1275:
scheme, is not complete, and is not even quasi-separated. This shows that although the quotient of an algebraic space by an infinite discrete group is an algebraic space, it can have strange properties and might not be the algebraic space one was "expecting". Similar examples are given by the quotient of the complex affine line by the integers, or the quotient of the complex affine line minus the origin by the powers of some number: again the corresponding analytic space is a variety, but the algebraic space is not.
1274:
by a lattice is an algebraic space, but is not an elliptic curve, even though the corresponding analytic space is an elliptic curve (or more precisely is the image of an elliptic curve under the functor from complex algebraic spaces to analytic spaces). In fact this algebraic space quotient is not a
1295:
Roughly speaking, the difference between complex algebraic spaces and analytic spaces is that complex algebraic spaces are formed by gluing affine pieces together using the étale topology, while analytic spaces are formed by gluing with the classical topology. In particular there is a functor from
1259:
can be used to give a non-singular 3-dimensional proper algebraic space that is not a scheme, given by the quotient of a scheme by a group of order 2 acting freely. This illustrates one difference between schemes and algebraic spaces: the quotient of an algebraic space by a discrete group acting
1300:
give examples of analytic surfaces that do not come from a proper algebraic space (though one can construct non-proper and non-separated algebraic spaces whose analytic space is the Hopf surface). It is also possible for different algebraic spaces to correspond to the same analytic space: for
1233:
Algebraic spaces are similar to schemes, and much of the theory of schemes extends to algebraic spaces. For example, most properties of morphisms of schemes also apply to algebraic spaces, one can define cohomology of quasicoherent sheaves, this has the usual finiteness properties for proper
403:
1064:
949:
288:
52:. Alternatively one can think of schemes as being locally isomorphic to affine schemes in the Zariski topology, while algebraic spaces are locally isomorphic to affine schemes in the étale topology.
1250:
Commutative-group objects in the category of algebraic spaces over an arbitrary scheme which are proper, locally finite presentation, flat, and cohomologically flat in dimension 0 are schemes.
1196:
1144:
518:
993:
877:
182:. Doing so means that the theory of algebraic spaces is not dependent on the full theory of schemes, and can indeed be used as a (more general) replacement of that theory.
1000:
1094:
1219:
83:
There are two common ways to define algebraic spaces: they can be defined as either quotients of schemes by étale equivalence relations, or as sheaves on a
1320:. In the category of stacks we can form even more quotients by group actions than in the category of algebraic spaces (the resulting quotient is called a
885:
1247:
group objects in the category of algebraic spaces over a field are schemes, though there are non quasi-separated group objects that are not schemes.
398:{\displaystyle \mathrm {Hom} (Y,X)\rightarrow \mathrm {Hom} (V,X){{{} \atop \longrightarrow } \atop {\longrightarrow \atop {}}}\mathrm {Hom} (S,X)}
59:
of algebraic spaces extends the category of schemes and allows one to carry out several natural constructions that are used in the construction of
1260:
freely is an algebraic space, but the quotient of a scheme by a discrete group acting freely need not be a scheme (even if the group is finite).
1438:
1400:
1263:
Every quasi-separated algebraic space contains a dense open affine subscheme, and the complement of such a subscheme always has
1305:
by the corresponding lattice are not isomorphic as algebraic spaces, but the corresponding analytic spaces are isomorphic.
1476:
1152:
1498:
1471:
1099:
1308:
Artin showed that proper algebraic spaces over the complex numbers are more or less the same as
Moishezon spaces.
280:
465:
700:
961:
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1371:
413:
56:
72:
1256:
1466:
1364:(1969), "The implicit function theorem in algebraic geometry", in Abhyankar, Shreeram Shankar (ed.),
858:
127:
25:
1221:. Note that some authors, such as Knutson, add an extra condition that an algebraic space has to be
1059:{\displaystyle \Delta _{{\mathfrak {X}}/S}:{\mathfrak {X}}\to {\mathfrak {X}}\times {\mathfrak {X}}}
1317:
412:
for surjective étale maps of affine schemes). With these definitions, the algebraic spaces form a
1289:
1241:
Non-singular proper algebraic spaces of dimension two over a field (smooth surfaces) are schemes.
528:
524:
217:), then the algebraic space will be a scheme in the usual sense. Since a general algebraic space
63:
but are not always possible in the smaller category of schemes, such as taking the quotient of a
37:
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that are locally isomorphic to schemes. These two definitions are essentially equivalent.
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Some authors, such as
Knutson, add an extra condition that an algebraic space has to be
1321:
1285:
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1204:
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48:, while algebraic spaces are given by gluing together affine schemes using the finer
41:
33:
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68:
60:
1365:
944:{\displaystyle {\mathfrak {X}}:({\text{Sch}}/S)_{\text{et}}^{op}\to {\text{Sets}}}
1370:, of Tata Institute of Fundamental Research studies in mathematics, vol. 4,
1264:
64:
17:
581:
221:
does not satisfy this requirement, it allows a single connected component of
1482:
1238:
Proper algebraic spaces over a field of dimension one (curves) are schemes.
1070:
The second condition is equivalent to the property that given any schemes
1430:
189:
is the trivial equivalence relation over each connected component of
1395:, Yale Mathematical Monographs, vol. 3, Yale University Press,
1267:≥ 1. Thus algebraic spaces are in a sense "close" to affine schemes.
1367:
Algebraic geometry: papers presented at the Bombay
Colloquium, 1968
1421:, Lecture Notes in Mathematics, vol. 203, Berlin, New York:
1284:
Algebraic spaces over the complex numbers are closely related to
232:
with many "sheets". The point set underlying the algebraic space
1316:
A far-reaching generalization of algebraic spaces is given by
1296:
complex algebraic spaces of finite type to analytic spaces.
408:
exact (this definition is motivated by a descent theorem of
255:
be an algebraic space defined by an equivalence relation
1207:
1155:
1102:
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1003:
964:
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468:
291:
40:. Intuitively, schemes are given by gluing together
279:is then defined by the condition that it makes the
1225:, meaning that the diagonal map is quasi-compact.
1213:
1190:
1138:
1088:
1058:
987:
943:
871:
836:in the sense just defined) to any algebraic space
512:
397:
167:, meaning that the diagonal map is quasi-compact.
1253:Not every singular algebraic surface is a scheme.
1191:{\displaystyle h_{Y}\times _{\mathfrak {X}}h_{Z}}
816:is defined by associating the ring of functions
1301:example, an elliptic curve and the quotient of
1139:{\displaystyle h_{Y},h_{Z}\to {\mathfrak {X}}}
201:belonging to the same connected component of
8:
664:A point on an algebraic space is said to be
504:
472:
513:{\displaystyle k\{x_{1},\ldots ,x_{n}\}\ }
1206:
1182:
1171:
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978:
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118:satisfying the following two conditions:
988:{\displaystyle h_{X}\to {\mathfrak {X}}}
91:Algebraic spaces as quotients of schemes
1333:
1344:
1340:
958:There is a surjective étale morphism
7:
1280:Algebraic spaces and analytic spaces
1172:
1131:
1051:
1041:
1031:
1011:
980:
891:
864:
622:is the local ring corresponding to
427:defined by a system of polynomials
1201:is representable by a scheme over
1005:
879:can be defined as a sheaf of sets
743:of algebraic spaces is said to be
584:of algebraic functions defined by
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14:
1146:, their fiber-product of sheaves
423:be an affine scheme over a field
767:)) if the induced map on stalks
872:{\displaystyle {\mathfrak {X}}}
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933:
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277:morphisms of algebraic spaces
24:form a generalization of the
1229:Algebraic spaces and schemes
828:(defined by étale maps from
1472:Encyclopedia of Mathematics
848:Algebraic spaces as sheaves
724:is then just defined to be
580:are then defined to be the
170:One can always assume that
1515:
657:of algebraic functions on
557:} be an algebraic space.
1417:Knutson, Donald (1971),
1465:Danilov, V.I. (2001) ,
1372:Oxford University Press
812:on the algebraic space
106:and a closed subscheme
1234:morphisms, and so on.
1215:
1192:
1140:
1090:
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997:the diagonal morphism
989:
945:
873:
605:is a point lying over
514:
399:
1485:in the stacks project
1216:
1193:
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990:
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515:
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155:onto each factor are
1270:The quotient of the
1205:
1153:
1100:
1074:
1001:
962:
886:
859:
840:which is étale over
466:
289:
128:equivalence relation
1290:Moishezon manifolds
1089:{\displaystyle Y,Z}
932:
832:to the affine line
799:is an isomorphism.
716:. The dimension of
529:algebraic functions
246:equivalence classes
140:2. The projections
102:comprises a scheme
1499:Algebraic geometry
1431:10.1007/BFb0059750
1374:, pp. 13–34,
1257:Hironaka's example
1211:
1188:
1136:
1086:
1056:
985:
941:
915:
869:
510:
395:
236:is then given by |
38:deformation theory
30:algebraic geometry
1467:"Algebraic space"
1440:978-3-540-05496-2
1402:978-0-300-01396-2
1214:{\displaystyle S}
1066:is representable.
939:
922:
905:
640:, ...,
509:
367:
365:
356:
73:Keel–Mori theorem
1506:
1479:
1451:
1419:Algebraic spaces
1413:
1392:Algebraic spaces
1382:
1348:
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1318:algebraic stacks
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560:The appropriate
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368:
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281:descent sequence
46:Zariski topology
32:, introduced by
22:algebraic spaces
1514:
1513:
1509:
1508:
1507:
1505:
1504:
1503:
1489:
1488:
1483:Algebraic space
1464:
1461:
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1423:Springer-Verlag
1416:
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1385:
1360:
1357:
1352:
1351:
1339:
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1286:analytic spaces
1282:
1272:complex numbers
1245:Quasi-separated
1231:
1223:quasi-separated
1203:
1202:
1178:
1166:
1156:
1151:
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1103:
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854:algebraic space
850:
810:
804:structure sheaf
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673:
647:} / (
645:
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621:
614:
596:
589:
575:
568:
494:
475:
464:
463:
457:
451:
287:
286:
267:. The set Hom(
209:if and only if
165:quasi-separated
145:
130:as a subset of
97:algebraic space
93:
81:
12:
11:
5:
1512:
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1491:
1490:
1487:
1486:
1480:
1460:
1459:External links
1457:
1453:
1452:
1439:
1414:
1401:
1387:Artin, Michael
1383:
1362:Artin, Michael
1356:
1353:
1350:
1349:
1332:
1331:
1329:
1326:
1322:quotient stack
1313:
1312:Generalization
1310:
1298:Hopf manifolds
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1119:
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1096:and morphisms
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701:indeterminates
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244:| as a set of
193:(i.e. for all
180:affine schemes
161:
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138:
92:
89:
85:big étale site
80:
77:
55:The resulting
50:étale topology
42:affine schemes
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61:moduli spaces
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34:Michael Artin
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626:of the ring
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954:such that
731:A morphism
699:} for some
582:local rings
523:denote the
65:free action
36:for use in
18:mathematics
1355:References
1345:Artin 1971
1341:Artin 1969
541:, and let
205:, we have
157:étale maps
79:Definition
44:using the
1477:EMS Press
1328:Citations
1168:×
1127:→
1047:×
1037:→
1006:Δ
976:→
934:→
489:…
361:⟶
354:⟶
319:→
71:(cf. the
1493:Category
1389:(1971),
597:, where
414:category
57:category
1449:0302647
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1380:0262237
755:(where
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692:, ...,
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452:, ...,
26:schemes
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666:smooth
562:stalks
508:
126:is an
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745:étale
537:over
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240:| / |
227:cover
67:by a
1435:ISBN
1397:ISBN
1288:and
938:Sets
802:The
609:and
525:ring
419:Let
251:Let
178:are
174:and
1427:doi
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