31:
171:, not only the usual points corresponding to maximal ideals, but also all (irreducible) algebraic varieties, which correspond to prime ideals. Thus the Zariski topology on the set of prime ideals (spectrum) of a commutative ring is the topology such that a set of prime ideals is closed if and only if it is the set of all prime ideals that contain a fixed ideal.
1385:
The projective
Zariski topology is defined for projective algebraic sets just as the affine one is defined for affine algebraic sets, by taking the subspace topology. Similarly, it may be shown that this topology is defined intrinsically by sets of elements of the projective coordinate ring, by the
1526:
in the
Zariski topology. In fact, the Zariski topology is the weakest topology (with the fewest open sets) in which this is true and in which points are closed. This is easily verified by noting that the Zariski-closed sets are simply the intersections of the inverse images of 0 by the polynomial
2066:
replaces projective varieties in modern algebraic geometry. Just as in the classical case, to move from the affine to the projective definition we need only replace "ideal" by "homogeneous ideal", though there is a complication involving the "irrelevant maximal ideal", which is discussed in the
1933:
2059:-tuples. Since as shown above, the classical definition is essentially the modern definition with only maximal ideals considered, this shows that the interpretation of the modern definition as "zero sets of functions" agrees with the classical definition where they both make sense.
2497:
Just as in classical algebraic geometry, any spectrum or projective spectrum is (quasi)compact, and if the ring in question is
Noetherian then the space is a Noetherian topological space. However, these facts are counterintuitive: we do not normally expect open sets, other than
1465:. In the old topological literature "compact" was taken to include the Hausdorff property, and this convention is still honored in algebraic geometry; therefore compactness in the modern sense is called "quasicompactness" in algebraic geometry. However, since every point (
2502:, to be compact, and for affine varieties (for example, Euclidean space) we do not even expect the space itself to be compact. This is one instance of the geometric unsuitability of the Zariski topology. Grothendieck solved this problem by defining the notion of
2076:
163:. This suggests defining the Zariski topology on the set of the maximal ideals of a commutative ring as the topology such that a set of maximal ideals is closed if and only if it is the set of all maximal ideals that contain a given ideal. Another basic idea of
1756:. Grothendieck's innovation in defining Spec was to replace maximal ideals with all prime ideals; in this formulation it is natural to simply generalize this observation to the definition of a closed set in the spectrum of a ring.
397:
1817:
1310:
1670:
996:
2010:
1415:. That these form a basis follows from the formula for the intersection of two Zariski-closed sets given above (apply it repeatedly to the principal ideals generated by the generators of
2166:
591:
520:
2442:
The most dramatic change in the topology from the classical picture to the new is that points are no longer necessarily closed; by expanding the definition, Grothendieck introduced
1116:
1557:
1361:
690:
648:
306:
263:
1204:
1081:
1037:
874:
786:
2404:
2362:
2328:
2192:
2118:
311:
1744:; moreover, these are maximal ideals and by the "weak" Nullstellensatz, an ideal of any affine coordinate ring is maximal if and only if it is of this form. Thus,
1210:
the condition of having zero or nonzero value on any given projective point is well-defined since the scalar multiple factors out of the polynomial. Therefore, if
1175:
845:
2680:. Lecture Notes in Mathematics. Vol. 1358 (expanded, Includes Michigan Lectures (1974) on Curves and their Jacobians ed.). Berlin, New York:
1928:{\displaystyle e_{a}\colon {\bigl (}P\in \operatorname {Spec} A{\bigr )}\mapsto \left({\frac {a\;{\bmod {P}}}{1}}\in \operatorname {Frac} (A/P)\right)}
913:
276:. The topology is defined by specifying its closed sets, rather than its open sets, and these are taken simply to be all the algebraic sets in
1220:
1610:
1450:, every affine or projective coordinate ring is Noetherian. As a consequence, affine or projective spaces with the Zariski topology are
2697:
2663:
2620:
2510:(actually, of a morphism of schemes), which recovers the intuitive idea of compactness: Proj is proper, but Spec is not.
1952:
1759:
Another way, perhaps more similar to the original, to interpret the modern definition is to realize that the elements of
606:) are also of this form, so that these sets form the closed sets of a topology (equivalently, their complements, denoted
144:, the Zariski topology is thus coarser than the usual topology, as every algebraic set is closed for the usual topology.
1315:
The same facts as above may be established for these sets, except that the word "ideal" must be replaced by the phrase "
2592:
702:
1519:
17:
207:, it is useful to make this definition more explicit in both cases. We assume that we are working over a fixed,
2137:
1523:
526:
458:
2655:
2172:(i.e., whose closure is the whole space) corresponding to the zero ideal (0). So the closed subsets of Spec
656:
is an affine algebraic set (irreducible or not) then the
Zariski topology on it is defined simply to be the
2742:
2031:
vanish, which is formally similar to the classical definition. In fact, they agree in the sense that when
1687:
1451:
1010:
This establishes that the above equation, clearly a generalization of the definition of the closed sets in
148:
2299:
irreducible polynomial, and a generic point corresponding to the zero ideal. For example, the spectrum of
147:
The generalization of the
Zariski topology to the set of prime ideals of a commutative ring follows from
2429:, and, in a principal ideal domain, the prime ideals that contain an ideal are the prime factors of the
1086:
1530:
1439:
1334:
663:
621:
279:
236:
208:
152:
1431:
open sets. The importance of this property results in particular from its use in the definition of an
1180:
1057:
1013:
850:
151:, that establishes a bijective correspondence between the points of an affine variety defined over an
2752:
2607:
1206:
because any point has many representatives that yield different values in a polynomial; however, for
179:
In classical algebraic geometry (that is, the part of algebraic geometry in which one does not use
2387:
2345:
2302:
2175:
2101:
2649:
1942:"), which assigns to each point its reflection in the residue field there, as a function on Spec
2747:
2426:
2292:
2217:
2213:
1207:
2275:
equipped with its
Zariski topology. Because of this homeomorphism, some authors use the term
191:. The Zariski topology, defined on the points of the variety, is the topology such that the
2707:
2630:
2507:
2233:
1575:
1567:
204:
180:
55:
2421:
are finite unions of closed points, and the whole space. (This results from the fact that
2264:
and a generic point, corresponding to the zero ideal, and the set of the closed points is
1566:
In modern algebraic geometry, an algebraic variety is often represented by its associated
8:
2447:
2430:
2206:
2089:
435:
188:
102:
47:
2055:, and the "evaluation" maps are actually evaluation of polynomials at the corresponding
1795:
has a reflection in this residue field. Furthermore, the elements that are actually in
1780:
1579:
1125:
795:
87:
43:
1946:(whose values, admittedly, lie in different fields at different points), then we have
2717:
2693:
2659:
2634:
2616:
2588:
2063:
1571:
1316:
657:
106:
98:
59:
2417:< 0, and finally a generic point (0). For any field, the closed subsets of Spec
2240:, a non-constant polynomial is irreducible if and only if it is linear, of the form
2685:
2602:
2296:
1052:
160:
122:
114:
83:
67:
1409:
for individual polynomials (or for projective varieties, homogeneous polynomials)
2703:
2681:
2626:
2612:
2503:
2499:
2237:
2202:
1462:
1443:
1395:
697:
392:{\displaystyle V(S)=\{x\in \mathbb {A} ^{n}\mid f(x)=0{\text{ for all }}f\in S\}}
71:
2519:
219:
200:
141:
2291:, the picture becomes more complicated because of the existence of non-linear
2736:
2673:
2443:
2221:
2169:
2132:
1776:
1455:
1447:
1432:
196:
156:
137:
110:
75:
63:
2638:
2407:
2265:
2125:
231:
184:
164:
126:
113:, which allows one to build general algebraic varieties by gluing together
35:
2645:
2288:
2269:
1305:{\displaystyle V(S)=\{x\in \mathbb {P} ^{n}\mid f(x)=0,\forall f\in S\}.}
79:
1682:
To see the connection with the classical picture, note that for any set
192:
133:
1461:
However, except for finite algebraic sets, no algebraic set is ever a
1083:
is defined to be the set of equivalence classes of non-zero points in
598:
It follows that finite unions and arbitrary intersections of the sets
2722:
2194:
are precisely the whole space and the finite unions of closed points.
1686:
of polynomials (over an algebraically closed field), it follows from
1665:{\displaystyle V(I)=\{P\in \operatorname {Spec} A\mid P\supseteq I\}}
998:(these notations are not standard) is equal to the intersection with
30:
2689:
2455:
2295:. In this case, the spectrum consists of one closed point for each
1508:
132:
The
Zariski topology of an algebraic variety is the topology whose
118:
94:
2256:. So, the spectrum consists of one closed point for every element
2121:
1214:
is any set of homogeneous polynomials we may reasonably speak of
62:. It is very different from topologies that are commonly used in
2035:
is the ring of polynomials over some algebraically closed field
1394:
An important property of
Zariski topologies is that they have a
199:
of the variety. As the most elementary algebraic varieties are
1763:
can actually be thought of as functions on the prime ideals of
266:
1118:
by identifying two points that differ by a scalar multiple in
618:, form the topology itself). This is the Zariski topology on
2043:
are (as discussed in the previous paragraph) identified with
1879:
140:
of the variety. In the case of an algebraic variety over the
2537:
2535:
1039:
above, defines the
Zariski topology on any affine variety.
2454:. However, the spectrum and projective spectrum are still
2287:
is not algebraically closed, for example the field of the
1454:, which implies that any closed subset of these spaces is
991:{\displaystyle V'(T)=\{x\in X\mid f(x)=0,\forall f\in T\}}
2714:
2446:, which are the points with maximal closure, that is the
2075:
2532:
1803:. So if we think of the map, associated to any element
1507:, points are closed and so every variety satisfies the
2559:
411:. It is a straightforward verification to show that:
2450:. The closed points correspond to maximal ideals of
2390:
2348:
2305:
2178:
2140:
2104:
1955:
1820:
1613:
1533:
1337:
1223:
1183:
1128:
1089:
1060:
1016:
916:
853:
798:
705:
666:
624:
529:
461:
314:
282:
239:
121:
theory, where manifolds are built by gluing together
2005:{\displaystyle e_{a}(P)=0\Leftrightarrow P\in V(a)}
1712:) such that the ideal generated by the polynomials
1363:As above the complements of these sets are denoted
2547:
2398:
2356:
2322:
2186:
2160:
2112:
2027:is the common set on which all the "functions" in
2004:
1927:
1664:
1551:
1355:
1304:
1198:
1169:
1110:
1075:
1031:
990:
868:
839:
780:
684:
642:
585:
514:
391:
300:
257:
1799:are precisely those whose reflection vanishes at
1752:) is "the same as" the maximal ideals containing
1331:of homogeneous polynomials, define a topology on
187:around 1960), the Zariski topology is defined on
2734:
174:
27:Topology on prime ideals and algebraic varieties
884:) is the ideal of all polynomials vanishing on
308:That is, the closed sets are those of the form
1574:(equipped with additional structures) that is
1858:
1836:
1698:) (in the old sense) are exactly the tuples (
2062:Just as Spec replaces affine varieties, the
1659:
1629:
1296:
1239:
985:
937:
386:
330:
78:and later generalized for making the set of
74:. This topology was introduced primarily by
2582:
2565:
1527:functions, considered as regular maps into
2601:
2212:: such a polynomial ring is known to be a
2095:is the topological space with one element.
1877:
1398:consisting of simple elements, namely the
2392:
2350:
2307:
2180:
2161:{\displaystyle (p)\subseteq \mathbb {Z} }
2154:
2106:
1604:, for which the closed sets are the sets
1536:
1423:). The open sets in this base are called
1371:), or, if confusion is likely to result,
1340:
1250:
1186:
1092:
1063:
1019:
856:
722:
718:
669:
627:
586:{\displaystyle V(I)\cap V(J)\,=\,V(I+J).}
561:
557:
493:
489:
341:
285:
242:
2074:
515:{\displaystyle V(I)\cup V(J)\,=\,V(IJ);}
29:
2672:
2553:
1122:. The elements of the polynomial ring
1042:
93:The Zariski topology allows tools from
38:, this graph of a polynomial is closed.
14:
2735:
692:Equivalently, it can be checked that:
2715:
2678:The Red Book of Varieties and Schemes
2644:
2587:(3 ed.). Wiley. pp. 71–72.
2541:
2437:
1561:
1479:) is the zero set of the polynomials
230:First, we define the topology on the
2478:, does not contain the other. Then
1596:, is the set of the prime ideals of
109:. This is one of the basic ideas of
58:defined on geometric objects called
660:induced by its inclusion into some
225:
167:'s scheme theory is to consider as
24:
1284:
1111:{\displaystyle \mathbb {A} ^{n+1}}
973:
445:For any two ideals of polynomials
214:(in classical algebraic geometry,
90:of the ring) a topological space.
25:
2764:
2583:Dummit, D. S.; Foote, R. (2004).
1552:{\displaystyle \mathbb {A} ^{1}.}
1356:{\displaystyle \mathbb {P} ^{n}.}
685:{\displaystyle \mathbb {A} ^{n}.}
643:{\displaystyle \mathbb {A} ^{n}.}
301:{\displaystyle \mathbb {A} ^{n}.}
258:{\displaystyle \mathbb {A} ^{n},}
125:, which are open subsets of real
2051:, their residue fields are just
1199:{\displaystyle \mathbb {P} ^{n}}
1076:{\displaystyle \mathbb {P} ^{n}}
1032:{\displaystyle \mathbb {A} ^{n}}
869:{\displaystyle \mathbb {A} ^{n}}
2330:consists of the closed points (
1767:; namely, as functions on Spec
1177:are not generally functions on
781:{\displaystyle A(X)\,=\,k/I(X)}
34:In the Zariski topology on the
2317:
2311:
2147:
2141:
1999:
1993:
1981:
1972:
1966:
1917:
1903:
1863:
1623:
1617:
1584:spectrum of a commutative ring
1272:
1266:
1233:
1227:
1164:
1132:
961:
955:
931:
925:
899:be the set of their images in
834:
802:
775:
769:
758:
726:
715:
709:
577:
565:
554:
548:
539:
533:
506:
497:
486:
480:
471:
465:
363:
357:
324:
318:
13:
1:
2651:Elementary Algebraic Geometry
2575:
2433:of a generator of the ideal).
2124:has a closed point for every
1452:Noetherian topological spaces
1389:
438:generated by the elements of
403:is any set of polynomials in
175:Zariski topology of varieties
2525:
2474:, at least one of them, say
2399:{\displaystyle \mathbb {R} }
2357:{\displaystyle \mathbb {R} }
2323:{\displaystyle \mathbb {R} }
2187:{\displaystyle \mathbb {Z} }
2113:{\displaystyle \mathbb {Z} }
117:in a way similar to that in
7:
2513:
2236:, for example the field of
2070:
1771:. Simply, any prime ideal
891:For any set of polynomials
696:The elements of the affine
183:, which were introduced by
101:, even when the underlying
70:; in particular, it is not
10:
2769:
209:algebraically closed field
153:algebraically closed field
2470:that are prime ideals of
1688:Hilbert's Nullstellensatz
149:Hilbert's Nullstellensatz
2039:, the maximal ideals of
792:just as the elements of
218:is usually the field of
2566:Dummit & Foote 2004
2293:irreducible polynomials
2218:irreducible polynomials
2047:-tuples of elements of
1440:Hilbert's basis theorem
1386:same formula as above.
1208:homogeneous polynomials
907:). Then the subset of
2427:principal ideal domain
2400:
2358:
2324:
2214:principal ideal domain
2201:, the spectrum of the
2188:
2162:
2120:, the spectrum of the
2114:
2080:
2006:
1929:
1666:
1553:
1357:
1306:
1200:
1171:
1112:
1077:
1033:
992:
870:
841:
782:
686:
644:
587:
516:
393:
302:
259:
39:
2544:, p. 19, 1.1.1..
2401:
2364:, the closed points (
2359:
2325:
2189:
2168:, and one non-closed
2163:
2131:corresponding to the
2115:
2078:
2007:
1930:
1791:, and any element of
1667:
1554:
1358:
1307:
1201:
1172:
1113:
1078:
1034:
993:
871:
842:
783:
687:
645:
588:
517:
394:
303:
260:
33:
2611:, Berlin, New York:
2500:connected components
2490:but, of course, not
2448:minimal prime ideals
2388:
2346:
2303:
2279:for the spectrum of
2234:algebraically closed
2176:
2138:
2102:
2088:, the spectrum of a
1953:
1818:
1775:has a corresponding
1611:
1600:, equipped with the
1576:locally homeomorphic
1531:
1335:
1221:
1181:
1126:
1087:
1058:
1043:Projective varieties
1014:
914:
851:
847:act as functions on
796:
788:act as functions on
703:
664:
622:
527:
459:
312:
280:
237:
205:projective varieties
97:to be used to study
2743:Algebraic varieties
2462:: given two points
2431:prime factorization
2248:, for some element
1690:that the points of
616:principal open sets
374: for all
189:algebraic varieties
159:of the ring of its
99:algebraic varieties
48:commutative algebra
2718:"Zariski Topology"
2608:Algebraic Geometry
2438:Further properties
2406:and with negative
2396:
2354:
2320:
2184:
2158:
2110:
2081:
2002:
1925:
1781:field of fractions
1662:
1580:spectrum of a ring
1562:Spectrum of a ring
1549:
1442:and the fact that
1353:
1302:
1196:
1167:
1108:
1073:
1029:
988:
866:
837:
778:
682:
640:
583:
512:
389:
298:
255:
44:algebraic geometry
40:
2699:978-3-540-63293-1
2665:978-0-8218-2952-3
2622:978-0-387-90244-9
2603:Hartshorne, Robin
2079:The spectrum of ℤ
2064:Proj construction
1892:
1572:topological space
1446:are closed under
1317:homogeneous ideal
1170:{\displaystyle k}
840:{\displaystyle k}
658:subspace topology
375:
197:algebraic subsets
161:regular functions
138:algebraic subsets
107:topological field
16:(Redirected from
2760:
2753:General topology
2728:
2727:
2711:
2669:
2641:
2598:
2585:Abstract Algebra
2569:
2563:
2557:
2551:
2545:
2539:
2405:
2403:
2402:
2397:
2395:
2363:
2361:
2360:
2355:
2353:
2329:
2327:
2326:
2321:
2310:
2193:
2191:
2190:
2185:
2183:
2167:
2165:
2164:
2159:
2157:
2119:
2117:
2116:
2111:
2109:
2023:) for any ideal
2015:More generally,
2011:
2009:
2008:
2003:
1965:
1964:
1938:("evaluation of
1934:
1932:
1931:
1926:
1924:
1920:
1913:
1893:
1888:
1887:
1886:
1872:
1862:
1861:
1840:
1839:
1830:
1829:
1783:of the quotient
1671:
1669:
1668:
1663:
1602:Zariski topology
1595:
1558:
1556:
1555:
1550:
1545:
1544:
1539:
1522:of varieties is
1444:Noetherian rings
1422:
1414:
1408:
1376:
1362:
1360:
1359:
1354:
1349:
1348:
1343:
1311:
1309:
1308:
1303:
1259:
1258:
1253:
1205:
1203:
1202:
1197:
1195:
1194:
1189:
1176:
1174:
1173:
1168:
1163:
1162:
1144:
1143:
1117:
1115:
1114:
1109:
1107:
1106:
1095:
1082:
1080:
1079:
1074:
1072:
1071:
1066:
1053:projective space
1038:
1036:
1035:
1030:
1028:
1027:
1022:
997:
995:
994:
989:
924:
875:
873:
872:
867:
865:
864:
859:
846:
844:
843:
838:
833:
832:
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813:
787:
785:
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779:
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738:
737:
691:
689:
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683:
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672:
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647:
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641:
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630:
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589:
584:
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519:
518:
513:
398:
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390:
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373:
350:
349:
344:
307:
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299:
294:
293:
288:
275:
269:
264:
262:
261:
256:
251:
250:
245:
226:Affine varieties
115:affine varieties
84:commutative ring
68:complex analysis
52:Zariski topology
21:
2768:
2767:
2763:
2762:
2761:
2759:
2758:
2757:
2733:
2732:
2731:
2700:
2682:Springer-Verlag
2666:
2623:
2613:Springer-Verlag
2595:
2578:
2573:
2572:
2564:
2560:
2552:
2548:
2540:
2533:
2528:
2516:
2459:
2440:
2391:
2389:
2386:
2385:
2349:
2347:
2344:
2343:
2306:
2304:
2301:
2300:
2238:complex numbers
2203:polynomial ring
2179:
2177:
2174:
2173:
2153:
2139:
2136:
2135:
2105:
2103:
2100:
2099:
2073:
2067:cited article.
1960:
1956:
1954:
1951:
1950:
1909:
1882:
1878:
1873:
1871:
1870:
1866:
1857:
1856:
1835:
1834:
1825:
1821:
1819:
1816:
1815:
1779:, which is the
1738:
1731:
1725:
1718:
1710:
1704:
1612:
1609:
1608:
1590:
1564:
1540:
1535:
1534:
1532:
1529:
1528:
1512:
1505:
1498:
1491:
1484:
1477:
1470:
1463:Hausdorff space
1416:
1410:
1399:
1392:
1374:
1344:
1339:
1338:
1336:
1333:
1332:
1319:", so that the
1254:
1249:
1248:
1222:
1219:
1218:
1190:
1185:
1184:
1182:
1179:
1178:
1158:
1154:
1139:
1135:
1127:
1124:
1123:
1096:
1091:
1090:
1088:
1085:
1084:
1067:
1062:
1061:
1059:
1056:
1055:
1045:
1023:
1018:
1017:
1015:
1012:
1011:
917:
915:
912:
911:
860:
855:
854:
852:
849:
848:
828:
824:
809:
805:
797:
794:
793:
761:
752:
748:
733:
729:
704:
701:
700:
698:coordinate ring
673:
668:
667:
665:
662:
661:
631:
626:
625:
623:
620:
619:
528:
525:
524:
460:
457:
456:
407:variables over
372:
345:
340:
339:
313:
310:
309:
289:
284:
283:
281:
278:
277:
273:
272:of elements of
267:
246:
241:
240:
238:
235:
234:
228:
220:complex numbers
177:
142:complex numbers
28:
23:
22:
18:Zariski closure
15:
12:
11:
5:
2766:
2756:
2755:
2750:
2745:
2730:
2729:
2716:Todd Rowland.
2712:
2698:
2690:10.1007/b62130
2674:Mumford, David
2670:
2664:
2642:
2621:
2599:
2593:
2579:
2577:
2574:
2571:
2570:
2558:
2546:
2530:
2529:
2527:
2524:
2523:
2522:
2520:Spectral space
2515:
2512:
2457:
2444:generic points
2439:
2436:
2435:
2434:
2394:
2352:
2319:
2316:
2313:
2309:
2222:prime elements
2195:
2182:
2156:
2152:
2149:
2146:
2143:
2108:
2096:
2072:
2069:
2013:
2012:
2001:
1998:
1995:
1992:
1989:
1986:
1983:
1980:
1977:
1974:
1971:
1968:
1963:
1959:
1936:
1935:
1923:
1919:
1916:
1912:
1908:
1905:
1902:
1899:
1896:
1891:
1885:
1881:
1876:
1869:
1865:
1860:
1855:
1852:
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1846:
1843:
1838:
1833:
1828:
1824:
1736:
1729:
1723:
1716:
1708:
1702:
1673:
1672:
1661:
1658:
1655:
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1649:
1646:
1643:
1640:
1637:
1634:
1631:
1628:
1625:
1622:
1619:
1616:
1563:
1560:
1548:
1543:
1538:
1510:
1503:
1496:
1489:
1482:
1475:
1468:
1391:
1388:
1352:
1347:
1342:
1313:
1312:
1301:
1298:
1295:
1292:
1289:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1262:
1257:
1252:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1193:
1188:
1166:
1161:
1157:
1153:
1150:
1147:
1142:
1138:
1134:
1131:
1105:
1102:
1099:
1094:
1070:
1065:
1044:
1041:
1026:
1021:
1008:
1007:
987:
984:
981:
978:
975:
972:
969:
966:
963:
960:
957:
954:
951:
948:
945:
942:
939:
936:
933:
930:
927:
923:
920:
889:
863:
858:
836:
831:
827:
823:
820:
817:
812:
808:
804:
801:
777:
774:
771:
768:
764:
760:
755:
751:
747:
744:
741:
736:
732:
728:
725:
721:
717:
714:
711:
708:
681:
676:
671:
639:
634:
629:
596:
595:
594:
593:
582:
579:
576:
573:
570:
567:
564:
560:
556:
553:
550:
547:
544:
541:
538:
535:
532:
522:
511:
508:
505:
502:
499:
496:
492:
488:
485:
482:
479:
476:
473:
470:
467:
464:
443:
388:
385:
382:
379:
371:
368:
365:
362:
359:
356:
353:
348:
343:
338:
335:
332:
329:
326:
323:
320:
317:
297:
292:
287:
265:formed by the
254:
249:
244:
227:
224:
176:
173:
157:maximal ideals
26:
9:
6:
4:
3:
2:
2765:
2754:
2751:
2749:
2748:Scheme theory
2746:
2744:
2741:
2740:
2738:
2725:
2724:
2719:
2713:
2709:
2705:
2701:
2695:
2691:
2687:
2683:
2679:
2675:
2671:
2667:
2661:
2657:
2653:
2652:
2647:
2643:
2640:
2636:
2632:
2628:
2624:
2618:
2614:
2610:
2609:
2604:
2600:
2596:
2594:9780471433347
2590:
2586:
2581:
2580:
2567:
2562:
2555:
2550:
2543:
2538:
2536:
2531:
2521:
2518:
2517:
2511:
2509:
2505:
2501:
2495:
2493:
2489:
2485:
2481:
2477:
2473:
2469:
2465:
2461:
2453:
2449:
2445:
2432:
2428:
2424:
2420:
2416:
2412:
2409:
2383:
2379:
2375:
2371:
2367:
2341:
2337:
2333:
2314:
2298:
2294:
2290:
2286:
2282:
2278:
2274:
2271:
2267:
2263:
2259:
2255:
2251:
2247:
2243:
2239:
2235:
2231:
2227:
2223:
2219:
2215:
2211:
2208:
2204:
2200:
2196:
2171:
2170:generic point
2150:
2144:
2134:
2133:maximal ideal
2130:
2127:
2123:
2097:
2094:
2091:
2087:
2083:
2082:
2077:
2068:
2065:
2060:
2058:
2054:
2050:
2046:
2042:
2038:
2034:
2030:
2026:
2022:
2018:
1996:
1990:
1987:
1984:
1978:
1975:
1969:
1961:
1957:
1949:
1948:
1947:
1945:
1941:
1921:
1914:
1910:
1906:
1900:
1897:
1894:
1889:
1883:
1874:
1867:
1853:
1850:
1847:
1844:
1841:
1831:
1826:
1822:
1814:
1813:
1812:
1810:
1806:
1802:
1798:
1794:
1790:
1786:
1782:
1778:
1777:residue field
1774:
1770:
1766:
1762:
1757:
1755:
1751:
1747:
1743:
1739:
1732:
1722:
1715:
1711:
1701:
1697:
1693:
1689:
1685:
1680:
1679:is an ideal.
1678:
1656:
1653:
1650:
1647:
1644:
1641:
1638:
1635:
1632:
1626:
1620:
1614:
1607:
1606:
1605:
1603:
1599:
1594:
1588:
1585:
1581:
1577:
1573:
1570:, which is a
1569:
1559:
1546:
1541:
1525:
1521:
1516:
1514:
1506:
1499:
1492:
1485:
1478:
1471:
1464:
1459:
1457:
1453:
1449:
1445:
1441:
1436:
1434:
1433:affine scheme
1430:
1426:
1425:distinguished
1420:
1413:
1406:
1402:
1397:
1387:
1383:
1381:
1377:
1370:
1366:
1350:
1345:
1330:
1326:
1322:
1318:
1299:
1293:
1290:
1287:
1281:
1278:
1275:
1269:
1263:
1260:
1255:
1245:
1242:
1236:
1230:
1224:
1217:
1216:
1215:
1213:
1209:
1191:
1159:
1155:
1151:
1148:
1145:
1140:
1136:
1129:
1121:
1103:
1100:
1097:
1068:
1054:
1051:-dimensional
1050:
1040:
1024:
1005:
1001:
982:
979:
976:
970:
967:
964:
958:
952:
949:
946:
943:
940:
934:
928:
921:
918:
910:
906:
902:
898:
894:
890:
887:
883:
879:
861:
829:
825:
821:
818:
815:
810:
806:
799:
791:
772:
766:
762:
753:
749:
745:
742:
739:
734:
730:
723:
719:
712:
706:
699:
695:
694:
693:
679:
674:
659:
655:
650:
637:
632:
617:
614:) and called
613:
609:
605:
601:
580:
574:
571:
568:
562:
558:
551:
545:
542:
536:
530:
523:
509:
503:
500:
494:
490:
483:
477:
474:
468:
462:
455:
454:
452:
448:
444:
441:
437:
433:
429:
425:
421:
417:
414:
413:
412:
410:
406:
402:
383:
380:
377:
369:
366:
360:
354:
351:
346:
336:
333:
327:
321:
315:
295:
290:
271:
252:
247:
233:
223:
221:
217:
213:
210:
206:
202:
198:
194:
190:
186:
182:
172:
170:
166:
162:
158:
154:
150:
145:
143:
139:
135:
130:
128:
127:affine spaces
124:
120:
116:
112:
111:scheme theory
108:
104:
100:
96:
91:
89:
85:
81:
77:
76:Oscar Zariski
73:
69:
65:
61:
57:
53:
49:
45:
37:
32:
19:
2721:
2677:
2650:
2646:Hulek, Klaus
2606:
2584:
2561:
2554:Mumford 1999
2549:
2496:
2491:
2487:
2483:
2479:
2475:
2471:
2467:
2463:
2451:
2441:
2422:
2418:
2414:
2410:
2408:discriminant
2381:
2377:
2373:
2369:
2365:
2339:
2335:
2331:
2289:real numbers
2284:
2280:
2276:
2272:
2266:homeomorphic
2261:
2257:
2253:
2249:
2245:
2241:
2229:
2225:
2209:
2198:
2128:
2126:prime number
2092:
2085:
2061:
2056:
2052:
2048:
2044:
2040:
2036:
2032:
2028:
2024:
2020:
2016:
2014:
1943:
1939:
1937:
1808:
1804:
1800:
1796:
1792:
1788:
1784:
1772:
1768:
1764:
1760:
1758:
1753:
1749:
1745:
1741:
1734:
1727:
1720:
1713:
1706:
1699:
1695:
1691:
1683:
1681:
1676:
1674:
1601:
1597:
1592:
1586:
1583:
1565:
1517:
1501:
1494:
1487:
1480:
1473:
1466:
1460:
1437:
1428:
1424:
1418:
1411:
1404:
1400:
1393:
1384:
1379:
1372:
1368:
1364:
1328:
1327:), for sets
1324:
1320:
1314:
1211:
1119:
1048:
1047:Recall that
1046:
1009:
1003:
999:
908:
904:
900:
896:
892:
885:
881:
877:
789:
653:
651:
615:
611:
607:
603:
599:
597:
450:
446:
439:
431:
427:
423:
419:
415:
408:
404:
400:
232:affine space
229:
215:
211:
185:Grothendieck
178:
168:
165:Grothendieck
146:
131:
92:
86:(called the
80:prime ideals
51:
41:
36:affine plane
2486:) contains
2277:affine line
2270:affine line
1520:regular map
430:)), where (
193:closed sets
134:closed sets
2737:Categories
2576:References
2542:Hulek 2003
2504:properness
1589:, denoted
1524:continuous
1390:Properties
453:, we have
2723:MathWorld
2676:(1999) .
2526:Citations
2268:with the
2151:⊆
1988:∈
1982:⇔
1901:
1895:∈
1864:↦
1851:
1845:∈
1832::
1740:contains
1654:⊇
1648:∣
1642:
1636:∈
1448:quotients
1291:∈
1285:∀
1261:∣
1246:∈
1149:…
980:∈
974:∀
950:∣
944:∈
819:…
743:…
543:∩
475:∪
434:) is the
381:∈
352:∣
337:∈
105:is not a
72:Hausdorff
60:varieties
2648:(2003).
2639:13348052
2605:(1977),
2514:See also
2376:) where
2244:−
2220:are the
2216:and the
2122:integers
2071:Examples
922:′
876:; here,
195:are the
155:and the
136:are the
119:manifold
95:topology
88:spectrum
56:topology
2708:1748380
2631:0463157
2384:are in
2338:), for
2205:over a
1726:, ...,
1705:, ...,
1582:. The
1578:to the
1493:, ...,
1472:, ...,
1456:compact
270:-tuples
181:schemes
2706:
2696:
2662:
2637:
2629:
2619:
2591:
2508:scheme
2460:spaces
1675:where
1568:scheme
1518:Every
895:, let
399:where
201:affine
169:points
123:charts
50:, the
2506:of a
2425:is a
2297:monic
2283:. If
2228:. If
2207:field
2197:Spec
2098:Spec
2090:field
2084:Spec
1591:Spec
1513:axiom
1429:basic
436:ideal
103:field
82:of a
54:is a
2694:ISBN
2660:ISBN
2635:OCLC
2617:ISBN
2589:ISBN
1898:Frac
1848:Spec
1639:Spec
1396:base
1004:V(S)
422:) =
203:and
64:real
46:and
2686:doi
2656:AMS
2413:− 4
2342:in
2260:of
2252:of
2232:is
2224:of
1880:mod
1807:of
1438:By
1427:or
1382:).
1002:of
652:If
222:).
66:or
42:In
2739::
2720:.
2704:MR
2702:.
2692:.
2684:.
2658:.
2654:.
2633:,
2627:MR
2625:,
2615:,
2534:^
2494:.
2466:,
2380:,
2372:+
2370:px
2368:+
2334:−
1811::
1733:−
1719:−
1515:.
1500:-
1486:-
1458:.
1435:.
449:,
426:((
129:.
2726:.
2710:.
2688::
2668:.
2597:.
2568:.
2556:.
2492:Q
2488:P
2484:Q
2482:(
2480:D
2476:P
2472:A
2468:Q
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2458:0
2456:T
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2423:k
2419:k
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2411:p
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2351:R
2340:a
2336:a
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2318:]
2315:t
2312:[
2308:R
2285:k
2281:k
2273:k
2262:k
2258:a
2254:k
2250:a
2246:a
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2210:k
2199:k
2181:Z
2155:Z
2148:)
2145:p
2142:(
2129:p
2107:Z
2093:k
2086:k
2057:n
2053:k
2049:k
2045:n
2041:A
2037:k
2033:A
2029:I
2025:I
2021:I
2019:(
2017:V
2000:)
1997:a
1994:(
1991:V
1985:P
1979:0
1976:=
1973:)
1970:P
1967:(
1962:a
1958:e
1944:A
1940:a
1922:)
1918:)
1915:P
1911:/
1907:A
1904:(
1890:1
1884:P
1875:a
1868:(
1859:)
1854:A
1842:P
1837:(
1827:a
1823:e
1809:A
1805:a
1801:P
1797:P
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1789:P
1787:/
1785:A
1773:P
1769:A
1765:A
1761:A
1754:S
1750:S
1748:(
1746:V
1742:S
1737:n
1735:a
1730:n
1728:x
1724:1
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1717:1
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1700:a
1696:S
1694:(
1692:V
1684:S
1677:I
1660:}
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1633:P
1630:{
1627:=
1624:)
1621:I
1618:(
1615:V
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1542:1
1537:A
1511:1
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1417:(
1412:f
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1401:D
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1375:′
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1367:(
1365:D
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1321:V
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1297:}
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1240:{
1237:=
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1228:(
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986:}
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903:(
901:A
897:T
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888:.
886:X
882:X
880:(
878:I
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830:n
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800:k
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763:/
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754:n
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720:=
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713:X
710:(
707:A
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581:.
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575:J
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559:=
555:)
552:J
549:(
546:V
540:)
537:I
534:(
531:V
510:;
507:)
504:J
501:I
498:(
495:V
491:=
487:)
484:J
481:(
478:V
472:)
469:I
466:(
463:V
451:J
447:I
442:;
440:S
432:S
428:S
424:V
420:S
418:(
416:V
409:k
405:n
401:S
387:}
384:S
378:f
370:0
367:=
364:)
361:x
358:(
355:f
347:n
342:A
334:x
331:{
328:=
325:)
322:S
319:(
316:V
296:.
291:n
286:A
274:k
268:n
253:,
248:n
243:A
216:k
212:k
20:)
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