
2903:
2808:
2744:
2664:
2599:
2535:
2498:
2419:
2241:
2152:
2024:
1966:
1865:
1731:
1622:
1542:
1407:
1337:
1193:
1137:
1061:
976:
942:
904:
873:
710:
608:
368:
172:
85:
2904:{\displaystyle V(f)=D_{f}=\{p\in k^{n}:f(p)=0\},}
642:is the ideal of all polynomials that are zero on
6898:
5923:
4899:. This corresponds to the morphism of varieties
4252:
1884:be an affine variety defined by the polynomials
1351:and all its coordinates are integers. The point
1201:Often, if the base field is the complex numbers
27:Algebraic variety defined within an affine space
6082:, which is a regular morphism that follows the
3914:
2745:{\displaystyle U_{f}=\{p\in k^{n}:f(p)\neq 0\}}
1745:-rational points, but has many complex points.
239:, and the points of the variety that belong to
6451:is fixed, this is equivalent to the group of
5866:), then it must be in the coordinate ring of
5197:sheaf of modules#Sheaf associated to a module
5187:is determined by its values on the open sets
4486:of affine varieties, there is a homomorphism
2816:These basic open sets are the complements in
5814:
5779:
5541:
5512:
5353:
5324:
3748:
3672:
3611:
3579:
3332:is radical if and only if the quotient ring
2895:
2855:
2739:
2699:
6754:, vol. 52, New York: Springer-Verlag,
2974:, the coordinate ring of an affine variety
188:, is an affine algebraic set such that the
6740:
6555:An affine variety is a special case of an
6030:
4521:be affine varieties with coordinate rings
3813:denotes the image in the quotient algebra
3523:is an affine variety with coordinate ring
2066:is the matrix of the partial derivatives
1660:-rational point has infinitely many other
1008:A drawing of the real points of the curve
492:
486:
5386:, there is some open affine neighborhood
4886:{\displaystyle f_{i}(X_{1},\dots ,X_{n})}
3806:{\displaystyle {\overline {x_{i}-a_{i}}}}
924:
892:
214:), it is useful to distinguish the field
192:generated by the defining polynomials is
5547:{\displaystyle V(J)\subset \{x|f(x)=0\}}
5206:in the essential way, is the following:
4610:is determined uniquely by the images of
1967:{\displaystyle f_{1},\dots ,f_{r}\in k,}
1003:
29:
6836:
6699:
5152:. They form a base for the topology of
2438:
738:is prime, so the coordinate ring is an
382:, they define an affine algebraic set
302:has no rational points for any integer
14:
6899:
6775:
6645:
6491:classification of finite simple groups
5913:{\displaystyle (X,{\mathcal {O}}_{X})}
5624:The claim, first of all, implies that
2453:form the closed sets of a topology on
2025:{\displaystyle a=(a_{1},\dots ,a_{n})}
1447:-rational points that are the points
1163:-rational points of an affine variety
742:. The elements of the coordinate ring
6802:
6662:
2911:zero loci of a single polynomial. If
376:are polynomials with coefficients in
283:is a point that is rational over the
232:). In this case, the variety is said
6870:
6674:
6651:
6461:invertible matrices with entries in
5359:{\displaystyle J=\{h\in A|hg\in A\}}
4745:{\displaystyle {\overline {f_{i}}},}
3958:respectively, so that their product
3202:{\displaystyle I(V(J))={\sqrt {J}}.}
2665:{\displaystyle V(S)\cap V(T)=V(S,T)}
2600:{\displaystyle V(S)\cup V(T)=V(ST),}
1427:-rational. This variety is called a
766:
210:In some contexts (see, for example,
6726:Representations on coordinate rings
6628:over an algebraically closed field
6047:over an algebraically closed field
5827:. Secondly, the claim implies that
5765:
5729:
4779:{\displaystyle {\overline {f_{i}}}}
4142:The product is irreducible if each
3418:). This is the case if and only if
3293:{\displaystyle V(J)\subseteq V(I).}
3215:correspond to algebraic subsets of
1647:, the sum of two squares cannot be
1069:over an algebraically closed field
369:{\displaystyle f_{1},\ldots ,f_{m}}
24:
6522:
5896:
5851:{\displaystyle {\mathcal {O}}_{X}}
5837:
5641:
5628:is a "locally ringed" space since
5246:
5222:
5180:{\displaystyle {\mathcal {O}}_{X}}
5166:
5089:
5074:
5051:
5034:{\displaystyle {\mathcal {O}}_{X}}
5020:
4986:
4690:a homomorphism can be constructed
4585:factors uniquely through the ring
4472:More precisely, for each morphism
3479:{\displaystyle V(J)\subseteq V(I)}
3411:{\displaystyle V(I)=V(J)\cup V(K)}
3158:is an algebraically closed field,
2527:
2461:. This follows from the fact that
2301:
2286:
2094:
2079:
1151:whose coordinates are elements of
999:
943:{\displaystyle \mathbb {C} ^{2}-0}
830:. The coordinate ring is thus the
25:
6918:
6594:then the scheme corresponding to
4249:but not in the product topology.
2982:be the set of all polynomials in
1876:Singular points and tangent space
1760:defined over the complex numbers
1138:{\displaystyle p\in V\cap k^{n}.}
782:Dimension of an algebraic variety
6878:Undergraduate Algebraic Geometry
6563:of a commutative ring (up to an
4088:is defined as the algebraic set
2536:{\displaystyle V(1)=\emptyset ,}
1850:
1719:
1610:
1322:
1062:{\displaystyle V\subseteq K^{n}}
761:on the variety, or, simply, the
636:is an affine algebraic set, and
86:{\displaystyle y^{2}=x^{2}(x+1)}
4259:Morphism of algebraic varieties
4220:Hence, polynomials that are in
2958:Geometry–algebra correspondence
2392:
1408:{\displaystyle (i,{\sqrt {2}})}
1266:-rational point of the variety
309:
6882:. Cambridge University Press.
6693:
6680:
6668:
6656:
5967:
5955:
5907:
5884:
5805:
5799:
5792:
5715:
5699:
5682:
5676:
5532:
5526:
5519:
5506:
5500:
5454:
5438:
5429:
5426:
5420:
5414:
5337:
5282:
5266:
5257:
5237:
5231:
5225:
5202:The key fact, which relies on
5100:
5077:
5068:
5062:
4880:
4848:
3669:
3666:
3634:
3571:
3539:
3473:
3467:
3458:
3452:
3405:
3399:
3390:
3384:
3375:
3369:
3324:(nilpotent-free), as an ideal
3284:
3278:
3269:
3263:
3183:
3180:
3174:
3168:
3138:
3106:
3024:
2992:
2886:
2880:
2836:
2830:
2800:
2768:
2730:
2724:
2659:
2647:
2638:
2632:
2623:
2617:
2591:
2582:
2573:
2567:
2558:
2552:
2521:
2515:
2477:
2471:
2380:
2354:
2351:
2319:
2144:
2112:
2019:
1987:
1958:
1926:
1842:
1794:
1711:
1679:
1602:
1570:
1402:
1386:
1314:
1282:
1185:
1179:
905:{\displaystyle \mathbb {C} -0}
868:
852:
849:
843:
754:on the variety. They form the
697:
665:
589:
557:
535:
503:
488:
470:
438:
427:
395:
164:
132:
80:
68:
13:
1:
6752:Graduate Texts in Mathematics
6731:
5930:Serre's theorem on affineness
5924:Serre's theorem on affineness
5311:be in the left-hand side and
4253:Morphisms of affine varieties
3508:{\displaystyle I\subseteq J.}
3219:. Indeed, for radical ideals
2449:The affine algebraic sets of
6831:Lectures on Étale cohomology
6702:, Ch. I, § 4. Proposition 1.
5979:{\displaystyle H^{i}(X,F)=0}
4786:is the equivalence class of
4771:
4734:
3915:Products of affine varieties
3853:{\displaystyle x_{i}-a_{i}.}
3798:
3743:
3702:
3246:{\displaystyle I\subseteq J}
300: − 1 = 0
271:-rational point is called a
7:
6709:
6608:the set of prime ideals of
5920:is a locally ringed space.
5595:{\displaystyle f^{n}g\in A}
3061:{\displaystyle {\sqrt {I}}}
2942:is an affine subvariety of
2499:{\displaystyle V(0)=k^{n},}
787:
734:is an affine variety, then
259:. In the common case where
203:for any algebraic set, and
10:
6923:
6688:integral domain#Properties
6034:
5927:
4654:. Then given any morphism
4256:
3997:be an algebraic subset of
2442:
1634:of degree two that has no
1243:) are often simply called
1029:
952:Hartogs' extension theorem
621:affine (algebraic) variety
110:algebraically closed field
6565:equivalence of categories
6282:Together, these define a
6086:axiom—that is, such that
4629:Hence, each homomorphism
3442:are radical ideals, then
3090:Hilbert's nullstellensatz
3076:, the set of polynomials
1441:. It has infinitely many
1431:, because the set of its
212:Hilbert's Nullstellensatz
104:is the set of the common
6638:
6573:is an affine variety in
6548:algebraic vector bundles
5615:{\displaystyle \square }
5460:{\displaystyle g\in k=A}
5366:, which is an ideal. If
4998:Given an affine variety
4157:The Zariski topology on
3875:Type of coordinate ring
3430:correspond to points of
3080:for which some power of
2946:the Zariski topology on
1754:is an affine variety in
1437:-rational points is the
199:Some texts use the term
186:affine algebraic variety
6484:linear algebraic groups
6031:Affine algebraic groups
5556:Hilbert nullstellensatz
5204:Hilbert nullstellensatz
4031:an algebraic subset of
3880:affine algebraic subset
2931:), then every ideal of
2809:{\displaystyle f\in k.}
6214:, a regular bijection
6055:affine algebraic group
6037:linear algebraic group
6006:
6005:{\displaystyle i>0}
5980:
5914:
5852:
5821:
5746:
5616:
5596:
5548:
5461:
5360:
5289:
5181:
5107:
5041:is defined by letting
5035:
4887:
4780:
4746:
3946:have coordinate rings
3854:
3807:
3758:
3621:
3509:
3480:
3412:
3294:
3247:
3203:
3148:
3062:
3034:
2929:principal ideal domain
2905:
2810:
2746:
2666:
2601:
2537:
2500:
2421:
2282:
2243:
2154:
2026:
1968:
1867:
1733:
1624:
1557:is a rational number.
1544:
1409:
1339:
1195:
1139:
1063:
1036:For an affine variety
1027:
978:
944:
906:
875:
824:the defining ideal of
778:dimension of a variety
765:; in other words (see
712:
610:
370:
174:
93:
87:
6579:with coordinate ring
6007:
5981:
5915:
5853:
5822:
5747:
5617:
5597:
5562:is in the radical of
5549:
5462:
5361:
5290:
5182:
5108:
5036:
5002:with coordinate ring
4888:
4781:
4747:
4418:of affine varieties.
4232:with a polynomial in
3869:Type of algebraic set
3855:
3808:
3759:
3622:
3510:
3481:
3413:
3295:
3248:
3204:
3149:
3063:
3035:
2906:
2811:
2747:
2667:
2602:
2538:
2501:
2433:Zariski tangent space
2422:
2262:
2244:
2242:{\displaystyle k^{n}}
2155:
2027:
1969:
1868:
1734:
1625:
1545:
1410:
1340:
1196:
1194:{\displaystyle V(k).}
1140:
1064:
1007:
979:
977:{\displaystyle k^{n}}
945:
907:
876:
713:
711:{\displaystyle R=k/I}
611:
371:
330:. More precisely, if
289:Fermat's Last Theorem
175:
88:
33:
6808:"Algebraic Geometry"
6544:projective varieties
6421:general linear group
6014:quasi-coherent sheaf
5990:
5942:
5881:
5831:
5759:
5635:
5606:
5570:
5494:
5402:
5315:
5219:
5160:
5045:
5014:
4993:locally ringed space
4835:
4756:
4719:
3964:has coordinate ring
3821:
3768:
3631:
3527:
3490:
3446:
3363:
3257:
3231:
3162:
3100:
3048:
2986:
2824:
2756:
2683:
2611:
2546:
2509:
2465:
2439:The Zariski topology
2259:
2226:
2073:
1978:
1888:
1788:
1784:-rational points of
1772:-rational points of
1673:
1669:The complex variety
1630:is an example of an
1564:
1454:
1383:
1270:
1227:of the variety, and
1173:
1157:. The collection of
1145:That is, a point of
1107:
1040:
961:
919:
888:
837:
810:for some polynomial
752:polynomial functions
746:are also called the
653:
389:
334:
316:affine algebraic set
279:is not specified, a
126:
102:affine algebraic set
42:
18:Affine algebraic set
6540:algebraic varieties
6519:is a finite field.
5490:). In other words,
5214: —
4592:and a homomorphism
4170:topological product
4049:is a polynomial in
2820:of the closed sets
1369:is a real point of
915:On the other hand,
763:ring of the variety
205:irreducible variety
6907:Algebraic geometry
6788:. Addison-Wesley.
6747:Algebraic Geometry
6495:groups of Lie type
6339:can be written as
6301:can be written as
6041:An affine variety
6025:Cartan's theorem B
6002:
5976:
5910:
5848:
5817:
5742:
5692:
5670:
5612:
5592:
5544:
5457:
5356:
5285:
5212:
5177:
5103:
5031:
4883:
4776:
4742:
4549:respectively. Let
3850:
3817:of the polynomial
3803:
3754:
3617:
3505:
3476:
3426:Maximal ideals of
3408:
3347:for proper ideals
3290:
3243:
3199:
3147:{\displaystyle k,}
3144:
3058:
3033:{\displaystyle k,}
3030:
2919:(for instance, if
2901:
2806:
2742:
2662:
2597:
2533:
2496:
2417:
2239:
2150:
2022:
1964:
1863:
1729:
1620:
1540:
1405:
1335:
1233:-rational points (
1207:, points that are
1191:
1135:
1059:
1028:
974:
940:
902:
871:
708:
606:
366:
306:greater than two.
173:{\displaystyle k.}
170:
115:of some family of
98:algebraic geometry
94:
83:
6761:978-0-387-90244-9
6742:Hartshorne, Robin
6716:Algebraic variety
5663:
5661:
5210:
5144:) ≠ 0 } for each
5115:regular functions
4774:
4737:
3912:
3911:
3891:affine subvariety
3801:
3746:
3705:
3194:
3056:
2317:
2110:
1533:
1500:
1400:
1260:-rational and an
1213:-rational (where
1169:is often denoted
1075:, and a subfield
874:{\displaystyle k}
758:regular functions
748:regular functions
275:. When the field
36:cubic plane curve
16:(Redirected from
6914:
6893:
6881:
6867:
6828:Milne, James S.
6825:
6823:
6821:
6812:
6799:
6787:
6782:Algebraic Curves
6772:
6703:
6697:
6691:
6684:
6678:
6672:
6666:
6660:
6654:
6649:
6634:
6614:
6607:
6599:
6593:
6578:
6572:
6518:
6507:
6497:are all sets of
6481:
6467:
6460:
6450:
6439:
6429:
6418:
6401:
6384:
6370:
6352:
6345:
6338:
6327:
6321:
6310:
6300:
6278:
6271:
6265:
6227:
6212:inverse morphism
6207:
6200:
6194:
6164:
6158:identity element
6153:
6146:
6140:
6134:
6128:
6081:
6052:
6046:
6011:
6009:
6008:
6003:
5985:
5983:
5982:
5977:
5954:
5953:
5936:theorem of Serre
5919:
5917:
5916:
5911:
5906:
5905:
5900:
5899:
5857:
5855:
5854:
5849:
5847:
5846:
5841:
5840:
5826:
5824:
5823:
5818:
5795:
5775:
5774:
5769:
5768:
5751:
5749:
5748:
5743:
5741:
5740:
5739:
5738:
5733:
5732:
5714:
5713:
5691:
5671:
5657:
5656:
5645:
5644:
5621:
5619:
5618:
5613:
5601:
5599:
5598:
5593:
5582:
5581:
5553:
5551:
5550:
5545:
5522:
5466:
5464:
5463:
5458:
5453:
5452:
5382:is regular near
5365:
5363:
5362:
5357:
5340:
5294:
5292:
5291:
5286:
5281:
5280:
5256:
5255:
5250:
5249:
5215:
5186:
5184:
5183:
5178:
5176:
5175:
5170:
5169:
5112:
5110:
5109:
5104:
5099:
5098:
5093:
5092:
5061:
5060:
5055:
5054:
5040:
5038:
5037:
5032:
5030:
5029:
5024:
5023:
4983:
4912:
4898:
4892:
4890:
4889:
4884:
4879:
4878:
4860:
4859:
4847:
4846:
4831:to a polynomial
4830:
4819:
4803:
4796:
4785:
4783:
4782:
4777:
4775:
4770:
4769:
4760:
4751:
4749:
4748:
4743:
4738:
4733:
4732:
4723:
4714:
4703:
4689:
4682:
4676:
4653:
4642:
4628:
4609:
4591:
4584:
4562:
4548:
4534:
4520:
4510:
4499:
4485:
4469:
4462:
4456:
4445:
4439:
4433:
4427:
4409:
4398:
4383:
4312:
4298:
4292:
4282:
4272:
4248:
4237:
4231:
4225:
4219:
4195:
4167:
4154:is irreducible.
4153:
4147:
4141:
4134:
4087:
4081:
4071:
4065:
4054:
4048:
4037:
4030:
4003:
3996:
3969:
3963:
3957:
3951:
3945:
3939:
3933:
3897:integral domain
3866:
3865:
3859:
3857:
3856:
3851:
3846:
3845:
3833:
3832:
3812:
3810:
3809:
3804:
3802:
3797:
3796:
3795:
3783:
3782:
3772:
3763:
3761:
3760:
3755:
3747:
3742:
3741:
3740:
3728:
3727:
3717:
3706:
3701:
3700:
3699:
3687:
3686:
3676:
3665:
3664:
3646:
3645:
3626:
3624:
3623:
3618:
3610:
3609:
3591:
3590:
3578:
3570:
3569:
3551:
3550:
3514:
3512:
3511:
3506:
3485:
3483:
3482:
3477:
3417:
3415:
3414:
3409:
3299:
3297:
3296:
3291:
3252:
3250:
3249:
3244:
3208:
3206:
3205:
3200:
3195:
3190:
3153:
3151:
3150:
3145:
3137:
3136:
3118:
3117:
3067:
3065:
3064:
3059:
3057:
3052:
3039:
3037:
3036:
3031:
3023:
3022:
3004:
3003:
2910:
2908:
2907:
2902:
2873:
2872:
2851:
2850:
2815:
2813:
2812:
2807:
2799:
2798:
2780:
2779:
2751:
2749:
2748:
2743:
2717:
2716:
2695:
2694:
2671:
2669:
2668:
2663:
2606:
2604:
2603:
2598:
2542:
2540:
2539:
2534:
2505:
2503:
2502:
2497:
2492:
2491:
2459:Zariski topology
2445:Zariski topology
2426:
2424:
2423:
2418:
2379:
2378:
2366:
2365:
2350:
2349:
2331:
2330:
2318:
2316:
2315:
2314:
2313:
2299:
2298:
2297:
2284:
2281:
2276:
2251:linear equations
2248:
2246:
2245:
2240:
2238:
2237:
2217:
2213:
2206:is regular, the
2205:
2194:
2186:
2166:
2159:
2157:
2156:
2151:
2143:
2142:
2124:
2123:
2111:
2109:
2108:
2107:
2106:
2092:
2091:
2090:
2077:
2065:
2061:
2057:
2035:
2031:
2029:
2028:
2023:
2018:
2017:
1999:
1998:
1973:
1971:
1970:
1965:
1957:
1956:
1938:
1937:
1919:
1918:
1900:
1899:
1883:
1872:
1870:
1869:
1864:
1859:
1858:
1853:
1832:
1831:
1819:
1818:
1806:
1805:
1783:
1777:
1771:
1765:
1759:
1753:
1744:
1738:
1736:
1735:
1730:
1728:
1727:
1722:
1704:
1703:
1691:
1690:
1665:
1659:
1650:
1646:
1639:
1629:
1627:
1626:
1621:
1619:
1618:
1613:
1595:
1594:
1582:
1581:
1556:
1549:
1547:
1546:
1541:
1539:
1535:
1534:
1532:
1531:
1530:
1514:
1506:
1501:
1499:
1498:
1497:
1481:
1480:
1479:
1463:
1446:
1436:
1426:
1420:
1414:
1412:
1411:
1406:
1401:
1396:
1379:-rational, and
1378:
1372:
1368:
1366:
1365:
1359:
1358:
1350:
1344:
1342:
1341:
1336:
1331:
1330:
1325:
1307:
1306:
1294:
1293:
1265:
1259:
1253:
1241:rational numbers
1238:
1232:
1218:
1212:
1206:
1200:
1198:
1197:
1192:
1168:
1162:
1156:
1150:
1144:
1142:
1141:
1136:
1131:
1130:
1102:
1092:
1086:
1080:
1074:
1068:
1066:
1065:
1060:
1058:
1057:
1026:
993:integral closure
983:
981:
980:
975:
973:
972:
949:
947:
946:
941:
933:
932:
927:
911:
909:
908:
903:
895:
884:In particular,
880:
878:
877:
872:
867:
866:
829:
823:
815:
809:
798:
767:#Structure sheaf
724:
723:
717:
715:
714:
709:
704:
696:
695:
677:
676:
645:
641:
635:
615:
613:
612:
607:
602:
598:
588:
587:
569:
568:
556:
555:
534:
533:
515:
514:
502:
501:
491:
485:
484:
469:
468:
450:
449:
426:
425:
407:
406:
381:
375:
373:
372:
367:
365:
364:
346:
345:
329:
323:
305:
301:
285:rational numbers
278:
270:
263:is the field of
262:
258:
249:
244:
238:
231:
225:
221:
217:
179:
177:
176:
171:
163:
162:
144:
143:
114:
92:
90:
89:
84:
67:
66:
54:
53:
21:
6922:
6921:
6917:
6916:
6915:
6913:
6912:
6911:
6897:
6896:
6890:
6864:
6846:Springer-Verlag
6819:
6817:
6810:
6804:Milne, James S.
6796:
6785:
6777:Fulton, William
6762:
6734:
6712:
6707:
6706:
6698:
6694:
6685:
6681:
6673:
6669:
6661:
6657:
6650:
6646:
6641:
6629:
6609:
6601:
6595:
6580:
6574:
6568:
6525:
6523:Generalizations
6517:
6509:
6506:
6498:
6475:
6469:
6462:
6452:
6445:
6434:
6424:
6412:
6406:
6386:
6372:
6362:) = (
6354:
6347:
6340:
6329:
6323:
6312:
6302:
6287:
6284:group structure
6273:
6267:
6261:)) =
6229:
6215:
6202:
6196:
6166:
6160:
6148:
6142:
6136:
6130:
6129:for all points
6087:
6065:
6048:
6042:
6039:
6033:
5991:
5988:
5987:
5949:
5945:
5943:
5940:
5939:
5932:
5926:
5901:
5895:
5894:
5893:
5882:
5879:
5878:
5842:
5836:
5835:
5834:
5832:
5829:
5828:
5791:
5770:
5764:
5763:
5762:
5760:
5757:
5756:
5734:
5728:
5727:
5726:
5725:
5721:
5706:
5702:
5672:
5662:
5646:
5640:
5639:
5638:
5636:
5633:
5632:
5607:
5604:
5603:
5577:
5573:
5571:
5568:
5567:
5518:
5495:
5492:
5491:
5445:
5441:
5403:
5400:
5399:
5378:), then, since
5336:
5316:
5313:
5312:
5305:
5273:
5269:
5251:
5245:
5244:
5243:
5220:
5217:
5216:
5213:
5171:
5165:
5164:
5163:
5161:
5158:
5157:
5113:be the ring of
5094:
5088:
5087:
5086:
5056:
5050:
5049:
5048:
5046:
5043:
5042:
5025:
5019:
5018:
5017:
5015:
5012:
5011:
5006:, the sheaf of
4989:
4987:Structure sheaf
4981:
4972:
4965:
4956:
4947:
4940:
4933:
4924:
4914:
4900:
4894:
4874:
4870:
4855:
4851:
4842:
4838:
4836:
4833:
4832:
4829:
4821:
4807:
4798:
4795:
4787:
4765:
4761:
4759:
4757:
4754:
4753:
4728:
4724:
4722:
4720:
4717:
4716:
4713:
4705:
4691:
4684:
4678:
4674:
4665:
4655:
4652:
4644:
4630:
4626:
4617:
4611:
4593:
4586:
4564:
4550:
4536:
4522:
4512:
4502:
4487:
4473:
4464:
4458:
4451:
4441:
4435:
4429:
4422:
4400:
4393:
4385:
4381:
4372:
4365:
4356:
4347:
4340:
4333:
4324:
4314:
4300:
4294:
4288:
4274:
4264:
4261:
4255:
4239:
4233:
4227:
4221:
4205:
4197:
4181:
4173:
4158:
4149:
4143:
4136:
4132:
4123:
4116:
4107:
4089:
4083:
4077:
4067:
4064:
4056:
4050:
4047:
4039:
4032:
4028:
4019:
4005:
3998:
3994:
3985:
3971:
3965:
3959:
3953:
3947:
3941:
3935:
3920:
3917:
3841:
3837:
3828:
3824:
3822:
3819:
3818:
3791:
3787:
3778:
3774:
3773:
3771:
3769:
3766:
3765:
3736:
3732:
3723:
3719:
3718:
3716:
3695:
3691:
3682:
3678:
3677:
3675:
3660:
3656:
3641:
3637:
3632:
3629:
3628:
3605:
3601:
3586:
3582:
3574:
3565:
3561:
3546:
3542:
3528:
3525:
3524:
3491:
3488:
3487:
3486:if and only if
3447:
3444:
3443:
3364:
3361:
3360:
3359:(in which case
3304:if and only if
3258:
3255:
3254:
3253:if and only if
3232:
3229:
3228:
3189:
3163:
3160:
3159:
3132:
3128:
3113:
3109:
3101:
3098:
3097:
3092:: for an ideal
3051:
3049:
3046:
3045:
3040:that vanish on
3018:
3014:
2999:
2995:
2987:
2984:
2983:
2960:
2868:
2864:
2846:
2842:
2825:
2822:
2821:
2794:
2790:
2775:
2771:
2757:
2754:
2753:
2712:
2708:
2690:
2686:
2684:
2681:
2680:
2677:basic open sets
2612:
2609:
2608:
2547:
2544:
2543:
2510:
2507:
2506:
2487:
2483:
2466:
2463:
2462:
2447:
2441:
2374:
2370:
2361:
2357:
2345:
2341:
2326:
2322:
2309:
2305:
2304:
2300:
2293:
2289:
2285:
2283:
2277:
2266:
2260:
2257:
2256:
2249:defined by the
2233:
2229:
2227:
2224:
2223:
2220:affine subspace
2215:
2211:
2203:
2192:
2180:
2172:
2171:if the rank of
2164:
2138:
2134:
2119:
2115:
2102:
2098:
2097:
2093:
2086:
2082:
2078:
2076:
2074:
2071:
2070:
2063:
2059:
2051:
2043:
2041:Jacobian matrix
2033:
2013:
2009:
1994:
1990:
1979:
1976:
1975:
1952:
1948:
1933:
1929:
1914:
1910:
1895:
1891:
1889:
1886:
1885:
1881:
1878:
1854:
1849:
1848:
1827:
1823:
1814:
1810:
1801:
1797:
1789:
1786:
1785:
1779:
1773:
1767:
1761:
1755:
1749:
1740:
1723:
1718:
1717:
1699:
1695:
1686:
1682:
1674:
1671:
1670:
1661:
1655:
1648:
1644:
1635:
1632:algebraic curve
1614:
1609:
1608:
1590:
1586:
1577:
1573:
1565:
1562:
1561:
1554:
1526:
1522:
1515:
1507:
1505:
1493:
1489:
1482:
1475:
1471:
1464:
1462:
1461:
1457:
1455:
1452:
1451:
1442:
1432:
1422:
1416:
1395:
1384:
1381:
1380:
1374:
1370:
1363:
1361:
1356:
1354:
1352:
1346:
1326:
1321:
1320:
1302:
1298:
1289:
1285:
1271:
1268:
1267:
1261:
1255:
1251:
1245:rational points
1234:
1228:
1214:
1208:
1202:
1174:
1171:
1170:
1164:
1158:
1152:
1146:
1126:
1122:
1108:
1105:
1104:
1098:
1088:
1082:
1076:
1070:
1053:
1049:
1041:
1038:
1037:
1034:
1021: − 16
1009:
1002:
1000:Rational points
968:
964:
962:
959:
958:
928:
923:
922:
920:
917:
916:
891:
889:
886:
885:
859:
855:
838:
835:
834:
825:
821:
811:
800:
794:
790:
740:integral domain
722:coordinate ring
721:
720:
700:
691:
687:
672:
668:
654:
651:
650:
643:
637:
631:
583:
579:
564:
560:
551:
547:
529:
525:
510:
506:
497:
493:
487:
480:
476:
464:
460:
445:
441:
437:
433:
421:
417:
402:
398:
390:
387:
386:
377:
360:
356:
341:
337:
335:
332:
331:
325:
319:
312:
303:
292:
287:. For example,
276:
268:
260:
256:
247:
240:
236:
227:
223:
219:
215:
158:
154:
139:
135:
127:
124:
123:
121:polynomial ring
112:
62:
58:
49:
45:
43:
40:
39:
28:
23:
22:
15:
12:
11:
5:
6920:
6910:
6909:
6895:
6894:
6888:
6868:
6862:
6854:10.1007/b62130
6838:Mumford, David
6834:
6826:
6815:www.jmilne.org
6800:
6794:
6773:
6760:
6733:
6730:
6729:
6728:
6723:
6718:
6711:
6708:
6705:
6704:
6692:
6679:
6667:
6655:
6643:
6642:
6640:
6637:
6636:
6635:
6552:
6551:
6535:
6534:
6524:
6521:
6513:
6502:
6471:
6408:
6328:; the inverse
6280:
6279:
6245:) =
6208:
6190:) =
6178:) =
6154:
6107:) =
6063:multiplication
6035:Main article:
6032:
6029:
6001:
5998:
5995:
5975:
5972:
5969:
5966:
5963:
5960:
5957:
5952:
5948:
5928:Main article:
5925:
5922:
5909:
5904:
5898:
5892:
5889:
5886:
5845:
5839:
5816:
5813:
5810:
5807:
5804:
5801:
5798:
5794:
5790:
5787:
5784:
5781:
5778:
5773:
5767:
5753:
5752:
5737:
5731:
5724:
5720:
5717:
5712:
5709:
5705:
5701:
5698:
5695:
5690:
5687:
5684:
5681:
5678:
5675:
5669:
5666:
5660:
5655:
5652:
5649:
5643:
5611:
5591:
5588:
5585:
5580:
5576:
5543:
5540:
5537:
5534:
5531:
5528:
5525:
5521:
5517:
5514:
5511:
5508:
5505:
5502:
5499:
5456:
5451:
5448:
5444:
5440:
5437:
5434:
5431:
5428:
5425:
5422:
5419:
5416:
5413:
5410:
5407:
5355:
5352:
5349:
5346:
5343:
5339:
5335:
5332:
5329:
5326:
5323:
5320:
5284:
5279:
5276:
5272:
5268:
5265:
5262:
5259:
5254:
5248:
5242:
5239:
5236:
5233:
5230:
5227:
5224:
5208:
5195:). (See also:
5174:
5168:
5102:
5097:
5091:
5085:
5082:
5079:
5076:
5073:
5070:
5067:
5064:
5059:
5053:
5028:
5022:
4988:
4985:
4977:
4970:
4961:
4952:
4945:
4938:
4929:
4922:
4882:
4877:
4873:
4869:
4866:
4863:
4858:
4854:
4850:
4845:
4841:
4825:
4791:
4773:
4768:
4764:
4741:
4736:
4731:
4727:
4709:
4670:
4663:
4648:
4622:
4615:
4410:These are the
4389:
4377:
4370:
4361:
4352:
4345:
4338:
4329:
4322:
4257:Main article:
4254:
4251:
4201:
4177:
4128:
4121:
4112:
4105:
4060:
4043:
4024:
4017:
3990:
3983:
3916:
3913:
3910:
3909:
3906:
3903:
3899:
3898:
3895:
3892:
3888:
3887:
3884:
3881:
3877:
3876:
3873:
3870:
3849:
3844:
3840:
3836:
3831:
3827:
3800:
3794:
3790:
3786:
3781:
3777:
3753:
3750:
3745:
3739:
3735:
3731:
3726:
3722:
3715:
3712:
3709:
3704:
3698:
3694:
3690:
3685:
3681:
3674:
3671:
3668:
3663:
3659:
3655:
3652:
3649:
3644:
3640:
3636:
3616:
3613:
3608:
3604:
3600:
3597:
3594:
3589:
3585:
3581:
3577:
3573:
3568:
3564:
3560:
3557:
3554:
3549:
3545:
3541:
3538:
3535:
3532:
3504:
3501:
3498:
3495:
3475:
3472:
3469:
3466:
3463:
3460:
3457:
3454:
3451:
3407:
3404:
3401:
3398:
3395:
3392:
3389:
3386:
3383:
3380:
3377:
3374:
3371:
3368:
3312:and returning
3289:
3286:
3283:
3280:
3277:
3274:
3271:
3268:
3265:
3262:
3242:
3239:
3236:
3198:
3193:
3188:
3185:
3182:
3179:
3176:
3173:
3170:
3167:
3143:
3140:
3135:
3131:
3127:
3124:
3121:
3116:
3112:
3108:
3105:
3055:
3029:
3026:
3021:
3017:
3013:
3010:
3007:
3002:
2998:
2994:
2991:
2959:
2956:
2900:
2897:
2894:
2891:
2888:
2885:
2882:
2879:
2876:
2871:
2867:
2863:
2860:
2857:
2854:
2849:
2845:
2841:
2838:
2835:
2832:
2829:
2805:
2802:
2797:
2793:
2789:
2786:
2783:
2778:
2774:
2770:
2767:
2764:
2761:
2741:
2738:
2735:
2732:
2729:
2726:
2723:
2720:
2715:
2711:
2707:
2704:
2701:
2698:
2693:
2689:
2661:
2658:
2655:
2652:
2649:
2646:
2643:
2640:
2637:
2634:
2631:
2628:
2625:
2622:
2619:
2616:
2596:
2593:
2590:
2587:
2584:
2581:
2578:
2575:
2572:
2569:
2566:
2563:
2560:
2557:
2554:
2551:
2532:
2529:
2526:
2523:
2520:
2517:
2514:
2495:
2490:
2486:
2482:
2479:
2476:
2473:
2470:
2443:Main article:
2440:
2437:
2428:
2427:
2416:
2413:
2410:
2407:
2404:
2401:
2398:
2395:
2391:
2388:
2385:
2382:
2377:
2373:
2369:
2364:
2360:
2356:
2353:
2348:
2344:
2340:
2337:
2334:
2329:
2325:
2321:
2312:
2308:
2303:
2296:
2292:
2288:
2280:
2275:
2272:
2269:
2265:
2236:
2232:
2176:
2161:
2160:
2149:
2146:
2141:
2137:
2133:
2130:
2127:
2122:
2118:
2114:
2105:
2101:
2096:
2089:
2085:
2081:
2047:
2032:be a point of
2021:
2016:
2012:
2008:
2005:
2002:
1997:
1993:
1989:
1986:
1983:
1963:
1960:
1955:
1951:
1947:
1944:
1941:
1936:
1932:
1928:
1925:
1922:
1917:
1913:
1909:
1906:
1903:
1898:
1894:
1877:
1874:
1862:
1857:
1852:
1847:
1844:
1841:
1838:
1835:
1830:
1826:
1822:
1817:
1813:
1809:
1804:
1800:
1796:
1793:
1726:
1721:
1716:
1713:
1710:
1707:
1702:
1698:
1694:
1689:
1685:
1681:
1678:
1617:
1612:
1607:
1604:
1601:
1598:
1593:
1589:
1585:
1580:
1576:
1572:
1569:
1551:
1550:
1538:
1529:
1525:
1521:
1518:
1513:
1510:
1504:
1496:
1492:
1488:
1485:
1478:
1474:
1470:
1467:
1460:
1415:is a point of
1404:
1399:
1394:
1391:
1388:
1334:
1329:
1324:
1319:
1316:
1313:
1310:
1305:
1301:
1297:
1292:
1288:
1284:
1281:
1278:
1275:
1250:For instance,
1190:
1187:
1184:
1181:
1178:
1134:
1129:
1125:
1121:
1118:
1115:
1112:
1095:rational point
1056:
1052:
1048:
1045:
1032:rational point
1030:Main article:
1001:
998:
997:
996:
985:
971:
967:
955:
939:
936:
931:
926:
913:
901:
898:
894:
882:
870:
865:
862:
858:
854:
851:
848:
845:
842:
789:
786:
718:is called the
707:
703:
699:
694:
690:
686:
683:
680:
675:
671:
667:
664:
661:
658:
617:
616:
605:
601:
597:
594:
591:
586:
582:
578:
575:
572:
567:
563:
559:
554:
550:
546:
543:
540:
537:
532:
528:
524:
521:
518:
513:
509:
505:
500:
496:
490:
483:
479:
475:
472:
467:
463:
459:
456:
453:
448:
444:
440:
436:
432:
429:
424:
420:
416:
413:
410:
405:
401:
397:
394:
363:
359:
355:
352:
349:
344:
340:
311:
308:
281:rational point
182:affine variety
169:
166:
161:
157:
153:
150:
147:
142:
138:
134:
131:
82:
79:
76:
73:
70:
65:
61:
57:
52:
48:
26:
9:
6:
4:
3:
2:
6919:
6908:
6905:
6904:
6902:
6891:
6889:0-521-35662-8
6885:
6880:
6879:
6873:
6869:
6865:
6859:
6855:
6851:
6847:
6843:
6839:
6835:
6833:
6832:
6827:
6816:
6809:
6805:
6801:
6797:
6791:
6784:
6783:
6778:
6774:
6771:
6767:
6763:
6757:
6753:
6749:
6748:
6743:
6739:
6738:
6737:
6727:
6724:
6722:
6721:Affine scheme
6719:
6717:
6714:
6713:
6701:
6696:
6689:
6683:
6677:, p. 94.
6676:
6671:
6664:
6659:
6653:
6648:
6644:
6632:
6627:
6623:
6619:
6612:
6605:
6598:
6591:
6587:
6583:
6577:
6571:
6566:
6562:
6558:
6557:affine scheme
6554:
6553:
6549:
6545:
6541:
6537:
6536:
6532:
6527:
6526:
6520:
6516:
6512:
6505:
6501:
6496:
6492:
6487:
6485:
6479:
6474:
6465:
6459:
6455:
6448:
6443:
6437:
6433:
6427:
6422:
6416:
6411:
6403:
6400:
6397: =
6396:
6393:
6390: =
6389:
6383:
6380: =
6379:
6376: =
6375:
6369:
6365:
6361:
6357:
6350:
6344:
6336:
6332:
6326:
6319:
6315:
6309:
6306: +
6305:
6298:
6294:
6290:
6285:
6276:
6270:
6264:
6260:
6256:
6252:
6248:
6244:
6240:
6236:
6232:
6226:
6223: →
6222:
6218:
6213:
6209:
6205:
6199:
6193:
6189:
6185:
6181:
6177:
6173:
6169:
6163:
6159:
6155:
6151:
6145:
6139:
6133:
6126:
6122:
6118:
6114:
6110:
6106:
6102:
6098:
6094:
6090:
6085:
6084:associativity
6080:
6077: →
6076:
6073: ×
6072:
6068:
6064:
6060:
6059:
6058:
6056:
6053:is called an
6051:
6045:
6038:
6028:
6026:
6022:
6018:
6015:
5999:
5996:
5993:
5973:
5970:
5964:
5961:
5958:
5950:
5946:
5937:
5931:
5921:
5902:
5890:
5887:
5875:
5873:
5869:
5865:
5861:
5843:
5811:
5808:
5802:
5796:
5788:
5785:
5782:
5776:
5771:
5735:
5722:
5718:
5710:
5707:
5703:
5696:
5693:
5688:
5685:
5679:
5673:
5667:
5664:
5658:
5653:
5650:
5647:
5631:
5630:
5629:
5627:
5622:
5609:
5589:
5586:
5583:
5578:
5574:
5565:
5561:
5557:
5554:and thus the
5538:
5535:
5529:
5523:
5515:
5509:
5503:
5497:
5489:
5485:
5481:
5477:
5473:
5470:
5449:
5446:
5442:
5435:
5432:
5423:
5417:
5411:
5408:
5405:
5397:
5393:
5389:
5385:
5381:
5377:
5373:
5369:
5350:
5347:
5344:
5341:
5333:
5330:
5327:
5321:
5318:
5310:
5304:
5302:
5298:
5277:
5274:
5270:
5263:
5260:
5252:
5240:
5234:
5228:
5207:
5205:
5200:
5198:
5194:
5190:
5172:
5155:
5151:
5147:
5143:
5139:
5135:
5131:
5127:
5122:
5120:
5116:
5095:
5083:
5080:
5071:
5065:
5057:
5026:
5009:
5005:
5001:
4996:
4994:
4984:
4980:
4976:
4969:
4964:
4960:
4955:
4951:
4944:
4937:
4932:
4928:
4921:
4917:
4911:
4907:
4903:
4897:
4875:
4871:
4867:
4864:
4861:
4856:
4852:
4843:
4839:
4828:
4824:
4818:
4814:
4810:
4804:
4801:
4794:
4790:
4766:
4762:
4739:
4729:
4725:
4712:
4708:
4702:
4698:
4694:
4687:
4681:
4673:
4669:
4662:
4658:
4651:
4647:
4641:
4637:
4633:
4625:
4621:
4614:
4608:
4604:
4600:
4596:
4589:
4583:
4579:
4575:
4571:
4567:
4561:
4557:
4553:
4547:
4543:
4539:
4533:
4529:
4525:
4519:
4515:
4509:
4505:
4498:
4494:
4490:
4484:
4480:
4476:
4470:
4467:
4461:
4454:
4449:
4444:
4438:
4432:
4425:
4419:
4417:
4413:
4407:
4403:
4397:
4392:
4388:
4380:
4376:
4369:
4364:
4360:
4355:
4351:
4344:
4337:
4332:
4328:
4321:
4317:
4311:
4307:
4303:
4297:
4291:
4286:
4281:
4277:
4271:
4267:
4260:
4250:
4246:
4243: ×
4242:
4236:
4230:
4224:
4217:
4213:
4210: −
4209:
4206: =
4204:
4200:
4193:
4189:
4186: −
4185:
4182: =
4180:
4176:
4171:
4165:
4162: ×
4161:
4155:
4152:
4146:
4139:
4131:
4127:
4120:
4115:
4111:
4104:
4100:
4097: =
4096:
4093: ×
4092:
4086:
4080:
4075:
4070:
4063:
4059:
4053:
4046:
4042:
4035:
4027:
4023:
4016:
4012:
4009: =
4008:
4001:
3993:
3989:
3982:
3978:
3975: =
3974:
3968:
3962:
3956:
3950:
3944:
3938:
3931:
3928: =
3927:
3924: ×
3923:
3907:
3905:maximal ideal
3904:
3901:
3900:
3896:
3893:
3890:
3889:
3886:reduced ring
3885:
3883:radical ideal
3882:
3879:
3878:
3874:
3872:Type of ideal
3871:
3868:
3867:
3864:
3861:
3847:
3842:
3838:
3834:
3829:
3825:
3816:
3792:
3788:
3784:
3779:
3775:
3751:
3737:
3733:
3729:
3724:
3720:
3713:
3710:
3707:
3696:
3692:
3688:
3683:
3679:
3661:
3657:
3653:
3650:
3647:
3642:
3638:
3614:
3606:
3602:
3598:
3595:
3592:
3587:
3583:
3575:
3566:
3562:
3558:
3555:
3552:
3547:
3543:
3536:
3533:
3530:
3522:
3518:
3502:
3499:
3496:
3493:
3470:
3464:
3461:
3455:
3449:
3441:
3437:
3433:
3429:
3424:
3421:
3402:
3396:
3393:
3387:
3381:
3378:
3372:
3366:
3358:
3355:not equal to
3354:
3350:
3346:
3342:
3337:
3335:
3331:
3327:
3323:
3319:
3315:
3311:
3307:
3303:
3287:
3281:
3275:
3272:
3266:
3260:
3240:
3237:
3234:
3226:
3222:
3218:
3214:
3209:
3196:
3191:
3186:
3177:
3171:
3165:
3157:
3141:
3133:
3129:
3125:
3122:
3119:
3114:
3110:
3103:
3095:
3091:
3087:
3083:
3079:
3075:
3072:of the ideal
3071:
3053:
3043:
3027:
3019:
3015:
3011:
3008:
3005:
3000:
2996:
2989:
2981:
2977:
2973:
2970:be ideals of
2969:
2965:
2955:
2953:
2949:
2945:
2941:
2936:
2934:
2930:
2926:
2922:
2918:
2914:
2898:
2892:
2889:
2883:
2877:
2874:
2869:
2865:
2861:
2858:
2852:
2847:
2843:
2839:
2833:
2827:
2819:
2803:
2795:
2791:
2787:
2784:
2781:
2776:
2772:
2765:
2762:
2759:
2736:
2733:
2727:
2721:
2718:
2713:
2709:
2705:
2702:
2696:
2691:
2687:
2678:
2673:
2656:
2653:
2650:
2644:
2641:
2635:
2629:
2626:
2620:
2614:
2594:
2588:
2585:
2579:
2576:
2570:
2564:
2561:
2555:
2549:
2530:
2524:
2518:
2512:
2493:
2488:
2484:
2480:
2474:
2468:
2460:
2457:, called the
2456:
2452:
2446:
2436:
2434:
2414:
2411:
2408:
2405:
2402:
2399:
2396:
2393:
2389:
2386:
2383:
2375:
2371:
2367:
2362:
2358:
2346:
2342:
2338:
2335:
2332:
2327:
2323:
2310:
2306:
2294:
2290:
2278:
2273:
2270:
2267:
2263:
2255:
2254:
2253:
2252:
2234:
2230:
2221:
2209:
2208:tangent space
2200:
2198:
2190:
2184:
2179:
2175:
2170:
2147:
2139:
2135:
2131:
2128:
2125:
2120:
2116:
2103:
2099:
2087:
2083:
2069:
2068:
2067:
2055:
2050:
2046:
2042:
2037:
2014:
2010:
2006:
2003:
2000:
1995:
1991:
1984:
1981:
1961:
1953:
1949:
1945:
1942:
1939:
1934:
1930:
1923:
1920:
1915:
1911:
1907:
1904:
1901:
1896:
1892:
1873:
1860:
1855:
1845:
1839:
1836:
1833:
1828:
1824:
1820:
1815:
1811:
1807:
1802:
1798:
1791:
1782:
1776:
1770:
1764:
1758:
1752:
1746:
1743:
1724:
1714:
1708:
1705:
1700:
1696:
1692:
1687:
1683:
1676:
1667:
1664:
1658:
1652:
1643:
1638:
1633:
1615:
1605:
1599:
1596:
1591:
1587:
1583:
1578:
1574:
1567:
1558:
1536:
1527:
1523:
1519:
1516:
1511:
1508:
1502:
1494:
1490:
1486:
1483:
1476:
1472:
1468:
1465:
1458:
1450:
1449:
1448:
1445:
1440:
1435:
1430:
1425:
1419:
1397:
1392:
1389:
1377:
1349:
1332:
1327:
1317:
1311:
1308:
1303:
1299:
1295:
1290:
1286:
1279:
1276:
1273:
1264:
1258:
1248:
1246:
1242:
1237:
1231:
1226:
1223:) are called
1222:
1217:
1211:
1205:
1188:
1182:
1176:
1167:
1161:
1155:
1149:
1132:
1127:
1123:
1119:
1116:
1113:
1110:
1101:
1096:
1091:
1085:
1079:
1073:
1054:
1050:
1046:
1043:
1033:
1024:
1020:
1017: −
1016:
1013: =
1012:
1006:
994:
990:
989:normalization
986:
969:
965:
956:
953:
937:
934:
929:
914:
899:
896:
883:
863:
860:
856:
846:
840:
833:
828:
819:
814:
807:
803:
797:
792:
791:
785:
783:
779:
774:
772:
768:
764:
760:
759:
753:
749:
745:
741:
737:
733:
729:
725:
705:
701:
692:
688:
684:
681:
678:
673:
669:
662:
659:
656:
649:
648:quotient ring
640:
634:
628:
626:
622:
603:
599:
595:
592:
584:
580:
576:
573:
570:
565:
561:
552:
548:
544:
541:
538:
530:
526:
522:
519:
516:
511:
507:
498:
494:
481:
477:
473:
465:
461:
457:
454:
451:
446:
442:
434:
430:
422:
418:
414:
411:
408:
403:
399:
392:
385:
384:
383:
380:
361:
357:
353:
350:
347:
342:
338:
328:
322:
317:
307:
299:
296: +
295:
290:
286:
282:
274:
266:
255:
254:rational over
251:
243:
235:
230:
213:
208:
206:
202:
197:
195:
191:
187:
183:
167:
159:
155:
151:
148:
145:
140:
136:
129:
122:
118:
111:
107:
103:
99:
77:
74:
71:
63:
59:
55:
50:
46:
37:
32:
19:
6877:
6841:
6829:
6818:. Retrieved
6814:
6795:0-201-510103
6781:
6745:
6735:
6700:Mumford 1999
6695:
6682:
6670:
6663:Milne (2017)
6658:
6647:
6630:
6610:
6603:
6596:
6589:
6585:
6581:
6575:
6569:
6531:real numbers
6514:
6510:
6503:
6499:
6488:
6483:
6477:
6472:
6463:
6457:
6453:
6446:
6435:
6432:vector space
6425:
6414:
6409:
6404:
6398:
6394:
6391:
6387:
6381:
6377:
6373:
6367:
6363:
6359:
6355:
6348:
6342:
6334:
6330:
6324:
6317:
6313:
6307:
6303:
6296:
6292:
6288:
6281:
6274:
6268:
6262:
6258:
6254:
6250:
6246:
6242:
6238:
6234:
6230:
6224:
6220:
6216:
6211:
6203:
6197:
6191:
6187:
6183:
6179:
6175:
6171:
6167:
6161:
6157:
6149:
6143:
6137:
6131:
6124:
6120:
6116:
6112:
6108:
6104:
6100:
6096:
6092:
6088:
6078:
6074:
6070:
6066:
6062:
6054:
6049:
6043:
6040:
6020:
6016:
5933:
5876:
5871:
5867:
5863:
5859:
5754:
5625:
5623:
5563:
5559:
5487:
5483:
5479:
5475:
5471:
5468:
5395:
5391:
5387:
5383:
5379:
5375:
5371:
5367:
5308:
5306:
5300:
5296:
5209:
5201:
5192:
5188:
5153:
5149:
5145:
5141:
5137:
5133:
5129:
5125:
5123:
5118:
5007:
5003:
4999:
4997:
4990:
4978:
4974:
4967:
4962:
4958:
4953:
4949:
4942:
4935:
4930:
4926:
4919:
4915:
4909:
4905:
4901:
4895:
4826:
4822:
4816:
4812:
4808:
4805:
4799:
4792:
4788:
4710:
4706:
4700:
4696:
4692:
4685:
4679:
4671:
4667:
4660:
4656:
4649:
4645:
4639:
4635:
4631:
4623:
4619:
4612:
4606:
4602:
4598:
4594:
4587:
4581:
4577:
4573:
4569:
4565:
4559:
4555:
4551:
4545:
4541:
4537:
4531:
4527:
4523:
4517:
4513:
4507:
4503:
4496:
4492:
4488:
4482:
4478:
4474:
4471:
4465:
4459:
4452:
4442:
4436:
4430:
4423:
4420:
4405:
4401:
4395:
4390:
4386:
4378:
4374:
4367:
4362:
4358:
4353:
4349:
4342:
4335:
4330:
4326:
4319:
4315:
4313:of the form
4309:
4305:
4301:
4295:
4289:
4284:
4279:
4275:
4269:
4265:
4262:
4244:
4240:
4234:
4228:
4222:
4215:
4211:
4207:
4202:
4198:
4191:
4187:
4183:
4178:
4174:
4163:
4159:
4156:
4150:
4144:
4137:
4129:
4125:
4118:
4113:
4109:
4102:
4098:
4094:
4090:
4084:
4078:
4073:
4068:
4061:
4057:
4051:
4044:
4040:
4033:
4025:
4021:
4014:
4010:
4006:
3999:
3991:
3987:
3980:
3976:
3972:
3966:
3960:
3954:
3948:
3942:
3936:
3929:
3925:
3921:
3918:
3862:
3814:
3520:
3516:
3439:
3435:
3431:
3427:
3425:
3419:
3356:
3352:
3348:
3344:
3340:
3338:
3336:is reduced.
3333:
3329:
3325:
3317:
3313:
3309:
3305:
3301:
3224:
3220:
3216:
3212:
3210:
3155:
3093:
3085:
3081:
3077:
3073:
3041:
2979:
2975:
2971:
2967:
2963:
2961:
2951:
2947:
2943:
2939:
2937:
2932:
2920:
2912:
2817:
2674:
2458:
2454:
2450:
2448:
2429:
2207:
2201:
2196:
2182:
2177:
2173:
2168:
2162:
2053:
2048:
2044:
2038:
1879:
1780:
1774:
1768:
1762:
1756:
1750:
1747:
1741:
1668:
1662:
1656:
1653:
1636:
1559:
1552:
1443:
1433:
1423:
1421:that is not
1417:
1375:
1373:that is not
1347:
1345:as it is in
1262:
1256:
1249:
1244:
1235:
1229:
1224:
1221:real numbers
1215:
1209:
1203:
1165:
1159:
1153:
1147:
1099:
1094:
1089:
1083:
1077:
1071:
1035:
1022:
1018:
1014:
1010:
832:localization
826:
812:
805:
801:
795:
777:
775:
770:
762:
755:
751:
747:
743:
735:
731:
727:
719:
638:
632:
629:
624:
620:
618:
378:
326:
320:
315:
313:
310:Introduction
297:
293:
280:
272:
265:real numbers
253:
246:
241:
234:defined over
233:
228:
222:(containing
209:
204:
200:
198:
185:
181:
101:
95:
6872:Reid, Miles
6675:Reid (1988)
6652:Reid (1988)
6626:finite type
6622:irreducible
6057:if it has:
5467:; that is,
4913:defined by
4704:that sends
4168:is not the
4124:,...,
4108:,...,
4055:, and each
4020:,...,
3986:,...,
3894:prime ideal
3068:denote the
2199:otherwise.
2189:codimension
2187:equals the
1560:The circle
1439:unit circle
1225:real points
1103:is a point
646:, then the
625:irreducible
117:polynomials
6863:354063293X
6732:References
6423:of degree
6266:for every
6228:such that
6195:for every
6165:such that
5482:is not in
5398:such that
5010:-algebras
4404:= 1, ...,
4038:Then each
3328:in a ring
3044:, and let
2917:Noetherian
2163:The point
818:saturating
273:real point
6624:, and of
6493:, as the
5786:∈
5708:−
5694:
5686:≠
5668:→
5610:◻
5587:∈
5510:⊂
5478:and thus
5447:−
5409:∈
5348:∈
5331:∈
5275:−
5223:Γ
5075:Γ
4865:…
4772:¯
4735:¯
4412:morphisms
4399:for each
4299:is a map
3835:−
3799:¯
3785:−
3749:⟩
3744:¯
3730:−
3711:…
3703:¯
3689:−
3673:⟨
3670:↦
3651:…
3612:⟩
3596:…
3580:⟨
3556:…
3497:⊆
3462:⊆
3394:∪
3302:V(I)=V(J)
3273:⊆
3238:⊆
3123:…
3009:…
2862:∈
2785:…
2763:∈
2734:≠
2706:∈
2627:∩
2562:∪
2528:∅
2406:…
2368:−
2336:…
2302:∂
2287:∂
2264:∑
2129:…
2095:∂
2080:∂
2004:…
1943:…
1921:∈
1905:…
1846:⊆
1808:−
1715:⊆
1606:⊆
1597:−
1469:−
1318:⊆
1309:−
1120:∩
1114:∈
1047:⊆
935:−
897:−
861:−
799:(that is
682:…
574:…
542:…
520:…
474:∈
455:…
412:…
351:…
250:-rational
245:are said
149:…
38:given by
6901:Category
6874:(1988).
6840:(1999).
6806:(2017).
6779:(1969).
6744:(1977),
6710:See also
6561:spectrum
6241:),
6103:),
6012:and any
5986:for any
5566:; i.e.,
5558:implies
5295:for any
4957:), ...,
4925:, ... ,
4904: :
4811: :
4695: :
4634: :
4597: :
4568: :
4554: :
4491: :
4477: :
4416:category
4357:), ...,
4304: :
4285:morphism
4218: ).
2197:singular
788:Examples
756:ring of
108:over an
6820:16 July
6770:0463157
6665:, Ch. 5
6618:reduced
6316:⋅
6295:,
6253:,
6219::
6186:,
6174:,
6123:,
6115:,
6099:,
6069::
6023:. (cf.
5877:Hence,
5156:and so
4973:, ...,
4948:, ...,
4666:, ...,
4618:, ...,
4414:in the
4373:, ...,
4348:, ...,
4325:, ...,
4247: ,
4214:(
4194: )
4190:(
4117:,
4101:(
4074:product
4013:(
3979:(
3322:reduced
3070:radical
2218:is the
2169:regular
1739:has no
1362:√
1355:√
1219:is the
750:or the
201:variety
119:in the
6886:
6860:
6792:
6768:
6758:
5755:where
5474:is in
5370:is in
5132:) = {
4820:sends
4752:where
4384:where
4166:
4072:. The
4066:is in
3970:. Let
3908:field
3764:where
3300:Hence
3154:where
3084:is in
2978:. Let
2195:, and
1766:, the
1642:modulo
1553:where
1429:circle
1252:(1, 0)
6811:(PDF)
6786:(PDF)
6639:Notes
6602:Spec(
6442:basis
6440:if a
5394:) of
5211:Claim
4934:) = (
4677:from
4334:) = (
4287:from
3902:point
3519:. If
3434:. If
2927:or a
2925:field
2923:is a
1254:is a
808:= 0 }
730:. If
194:prime
190:ideal
106:zeros
100:, an
6884:ISBN
6858:ISBN
6822:2021
6790:ISBN
6756:ISBN
6597:V(I)
6570:V(I)
6419:the
6385:and
6141:and
5997:>
5124:Let
4535:and
4511:and
4448:dual
4283:, a
4273:and
4196:and
4082:and
4004:and
3952:and
3940:and
3438:and
3351:and
3345:I=JK
3341:V(I)
3314:I(W)
3223:and
2980:I(V)
2966:and
2752:for
2607:and
2039:The
1974:and
1880:Let
1360:/2,
1239:the
1087:, a
987:The
804:\ {
776:The
267:, a
6850:doi
6600:is
6486:.
6444:of
6346:or
6322:or
6272:in
6210:An
6201:in
6156:An
6147:in
6019:on
5665:lim
5299:in
5199:.)
5148:in
5117:on
4982:)).
4893:in
4797:in
4715:to
4683:to
4659:= (
4446:is
4382:)),
4293:to
4135:in
4076:of
3334:R/I
3306:I=J
3096:in
2938:If
2915:is
2222:of
2214:at
2210:to
2202:If
2191:of
2167:is
2062:at
2058:of
1748:If
1367:/2)
1097:of
1081:of
820:by
784:).
726:of
630:If
627:.
619:An
314:An
252:or
184:or
180:An
96:In
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