885:
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1926:
The category of schemes admits finite pullbacks and in some cases finite pushouts; they both are constructed by gluing affine schemes. For affine schemes, fiber products and pushouts correspond to tensor products and fiber squares of algebras.
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is obtained by identifying two parallel lines except the origin; i.e., it is an affine line with the doubled origin. (It can be shown that
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1035:{\displaystyle X=\operatorname {Spec} (k)\simeq \mathbb {A} ^{1},Y=\operatorname {Spec} (k)\simeq \mathbb {A} ^{1}}
616:
1998:
1921:
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1843:.) In contrast, if two lines are glued so that origin on the one line corresponds to the (illusionary)
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The projective line is obtained by gluing two affine lines so that the origin and illusionary
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is covered by the two open affine charts whose affine rings are of the above form.
1981:"Section 37.14 (07RS): Pushouts in the category of schemes, I—The Stacks project"
2058:
17:
2047:
1980:
2029:
1386:{\displaystyle \Gamma (X,{\mathcal {O}}_{Z}),\Gamma (Y,{\mathcal {O}}_{Z})}
776:{\displaystyle \psi _{i}(U_{ij})=\psi _{i}(X_{i})\cap \psi _{j}(X_{j}),}
1883:, then the resulting scheme is, at least visually, the projective line
883:
406:{\displaystyle \varphi _{ij}(U_{ij}\cap U_{ik})=U_{ji}\cap U_{jk}}
218:{\displaystyle \varphi _{ij}:U_{ij}{\overset {\sim }{\to }}U_{ji}}
1531:
where the two rings are viewed as subrings of the function field
225:. Now, if the isomorphisms are compatible in the sense: for each
470:{\displaystyle \varphi _{jk}\circ \varphi _{ij}=\varphi _{ik}}
1479:
1418:
1369:
1337:
1127:{\displaystyle X_{t}=\{t\neq 0\}=\operatorname {Spec} (k)}
54:
Suppose there is a (possibly infinite) family of schemes
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are both polynomial rings in one variable in such a way
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833:{\displaystyle \psi _{i}=\psi _{j}\circ \varphi _{ij}}
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92:
307:{\displaystyle \varphi _{ij}=\varphi _{ji}^{-1}}
1521:{\displaystyle \Gamma (Y,{\mathcal {O}}_{Z})=k}
1450:{\displaystyle \Gamma (X,{\mathcal {O}}_{Z})=k}
928:and the origin on the other line, respectively.
1699:be as in the above example. But this time let
1042:be two copies of the affine line over a field
2094:
2030:"Math 216: Foundations of algebraic geometry"
1847:for the other line; i.e, use the isomrophism
8:
1166:
1154:
1078:
1066:
668:{\displaystyle X=\cup _{i}\psi _{i}(X_{i}),}
75:
61:
2006:, vol. 52, New York: Springer-Verlag,
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795:
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732:
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607:is an isomorphism onto an open subset of
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68:
59:
1936:
1876:{\displaystyle t^{-1}\leftrightarrow u}
1278:{\displaystyle t^{-1}\leftrightarrow u}
908:on one line corresponds to illusionary
1916:Fiber products and pushouts of schemes
1967:
1955:
1719:denote the scheme obtained by gluing
1183:denote the scheme obtained by gluing
7:
2055:
2053:
1134:be the complement of the origin and
570:{\displaystyle \psi _{i}:X_{i}\to X}
2073:. You can help Knowledge (XXG) by
1804:{\displaystyle t\leftrightarrow u}
1603:{\displaystyle Z=\mathbb {P} ^{1}}
1464:
1403:
1354:
1322:
915:
895:
93:{\displaystyle \{X_{i}\}_{i\in I}}
25:
2028:Vakil, Ravi (November 18, 2017).
1778:{\displaystyle X_{t}\simeq Y_{u}}
1242:{\displaystyle X_{t}\simeq Y_{u}}
1172:{\displaystyle Y_{u}=\{u\neq 0\}}
516:{\displaystyle U_{ij}\cap U_{ik}}
2057:
1905:{\displaystyle \mathbb {P} ^{1}}
1632:{\displaystyle \mathbb {P} ^{1}}
1692:{\displaystyle X,Y,X_{t},Y_{u}}
1643:Affine line with doubled origin
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531:, together with the morphisms
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336:
194:
1:
2004:Graduate Texts in Mathematics
30:In algebraic geometry, a new
527:then there exists a scheme
2141:
2052:
1919:
1610:; because, by definition,
1568:{\displaystyle k(Z)=k(s)}
1311:with the open subsets of
600:{\displaystyle \psi _{i}}
126:, there are open subsets
2120:Algebraic geometry stubs
1946:, Ch. II, Exercise 2.12.
1922:Fiber product of schemes
1315:. Now, the affine rings
1179:defined similarly. Let
921:{\displaystyle \infty }
901:{\displaystyle \infty }
2069:–related article is a
1906:
1877:
1825:
1805:
1779:
1745:along the isomorphism
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1713:
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1604:
1575:. But this means that
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1209:along the isomorphism
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863:{\displaystyle U_{ij}}
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251:
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150:
149:{\displaystyle U_{ij}}
120:
94:
2048:26.14 Glueing schemes
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1811:. So, geometrically,
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408:
309:
252:
250:{\displaystyle i,j,k}
220:
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121:
95:
46:through gluing maps.
38:) can be obtained by
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58:
27:Mathematical concept
1738:{\displaystyle X,Y}
1304:{\displaystyle X,Y}
1202:{\displaystyle X,Y}
303:
119:{\displaystyle i,j}
2067:algebraic geometry
1999:Algebraic Geometry
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90:
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2081:
2013:978-0-387-90244-9
1994:Hartshorne, Robin
1845:point at infinity
1824:{\displaystyle Z}
1712:{\displaystyle Z}
200:
156:and isomorphisms
36:algebraic variety
16:(Redirected from
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1841:separated scheme
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21:
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2110:
2109:
2108:
2107:
2040:
2038:Further reading
2027:
2014:
1992:
1989:
1988:
1979:
1978:
1974:
1966:
1962:
1954:
1950:
1944:Hartshorne 1977
1942:
1938:
1933:
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1812:
1787:
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1105:
1053:
1048:
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972:
934:
933:
910:
909:
890:
889:
882:
880:Projective line
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683:
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56:
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52:
28:
23:
22:
15:
12:
11:
5:
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2050:
2044:Stacks Project
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2036:
2035:
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2025:
2012:
1987:
1986:
1972:
1960:
1948:
1935:
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1932:
1929:
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1408:
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1382:
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1371:
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1356:
1353:
1350:
1345:
1339:
1333:
1330:
1327:
1324:
1300:
1297:
1294:
1285:; we identify
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1271:
1266:
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1259:
1236:
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1228:
1223:
1219:
1198:
1195:
1192:
1168:
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1162:
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1156:
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1148:
1144:
1123:
1120:
1115:
1112:
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1101:
1098:
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1086:
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1080:
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1029:
1024:
1019:
1016:
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1010:
1007:
1004:
1001:
998:
995:
992:
989:
986:
981:
976:
971:
968:
965:
962:
959:
956:
953:
950:
947:
944:
941:
917:
897:
881:
878:
876:
873:
872:
871:
857:
854:
850:
827:
824:
820:
816:
811:
807:
803:
798:
794:
783:
772:
769:
764:
760:
756:
751:
747:
743:
740:
735:
731:
727:
722:
718:
714:
711:
706:
703:
699:
695:
690:
686:
675:
664:
661:
656:
652:
648:
643:
639:
633:
629:
625:
622:
612:
594:
590:
566:
563:
558:
554:
550:
545:
541:
525:
524:
510:
507:
503:
499:
494:
491:
487:
464:
461:
457:
453:
448:
445:
441:
437:
432:
429:
425:
414:
400:
397:
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389:
384:
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365:
362:
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315:
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293:
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282:
277:
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246:
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205:
199:
196:
189:
186:
182:
178:
173:
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166:
143:
140:
136:
115:
112:
109:
100:and for pairs
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84:
81:
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71:
67:
63:
51:
48:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2137:
2126:
2125:Scheme theory
2123:
2121:
2118:
2117:
2115:
2104:
2099:
2097:
2092:
2090:
2085:
2084:
2078:
2076:
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2068:
2063:
2060:
2056:
2049:
2045:
2042:
2041:
2037:
2031:
2026:
2023:
2019:
2015:
2009:
2005:
2001:
2000:
1995:
1991:
1990:
1982:
1976:
1973:
1969:
1964:
1961:
1957:
1952:
1949:
1945:
1940:
1937:
1930:
1928:
1923:
1915:
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1726:
1706:
1684:
1680:
1676:
1671:
1667:
1663:
1660:
1657:
1654:
1642:
1640:
1624:
1595:
1585:
1582:
1559:
1553:
1550:
1544:
1538:
1510:
1507:
1503:
1496:
1493:
1485:
1473:
1470:
1441:
1435:
1432:
1424:
1412:
1409:
1396:
1395:
1394:
1375:
1363:
1360:
1351:
1343:
1331:
1328:
1314:
1298:
1295:
1292:
1272:
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1163:
1160:
1157:
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1146:
1142:
1113:
1110:
1106:
1102:
1099:
1093:
1087:
1084:
1081:
1075:
1072:
1069:
1063:
1058:
1054:
1045:
1027:
1017:
1008:
1002:
996:
993:
990:
987:
984:
979:
969:
960:
954:
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945:
942:
939:
886:
879:
874:
855:
852:
848:
825:
822:
818:
814:
809:
805:
801:
796:
792:
784:
770:
762:
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745:
741:
733:
729:
720:
716:
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704:
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697:
688:
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676:
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650:
641:
637:
631:
627:
623:
620:
613:
610:
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588:
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579:
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564:
556:
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489:
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451:
446:
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439:
435:
430:
427:
423:
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398:
395:
391:
387:
382:
379:
375:
371:
363:
360:
356:
352:
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344:
340:
331:
328:
324:
316:
299:
296:
291:
288:
284:
280:
275:
272:
268:
260:
259:
258:
244:
241:
238:
235:
232:
210:
207:
203:
197:
187:
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141:
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113:
110:
107:
85:
82:
79:
69:
65:
49:
47:
45:
41:
37:
33:
19:
18:Gluing scheme
2075:expanding it
2064:
1997:
1975:
1963:
1951:
1939:
1925:
1836:
1832:
1646:
1530:
1312:
1180:
1043:
931:
608:
528:
526:
53:
43:
39:
29:
2114:Categories
1970:, § 4.4.5.
1968:Vakil 2017
1958:, § 4.4.6.
1956:Vakil 2017
1931:References
1920:See also:
577:such that
1868:↔
1860:−
1796:↔
1785:given by
1763:≃
1508:−
1465:Γ
1404:Γ
1355:Γ
1323:Γ
1270:↔
1262:−
1249:given by
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1018:≃
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970:≃
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42:existing
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875:Examples
2022:0463157
44:schemes
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32:scheme
2065:This
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2008:ISBN
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