9033:
5463:
3450:
10585:). These sheaves admit algebraic operations that are associative and commutative only up to an equivalence relation. Taking the quotient by this equivalence relation yields the structure sheaf of an ordinary scheme. Not taking the quotient, however, leads to a theory that can remember higher information, in the same way that
520:
most often over the complex numbers. Grothendieck developed a large body of theory for arbitrary schemes extending much of the geometric intuition for varieties. For example, it is common to construct a moduli space first as a scheme, and only later study whether it is a more concrete object such as
321:
developed commutative algebra in the 1920s and 1930s. Their work generalizes algebraic geometry in a purely algebraic direction, generalizing the study of points (maximal ideals in a polynomial ring) to the study of prime ideals in any commutative ring. For example, Krull defined the
10484:
Considered as its functor of points, a scheme is a functor that is a sheaf of sets for the
Zariski topology on the category of commutative rings, and that, locally in the Zariski topology, is an affine scheme. This can be generalized in several ways. One is to use the
9457:
2286:
3677:
6811:
1868:
2101:
7707:
3763:
10554:. In that formulation, stacks are (informally speaking) sheaves of categories. From this general notion, Artin defined the narrower class of algebraic stacks (or "Artin stacks"), which can be considered geometric objects. These include
9557:
10499:
as a functor that is a sheaf in the étale topology and that, locally in the étale topology, is an affine scheme. Equivalently, an algebraic space is the quotient of a scheme by an étale equivalence relation. A powerful result, the
7974:
4512:
5156:
8086:
7161:
4021:
1755:
9354:
8953:
7089:
1633:
613:. One can think of a scheme as being covered by "coordinate charts" that are affine schemes. The definition means exactly that schemes are obtained by gluing together affine schemes using the Zariski topology.
10161:
2716:
9063:
It is also fruitful to consider examples of morphisms as examples of schemes since they demonstrate their technical effectiveness for encapsulating many objects of study in algebraic and arithmetic geometry.
3538:
3256:
4344:
1975:
7594:
4108:
2951:
8585:
5236:
3326:
9259:, as a scheme, are its complex points (one for each complex number) together with one generic point (whose closure is the whole scheme). The generic point is the image of a natural morphism Spec(
5277:
3862:
260:. The early 20th century saw analogies between algebraic geometry and number theory, suggesting the question: can algebraic geometry be developed over other fields, such as those with positive
7622:
2138:
8844:
3543:
8512:
6232:
4742:
1353:
7828:
7427:
5078:
3047:
8677:
6683:
4198:
2411:
1784:
8290:
8462:
6465:
145:
of open sets, each endowed with a coordinate ring of regular functions, with specified coordinate changes between the functions over intersecting open sets. Such a structure is called a
9942:
6960:
6384:
4927:
6678:
6139:
5603:
5328:
4154:
2889:
1683:
8763:
3440:
3369:
1411:
2796:
2611:
2006:
5775:
10501:
9702:
8376:
8245:
8199:
7207:
7006:
6302:
4999:
3160:
1538:
5552:
4843:
4812:
4670:
4624:
4571:
4378:
4249:
3944:
2642:
1899:
7631:
7240:
6501:
6050:
2756:
2553:
2321:
1253:
7525:
3893:
3482:
3397:
2516:
372:
For applications to number theory, van der
Waerden and Weil formulated algebraic geometry over any field, not necessarily algebraically closed. Weil was the first to define an
9466:
8714:
8319:
7736:
6840:
6530:
3122:
1214:
1185:
7837:
2484:
1934:
1779:
10049:
9559:
Thus, having an actual generic point of a variety yields a geometric relation between a degree-2 morphism of algebraic varieties and the corresponding degree-2 extension of
8997:
415:. This worked awkwardly: there were many different generic points for the same variety. (In the later theory of schemes, each algebraic variety has a single generic point.)
1125:
8607:
6582:
5187:
4949:
3286:
2986:
9996:
10547:. More generally, moduli spaces in algebraic geometry are often best viewed as stacks, thereby keeping track of the automorphism groups of the objects being classified.
9865:
6624:
5961:
2440:
6866:
8119:
3682:
9823:
7342:
6014:
5813:
7304:
5733:
2130:
7999:
6922:
5847:
9790:
8873:
7391:
7269:
6895:
5876:
5632:
5516:
5377:
2825:
8145:
6410:
6258:
6096:
5698:
4540:
4404:
1492:
10224:
10204:
9757:
9722:
8213:
along open subsets. This is the fundamental example that motivates going beyond affine schemes. The key advantage of projective space over affine space is that
7994:
7450:
7362:
7026:
6070:
5981:
5916:
5896:
5672:
5652:
5487:
5457:
5437:
5417:
5397:
5348:
5023:
4863:
4766:
4444:
4424:
4289:
4269:
4218:
4045:
3913:
3067:
2573:
2460:
2341:
1998:
1653:
1431:
1304:
434:
relating number theory and algebraic geometry, further extended the objects of algebraic geometry, for example by generalizing the base rings allowed. The word
4449:
521:
a projective variety. Applying
Grothendieck's theory to schemes over the integers and other number fields led to powerful new perspectives in number theory.
5091:
624:
prescheme. The term prescheme has fallen out of use, but can still be found in older books, such as
Grothendieck's "Éléments de géométrie algébrique" and
8537:, and gluing together the two open subsets A − 0 by the identity map. This is a simple example of a non-separated scheme. In particular, it is not affine.
365:
is a useful topology on a variety over any algebraically closed field, replacing to some extent the classical topology on a complex variety (based on the
10058:
10581:
or "spectral algebraic geometry", the structure sheaf is replaced by a homotopical analog of a sheaf of commutative rings (for example, a sheaf of
7094:
770:
with certain properties. There are different conventions about exactly which schemes should be called varieties. One standard choice is that a
3949:
1688:
11018:
10692:
86:
11134:
8901:
7031:
1543:
119:
3125:
2650:
17:
8959:
of the positive integers (with the discrete topology). In fact, the prime ideals of this ring are in one-to-one correspondence with the
446:
who suggested to Serre the possibility of using the spectrum of an arbitrary commutative ring as a foundation for algebraic geometry.
3487:
184:
10562:
in topology), for which the stabilizer groups are finite, and algebraic spaces, for which the stabilizer groups are trivial. The
8389:
3165:
4294:
1939:
9560:
9452:{\displaystyle \operatorname {Spec} \mathbf {C} (x)\left({\sqrt {x(x-1)(x-5)}}\right)\to \operatorname {Spec} \mathbf {C} (x).}
7530:
4053:
2898:
645:
which are not necessarily functions, can neverthess be patched together from their restrictions in the same way as functions.
11273:
11235:
11177:
11103:
11064:
9638:
but not zero. To indicate the meaning of this scheme: two regular functions on the affine line have the same restriction to
8543:
8540:
A simple reason to go beyond affine schemes is that an open subset of an affine scheme need not be affine. For example, let
3446:. Closed sets are finite sets, and open sets are their complements, the cofinite sets; any infinite set of points is dense.
462:(SGA), bringing to a conclusion a generation of experimental suggestions and partial developments. Grothendieck defined the
10440:
of a smooth variety over a field. However, coherent sheaves are richer; for example, a vector bundle on a closed subscheme
9015:, this is an example of a non-Noetherian quasi-compact scheme with infinitely many irreducible components. (By contrast, a
5192:
3291:
400:
2281:{\displaystyle {\mathcal {O}}_{X}(U_{f})=R=\{{\tfrac {r}{f^{m}}}\ \ {\text{for}}\ \ r\in R,\ m\in \mathbb {Z} _{\geq 0}\}}
11016:(2001), "A mad day's work: from Grothendieck to Connes and Kontsevich. The evolution of concepts of space and symmetry",
9290:
in one variable. To see why it is useful to have an actual "generic point" in the scheme, consider the following example.
5241:
3672:{\displaystyle {\mathcal {O}}_{Z}(U_{p})=\mathbb {Z} =\{{\tfrac {n}{p^{m}}}\ {\text{for}}\ n\in \mathbb {Z} ,\ m\geq 0\}}
3768:
11204:
7599:
177:
10274:. This again indicates that non-reduced subschemes have geometric meaning, related to derivatives and tangent vectors.
8956:
8768:
6806:{\displaystyle k({\mathfrak {m}})=\mathbb {Z} /{\mathfrak {m}}=\mathbb {F} _{p}/(f(x))\cong \mathbb {F} _{p}(\alpha )}
1863:{\displaystyle V({\mathfrak {p}})=\{{\mathfrak {q}}\in X\ \ {\text{with}}\ \ {\mathfrak {p}}\subset {\mathfrak {q}}\}}
8481:
6144:
4675:
1309:
213:. In many cases, the family of all varieties of a given type can itself be viewed as a variety or scheme, known as a
7741:
7396:
5028:
11095:
9072:
Here are some of the ways in which schemes go beyond older notions of algebraic varieties, and their significance.
8620:
4159:
2991:
2349:
8261:
10973:
10582:
6415:
346:
9103:
is algebraically closed (for example the complex numbers), then one can base algebraic geometry on sets such as
919:. In examples, this definition reconstructs the old notion of the set of solutions of the defining equations of
407:
of an algebraic variety. What is true for the generic point is true for "most" points of the variety. In Weil's
11013:
9870:
7472:
overcomes this obstacle by compactifying affine arithmetic schemes, adding points at infinity corresponding to
6927:
6307:
4868:
2096:{\displaystyle U_{f}=X\smallsetminus V(f)=\{{\mathfrak {p}}\in X\ \ {\text{with}}\ \ f\notin {\mathfrak {p}}\}}
779:
439:
357:
applied commutative algebra as a new foundation for algebraic geometry in the richer setting of projective (or
6632:
6101:
5557:
5282:
4113:
2833:
1658:
11296:
8727:
3402:
3331:
1358:
857:
829:). In the case of affine schemes, this construction gives a one-to-one correspondence between morphisms Spec(
411:(1946), generic points are constructed by taking points in a very large algebraically closed field, called a
2761:
2578:
137:
consisting of closed points which correspond to geometric points, together with non-closed points which are
10345:
10229:
For a more elaborate example, one can describe all the zero-dimensional closed subschemes of degree 2 in a
9945:
9044:
7702:{\displaystyle X_{p}=X\times _{\operatorname {Spec} (\mathbb {Z} )}\operatorname {Spec} (\mathbb {F} _{p})}
5738:
4745:
275:
176:. The coordinate ring of an affine scheme is the ring itself, and the coordinate rings of open subsets are
11258:
The Red Book of
Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians
9673:
8347:
8216:
8170:
7166:
6965:
6267:
4954:
3131:
1497:
783:
9460:
5521:
4817:
4771:
4629:
4583:
4545:
4352:
4223:
3918:
2616:
1873:
129:
of regular algebraic functions defined on it (or on its subsets), and each subvariety corresponds to the
10849:
7212:
6470:
6019:
4271:
as a kind of "regular function" on the closed points, a very special type among the arbitrary functions
2721:
2525:
2295:
1223:
11085:
10667:
10578:
10573:
Another type of generalization is to enrich the structure sheaf, bringing algebraic geometry closer to
10522:
7487:
3870:
3459:
3374:
2489:
1092:
438:
was first used in the 1956 Chevalley
Seminar, in which Chevalley pursued Zariski's ideas. According to
257:
62:= 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any
51:
11355:- the comment section contains some interesting discussion on scheme theory (including the posts from
11337:
9580:
8690:
8295:
7712:
6816:
6506:
3072:
1190:
1161:
11167:
10555:
10473:
10418:
2465:
1904:
1760:
892:
221:
10001:
330:, he proved that this definition satisfies many of the intuitive properties of geometric dimension.
11352:
8966:
8256:
1097:
1076:
261:
10521:
attached to each point, which is viewed as the automorphism group of that point. For example, any
8590:
6539:
5161:
4932:
3269:
2959:
11227:
11089:
9951:
8379:
3758:{\displaystyle U=Z\smallsetminus \{{\mathfrak {m}}_{p_{1}},\ldots ,{\mathfrak {m}}_{p_{\ell }}\}}
1267:
513:) are the affine schemes; a general scheme is then obtained by "gluing together" affine schemes.
10253:) as in the previous paragraph. Subschemes of the latter type are determined by a complex point
9828:
6587:
5924:
2416:
11372:
11121:
10658:
8325:
6845:
81:
43:
10971:
Arapura, Donu (2011), "Frobenius amplitude, ultraproducts, and vanishing on singular spaces",
8091:
7457:
125:
Schemes elaborate the fundamental idea that an algebraic variety is best analyzed through the
10563:
10551:
9795:
9008:
7461:
7309:
5986:
5780:
1540:
is a topological space with the
Zariski topology, whose closed points are the maximal ideals
7274:
5703:
2109:
516:
Much of algebraic geometry focuses on projective or quasi-projective varieties over a field
388:
of a curve over any field. (Later, Jacobians were shown to be projective varieties by Weil,
11322:
11310:
11283:
11245:
11214:
11187:
11155:
11113:
11074:
11041:
11006:
10379:
10324:
7465:
6900:
5818:
534:
338:
241:
11291:
Vistoli, Angelo (2005), "Grothendieck topologies, fibered categories and descent theory",
9766:
8849:
7367:
7245:
6871:
5852:
5608:
5492:
5353:
2801:
8:
10632:
10567:
10540:
10508:
10164:
10055:
axis tangent direction (the common tangent of the two curves) and having coordinate ring:
9667:
9552:{\displaystyle \mathbf {C} (x)\subset \mathbf {C} (x)\left({\sqrt {x(x-1)(x-5)}}\right).}
8124:
7473:
6389:
6237:
6075:
5677:
4517:
4383:
1436:
971:
707:
485:
249:
150:
130:
111:
103:
11314:
10637:
7969:{\displaystyle \Delta _{f}=-4a^{3}c+a^{2}b^{2}+18abc-4b^{3}-27c^{2}=0\ {\text{mod}}\ p,}
164:
Formally, a scheme is a ringed space covered by affine schemes. An affine scheme is the
11300:
11053:
10982:
10745:
10472:
has good properties for coherent (and quasi-coherent) sheaves. The resulting theory of
10209:
10174:
10168:
9727:
9707:
9119:), consider polynomial mappings between different sets of this type, and so on. But if
8721:
7979:
7625:
7435:
7347:
7011:
6261:
6055:
5966:
5901:
5881:
5657:
5637:
5472:
5442:
5422:
5402:
5382:
5333:
5008:
5002:
4848:
4751:
4507:{\displaystyle k({\mathfrak {p}}_{0})=\operatorname {Frac} (\mathbb {Z} )=\mathbb {Q} }
4429:
4409:
4274:
4254:
4203:
4030:
3898:
3052:
2558:
2445:
2326:
1983:
1638:
1416:
1289:
538:
463:
393:
165:
91:
35:
5151:{\displaystyle \mathbb {A} _{\mathbb {Z} }^{1}=\{a\ {\text{for}}\ a\in \mathbb {Z} \}}
443:
11269:
11231:
11200:
11173:
11099:
11060:
9564:
9325:
9287:
9016:
8383:
4577:
1069:
427:
206:
154:
134:
90:(EGA); one of its aims was developing the formalism needed to solve deep problems of
47:
11125:
11048:
10622:
10486:
5379:
is a non-constant polynomial with no integer factor and which is irreducible modulo
845:. In this sense, scheme theory completely subsumes the theory of commutative rings.
11261:
11163:
11143:
11027:
10992:
10469:
10307:
10302:
8617:≥ 2. (However, the affine line minus the origin is isomorphic to the affine scheme
6626:
is a kind of fusion of two Galois-symmetric horizonal lines, a curve of degree 2.
621:
470:
431:
423:
419:
385:
377:
362:
358:
337:
algebraic varieties. However, many arguments in algebraic geometry work better for
269:
245:
142:
95:
63:
11032:
10766:
333:
Noether and Krull's commutative algebra can be viewed as an algebraic approach to
248:) that algebraic geometry over the real numbers is simplified by working over the
11318:
11279:
11241:
11210:
11183:
11151:
11129:
11109:
11070:
11037:
11002:
10871:
10662:
10654:
10618:
10614:
10610:
10586:
10574:
10518:
10513:
10504:, gives simple conditions for a functor to be represented by an algebraic space.
10495:
9576:
9004:
8519:
8248:
7469:
6532:; since we cannot distinguish between these values (they are symmetric under the
5419:
as two-dimensional, with a "characteristic direction" measured by the coordinate
2954:
1262:
Since the category of schemes has fiber products and also a terminal object Spec(
1022:
865:
703:
484:
with a natural topology (known as the
Zariski topology), but augmented it with a
366:
327:
323:
287:
158:
126:
11260:. Lecture Notes in Mathematics. Vol. 1358 (2nd ed.). Springer-Verlag.
11081:
10907:
10672:
10590:
10535:
10437:
10396:-module that is the sheaf associated to a module on each affine open subset of
10290:
10284:
9651:
9568:
8515:
7831:
1655:. The scheme also contains a non-closed point for each non-maximal prime ideal
1017:. (This generalizes the old observation that given some equations over a field
948:
853:
661:
389:
318:
278:
suggests an approach to algebraic geometry over any algebraically closed field
253:
99:
8999:
corresponding to the principal ultrafilter associated to the positive integer
8081:{\displaystyle X=\operatorname {Spec} {\frac {\mathbb {Z} }{(y^{2}-x^{3}-p)}}}
326:
of a commutative ring in terms of prime ideals and, at least when the ring is
209:, it can be useful to consider families of algebraic surfaces over any scheme
11366:
11253:
10997:
10606:
10490:
10294:
10262:
10230:
9660:
9012:
7453:
2323:
which gives the usual ring of rational functions regular on a given open set
625:
404:
354:
342:
283:
173:
138:
115:
10710:
8533:) is the scheme defined by starting with two copies of the affine line over
4426:
is equal to zero in the residue field. The field of "rational functions" on
350:
10938:
10809:
10650:
10645:
10627:
10457:
9572:
6533:
4024:
2892:
706:, with morphisms defined as morphisms of locally ringed spaces. (See also:
380:), by gluing affine varieties along open subsets, on the model of abstract
314:
214:
146:
67:
10830:
7156:{\displaystyle r({\mathfrak {m}})=r(\alpha )\in \mathbb {F} _{p}(\alpha )}
505:), which may be thought of as the coordinate ring of regular functions on
11356:
10594:
10589:
in homological algebra yield higher information about operations such as
10433:
8960:
5189:, the polynomials with integer coefficients. The corresponding scheme is
477:
306:
265:
237:
169:
31:
10818:
9032:
4016:{\displaystyle k({\mathfrak {m}}_{p})=\mathbb {Z} /(p)=\mathbb {F} _{p}}
1750:{\displaystyle {\bar {V}}={\bar {V}}({\mathfrak {p}})\subset {\bar {X}}}
133:
of functions which vanish on the subvariety. Intuitively, a scheme is a
11147:
10895:
10641:
9645:
2000:
has a basis of open subsets given by the complements of hypersurfaces,
233:
220:
For some of the detailed definitions in the theory of schemes, see the
10859:
10550:
Grothendieck originally introduced stacks as a tool for the theory of
9583:
treats the fundamental group and the Galois group on the same footing.
8948:{\textstyle \operatorname {Spec} \left(\prod _{n=1}^{\infty }k\right)}
8887:
is not surjective and hence not an isomorphism. Therefore, the scheme
187:
is that much of algebraic geometry should be developed for a morphism
11305:
10698:
10237:. Such a subscheme consists of either two distinct complex points of
9635:
7084:{\displaystyle r({\mathfrak {m}})=r\ \mathrm {mod} \ {\mathfrak {m}}}
3443:
1628:{\displaystyle {\mathfrak {m}}_{a}=(x_{1}-a_{1},\ldots ,x_{n}-a_{n})}
10727:
10725:
8724:
in complex analysis, though easier to prove. That is, the inclusion
2132:. This set is endowed with its coordinate ring of regular functions
11344:
11265:
10559:
10156:{\displaystyle {\frac {k}{(x^{2},\,y)}}\cong {\frac {k}{(x^{2})}}.}
9656:
2711:{\displaystyle k({\mathfrak {m}}_{a})=R/{\mathfrak {m}}_{a}\cong k}
384:
in topology. He needed this generality for his construction of the
381:
107:
10987:
10432:, which are the sheaves that locally come from finitely generated
205:), rather than for an individual scheme. For example, in studying
10947:
10919:
10778:
10722:
10566:
says that an algebraic stack with finite stabilizer groups has a
1053:
71:
11353:
https://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/
9347:→ A over the generic point of A is exactly the generic point of
1060:-algebras to sets. It is an important observation that a scheme
1021:, one can consider the set of solutions of the equations in any
575:(as a locally ringed space) is an affine scheme. In particular,
8255:; this is an algebro-geometric version of compactness. Indeed,
240:. By the 19th century, it became clear (notably in the work of
9867:. Their scheme-theoretic intersection is defined by the ideal
8955:
is an affine scheme whose underlying topological space is the
11130:"Éléments de géométrie algébrique: I. Le langage des schémas"
9313:−5) over the complex numbers. This is a closed subscheme of A
3533:{\displaystyle U_{p}=Z\smallsetminus \{{\mathfrak {m}}_{p}\}}
232:
The origins of algebraic geometry mostly lie in the study of
114:. Scheme theory also unifies algebraic geometry with much of
10883:
10790:
3456:
The basis open set corresponding to the irreducible element
8683:
is not affine, one computes that every regular function on
3251:{\displaystyle V=\operatorname {Spec} k/(x^{2}-y^{2}(y+1))}
10476:
is perhaps the main technical tool in algebraic geometry.
9131:) is not rich enough. Indeed, one can study the solutions
4339:{\displaystyle f({\mathfrak {m}}_{p})\in \mathbb {F} _{p}}
3049:, a closed subscheme of affine space. For example, taking
1970:{\displaystyle {\mathfrak {p}}\subset {\mathfrak {m}}_{a}}
1870:, including all the closed points of the subvariety, i.e.
1278:
Here and below, all the rings considered are commutative.
11197:
Basic
Algebraic Geometry 2: Schemes and Complex Manifolds
10754:
10460:
construction). In this way, coherent sheaves on a scheme
7589:{\displaystyle X=\operatorname {Spec} (\mathbb {Z} /(f))}
5462:
4103:{\displaystyle n({\mathfrak {m}}_{p})=n\ {\text{mod}}\ p}
2946:{\displaystyle {\bar {V}}(f)\subset \mathbb {A} _{k}^{n}}
9155:) in any reasonable sense. For example, the plane curve
3449:
686:-space can in fact be defined over any commutative ring
141:
of irreducible subvarieties. The space is covered by an
3288:
can be considered as the coordinate ring of the scheme
313:, known as affine varieties. Motivated by these ideas,
11342:
8969:
8904:
8580:{\displaystyle X=\mathbb {A} ^{n}\smallsetminus \{0\}}
8292:
is a compact space in the classical topology, whereas
7834:, then the fibers over its discriminant locus, where
3611:
2994:
2442:, also defines a function on the points of the scheme
2204:
454:
272:
used to study complex varieties do not seem to apply.
106:, scheme theory allows a systematic use of methods of
11120:
10212:
10177:
10061:
10004:
9954:
9873:
9831:
9798:
9769:
9730:
9710:
9676:
9469:
9357:
8852:
8771:
8730:
8693:
8623:
8593:
8546:
8484:
8392:
8350:
8298:
8264:
8219:
8173:
8127:
8094:
8002:
7982:
7840:
7744:
7715:
7634:
7602:
7533:
7490:
7438:
7399:
7370:
7350:
7312:
7277:
7248:
7215:
7169:
7097:
7034:
7014:
6968:
6930:
6903:
6874:
6848:
6819:
6686:
6635:
6590:
6542:
6509:
6473:
6418:
6392:
6310:
6270:
6240:
6147:
6104:
6078:
6058:
6022:
5989:
5969:
5927:
5904:
5884:
5855:
5821:
5783:
5741:
5706:
5680:
5660:
5640:
5611:
5560:
5524:
5495:
5475:
5445:
5425:
5405:
5385:
5356:
5336:
5285:
5244:
5231:{\displaystyle Y=\operatorname {Spec} (\mathbb {Z} )}
5195:
5164:
5094:
5031:
5011:
4957:
4935:
4871:
4851:
4820:
4774:
4754:
4678:
4632:
4626:
have no common prime factor, then there are integers
4586:
4573:
corresponding to prime divisors of the denominator.
4548:
4520:
4452:
4432:
4412:
4386:
4355:
4297:
4277:
4257:
4226:
4206:
4162:
4116:
4056:
4033:
3952:
3921:
3901:
3873:
3771:
3685:
3546:
3490:
3462:
3405:
3377:
3334:
3321:{\displaystyle Z=\operatorname {Spec} (\mathbb {Z} )}
3294:
3272:
3168:
3134:
3075:
3055:
2962:
2901:
2836:
2804:
2764:
2724:
2653:
2619:
2581:
2561:
2528:
2492:
2468:
2448:
2419:
2352:
2329:
2298:
2141:
2112:
2009:
1986:
1942:
1907:
1876:
1787:
1763:
1691:
1661:
1641:
1546:
1500:
1439:
1419:
1361:
1312:
1292:
1226:
1193:
1164:
1100:
1685:, whose vanishing defines an irreducible subvariety
1158:-schemes. For example, the product of affine spaces
11194:
10464:include information about all closed subschemes of
9095:) of solutions of the equations in the product set
5279:. The closed points are maximal ideals of the form
5272:{\displaystyle {\mathfrak {p}}\subset \mathbb {Z} }
4446:is the fraction field of the generic residue ring,
1306:be an algebraically closed field. The affine space
268:like the integers, where the tools of topology and
11052:
10218:
10198:
10155:
10043:
9990:
9936:
9859:
9817:
9784:
9751:
9716:
9696:
9551:
9451:
8991:
8947:
8867:
8838:
8757:
8708:
8671:
8601:
8579:
8506:
8456:
8370:
8313:
8284:
8239:
8193:
8139:
8113:
8080:
7988:
7968:
7822:
7730:
7701:
7616:
7588:
7519:
7444:
7421:
7385:
7356:
7336:
7298:
7271:is the vanishing locus of the constant polynomial
7263:
7234:
7201:
7155:
7083:
7020:
7000:
6954:
6916:
6889:
6860:
6834:
6805:
6672:
6618:
6576:
6524:
6495:
6459:
6404:
6378:
6296:
6252:
6226:
6133:
6090:
6064:
6044:
6008:
5975:
5955:
5910:
5890:
5870:
5841:
5807:
5769:
5727:
5692:
5666:
5646:
5626:
5597:
5546:
5510:
5481:
5451:
5431:
5411:
5391:
5371:
5342:
5322:
5271:
5230:
5181:
5150:
5072:
5017:
4993:
4943:
4921:
4857:
4837:
4806:
4760:
4736:
4664:
4618:
4565:
4534:
4506:
4438:
4418:
4398:
4372:
4338:
4283:
4263:
4243:
4212:
4192:
4148:
4102:
4039:
4015:
3938:
3907:
3887:
3857:{\displaystyle {\mathcal {O}}_{Z}(U)=\mathbb {Z} }
3856:
3757:
3671:
3532:
3476:
3434:
3391:
3363:
3320:
3280:
3250:
3154:
3116:
3061:
3041:
2980:
2945:
2883:
2819:
2790:
2750:
2710:
2636:
2605:
2567:
2547:
2510:
2478:
2454:
2434:
2405:
2335:
2315:
2280:
2124:
2095:
1992:
1969:
1928:
1893:
1862:
1773:
1749:
1677:
1647:
1627:
1532:
1486:
1425:
1405:
1347:
1298:
1247:
1208:
1179:
1119:
10289:A central part of scheme theory is the notion of
7709:are then algebraic curves over the finite fields
7617:{\displaystyle \operatorname {Spec} \mathbb {Z} }
403:had often used the somewhat foggy concept of the
11364:
10428:Coherent sheaves include the important class of
9139:) of the given equations in any field extension
8839:{\displaystyle O(\mathbb {A} ^{n})=\mathbb {C} }
8121:. This curve is singular over the prime numbers
5083:
3069:to be the complex or real numbers, the equation
309:correspond to the irreducible algebraic sets in
9019:has only finitely many irreducible components.)
8507:{\displaystyle \mathbb {P} _{\mathbb {Q} }^{2}}
6227:{\displaystyle (5,x^{2}+1)=(5,x-2)\cap (5,x+2)}
4744:. Geometrically, this is a version of the weak
4737:{\displaystyle a_{1}n_{1}+\cdots +a_{r}n_{r}=1}
1348:{\displaystyle {\bar {X}}=\mathbb {A} _{k}^{n}}
11080:
10901:
10836:
10743:
7823:{\displaystyle f(x,y)=y^{2}-x^{3}+ax^{2}+bx+c}
7422:{\displaystyle {\overline {\mathbb {F} }}_{p}}
5073:{\displaystyle U_{i}=Z\smallsetminus V(n_{i})}
3042:{\textstyle V(f)=\operatorname {Spec} (R/(f))}
1757:; the topological closure of the scheme point
293:are in one-to-one correspondence with the set
11019:Bulletin of the American Mathematical Society
9563:. This generalizes to a relation between the
8672:{\displaystyle \mathrm {Spec} \,\mathbb {C} }
4576:This also gives a geometric interpretaton of
4193:{\displaystyle \mathbb {Z} /(0)=\mathbb {Z} }
2406:{\displaystyle r=r(x_{1},\ldots ,x_{n})\in R}
1433:; its coordinate ring is the polynomial ring
153:of rings. The cases of main interest are the
9595:be the closed subscheme of the affine line A
9203:has enough information to determine the set
8574:
8568:
8285:{\displaystyle \mathbb {C} \mathbb {P} ^{n}}
5921:A higher degree "horizontal" subscheme like
5439:, and a "spatial direction" with coordinate
5145:
5117:
3752:
3698:
3666:
3607:
3527:
3510:
3371:, the principal ideals of the prime numbers
2275:
2200:
2090:
2044:
1857:
1807:
9223:. (In particular, the closed subscheme of A
9215:-rational points for every extension field
8457:{\displaystyle \operatorname {Proj} R/(f).}
7364:corresponding to Galois orbits of roots of
7344:contains the points in each characteristic
6460:{\displaystyle {\mathfrak {m}}=(3,x^{2}+1)}
6052:. This behaves differently under different
5674:-coordinate, we have the "horizontal line"
5238:, whose points are all of the prime ideals
50:in several ways, such as taking account of
11162:
10925:
10877:
10865:
10824:
10784:
10507:A further generalization is the idea of a
10417:-module that is the sheaf associated to a
9123:is not algebraically closed, then the set
8331:of positive degree in the polynomial ring
7976:are all singular schemes. For example, if
4220:is determined by its values at the points
697:
168:of a commutative ring; its points are the
11304:
11047:
11031:
10996:
10986:
10913:
10772:
10731:
10716:
10704:
10517:generalize algebraic spaces by having an
10319:, which are sheaves of abelian groups on
10293:, generalizing the notion of (algebraic)
10101:
9937:{\displaystyle (y)+(x^{2}-y)=(x^{2},\,y)}
9927:
9679:
9067:
8963:on the positive integers, with the ideal
8797:
8780:
8745:
8696:
8640:
8638:
8595:
8555:
8493:
8487:
8353:
8301:
8272:
8266:
8222:
8201:can be constructed as a scheme by gluing
8176:
8019:
7718:
7686:
7666:
7610:
7550:
7498:
7404:
7186:
7134:
6985:
6955:{\displaystyle d=\operatorname {deg} (f)}
6822:
6784:
6737:
6707:
6512:
6379:{\displaystyle (2,x^{2}+1)=(2,(x-1)^{2})}
5815:corresponding to the rational coordinate
5489:defines a "vertical line", the subscheme
5256:
5212:
5166:
5141:
5103:
5097:
4937:
4922:{\displaystyle (n_{1},\ldots ,n_{r})=(1)}
4500:
4489:
4326:
4186:
4164:
4003:
3980:
3881:
3799:
3647:
3581:
3470:
3385:
3328:. The Zariski topology has closed points
3311:
3274:
3137:
2928:
2262:
1330:
1229:
1196:
1167:
682:. In the spirit of scheme theory, affine
11224:Algebraic Geometry and Arithmetic Curves
11221:
10051:, but rather a fat point containing the
9642:if and only if they have the same value
9579:). Indeed, Grothendieck's theory of the
9023:
7460:. This is a major obstacle to analyzing
7458:intersect with the expected multiplicity
7008:corresponds to a function on the scheme
6673:{\displaystyle {\mathfrak {m}}=(p,f(x))}
6134:{\displaystyle x=\pm 2\ {\text{mod}}\ 5}
5598:{\displaystyle {\mathfrak {m}}=(p,f(x))}
5323:{\displaystyle {\mathfrak {m}}=(p,f(x))}
4149:{\displaystyle n({\mathfrak {p}}_{0})=n}
3895:corresponds to a function on the scheme
2884:{\displaystyle f=f(x_{1},\ldots ,x_{n})}
1678:{\displaystyle {\mathfrak {p}}\subset R}
1079:always exists. That is, for any schemes
11290:
11252:
11012:
10970:
10953:
10889:
10796:
10760:
9626:); in particular, the regular function
9343:-coordinate. The fiber of the morphism
9147:, but these sets are not determined by
8758:{\displaystyle f:X\to \mathbb {A} ^{n}}
4380:is the vanishing locus of the function
3435:{\displaystyle {\mathfrak {p}}_{0}=(0)}
3364:{\displaystyle {\mathfrak {m}}_{p}=(p)}
3261:
1406:{\displaystyle a=(a_{1},\ldots ,a_{n})}
1355:is the algebraic variety of all points
678:is the spectrum of the polynomial ring
648:A basic example of an affine scheme is
628:'s "Red Book". The sheaf properties of
172:of the ring, and its closed points are
14:
11365:
11338:Can one explain schemes to biologists?
10931:
10802:
10691:Introduction of the first edition of "
9666:Nilpotent elements arise naturally in
9242:= −1 is a nonempty topological space.)
9007:, and in particular, each point is an
7479:
7456:, so that pairs of curves may fail to
2791:{\displaystyle r({\mathfrak {m}}_{a})}
2606:{\displaystyle R\to R/{\mathfrak {p}}}
1635:, the set of polynomials vanishing at
1127:exists in the category of schemes. If
120:Wiles's proof of Fermat's Last Theorem
10750:, Séminaire Henri Cartan, vol. 8
10448:can be viewed as a coherent sheaf on
8150:
5770:{\displaystyle {\mathfrak {p}}=(x-a)}
616:In the early days, this was called a
588:, which assigns to every open subset
11135:Publications Mathématiques de l'IHÉS
10327:over the sheaf of regular functions
10241:, or else a subscheme isomorphic to
9697:{\displaystyle \mathbb {A} _{k}^{2}}
9027:
8371:{\displaystyle \mathbb {P} _{R}^{n}}
8240:{\displaystyle \mathbb {P} _{R}^{n}}
8194:{\displaystyle \mathbb {P} _{R}^{n}}
7202:{\displaystyle r(x)\in \mathbb {Z} }
7001:{\displaystyle r(x)\in \mathbb {Z} }
6297:{\displaystyle x=1\ {\text{mod}}\ 2}
4994:{\displaystyle \rho _{i}=a_{i}n_{i}}
4951:. Indeed, we may consider the terms
4865:, then they generate the unit ideal
3155:{\displaystyle \mathbb {A} _{k}^{2}}
2830:The vanishing locus of a polynomial
1533:{\displaystyle X=\mathrm {Spec} (R)}
837:) of schemes and ring homomorphisms
449:
157:, in which the coordinate rings are
10278:
9614:. The ring of regular functions on
9079:Given some polynomial equations in
8386:, this subscheme can be written as
7224:
7106:
7076:
7043:
6727:
6695:
6638:
6421:
5744:
5563:
5547:{\displaystyle {\mathfrak {p}}=(p)}
5527:
5288:
5247:
4838:{\displaystyle {\mathfrak {m}}_{p}}
4824:
4807:{\displaystyle n_{1},\ldots ,n_{r}}
4665:{\displaystyle a_{1},\ldots ,a_{r}}
4619:{\displaystyle n_{1},\ldots ,n_{r}}
4566:{\displaystyle {\mathfrak {m}}_{p}}
4552:
4462:
4373:{\displaystyle {\mathfrak {m}}_{p}}
4359:
4307:
4244:{\displaystyle {\mathfrak {m}}_{p}}
4230:
4126:
4066:
3962:
3939:{\displaystyle {\mathfrak {m}}_{p}}
3925:
3734:
3704:
3516:
3409:
3338:
2774:
2691:
2663:
2637:{\displaystyle {\mathfrak {m}}_{a}}
2623:
2598:
2537:
2503:
2471:
2085:
2049:
1956:
1945:
1894:{\displaystyle {\mathfrak {m}}_{a}}
1880:
1852:
1842:
1812:
1796:
1766:
1724:
1664:
1550:
860:, the category of schemes has Spec(
809:on the rings of regular functions,
620:, and a scheme was defined to be a
256:, which has the advantage of being
27:Generalization of algebraic variety
24:
10479:
9459:This in turn is equivalent to the
9111:): define the Zariski topology on
8932:
8879:were affine, it would follow that
8634:
8631:
8628:
8625:
8464:For example, the closed subscheme
7842:
7235:{\displaystyle r({\mathfrak {m}})}
7067:
7064:
7061:
6496:{\displaystyle x=\pm {\sqrt {-1}}}
6467:is a prime ideal corresponding to
6045:{\displaystyle x=\pm {\sqrt {-1}}}
5654:points" of the scheme. Fixing the
5461:
5158:is a variety with coordinate ring
3775:
3550:
3448:
2798:corresponds to the original value
2751:{\displaystyle x_{i}\mapsto a_{i}}
2548:{\displaystyle r({\mathfrak {p}})}
2316:{\displaystyle {\mathcal {O}}_{X}}
2302:
2145:
1517:
1514:
1511:
1508:
1248:{\displaystyle \mathbb {A} ^{m+n}}
762:An algebraic variety over a field
557:admitting a covering by open sets
25:
11384:
11330:
10775:, sections VII.4, VIII.2, VIII.3.
10206:-module, i.e. its dimension as a
9649:at the origin. Allowing such non-
9328:double cover of the affine line A
9159:over the real numbers defined by
8687:extends to a regular function on
7520:{\displaystyle f\in \mathbb {Z} }
3888:{\displaystyle n\in \mathbb {Z} }
3477:{\displaystyle p\in \mathbb {Z} }
3392:{\displaystyle p\in \mathbb {Z} }
2511:{\displaystyle R/{\mathfrak {p}}}
460:Séminaire de géométrie algébrique
409:Foundations of Algebraic Geometry
98:(the last of which was proved by
10693:Éléments de géométrie algébrique
10301:, one starts by considering the
9944:. Since the intersection is not
9488:
9471:
9433:
9365:
9031:
8709:{\displaystyle \mathbb {A} ^{n}}
8314:{\displaystyle \mathbb {C} ^{n}}
7731:{\displaystyle \mathbb {F} _{p}}
6835:{\displaystyle \mathbb {F} _{p}}
6525:{\displaystyle \mathbb {F} _{3}}
4814:have no common vanishing points
3117:{\displaystyle x^{2}=y^{2}(y+1)}
1209:{\displaystyle \mathbb {A} ^{n}}
1180:{\displaystyle \mathbb {A} ^{m}}
1139:, their fiber product over Spec(
766:can be defined as a scheme over
456:Éléments de géométrie algébrique
361:) varieties. In particular, the
87:Éléments de géométrie algébrique
10974:Illinois Journal of Mathematics
10842:
9948:, this is not merely the point
9248:The points of the affine line A
8587:, say over the complex numbers
5005:subordinate to the covering of
4406:, the point where the value of
3399:; as well as the generic point
2718:, with the natural isomorphism
2479:{\displaystyle {\mathfrak {p}}}
1929:{\displaystyle a\in {\bar {V}}}
1774:{\displaystyle {\mathfrak {p}}}
1281:
492:he assigned a commutative ring
488:of rings: to every open subset
399:The algebraic geometers of the
345:. From the 1920s to the 1940s,
341:, essentially because they are
11293:Fundamental Algebraic Geometry
10880:, Exercises I.3.6 and III.4.3.
10737:
10685:
10502:Artin representability theorem
10421:on each affine open subset of
10193:
10181:
10144:
10131:
10126:
10120:
10105:
10085:
10080:
10068:
10044:{\displaystyle (x,y)\subset k}
10038:
10026:
10017:
10005:
9985:
9973:
9967:
9955:
9931:
9911:
9905:
9886:
9880:
9874:
9854:
9835:
9779:
9773:
9746:
9734:
9537:
9525:
9522:
9510:
9498:
9492:
9481:
9475:
9443:
9437:
9423:
9414:
9402:
9399:
9387:
9375:
9369:
8992:{\textstyle \prod _{m\neq n}k}
8862:
8856:
8833:
8801:
8790:
8775:
8740:
8666:
8644:
8448:
8442:
8434:
8402:
8337:determines a closed subscheme
8072:
8040:
8035:
8023:
7760:
7748:
7696:
7681:
7670:
7662:
7583:
7580:
7574:
7566:
7554:
7546:
7514:
7502:
7380:
7374:
7331:
7328:
7322:
7316:
7287:
7281:
7258:
7252:
7229:
7219:
7196:
7190:
7179:
7173:
7150:
7144:
7126:
7120:
7111:
7101:
7048:
7038:
6995:
6989:
6978:
6972:
6949:
6943:
6897:; this is a finite field with
6884:
6878:
6800:
6794:
6776:
6773:
6767:
6761:
6753:
6747:
6717:
6711:
6700:
6690:
6667:
6664:
6658:
6646:
6613:
6594:
6584:as two fused points. Overall,
6571:
6546:
6454:
6429:
6373:
6364:
6351:
6342:
6336:
6311:
6221:
6203:
6197:
6179:
6173:
6148:
5950:
5931:
5865:
5859:
5802:
5787:
5764:
5752:
5722:
5710:
5621:
5615:
5592:
5589:
5583:
5571:
5541:
5535:
5505:
5499:
5366:
5360:
5317:
5314:
5308:
5296:
5266:
5260:
5225:
5222:
5216:
5208:
5176:
5170:
5067:
5054:
4916:
4910:
4904:
4872:
4493:
4485:
4473:
4456:
4318:
4301:
4179:
4173:
4137:
4120:
4077:
4060:
3995:
3989:
3973:
3956:
3851:
3803:
3792:
3786:
3601:
3585:
3574:
3561:
3429:
3423:
3358:
3352:
3315:
3307:
3245:
3242:
3230:
3204:
3196:
3184:
3162:, corresponding to the scheme
3111:
3099:
3036:
3033:
3027:
3016:
3004:
2998:
2988:. The corresponding scheme is
2969:
2963:
2920:
2914:
2908:
2878:
2846:
2814:
2808:
2785:
2768:
2735:
2674:
2657:
2585:
2542:
2532:
2426:
2394:
2362:
2194:
2178:
2169:
2156:
2038:
2032:
1920:
1801:
1791:
1741:
1729:
1719:
1713:
1698:
1622:
1564:
1527:
1521:
1481:
1449:
1400:
1368:
1319:
227:
13:
1:
11297:American Mathematical Society
11055:History of Algebraic Geometry
11033:10.1090/S0273-0979-01-00913-2
10963:
10916:, sections VIII.2 and VIII.3.
5084:Affine line over the integers
4580:stating that if the integers
1120:{\displaystyle X\times _{Y}Z}
958:More generally, for a scheme
858:category of commutative rings
524:
11343:The Stacks Project Authors,
11195:Igor R. Shafarevich (2013).
10678:
10577:. In this setting, known as
10570:that is an algebraic space.
10261:together with a line in the
9655:schemes brings the ideas of
9195:− 0.) By contrast, a scheme
9003:. This topological space is
8898:be a field. Then the scheme
8765:induces an isomorphism from
8602:{\displaystyle \mathbb {C} }
7596:has a canonical morphism to
7484:If we consider a polynomial
7408:
7209:is determined by its values
6577:{\displaystyle V(3,x^{2}+1)}
5182:{\displaystyle \mathbb {Z} }
4944:{\displaystyle \mathbb {Z} }
4156:in the generic residue ring
3915:, a function whose value at
3281:{\displaystyle \mathbb {Z} }
2981:{\displaystyle (f)\subset R}
2292:This induces a unique sheaf
2106:for irreducible polynomials
943:) is also called the set of
195:of schemes (called a scheme
46:that enlarges the notion of
7:
10744:Chevalley, C. (1955–1956),
10719:, sections VII.2 and VII.5.
10600:
9991:{\displaystyle (x,y)=(0,0)}
9763:axis, which is the variety
9670:. For example in the plane
9011:. Since affine schemes are
8957:Stone–Čech compactification
5983:-values which are roots of
5849:, which does not intersect
4050:the function is defined by
3444:closure is the whole scheme
2413:, a polynomial function on
1494:. The corresponding scheme
1273:
1087:with morphisms to a scheme
430:, motivated in part by the
18:Scheme (algebraic geometry)
10:
11389:
10902:Eisenbud & Harris 1998
10837:Eisenbud & Harris 1998
10668:Moduli of algebraic curves
10579:derived algebraic geometry
10336:. In particular, a module
10282:
9860:{\displaystyle V(x^{2}-y)}
9575:(which classifies certain
8720:≥ 2: this is analogous to
6619:{\displaystyle V(x^{2}+1)}
5956:{\displaystyle V(x^{2}+1)}
4542:has "poles" at the points
3946:lies in the residue field
2486:lies in the quotient ring
2435:{\displaystyle {\bar {X}}}
553:is a locally ringed space
264:, and more generally over
118:, which eventually led to
10474:coherent sheaf cohomology
10419:finitely generated module
10165:intersection multiplicity
9191:) can be identified with
8156:For any commutative ring
8088:then its discriminant is
7393:in the algebraic closure
6861:{\displaystyle x=\alpha }
6503:in an extension field of
4251:only, so we can think of
1135:are schemes over a field
1093:categorical fiber product
607:ring of regular functions
369:of the complex numbers).
276:Hilbert's Nullstellensatz
222:glossary of scheme theory
10940:Stacks Project, Tag 07Y1
10811:Stacks Project, Tag 020D
10529:on an algebraic variety
10404:(on a Noetherian scheme
10340:over a commutative ring
9663:into algebraic geometry.
9610:= 0, sometimes called a
9351:, yielding the morphism
9324:. It can be viewed as a
9087:, one can study the set
8257:complex projective space
8114:{\displaystyle -27p^{2}}
5777:. We also have the line
3442:, the zero ideal, whose
1077:fiber product of schemes
962:over a commutative ring
875:over a commutative ring
805:of schemes determines a
641:) mean that its elements
545:) of a commutative ring
84:in 1960 in his treatise
11228:Oxford University Press
11122:Grothendieck, Alexandre
11091:The Geometry of Schemes
10583:E-infinity ring spectra
10167:of 2 is defined as the
9818:{\displaystyle y=x^{2}}
9724:, with coordinate ring
9581:étale fundamental group
9463:-2 extension of fields
9183:) not empty. (In fact,
9083:variables over a field
8883:is an isomorphism, but
8380:projective hypersurface
7527:then the affine scheme
7337:{\displaystyle V(f(x))}
6813:, a field extension of
6009:{\displaystyle x^{2}+1}
5808:{\displaystyle V(bx-a)}
5399:. Thus, we may picture
5350:is a prime number, and
4929:in the coordinate ring
4746:Hilbert Nullstellensatz
3540:, with coordinate ring
2953:, corresponding to the
698:The category of schemes
301:-tuples of elements of
10998:10.1215/ijm/1373636688
10659:Linear algebraic group
10556:Deligne–Mumford stacks
10539:, which remembers the
10525:of an algebraic group
10220:
10200:
10157:
10045:
9992:
9938:
9861:
9819:
9786:
9753:
9718:
9698:
9553:
9453:
9068:Motivation for schemes
8993:
8949:
8936:
8869:
8840:
8759:
8710:
8673:
8603:
8581:
8508:
8458:
8372:
8326:homogeneous polynomial
8315:
8286:
8241:
8195:
8141:
8115:
8082:
7996:is a prime number and
7990:
7970:
7824:
7732:
7703:
7618:
7590:
7521:
7446:
7423:
7387:
7358:
7338:
7300:
7299:{\displaystyle r(x)=p}
7265:
7236:
7203:
7157:
7085:
7022:
7002:
6956:
6918:
6891:
6862:
6836:
6807:
6674:
6620:
6578:
6526:
6497:
6461:
6406:
6380:
6298:
6254:
6228:
6135:
6092:
6066:
6046:
6010:
5977:
5957:
5912:
5892:
5872:
5843:
5809:
5771:
5729:
5728:{\displaystyle V(x-a)}
5694:
5668:
5648:
5634:, the "characteristic
5628:
5599:
5548:
5512:
5483:
5466:
5453:
5433:
5413:
5393:
5373:
5344:
5324:
5273:
5232:
5183:
5152:
5074:
5019:
4995:
4945:
4923:
4859:
4839:
4808:
4762:
4738:
4666:
4620:
4567:
4536:
4508:
4440:
4420:
4400:
4374:
4340:
4285:
4265:
4245:
4214:
4194:
4150:
4104:
4041:
4017:
3940:
3909:
3889:
3858:
3759:
3673:
3534:
3478:
3453:
3436:
3393:
3365:
3322:
3282:
3252:
3156:
3118:
3063:
3043:
2982:
2947:
2885:
2821:
2792:
2752:
2712:
2638:
2607:
2575:under the natural map
2569:
2549:
2512:
2480:
2456:
2436:
2407:
2337:
2317:
2290:
2282:
2126:
2125:{\displaystyle f\in R}
2104:
2097:
1994:
1971:
1930:
1895:
1864:
1775:
1751:
1679:
1649:
1629:
1534:
1488:
1427:
1407:
1349:
1300:
1249:
1210:
1181:
1121:
1068:is determined by this
989:means a morphism Spec(
185:relative point of view
82:Alexander Grothendieck
10827:, Proposition II.2.3.
10452:that is zero outside
10221:
10201:
10158:
10046:
9998:defined by the ideal
9993:
9939:
9862:
9820:
9787:
9754:
9719:
9699:
9571:in topology) and the
9554:
9454:
9339:by projecting to the
9024:Examples of morphisms
9009:irreducible component
8994:
8950:
8916:
8870:
8841:
8760:
8711:
8674:
8604:
8582:
8527:line with two origins
8509:
8459:
8373:
8316:
8287:
8242:
8205:+ 1 copies of affine
8196:
8142:
8116:
8083:
7991:
7971:
7825:
7733:
7704:
7619:
7591:
7522:
7462:Diophantine equations
7447:
7424:
7388:
7359:
7339:
7301:
7266:
7237:
7204:
7158:
7086:
7023:
7003:
6957:
6919:
6917:{\displaystyle p^{d}}
6892:
6863:
6837:
6808:
6675:
6629:The residue field at
6621:
6579:
6536:), we should picture
6527:
6498:
6462:
6407:
6381:
6299:
6255:
6229:
6136:
6093:
6067:
6047:
6011:
5978:
5958:
5913:
5893:
5873:
5844:
5842:{\displaystyle x=a/b}
5810:
5772:
5730:
5695:
5669:
5649:
5629:
5600:
5549:
5513:
5484:
5469:A given prime number
5465:
5454:
5434:
5414:
5394:
5374:
5345:
5325:
5274:
5233:
5184:
5153:
5075:
5020:
5001:as forming a kind of
4996:
4946:
4924:
4860:
4840:
4809:
4763:
4739:
4667:
4621:
4568:
4537:
4509:
4441:
4421:
4401:
4375:
4341:
4286:
4266:
4246:
4215:
4195:
4151:
4105:
4042:
4018:
3941:
3910:
3890:
3859:
3760:
3674:
3535:
3479:
3452:
3437:
3394:
3366:
3323:
3283:
3266:The ring of integers
3253:
3157:
3119:
3064:
3044:
2983:
2948:
2886:
2822:
2793:
2753:
2713:
2639:
2608:
2570:
2550:
2513:
2481:
2457:
2437:
2408:
2338:
2318:
2283:
2134:
2127:
2098:
2002:
1995:
1972:
1931:
1896:
1865:
1776:
1752:
1680:
1650:
1630:
1535:
1489:
1428:
1408:
1350:
1301:
1266:), it has all finite
1250:
1211:
1182:
1122:
807:pullback homomorphism
740:of schemes. A scheme
509:. These objects Spec(
347:B. L. van der Waerden
102:). Strongly based on
70:are defined over the
10707:, Chapters IV and V.
10541:stabilizer subgroups
10511:. Crudely speaking,
10436:. An example is the
10380:quasi-coherent sheaf
10210:
10175:
10059:
10002:
9952:
9871:
9829:
9796:
9785:{\displaystyle V(y)}
9767:
9728:
9708:
9674:
9467:
9355:
8967:
8902:
8868:{\displaystyle O(X)}
8850:
8769:
8728:
8691:
8621:
8591:
8544:
8482:
8390:
8348:
8344:in projective space
8296:
8262:
8217:
8171:
8125:
8092:
8000:
7980:
7838:
7742:
7713:
7632:
7600:
7531:
7488:
7436:
7397:
7386:{\displaystyle f(x)}
7368:
7348:
7310:
7275:
7264:{\displaystyle V(p)}
7246:
7213:
7167:
7095:
7032:
7012:
6966:
6928:
6901:
6890:{\displaystyle f(x)}
6872:
6846:
6817:
6684:
6633:
6588:
6540:
6507:
6471:
6416:
6390:
6308:
6268:
6238:
6145:
6102:
6098:, we get two points
6076:
6056:
6020:
5987:
5967:
5925:
5902:
5882:
5871:{\displaystyle V(p)}
5853:
5819:
5781:
5739:
5704:
5678:
5658:
5638:
5627:{\displaystyle f(x)}
5609:
5558:
5522:
5511:{\displaystyle V(p)}
5493:
5473:
5443:
5423:
5403:
5383:
5372:{\displaystyle f(x)}
5354:
5334:
5283:
5242:
5193:
5162:
5092:
5029:
5009:
4955:
4933:
4869:
4849:
4818:
4772:
4752:
4676:
4630:
4584:
4546:
4518:
4450:
4430:
4410:
4384:
4353:
4349:Note that the point
4295:
4275:
4255:
4224:
4204:
4160:
4114:
4054:
4031:
3950:
3919:
3899:
3871:
3769:
3683:
3544:
3488:
3460:
3403:
3375:
3332:
3292:
3270:
3262:Spec of the integers
3166:
3132:
3128:in the affine plane
3073:
3053:
2992:
2960:
2899:
2834:
2820:{\displaystyle r(a)}
2802:
2762:
2722:
2651:
2617:
2579:
2559:
2526:
2490:
2466:
2446:
2417:
2350:
2327:
2296:
2139:
2110:
2007:
1984:
1940:
1905:
1874:
1785:
1761:
1689:
1659:
1639:
1544:
1498:
1437:
1417:
1413:with coordinates in
1359:
1310:
1290:
1224:
1191:
1162:
1143:) may be called the
1098:
966:and any commutative
535:locally ringed space
458:(EGA) and the later
339:projective varieties
258:algebraically closed
242:Jean-Victor Poncelet
11315:2004math.....12512V
11199:. Springer-Verlag.
11172:. Springer-Verlag.
10868:, Example II.4.0.1.
10839:, Proposition VI-2.
10633:Birational geometry
10568:coarse moduli space
9792:, and the parabola
9693:
9668:intersection theory
9297:be the plane curve
8503:
8367:
8236:
8190:
8160:and natural number
8140:{\displaystyle 3,p}
7480:Arithmetic surfaces
6405:{\displaystyle p=3}
6253:{\displaystyle p=2}
6091:{\displaystyle p=5}
5735:of the prime ideal
5693:{\displaystyle x=a}
5518:of the prime ideal
5113:
4768:: if the functions
4535:{\displaystyle a/b}
4399:{\displaystyle n=p}
4027:of integers modulo
3850:
3823:
3679:. For the open set
3151:
2942:
1487:{\displaystyle R=k}
1344:
1154:in the category of
747:a commutative ring
732:) means a morphism
708:morphism of schemes
592:a commutative ring
579:comes with a sheaf
236:equations over the
112:homological algebra
104:commutative algebra
11346:The Stacks Project
11299:, pp. 1–104,
11295:, Providence, RI:
11169:Algebraic Geometry
11148:10.1007/bf02684778
10543:for the action of
10216:
10196:
10153:
10041:
9988:
9934:
9857:
9815:
9782:
9749:
9714:
9694:
9677:
9589:Nilpotent elements
9567:(which classifies
9549:
9449:
9288:rational functions
9286:) is the field of
9043:. You can help by
8989:
8985:
8945:
8865:
8836:
8755:
8706:
8669:
8613:is not affine for
8599:
8577:
8504:
8485:
8454:
8382:. In terms of the
8368:
8351:
8311:
8282:
8237:
8220:
8191:
8174:
8151:Non-affine schemes
8137:
8111:
8078:
7986:
7966:
7820:
7728:
7699:
7626:arithmetic surface
7614:
7586:
7517:
7442:
7419:
7383:
7354:
7334:
7296:
7261:
7242:at closed points;
7232:
7199:
7153:
7081:
7018:
6998:
6952:
6914:
6887:
6858:
6832:
6803:
6670:
6616:
6574:
6522:
6493:
6457:
6402:
6376:
6294:
6250:
6224:
6131:
6088:
6062:
6042:
6006:
5973:
5953:
5908:
5888:
5868:
5839:
5805:
5767:
5725:
5690:
5664:
5644:
5624:
5595:
5544:
5508:
5479:
5467:
5449:
5429:
5409:
5389:
5369:
5340:
5320:
5269:
5228:
5179:
5148:
5095:
5070:
5015:
5003:partition of unity
4991:
4941:
4919:
4855:
4835:
4804:
4758:
4734:
4662:
4616:
4563:
4532:
4504:
4436:
4416:
4396:
4370:
4336:
4281:
4261:
4241:
4210:
4190:
4146:
4100:
4037:
4013:
3936:
3905:
3885:
3854:
3833:
3806:
3755:
3669:
3627:
3530:
3474:
3454:
3432:
3389:
3361:
3318:
3278:
3248:
3152:
3135:
3114:
3059:
3039:
2978:
2943:
2926:
2881:
2817:
2788:
2748:
2708:
2634:
2613:. A maximal ideal
2603:
2565:
2545:
2508:
2476:
2452:
2432:
2403:
2346:Each ring element
2333:
2313:
2278:
2220:
2122:
2093:
1990:
1967:
1936:, or equivalently
1926:
1891:
1860:
1771:
1747:
1675:
1645:
1625:
1530:
1484:
1423:
1403:
1345:
1328:
1296:
1245:
1206:
1177:
1117:
780:integral separated
667:. By definition, A
537:isomorphic to the
207:algebraic surfaces
178:rings of fractions
155:Noetherian schemes
92:algebraic geometry
80:was introduced by
36:algebraic geometry
11275:978-3-540-63293-1
11237:978-0-19-850284-5
11222:Qing Liu (2002).
11179:978-0-387-90244-9
11164:Hartshorne, Robin
11105:978-0-387-98637-1
11066:978-0-534-03723-9
10956:, Definition 4.6.
10850:"Elliptic curves"
10564:Keel–Mori theorem
10219:{\displaystyle k}
10199:{\displaystyle k}
10148:
10109:
9752:{\displaystyle k}
9717:{\displaystyle k}
9565:fundamental group
9540:
9417:
9077:Field extensions.
9061:
9060:
9017:Noetherian scheme
8970:
8384:Proj construction
8076:
7989:{\displaystyle p}
7959:
7955:
7951:
7624:and is called an
7445:{\displaystyle Y}
7411:
7357:{\displaystyle p}
7073:
7059:
7021:{\displaystyle Y}
6842:adjoining a root
6491:
6290:
6286:
6282:
6127:
6123:
6119:
6072:-coordinates. At
6065:{\displaystyle p}
6040:
5976:{\displaystyle x}
5911:{\displaystyle b}
5891:{\displaystyle p}
5667:{\displaystyle x}
5647:{\displaystyle p}
5482:{\displaystyle p}
5452:{\displaystyle x}
5432:{\displaystyle p}
5412:{\displaystyle Y}
5392:{\displaystyle p}
5343:{\displaystyle p}
5133:
5129:
5125:
5088:The affine space
5025:by the open sets
5018:{\displaystyle Z}
4858:{\displaystyle Z}
4761:{\displaystyle Z}
4439:{\displaystyle Z}
4419:{\displaystyle p}
4284:{\displaystyle f}
4264:{\displaystyle n}
4213:{\displaystyle n}
4096:
4092:
4088:
4040:{\displaystyle p}
3908:{\displaystyle Z}
3656:
3639:
3635:
3631:
3626:
3126:nodal cubic curve
3062:{\displaystyle k}
2911:
2568:{\displaystyle r}
2455:{\displaystyle X}
2429:
2336:{\displaystyle U}
2253:
2238:
2235:
2231:
2227:
2224:
2219:
2076:
2073:
2069:
2065:
2062:
1993:{\displaystyle X}
1923:
1839:
1836:
1832:
1828:
1825:
1781:is the subscheme
1744:
1716:
1701:
1648:{\displaystyle a}
1426:{\displaystyle k}
1322:
1299:{\displaystyle k}
1070:functor of points
1056:from commutative
1040:, the assignment
1009:) for the set of
911:) for the set of
751:means a morphism
566:, such that each
450:Origin of schemes
428:Jean-Pierre Serre
376:(not embedded in
135:topological space
48:algebraic variety
16:(Redirected from
11380:
11349:
11325:
11308:
11287:
11249:
11218:
11191:
11159:
11117:
11077:
11058:
11044:
11035:
11009:
11000:
10990:
10981:(4): 1367–1384,
10957:
10951:
10945:
10943:
10935:
10929:
10923:
10917:
10911:
10905:
10904:, Example II-10.
10899:
10893:
10887:
10881:
10875:
10869:
10863:
10857:
10856:
10854:
10846:
10840:
10834:
10828:
10822:
10816:
10814:
10806:
10800:
10794:
10788:
10782:
10776:
10770:
10764:
10758:
10752:
10751:
10741:
10735:
10734:, section VII.4.
10729:
10720:
10714:
10708:
10702:
10696:
10689:
10638:Étale cohomology
10587:derived functors
10514:algebraic stacks
10470:sheaf cohomology
10368:
10367:
10366:
10361:
10303:abelian category
10291:coherent sheaves
10279:Coherent sheaves
10233:complex variety
10225:
10223:
10222:
10217:
10205:
10203:
10202:
10197:
10162:
10160:
10159:
10154:
10149:
10147:
10143:
10142:
10129:
10115:
10110:
10108:
10097:
10096:
10083:
10063:
10050:
10048:
10047:
10042:
9997:
9995:
9994:
9989:
9943:
9941:
9940:
9935:
9923:
9922:
9898:
9897:
9866:
9864:
9863:
9858:
9847:
9846:
9824:
9822:
9821:
9816:
9814:
9813:
9791:
9789:
9788:
9783:
9758:
9756:
9755:
9750:
9723:
9721:
9720:
9715:
9703:
9701:
9700:
9695:
9692:
9687:
9682:
9605:
9604:
9577:field extensions
9558:
9556:
9555:
9550:
9545:
9541:
9506:
9491:
9474:
9458:
9456:
9455:
9450:
9436:
9422:
9418:
9383:
9368:
9338:
9337:
9323:
9322:
9277:
9276:
9258:
9257:
9233:
9232:
9056:
9053:
9035:
9028:
9005:zero-dimensional
8998:
8996:
8995:
8990:
8984:
8954:
8952:
8951:
8946:
8944:
8940:
8935:
8930:
8874:
8872:
8871:
8866:
8845:
8843:
8842:
8837:
8832:
8831:
8813:
8812:
8800:
8789:
8788:
8783:
8764:
8762:
8761:
8756:
8754:
8753:
8748:
8715:
8713:
8712:
8707:
8705:
8704:
8699:
8678:
8676:
8675:
8670:
8665:
8664:
8643:
8637:
8608:
8606:
8605:
8600:
8598:
8586:
8584:
8583:
8578:
8564:
8563:
8558:
8520:rational numbers
8513:
8511:
8510:
8505:
8502:
8497:
8496:
8490:
8477:
8463:
8461:
8460:
8455:
8441:
8433:
8432:
8414:
8413:
8377:
8375:
8374:
8369:
8366:
8361:
8356:
8343:
8336:
8320:
8318:
8317:
8312:
8310:
8309:
8304:
8291:
8289:
8288:
8283:
8281:
8280:
8275:
8269:
8246:
8244:
8243:
8238:
8235:
8230:
8225:
8200:
8198:
8197:
8192:
8189:
8184:
8179:
8166:projective space
8146:
8144:
8143:
8138:
8120:
8118:
8117:
8112:
8110:
8109:
8087:
8085:
8084:
8079:
8077:
8075:
8065:
8064:
8052:
8051:
8038:
8022:
8016:
7995:
7993:
7992:
7987:
7975:
7973:
7972:
7967:
7957:
7956:
7953:
7949:
7942:
7941:
7926:
7925:
7895:
7894:
7885:
7884:
7869:
7868:
7850:
7849:
7829:
7827:
7826:
7821:
7804:
7803:
7788:
7787:
7775:
7774:
7737:
7735:
7734:
7729:
7727:
7726:
7721:
7708:
7706:
7705:
7700:
7695:
7694:
7689:
7674:
7673:
7669:
7644:
7643:
7623:
7621:
7620:
7615:
7613:
7595:
7593:
7592:
7587:
7573:
7553:
7526:
7524:
7523:
7518:
7501:
7451:
7449:
7448:
7443:
7428:
7426:
7425:
7420:
7418:
7417:
7412:
7407:
7402:
7392:
7390:
7389:
7384:
7363:
7361:
7360:
7355:
7343:
7341:
7340:
7335:
7305:
7303:
7302:
7297:
7270:
7268:
7267:
7262:
7241:
7239:
7238:
7233:
7228:
7227:
7208:
7206:
7205:
7200:
7189:
7162:
7160:
7159:
7154:
7143:
7142:
7137:
7110:
7109:
7090:
7088:
7087:
7082:
7080:
7079:
7071:
7070:
7057:
7047:
7046:
7027:
7025:
7024:
7019:
7007:
7005:
7004:
6999:
6988:
6961:
6959:
6958:
6953:
6923:
6921:
6920:
6915:
6913:
6912:
6896:
6894:
6893:
6888:
6867:
6865:
6864:
6859:
6841:
6839:
6838:
6833:
6831:
6830:
6825:
6812:
6810:
6809:
6804:
6793:
6792:
6787:
6760:
6746:
6745:
6740:
6731:
6730:
6724:
6710:
6699:
6698:
6679:
6677:
6676:
6671:
6642:
6641:
6625:
6623:
6622:
6617:
6606:
6605:
6583:
6581:
6580:
6575:
6564:
6563:
6531:
6529:
6528:
6523:
6521:
6520:
6515:
6502:
6500:
6499:
6494:
6492:
6484:
6466:
6464:
6463:
6458:
6447:
6446:
6425:
6424:
6411:
6409:
6408:
6403:
6385:
6383:
6382:
6377:
6372:
6371:
6329:
6328:
6303:
6301:
6300:
6295:
6288:
6287:
6284:
6280:
6259:
6257:
6256:
6251:
6233:
6231:
6230:
6225:
6166:
6165:
6140:
6138:
6137:
6132:
6125:
6124:
6121:
6117:
6097:
6095:
6094:
6089:
6071:
6069:
6068:
6063:
6051:
6049:
6048:
6043:
6041:
6033:
6015:
6013:
6012:
6007:
5999:
5998:
5982:
5980:
5979:
5974:
5962:
5960:
5959:
5954:
5943:
5942:
5917:
5915:
5914:
5909:
5897:
5895:
5894:
5889:
5877:
5875:
5874:
5869:
5848:
5846:
5845:
5840:
5835:
5814:
5812:
5811:
5806:
5776:
5774:
5773:
5768:
5748:
5747:
5734:
5732:
5731:
5726:
5700:, the subscheme
5699:
5697:
5696:
5691:
5673:
5671:
5670:
5665:
5653:
5651:
5650:
5645:
5633:
5631:
5630:
5625:
5604:
5602:
5601:
5596:
5567:
5566:
5554:: this contains
5553:
5551:
5550:
5545:
5531:
5530:
5517:
5515:
5514:
5509:
5488:
5486:
5485:
5480:
5458:
5456:
5455:
5450:
5438:
5436:
5435:
5430:
5418:
5416:
5415:
5410:
5398:
5396:
5395:
5390:
5378:
5376:
5375:
5370:
5349:
5347:
5346:
5341:
5329:
5327:
5326:
5321:
5292:
5291:
5278:
5276:
5275:
5270:
5259:
5251:
5250:
5237:
5235:
5234:
5229:
5215:
5188:
5186:
5185:
5180:
5169:
5157:
5155:
5154:
5149:
5144:
5131:
5130:
5127:
5123:
5112:
5107:
5106:
5100:
5079:
5077:
5076:
5071:
5066:
5065:
5041:
5040:
5024:
5022:
5021:
5016:
5000:
4998:
4997:
4992:
4990:
4989:
4980:
4979:
4967:
4966:
4950:
4948:
4947:
4942:
4940:
4928:
4926:
4925:
4920:
4903:
4902:
4884:
4883:
4864:
4862:
4861:
4856:
4844:
4842:
4841:
4836:
4834:
4833:
4828:
4827:
4813:
4811:
4810:
4805:
4803:
4802:
4784:
4783:
4767:
4765:
4764:
4759:
4743:
4741:
4740:
4735:
4727:
4726:
4717:
4716:
4698:
4697:
4688:
4687:
4671:
4669:
4668:
4663:
4661:
4660:
4642:
4641:
4625:
4623:
4622:
4617:
4615:
4614:
4596:
4595:
4572:
4570:
4569:
4564:
4562:
4561:
4556:
4555:
4541:
4539:
4538:
4533:
4528:
4513:
4511:
4510:
4505:
4503:
4492:
4472:
4471:
4466:
4465:
4445:
4443:
4442:
4437:
4425:
4423:
4422:
4417:
4405:
4403:
4402:
4397:
4379:
4377:
4376:
4371:
4369:
4368:
4363:
4362:
4345:
4343:
4342:
4337:
4335:
4334:
4329:
4317:
4316:
4311:
4310:
4290:
4288:
4287:
4282:
4270:
4268:
4267:
4262:
4250:
4248:
4247:
4242:
4240:
4239:
4234:
4233:
4219:
4217:
4216:
4211:
4199:
4197:
4196:
4191:
4189:
4172:
4167:
4155:
4153:
4152:
4147:
4136:
4135:
4130:
4129:
4109:
4107:
4106:
4101:
4094:
4093:
4090:
4086:
4076:
4075:
4070:
4069:
4046:
4044:
4043:
4038:
4022:
4020:
4019:
4014:
4012:
4011:
4006:
3988:
3983:
3972:
3971:
3966:
3965:
3945:
3943:
3942:
3937:
3935:
3934:
3929:
3928:
3914:
3912:
3911:
3906:
3894:
3892:
3891:
3886:
3884:
3863:
3861:
3860:
3855:
3849:
3841:
3822:
3814:
3802:
3785:
3784:
3779:
3778:
3764:
3762:
3761:
3756:
3751:
3750:
3749:
3748:
3738:
3737:
3721:
3720:
3719:
3718:
3708:
3707:
3678:
3676:
3675:
3670:
3654:
3650:
3637:
3636:
3633:
3629:
3628:
3625:
3624:
3612:
3600:
3599:
3584:
3573:
3572:
3560:
3559:
3554:
3553:
3539:
3537:
3536:
3531:
3526:
3525:
3520:
3519:
3500:
3499:
3483:
3481:
3480:
3475:
3473:
3441:
3439:
3438:
3433:
3419:
3418:
3413:
3412:
3398:
3396:
3395:
3390:
3388:
3370:
3368:
3367:
3362:
3348:
3347:
3342:
3341:
3327:
3325:
3324:
3319:
3314:
3287:
3285:
3284:
3279:
3277:
3257:
3255:
3254:
3249:
3229:
3228:
3216:
3215:
3203:
3161:
3159:
3158:
3153:
3150:
3145:
3140:
3123:
3121:
3120:
3115:
3098:
3097:
3085:
3084:
3068:
3066:
3065:
3060:
3048:
3046:
3045:
3040:
3026:
2987:
2985:
2984:
2979:
2952:
2950:
2949:
2944:
2941:
2936:
2931:
2913:
2912:
2904:
2890:
2888:
2887:
2882:
2877:
2876:
2858:
2857:
2826:
2824:
2823:
2818:
2797:
2795:
2794:
2789:
2784:
2783:
2778:
2777:
2757:
2755:
2754:
2749:
2747:
2746:
2734:
2733:
2717:
2715:
2714:
2709:
2701:
2700:
2695:
2694:
2687:
2673:
2672:
2667:
2666:
2643:
2641:
2640:
2635:
2633:
2632:
2627:
2626:
2612:
2610:
2609:
2604:
2602:
2601:
2595:
2574:
2572:
2571:
2566:
2555:as the image of
2554:
2552:
2551:
2546:
2541:
2540:
2517:
2515:
2514:
2509:
2507:
2506:
2500:
2485:
2483:
2482:
2477:
2475:
2474:
2461:
2459:
2458:
2453:
2441:
2439:
2438:
2433:
2431:
2430:
2422:
2412:
2410:
2409:
2404:
2393:
2392:
2374:
2373:
2342:
2340:
2339:
2334:
2322:
2320:
2319:
2314:
2312:
2311:
2306:
2305:
2287:
2285:
2284:
2279:
2274:
2273:
2265:
2251:
2236:
2233:
2232:
2229:
2225:
2222:
2221:
2218:
2217:
2205:
2193:
2192:
2168:
2167:
2155:
2154:
2149:
2148:
2131:
2129:
2128:
2123:
2102:
2100:
2099:
2094:
2089:
2088:
2074:
2071:
2070:
2067:
2063:
2060:
2053:
2052:
2019:
2018:
1999:
1997:
1996:
1991:
1976:
1974:
1973:
1968:
1966:
1965:
1960:
1959:
1949:
1948:
1935:
1933:
1932:
1927:
1925:
1924:
1916:
1900:
1898:
1897:
1892:
1890:
1889:
1884:
1883:
1869:
1867:
1866:
1861:
1856:
1855:
1846:
1845:
1837:
1834:
1833:
1830:
1826:
1823:
1816:
1815:
1800:
1799:
1780:
1778:
1777:
1772:
1770:
1769:
1756:
1754:
1753:
1748:
1746:
1745:
1737:
1728:
1727:
1718:
1717:
1709:
1703:
1702:
1694:
1684:
1682:
1681:
1676:
1668:
1667:
1654:
1652:
1651:
1646:
1634:
1632:
1631:
1626:
1621:
1620:
1608:
1607:
1589:
1588:
1576:
1575:
1560:
1559:
1554:
1553:
1539:
1537:
1536:
1531:
1520:
1493:
1491:
1490:
1485:
1480:
1479:
1461:
1460:
1432:
1430:
1429:
1424:
1412:
1410:
1409:
1404:
1399:
1398:
1380:
1379:
1354:
1352:
1351:
1346:
1343:
1338:
1333:
1324:
1323:
1315:
1305:
1303:
1302:
1297:
1254:
1252:
1251:
1246:
1244:
1243:
1232:
1220:is affine space
1215:
1213:
1212:
1207:
1205:
1204:
1199:
1186:
1184:
1183:
1178:
1176:
1175:
1170:
1126:
1124:
1123:
1118:
1113:
1112:
1032:.) For a scheme
895:of the morphism
710:.) For a scheme
677:
676:
476:as the space of
471:commutative ring
432:Weil conjectures
424:Masayoshi Nagata
420:Claude Chevalley
413:universal domain
386:Jacobian variety
378:projective space
374:abstract variety
363:Zariski topology
359:quasi-projective
270:complex analysis
246:Bernhard Riemann
159:Noetherian rings
96:Weil conjectures
64:commutative ring
21:
11388:
11387:
11383:
11382:
11381:
11379:
11378:
11377:
11363:
11362:
11336:David Mumford,
11333:
11328:
11276:
11238:
11207:
11180:
11126:Dieudonné, Jean
11106:
11096:Springer-Verlag
11082:Eisenbud, David
11067:
11049:Dieudonné, Jean
11014:Cartier, Pierre
10966:
10961:
10960:
10952:
10948:
10937:
10936:
10932:
10926:Hartshorne 1997
10924:
10920:
10912:
10908:
10900:
10896:
10888:
10884:
10878:Hartshorne 1997
10876:
10872:
10866:Hartshorne 1997
10864:
10860:
10852:
10848:
10847:
10843:
10835:
10831:
10825:Hartshorne 1997
10823:
10819:
10808:
10807:
10803:
10795:
10791:
10787:, section II.2.
10785:Hartshorne 1997
10783:
10779:
10771:
10767:
10759:
10755:
10742:
10738:
10730:
10723:
10715:
10711:
10703:
10699:
10690:
10686:
10681:
10663:Reductive group
10655:Abelian variety
10619:Finite morphism
10615:Proper morphism
10611:Smooth morphism
10603:
10575:homotopy theory
10519:algebraic group
10496:algebraic space
10482:
10480:Generalizations
10416:
10395:
10362:
10359:
10358:
10357:
10355:
10335:
10315:
10297:. For a scheme
10287:
10281:
10270:
10211:
10208:
10207:
10176:
10173:
10172:
10138:
10134:
10130:
10116:
10114:
10092:
10088:
10084:
10064:
10062:
10060:
10057:
10056:
10003:
10000:
9999:
9953:
9950:
9949:
9918:
9914:
9893:
9889:
9872:
9869:
9868:
9842:
9838:
9830:
9827:
9826:
9809:
9805:
9797:
9794:
9793:
9768:
9765:
9764:
9759:, consider the
9729:
9726:
9725:
9709:
9706:
9705:
9688:
9683:
9678:
9675:
9672:
9671:
9603:
9598:
9597:
9596:
9569:covering spaces
9561:function fields
9505:
9501:
9487:
9470:
9468:
9465:
9464:
9432:
9382:
9378:
9364:
9356:
9353:
9352:
9336:
9331:
9330:
9329:
9321:
9316:
9315:
9314:
9275:
9270:
9269:
9268:
9256:
9251:
9250:
9249:
9231:
9226:
9225:
9224:
9099:. If the field
9070:
9057:
9051:
9048:
9041:needs expansion
9026:
8974:
8968:
8965:
8964:
8931:
8920:
8915:
8911:
8903:
8900:
8899:
8851:
8848:
8847:
8827:
8823:
8808:
8804:
8796:
8784:
8779:
8778:
8770:
8767:
8766:
8749:
8744:
8743:
8729:
8726:
8725:
8722:Hartogs's lemma
8700:
8695:
8694:
8692:
8689:
8688:
8657:
8653:
8639:
8624:
8622:
8619:
8618:
8594:
8592:
8589:
8588:
8559:
8554:
8553:
8545:
8542:
8541:
8498:
8492:
8491:
8486:
8483:
8480:
8479:
8465:
8437:
8428:
8424:
8409:
8405:
8391:
8388:
8387:
8362:
8357:
8352:
8349:
8346:
8345:
8338:
8332:
8305:
8300:
8299:
8297:
8294:
8293:
8276:
8271:
8270:
8265:
8263:
8260:
8259:
8231:
8226:
8221:
8218:
8215:
8214:
8185:
8180:
8175:
8172:
8169:
8168:
8153:
8126:
8123:
8122:
8105:
8101:
8093:
8090:
8089:
8060:
8056:
8047:
8043:
8039:
8018:
8017:
8015:
8001:
7998:
7997:
7981:
7978:
7977:
7952:
7937:
7933:
7921:
7917:
7890:
7886:
7880:
7876:
7864:
7860:
7845:
7841:
7839:
7836:
7835:
7799:
7795:
7783:
7779:
7770:
7766:
7743:
7740:
7739:
7722:
7717:
7716:
7714:
7711:
7710:
7690:
7685:
7684:
7665:
7655:
7651:
7639:
7635:
7633:
7630:
7629:
7609:
7601:
7598:
7597:
7569:
7549:
7532:
7529:
7528:
7497:
7489:
7486:
7485:
7482:
7470:Arakelov theory
7466:geometric tools
7437:
7434:
7433:
7413:
7403:
7401:
7400:
7398:
7395:
7394:
7369:
7366:
7365:
7349:
7346:
7345:
7311:
7308:
7307:
7276:
7273:
7272:
7247:
7244:
7243:
7223:
7222:
7214:
7211:
7210:
7185:
7168:
7165:
7164:
7138:
7133:
7132:
7105:
7104:
7096:
7093:
7092:
7075:
7074:
7060:
7042:
7041:
7033:
7030:
7029:
7013:
7010:
7009:
6984:
6967:
6964:
6963:
6962:. A polynomial
6929:
6926:
6925:
6908:
6904:
6902:
6899:
6898:
6873:
6870:
6869:
6847:
6844:
6843:
6826:
6821:
6820:
6818:
6815:
6814:
6788:
6783:
6782:
6756:
6741:
6736:
6735:
6726:
6725:
6720:
6706:
6694:
6693:
6685:
6682:
6681:
6637:
6636:
6634:
6631:
6630:
6601:
6597:
6589:
6586:
6585:
6559:
6555:
6541:
6538:
6537:
6516:
6511:
6510:
6508:
6505:
6504:
6483:
6472:
6469:
6468:
6442:
6438:
6420:
6419:
6417:
6414:
6413:
6391:
6388:
6387:
6367:
6363:
6324:
6320:
6309:
6306:
6305:
6283:
6269:
6266:
6265:
6239:
6236:
6235:
6161:
6157:
6146:
6143:
6142:
6120:
6103:
6100:
6099:
6077:
6074:
6073:
6057:
6054:
6053:
6032:
6021:
6018:
6017:
5994:
5990:
5988:
5985:
5984:
5968:
5965:
5964:
5963:corresponds to
5938:
5934:
5926:
5923:
5922:
5903:
5900:
5899:
5883:
5880:
5879:
5854:
5851:
5850:
5831:
5820:
5817:
5816:
5782:
5779:
5778:
5743:
5742:
5740:
5737:
5736:
5705:
5702:
5701:
5679:
5676:
5675:
5659:
5656:
5655:
5639:
5636:
5635:
5610:
5607:
5606:
5562:
5561:
5559:
5556:
5555:
5526:
5525:
5523:
5520:
5519:
5494:
5491:
5490:
5474:
5471:
5470:
5444:
5441:
5440:
5424:
5421:
5420:
5404:
5401:
5400:
5384:
5381:
5380:
5355:
5352:
5351:
5335:
5332:
5331:
5287:
5286:
5284:
5281:
5280:
5255:
5246:
5245:
5243:
5240:
5239:
5211:
5194:
5191:
5190:
5165:
5163:
5160:
5159:
5140:
5126:
5108:
5102:
5101:
5096:
5093:
5090:
5089:
5086:
5061:
5057:
5036:
5032:
5030:
5027:
5026:
5010:
5007:
5006:
4985:
4981:
4975:
4971:
4962:
4958:
4956:
4953:
4952:
4936:
4934:
4931:
4930:
4898:
4894:
4879:
4875:
4870:
4867:
4866:
4850:
4847:
4846:
4829:
4823:
4822:
4821:
4819:
4816:
4815:
4798:
4794:
4779:
4775:
4773:
4770:
4769:
4753:
4750:
4749:
4748:for the scheme
4722:
4718:
4712:
4708:
4693:
4689:
4683:
4679:
4677:
4674:
4673:
4656:
4652:
4637:
4633:
4631:
4628:
4627:
4610:
4606:
4591:
4587:
4585:
4582:
4581:
4557:
4551:
4550:
4549:
4547:
4544:
4543:
4524:
4519:
4516:
4515:
4499:
4488:
4467:
4461:
4460:
4459:
4451:
4448:
4447:
4431:
4428:
4427:
4411:
4408:
4407:
4385:
4382:
4381:
4364:
4358:
4357:
4356:
4354:
4351:
4350:
4330:
4325:
4324:
4312:
4306:
4305:
4304:
4296:
4293:
4292:
4276:
4273:
4272:
4256:
4253:
4252:
4235:
4229:
4228:
4227:
4225:
4222:
4221:
4205:
4202:
4201:
4200:. The function
4185:
4168:
4163:
4161:
4158:
4157:
4131:
4125:
4124:
4123:
4115:
4112:
4111:
4089:
4071:
4065:
4064:
4063:
4055:
4052:
4051:
4032:
4029:
4028:
4007:
4002:
4001:
3984:
3979:
3967:
3961:
3960:
3959:
3951:
3948:
3947:
3930:
3924:
3923:
3922:
3920:
3917:
3916:
3900:
3897:
3896:
3880:
3872:
3869:
3868:
3842:
3837:
3815:
3810:
3798:
3780:
3774:
3773:
3772:
3770:
3767:
3766:
3765:, this induces
3744:
3740:
3739:
3733:
3732:
3731:
3714:
3710:
3709:
3703:
3702:
3701:
3684:
3681:
3680:
3646:
3632:
3620:
3616:
3610:
3592:
3588:
3580:
3568:
3564:
3555:
3549:
3548:
3547:
3545:
3542:
3541:
3521:
3515:
3514:
3513:
3495:
3491:
3489:
3486:
3485:
3469:
3461:
3458:
3457:
3414:
3408:
3407:
3406:
3404:
3401:
3400:
3384:
3376:
3373:
3372:
3343:
3337:
3336:
3335:
3333:
3330:
3329:
3310:
3293:
3290:
3289:
3273:
3271:
3268:
3267:
3264:
3224:
3220:
3211:
3207:
3199:
3167:
3164:
3163:
3146:
3141:
3136:
3133:
3130:
3129:
3093:
3089:
3080:
3076:
3074:
3071:
3070:
3054:
3051:
3050:
3022:
2993:
2990:
2989:
2961:
2958:
2957:
2955:principal ideal
2937:
2932:
2927:
2903:
2902:
2900:
2897:
2896:
2872:
2868:
2853:
2849:
2835:
2832:
2831:
2803:
2800:
2799:
2779:
2773:
2772:
2771:
2763:
2760:
2759:
2742:
2738:
2729:
2725:
2723:
2720:
2719:
2696:
2690:
2689:
2688:
2683:
2668:
2662:
2661:
2660:
2652:
2649:
2648:
2628:
2622:
2621:
2620:
2618:
2615:
2614:
2597:
2596:
2591:
2580:
2577:
2576:
2560:
2557:
2556:
2536:
2535:
2527:
2524:
2523:
2502:
2501:
2496:
2491:
2488:
2487:
2470:
2469:
2467:
2464:
2463:
2462:whose value at
2447:
2444:
2443:
2421:
2420:
2418:
2415:
2414:
2388:
2384:
2369:
2365:
2351:
2348:
2347:
2328:
2325:
2324:
2307:
2301:
2300:
2299:
2297:
2294:
2293:
2266:
2261:
2260:
2228:
2213:
2209:
2203:
2185:
2181:
2163:
2159:
2150:
2144:
2143:
2142:
2140:
2137:
2136:
2111:
2108:
2107:
2084:
2083:
2066:
2048:
2047:
2014:
2010:
2008:
2005:
2004:
1985:
1982:
1981:
1961:
1955:
1954:
1953:
1944:
1943:
1941:
1938:
1937:
1915:
1914:
1906:
1903:
1902:
1885:
1879:
1878:
1877:
1875:
1872:
1871:
1851:
1850:
1841:
1840:
1829:
1811:
1810:
1795:
1794:
1786:
1783:
1782:
1765:
1764:
1762:
1759:
1758:
1736:
1735:
1723:
1722:
1708:
1707:
1693:
1692:
1690:
1687:
1686:
1663:
1662:
1660:
1657:
1656:
1640:
1637:
1636:
1616:
1612:
1603:
1599:
1584:
1580:
1571:
1567:
1555:
1549:
1548:
1547:
1545:
1542:
1541:
1507:
1499:
1496:
1495:
1475:
1471:
1456:
1452:
1438:
1435:
1434:
1418:
1415:
1414:
1394:
1390:
1375:
1371:
1360:
1357:
1356:
1339:
1334:
1329:
1314:
1313:
1311:
1308:
1307:
1291:
1288:
1287:
1284:
1276:
1233:
1228:
1227:
1225:
1222:
1221:
1200:
1195:
1194:
1192:
1189:
1188:
1171:
1166:
1165:
1163:
1160:
1159:
1108:
1104:
1099:
1096:
1095:
1023:field extension
949:rational points
923:with values in
866:terminal object
702:Schemes form a
700:
690:, meaning Spec(
675:
670:
669:
668:
636:
600:
587:
574:
565:
527:
500:
452:
444:André Martineau
367:metric topology
288:polynomial ring
254:complex numbers
230:
127:coordinate ring
54:(the equations
34:, specifically
28:
23:
22:
15:
12:
11:
5:
11386:
11376:
11375:
11361:
11360:
11350:
11340:
11332:
11331:External links
11329:
11327:
11326:
11288:
11274:
11266:10.1007/b62130
11254:Mumford, David
11250:
11236:
11219:
11206:978-3642380099
11205:
11192:
11178:
11160:
11118:
11104:
11078:
11065:
11045:
11026:(4): 389–408,
11010:
10967:
10965:
10962:
10959:
10958:
10946:
10930:
10928:, Chapter III.
10918:
10914:Dieudonné 1985
10906:
10894:
10882:
10870:
10858:
10841:
10829:
10817:
10801:
10789:
10777:
10773:Dieudonné 1985
10765:
10753:
10736:
10732:Dieudonné 1985
10721:
10717:Dieudonné 1985
10709:
10705:Dieudonné 1985
10697:
10683:
10682:
10680:
10677:
10676:
10675:
10673:Gluing schemes
10670:
10665:
10648:
10635:
10630:
10625:
10623:Étale morphism
10602:
10599:
10591:tensor product
10536:quotient stack
10487:étale topology
10481:
10478:
10438:tangent bundle
10430:vector bundles
10412:
10402:coherent sheaf
10391:
10351:
10344:determines an
10331:
10311:
10295:vector bundles
10285:Coherent sheaf
10283:Main article:
10280:
10277:
10276:
10275:
10266:
10227:
10226:-vector space.
10215:
10195:
10192:
10189:
10186:
10183:
10180:
10152:
10146:
10141:
10137:
10133:
10128:
10125:
10122:
10119:
10113:
10107:
10104:
10100:
10095:
10091:
10087:
10082:
10079:
10076:
10073:
10070:
10067:
10040:
10037:
10034:
10031:
10028:
10025:
10022:
10019:
10016:
10013:
10010:
10007:
9987:
9984:
9981:
9978:
9975:
9972:
9969:
9966:
9963:
9960:
9957:
9933:
9930:
9926:
9921:
9917:
9913:
9910:
9907:
9904:
9901:
9896:
9892:
9888:
9885:
9882:
9879:
9876:
9856:
9853:
9850:
9845:
9841:
9837:
9834:
9812:
9808:
9804:
9801:
9781:
9778:
9775:
9772:
9748:
9745:
9742:
9739:
9736:
9733:
9713:
9691:
9686:
9681:
9664:
9661:infinitesimals
9599:
9585:
9584:
9548:
9544:
9539:
9536:
9533:
9530:
9527:
9524:
9521:
9518:
9515:
9512:
9509:
9504:
9500:
9497:
9494:
9490:
9486:
9483:
9480:
9477:
9473:
9448:
9445:
9442:
9439:
9435:
9431:
9428:
9425:
9421:
9416:
9413:
9410:
9407:
9404:
9401:
9398:
9395:
9392:
9389:
9386:
9381:
9377:
9374:
9371:
9367:
9363:
9360:
9332:
9317:
9291:
9271:
9252:
9246:Generic point.
9243:
9227:
9069:
9066:
9059:
9058:
9038:
9036:
9025:
9022:
9021:
9020:
8988:
8983:
8980:
8977:
8973:
8943:
8939:
8934:
8929:
8926:
8923:
8919:
8914:
8910:
8907:
8892:
8891:is not affine.
8864:
8861:
8858:
8855:
8835:
8830:
8826:
8822:
8819:
8816:
8811:
8807:
8803:
8799:
8795:
8792:
8787:
8782:
8777:
8774:
8752:
8747:
8742:
8739:
8736:
8733:
8703:
8698:
8668:
8663:
8660:
8656:
8652:
8649:
8646:
8642:
8636:
8633:
8630:
8627:
8597:
8576:
8573:
8570:
8567:
8562:
8557:
8552:
8549:
8538:
8529:(over a field
8523:
8516:elliptic curve
8501:
8495:
8489:
8453:
8450:
8447:
8444:
8440:
8436:
8431:
8427:
8423:
8420:
8417:
8412:
8408:
8404:
8401:
8398:
8395:
8365:
8360:
8355:
8322:
8308:
8303:
8279:
8274:
8268:
8234:
8229:
8224:
8188:
8183:
8178:
8152:
8149:
8136:
8133:
8130:
8108:
8104:
8100:
8097:
8074:
8071:
8068:
8063:
8059:
8055:
8050:
8046:
8042:
8037:
8034:
8031:
8028:
8025:
8021:
8014:
8011:
8008:
8005:
7985:
7965:
7962:
7948:
7945:
7940:
7936:
7932:
7929:
7924:
7920:
7916:
7913:
7910:
7907:
7904:
7901:
7898:
7893:
7889:
7883:
7879:
7875:
7872:
7867:
7863:
7859:
7856:
7853:
7848:
7844:
7832:elliptic curve
7819:
7816:
7813:
7810:
7807:
7802:
7798:
7794:
7791:
7786:
7782:
7778:
7773:
7769:
7765:
7762:
7759:
7756:
7753:
7750:
7747:
7725:
7720:
7698:
7693:
7688:
7683:
7680:
7677:
7672:
7668:
7664:
7661:
7658:
7654:
7650:
7647:
7642:
7638:
7612:
7608:
7605:
7585:
7582:
7579:
7576:
7572:
7568:
7565:
7562:
7559:
7556:
7552:
7548:
7545:
7542:
7539:
7536:
7516:
7513:
7510:
7507:
7504:
7500:
7496:
7493:
7481:
7478:
7441:
7416:
7410:
7406:
7382:
7379:
7376:
7373:
7353:
7333:
7330:
7327:
7324:
7321:
7318:
7315:
7295:
7292:
7289:
7286:
7283:
7280:
7260:
7257:
7254:
7251:
7231:
7226:
7221:
7218:
7198:
7195:
7192:
7188:
7184:
7181:
7178:
7175:
7172:
7152:
7149:
7146:
7141:
7136:
7131:
7128:
7125:
7122:
7119:
7116:
7113:
7108:
7103:
7100:
7078:
7069:
7066:
7063:
7056:
7053:
7050:
7045:
7040:
7037:
7017:
6997:
6994:
6991:
6987:
6983:
6980:
6977:
6974:
6971:
6951:
6948:
6945:
6942:
6939:
6936:
6933:
6911:
6907:
6886:
6883:
6880:
6877:
6857:
6854:
6851:
6829:
6824:
6802:
6799:
6796:
6791:
6786:
6781:
6778:
6775:
6772:
6769:
6766:
6763:
6759:
6755:
6752:
6749:
6744:
6739:
6734:
6729:
6723:
6719:
6716:
6713:
6709:
6705:
6702:
6697:
6692:
6689:
6669:
6666:
6663:
6660:
6657:
6654:
6651:
6648:
6645:
6640:
6615:
6612:
6609:
6604:
6600:
6596:
6593:
6573:
6570:
6567:
6562:
6558:
6554:
6551:
6548:
6545:
6519:
6514:
6490:
6487:
6482:
6479:
6476:
6456:
6453:
6450:
6445:
6441:
6437:
6434:
6431:
6428:
6423:
6412:, we get that
6401:
6398:
6395:
6375:
6370:
6366:
6362:
6359:
6356:
6353:
6350:
6347:
6344:
6341:
6338:
6335:
6332:
6327:
6323:
6319:
6316:
6313:
6293:
6279:
6276:
6273:
6249:
6246:
6243:
6223:
6220:
6217:
6214:
6211:
6208:
6205:
6202:
6199:
6196:
6193:
6190:
6187:
6184:
6181:
6178:
6175:
6172:
6169:
6164:
6160:
6156:
6153:
6150:
6130:
6116:
6113:
6110:
6107:
6087:
6084:
6081:
6061:
6039:
6036:
6031:
6028:
6025:
6005:
6002:
5997:
5993:
5972:
5952:
5949:
5946:
5941:
5937:
5933:
5930:
5907:
5887:
5867:
5864:
5861:
5858:
5838:
5834:
5830:
5827:
5824:
5804:
5801:
5798:
5795:
5792:
5789:
5786:
5766:
5763:
5760:
5757:
5754:
5751:
5746:
5724:
5721:
5718:
5715:
5712:
5709:
5689:
5686:
5683:
5663:
5643:
5623:
5620:
5617:
5614:
5594:
5591:
5588:
5585:
5582:
5579:
5576:
5573:
5570:
5565:
5543:
5540:
5537:
5534:
5529:
5507:
5504:
5501:
5498:
5478:
5448:
5428:
5408:
5388:
5368:
5365:
5362:
5359:
5339:
5319:
5316:
5313:
5310:
5307:
5304:
5301:
5298:
5295:
5290:
5268:
5265:
5262:
5258:
5254:
5249:
5227:
5224:
5221:
5218:
5214:
5210:
5207:
5204:
5201:
5198:
5178:
5175:
5172:
5168:
5147:
5143:
5139:
5136:
5122:
5119:
5116:
5111:
5105:
5099:
5085:
5082:
5069:
5064:
5060:
5056:
5053:
5050:
5047:
5044:
5039:
5035:
5014:
4988:
4984:
4978:
4974:
4970:
4965:
4961:
4939:
4918:
4915:
4912:
4909:
4906:
4901:
4897:
4893:
4890:
4887:
4882:
4878:
4874:
4854:
4832:
4826:
4801:
4797:
4793:
4790:
4787:
4782:
4778:
4757:
4733:
4730:
4725:
4721:
4715:
4711:
4707:
4704:
4701:
4696:
4692:
4686:
4682:
4659:
4655:
4651:
4648:
4645:
4640:
4636:
4613:
4609:
4605:
4602:
4599:
4594:
4590:
4578:Bezout's lemma
4560:
4554:
4531:
4527:
4523:
4502:
4498:
4495:
4491:
4487:
4484:
4481:
4478:
4475:
4470:
4464:
4458:
4455:
4435:
4415:
4395:
4392:
4389:
4367:
4361:
4333:
4328:
4323:
4320:
4315:
4309:
4303:
4300:
4280:
4260:
4238:
4232:
4209:
4188:
4184:
4181:
4178:
4175:
4171:
4166:
4145:
4142:
4139:
4134:
4128:
4122:
4119:
4099:
4085:
4082:
4079:
4074:
4068:
4062:
4059:
4036:
4010:
4005:
4000:
3997:
3994:
3991:
3987:
3982:
3978:
3975:
3970:
3964:
3958:
3955:
3933:
3927:
3904:
3883:
3879:
3876:
3853:
3848:
3845:
3840:
3836:
3832:
3829:
3826:
3821:
3818:
3813:
3809:
3805:
3801:
3797:
3794:
3791:
3788:
3783:
3777:
3754:
3747:
3743:
3736:
3730:
3727:
3724:
3717:
3713:
3706:
3700:
3697:
3694:
3691:
3688:
3668:
3665:
3662:
3659:
3653:
3649:
3645:
3642:
3623:
3619:
3615:
3609:
3606:
3603:
3598:
3595:
3591:
3587:
3583:
3579:
3576:
3571:
3567:
3563:
3558:
3552:
3529:
3524:
3518:
3512:
3509:
3506:
3503:
3498:
3494:
3472:
3468:
3465:
3431:
3428:
3425:
3422:
3417:
3411:
3387:
3383:
3380:
3360:
3357:
3354:
3351:
3346:
3340:
3317:
3313:
3309:
3306:
3303:
3300:
3297:
3276:
3263:
3260:
3247:
3244:
3241:
3238:
3235:
3232:
3227:
3223:
3219:
3214:
3210:
3206:
3202:
3198:
3195:
3192:
3189:
3186:
3183:
3180:
3177:
3174:
3171:
3149:
3144:
3139:
3113:
3110:
3107:
3104:
3101:
3096:
3092:
3088:
3083:
3079:
3058:
3038:
3035:
3032:
3029:
3025:
3021:
3018:
3015:
3012:
3009:
3006:
3003:
3000:
2997:
2977:
2974:
2971:
2968:
2965:
2940:
2935:
2930:
2925:
2922:
2919:
2916:
2910:
2907:
2880:
2875:
2871:
2867:
2864:
2861:
2856:
2852:
2848:
2845:
2842:
2839:
2816:
2813:
2810:
2807:
2787:
2782:
2776:
2770:
2767:
2745:
2741:
2737:
2732:
2728:
2707:
2704:
2699:
2693:
2686:
2682:
2679:
2676:
2671:
2665:
2659:
2656:
2631:
2625:
2600:
2594:
2590:
2587:
2584:
2564:
2544:
2539:
2534:
2531:
2505:
2499:
2495:
2473:
2451:
2428:
2425:
2402:
2399:
2396:
2391:
2387:
2383:
2380:
2377:
2372:
2368:
2364:
2361:
2358:
2355:
2332:
2310:
2304:
2277:
2272:
2269:
2264:
2259:
2256:
2250:
2247:
2244:
2241:
2216:
2212:
2208:
2202:
2199:
2196:
2191:
2188:
2184:
2180:
2177:
2174:
2171:
2166:
2162:
2158:
2153:
2147:
2121:
2118:
2115:
2092:
2087:
2082:
2079:
2059:
2056:
2051:
2046:
2043:
2040:
2037:
2034:
2031:
2028:
2025:
2022:
2017:
2013:
1989:
1964:
1958:
1952:
1947:
1922:
1919:
1913:
1910:
1888:
1882:
1859:
1854:
1849:
1844:
1822:
1819:
1814:
1809:
1806:
1803:
1798:
1793:
1790:
1768:
1743:
1740:
1734:
1731:
1726:
1721:
1715:
1712:
1706:
1700:
1697:
1674:
1671:
1666:
1644:
1624:
1619:
1615:
1611:
1606:
1602:
1598:
1595:
1592:
1587:
1583:
1579:
1574:
1570:
1566:
1563:
1558:
1552:
1529:
1526:
1523:
1519:
1516:
1513:
1510:
1506:
1503:
1483:
1478:
1474:
1470:
1467:
1464:
1459:
1455:
1451:
1448:
1445:
1442:
1422:
1402:
1397:
1393:
1389:
1386:
1383:
1378:
1374:
1370:
1367:
1364:
1342:
1337:
1332:
1327:
1321:
1318:
1295:
1283:
1280:
1275:
1272:
1242:
1239:
1236:
1231:
1203:
1198:
1174:
1169:
1116:
1111:
1107:
1103:
903:). One writes
854:initial object
699:
696:
671:
662:natural number
632:
596:
583:
570:
561:
526:
523:
496:
451:
448:
440:Pierre Cartier
418:In the 1950s,
401:Italian school
319:Wolfgang Krull
284:maximal ideals
262:characteristic
229:
226:
174:maximal ideals
139:generic points
100:Pierre Deligne
94:, such as the
66:(for example,
52:multiplicities
26:
9:
6:
4:
3:
2:
11385:
11374:
11373:Scheme theory
11371:
11370:
11368:
11358:
11354:
11351:
11348:
11347:
11341:
11339:
11335:
11334:
11324:
11320:
11316:
11312:
11307:
11302:
11298:
11294:
11289:
11285:
11281:
11277:
11271:
11267:
11263:
11259:
11255:
11251:
11247:
11243:
11239:
11233:
11229:
11225:
11220:
11216:
11212:
11208:
11202:
11198:
11193:
11189:
11185:
11181:
11175:
11171:
11170:
11165:
11161:
11157:
11153:
11149:
11145:
11141:
11137:
11136:
11131:
11127:
11123:
11119:
11115:
11111:
11107:
11101:
11097:
11093:
11092:
11087:
11083:
11079:
11076:
11072:
11068:
11062:
11059:, Wadsworth,
11057:
11056:
11050:
11046:
11043:
11039:
11034:
11029:
11025:
11021:
11020:
11015:
11011:
11008:
11004:
10999:
10994:
10989:
10984:
10980:
10976:
10975:
10969:
10968:
10955:
10950:
10942:
10941:
10934:
10927:
10922:
10915:
10910:
10903:
10898:
10891:
10886:
10879:
10874:
10867:
10862:
10855:. p. 20.
10851:
10845:
10838:
10833:
10826:
10821:
10813:
10812:
10805:
10799:, Chapter II.
10798:
10793:
10786:
10781:
10774:
10769:
10762:
10757:
10749:
10748:
10740:
10733:
10728:
10726:
10718:
10713:
10706:
10701:
10694:
10688:
10684:
10674:
10671:
10669:
10666:
10664:
10660:
10656:
10652:
10649:
10647:
10643:
10639:
10636:
10634:
10631:
10629:
10626:
10624:
10620:
10616:
10612:
10608:
10607:Flat morphism
10605:
10604:
10598:
10596:
10592:
10588:
10584:
10580:
10576:
10571:
10569:
10565:
10561:
10557:
10553:
10548:
10546:
10542:
10538:
10537:
10533:determines a
10532:
10528:
10524:
10520:
10516:
10515:
10510:
10505:
10503:
10498:
10497:
10492:
10491:Michael Artin
10488:
10477:
10475:
10471:
10467:
10463:
10459:
10455:
10451:
10447:
10443:
10439:
10435:
10431:
10426:
10424:
10420:
10415:
10411:
10408:, say) is an
10407:
10403:
10400:. Finally, a
10399:
10394:
10390:
10386:
10382:
10381:
10376:
10372:
10365:
10354:
10350:
10347:
10343:
10339:
10334:
10330:
10326:
10322:
10318:
10317:
10314:
10310:
10304:
10300:
10296:
10292:
10286:
10273:
10269:
10264:
10263:tangent space
10260:
10256:
10252:
10248:
10244:
10240:
10236:
10232:
10228:
10213:
10190:
10187:
10184:
10178:
10170:
10166:
10150:
10139:
10135:
10123:
10117:
10111:
10102:
10098:
10093:
10089:
10077:
10074:
10071:
10065:
10054:
10035:
10032:
10029:
10023:
10020:
10014:
10011:
10008:
9982:
9979:
9976:
9970:
9964:
9961:
9958:
9947:
9928:
9924:
9919:
9915:
9908:
9902:
9899:
9894:
9890:
9883:
9877:
9851:
9848:
9843:
9839:
9832:
9810:
9806:
9802:
9799:
9776:
9770:
9762:
9743:
9740:
9737:
9731:
9711:
9704:over a field
9689:
9684:
9669:
9665:
9662:
9658:
9654:
9653:
9648:
9647:
9641:
9637:
9633:
9629:
9625:
9621:
9617:
9613:
9609:
9602:
9594:
9590:
9587:
9586:
9582:
9578:
9574:
9570:
9566:
9562:
9546:
9542:
9534:
9531:
9528:
9519:
9516:
9513:
9507:
9502:
9495:
9484:
9478:
9462:
9446:
9440:
9429:
9426:
9419:
9411:
9408:
9405:
9396:
9393:
9390:
9384:
9379:
9372:
9361:
9358:
9350:
9346:
9342:
9335:
9327:
9320:
9312:
9308:
9304:
9300:
9296:
9292:
9289:
9285:
9281:
9274:
9266:
9262:
9255:
9247:
9244:
9241:
9237:
9230:
9222:
9218:
9214:
9210:
9206:
9202:
9199:over a field
9198:
9194:
9190:
9186:
9182:
9178:
9175:) empty, but
9174:
9170:
9166:
9162:
9158:
9154:
9150:
9146:
9142:
9138:
9134:
9130:
9126:
9122:
9118:
9114:
9110:
9106:
9102:
9098:
9094:
9090:
9086:
9082:
9078:
9075:
9074:
9073:
9065:
9055:
9046:
9042:
9039:This section
9037:
9034:
9030:
9029:
9018:
9014:
9013:quasi-compact
9010:
9006:
9002:
8986:
8981:
8978:
8975:
8971:
8962:
8958:
8941:
8937:
8927:
8924:
8921:
8917:
8912:
8908:
8905:
8897:
8893:
8890:
8886:
8882:
8878:
8859:
8853:
8828:
8824:
8820:
8817:
8814:
8809:
8805:
8793:
8785:
8772:
8750:
8737:
8734:
8731:
8723:
8719:
8701:
8686:
8682:
8661:
8658:
8654:
8650:
8647:
8616:
8612:
8571:
8565:
8560:
8550:
8547:
8539:
8536:
8532:
8528:
8524:
8521:
8517:
8499:
8476:
8472:
8468:
8451:
8445:
8438:
8429:
8425:
8421:
8418:
8415:
8410:
8406:
8399:
8396:
8393:
8385:
8381:
8363:
8358:
8341:
8335:
8330:
8327:
8323:
8306:
8277:
8258:
8254:
8250:
8232:
8227:
8212:
8208:
8204:
8186:
8181:
8167:
8163:
8159:
8155:
8154:
8148:
8134:
8131:
8128:
8106:
8102:
8098:
8095:
8069:
8066:
8061:
8057:
8053:
8048:
8044:
8032:
8029:
8026:
8012:
8009:
8006:
8003:
7983:
7963:
7960:
7946:
7943:
7938:
7934:
7930:
7927:
7922:
7918:
7914:
7911:
7908:
7905:
7902:
7899:
7896:
7891:
7887:
7881:
7877:
7873:
7870:
7865:
7861:
7857:
7854:
7851:
7846:
7833:
7817:
7814:
7811:
7808:
7805:
7800:
7796:
7792:
7789:
7784:
7780:
7776:
7771:
7767:
7763:
7757:
7754:
7751:
7745:
7723:
7691:
7678:
7675:
7659:
7656:
7652:
7648:
7645:
7640:
7636:
7628:. The fibers
7627:
7606:
7603:
7577:
7570:
7563:
7560:
7557:
7543:
7540:
7537:
7534:
7511:
7508:
7505:
7494:
7491:
7477:
7475:
7471:
7467:
7463:
7459:
7455:
7439:
7430:
7414:
7377:
7371:
7351:
7325:
7319:
7313:
7293:
7290:
7284:
7278:
7255:
7249:
7216:
7193:
7182:
7176:
7170:
7163:. Again each
7147:
7139:
7129:
7123:
7117:
7114:
7098:
7054:
7051:
7035:
7015:
6992:
6981:
6975:
6969:
6946:
6940:
6937:
6934:
6931:
6909:
6905:
6881:
6875:
6855:
6852:
6849:
6827:
6797:
6789:
6779:
6770:
6764:
6757:
6750:
6742:
6732:
6721:
6714:
6703:
6687:
6661:
6655:
6652:
6649:
6643:
6627:
6610:
6607:
6602:
6598:
6591:
6568:
6565:
6560:
6556:
6552:
6549:
6543:
6535:
6517:
6488:
6485:
6480:
6477:
6474:
6451:
6448:
6443:
6439:
6435:
6432:
6426:
6399:
6396:
6393:
6368:
6360:
6357:
6354:
6348:
6345:
6339:
6333:
6330:
6325:
6321:
6317:
6314:
6291:
6277:
6274:
6271:
6264:double-point
6263:
6260:, we get one
6247:
6244:
6241:
6218:
6215:
6212:
6209:
6206:
6200:
6194:
6191:
6188:
6185:
6182:
6176:
6170:
6167:
6162:
6158:
6154:
6151:
6128:
6114:
6111:
6108:
6105:
6085:
6082:
6079:
6059:
6037:
6034:
6029:
6026:
6023:
6003:
6000:
5995:
5991:
5970:
5947:
5944:
5939:
5935:
5928:
5919:
5905:
5898:which divide
5885:
5862:
5856:
5836:
5832:
5828:
5825:
5822:
5799:
5796:
5793:
5790:
5784:
5761:
5758:
5755:
5749:
5719:
5716:
5713:
5707:
5687:
5684:
5681:
5661:
5641:
5618:
5612:
5586:
5580:
5577:
5574:
5568:
5538:
5532:
5502:
5496:
5476:
5464:
5460:
5446:
5426:
5406:
5386:
5363:
5357:
5337:
5311:
5305:
5302:
5299:
5293:
5263:
5252:
5219:
5205:
5202:
5199:
5196:
5173:
5137:
5134:
5120:
5114:
5109:
5081:
5062:
5058:
5051:
5048:
5045:
5042:
5037:
5033:
5012:
5004:
4986:
4982:
4976:
4972:
4968:
4963:
4959:
4913:
4907:
4899:
4895:
4891:
4888:
4885:
4880:
4876:
4852:
4830:
4799:
4795:
4791:
4788:
4785:
4780:
4776:
4755:
4747:
4731:
4728:
4723:
4719:
4713:
4709:
4705:
4702:
4699:
4694:
4690:
4684:
4680:
4657:
4653:
4649:
4646:
4643:
4638:
4634:
4611:
4607:
4603:
4600:
4597:
4592:
4588:
4579:
4574:
4558:
4529:
4525:
4521:
4514:. A fraction
4496:
4482:
4479:
4476:
4468:
4453:
4433:
4413:
4393:
4390:
4387:
4365:
4347:
4331:
4321:
4313:
4298:
4278:
4258:
4236:
4207:
4182:
4176:
4169:
4143:
4140:
4132:
4117:
4097:
4083:
4080:
4072:
4057:
4049:
4034:
4026:
4008:
3998:
3992:
3985:
3976:
3968:
3953:
3931:
3902:
3877:
3874:
3865:
3846:
3843:
3838:
3834:
3830:
3827:
3824:
3819:
3816:
3811:
3807:
3795:
3789:
3781:
3745:
3741:
3728:
3725:
3722:
3715:
3711:
3695:
3692:
3689:
3686:
3663:
3660:
3657:
3651:
3643:
3640:
3621:
3617:
3613:
3604:
3596:
3593:
3589:
3577:
3569:
3565:
3556:
3522:
3507:
3504:
3501:
3496:
3492:
3466:
3463:
3451:
3447:
3445:
3426:
3420:
3415:
3381:
3378:
3355:
3349:
3344:
3304:
3301:
3298:
3295:
3259:
3239:
3236:
3233:
3225:
3221:
3217:
3212:
3208:
3200:
3193:
3190:
3187:
3181:
3178:
3175:
3172:
3169:
3147:
3142:
3127:
3108:
3105:
3102:
3094:
3090:
3086:
3081:
3077:
3056:
3030:
3023:
3019:
3013:
3010:
3007:
3001:
2995:
2975:
2972:
2966:
2956:
2938:
2933:
2923:
2917:
2905:
2894:
2873:
2869:
2865:
2862:
2859:
2854:
2850:
2843:
2840:
2837:
2828:
2811:
2805:
2780:
2765:
2743:
2739:
2730:
2726:
2705:
2702:
2697:
2684:
2680:
2677:
2669:
2654:
2647:
2646:residue field
2629:
2592:
2588:
2582:
2562:
2529:
2521:
2497:
2493:
2449:
2423:
2400:
2397:
2389:
2385:
2381:
2378:
2375:
2370:
2366:
2359:
2356:
2353:
2344:
2330:
2308:
2289:
2270:
2267:
2257:
2254:
2248:
2245:
2242:
2239:
2214:
2210:
2206:
2197:
2189:
2186:
2182:
2175:
2172:
2164:
2160:
2151:
2133:
2119:
2116:
2113:
2103:
2080:
2077:
2057:
2054:
2041:
2035:
2029:
2026:
2023:
2020:
2015:
2011:
2001:
1987:
1978:
1962:
1950:
1917:
1911:
1908:
1886:
1847:
1820:
1817:
1804:
1788:
1738:
1732:
1710:
1704:
1695:
1672:
1669:
1642:
1617:
1613:
1609:
1604:
1600:
1596:
1593:
1590:
1585:
1581:
1577:
1572:
1568:
1561:
1556:
1524:
1504:
1501:
1476:
1472:
1468:
1465:
1462:
1457:
1453:
1446:
1443:
1440:
1420:
1395:
1391:
1387:
1384:
1381:
1376:
1372:
1365:
1362:
1340:
1335:
1325:
1316:
1293:
1279:
1271:
1269:
1265:
1260:
1258:
1240:
1237:
1234:
1219:
1201:
1172:
1157:
1153:
1149:
1146:
1142:
1138:
1134:
1130:
1114:
1109:
1105:
1101:
1094:
1090:
1086:
1082:
1078:
1073:
1071:
1067:
1063:
1059:
1055:
1051:
1047:
1043:
1039:
1035:
1031:
1027:
1024:
1020:
1016:
1012:
1008:
1004:
1001:. One writes
1000:
996:
992:
988:
984:
980:
976:
973:
969:
965:
961:
956:
954:
950:
946:
942:
938:
934:
930:
926:
922:
918:
914:
910:
906:
902:
898:
894:
890:
886:
882:
878:
874:
871:For a scheme
869:
867:
863:
859:
855:
851:
846:
844:
840:
836:
832:
828:
824:
820:
816:
812:
808:
804:
800:
796:
791:
789:
785:
781:
777:
773:
769:
765:
760:
758:
754:
750:
746:
743:
739:
735:
731:
727:
723:
720:
717:
713:
709:
705:
695:
693:
689:
685:
681:
674:
666:
663:
659:
656:over a field
655:
653:
646:
644:
640:
635:
631:
627:
623:
619:
614:
612:
608:
605:) called the
604:
599:
595:
591:
586:
582:
578:
573:
569:
564:
560:
556:
552:
548:
544:
540:
536:
532:
531:affine scheme
522:
519:
514:
512:
508:
504:
499:
495:
491:
487:
483:
479:
475:
472:
468:
465:
461:
457:
447:
445:
441:
437:
433:
429:
425:
421:
416:
414:
410:
406:
405:generic point
402:
397:
395:
391:
387:
383:
379:
375:
370:
368:
364:
360:
356:
355:Oscar Zariski
352:
348:
344:
340:
336:
331:
329:
325:
320:
316:
312:
308:
304:
300:
296:
292:
289:
285:
281:
277:
273:
271:
267:
263:
259:
255:
251:
247:
243:
239:
235:
225:
223:
218:
216:
212:
208:
204:
201:
200:over the base
198:
194:
190:
186:
181:
179:
175:
171:
167:
162:
160:
156:
152:
148:
144:
140:
136:
132:
128:
123:
121:
117:
116:number theory
113:
109:
105:
101:
97:
93:
89:
88:
83:
79:
78:Scheme theory
75:
73:
69:
68:Fermat curves
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
11345:
11306:math/0412512
11292:
11257:
11223:
11196:
11168:
11139:
11133:
11090:
11054:
11023:
11017:
10978:
10972:
10954:Vistoli 2005
10949:
10939:
10933:
10921:
10909:
10897:
10892:, section 1.
10890:Arapura 2011
10885:
10873:
10861:
10844:
10832:
10820:
10810:
10804:
10797:Mumford 1999
10792:
10780:
10768:
10761:Cartier 2001
10756:
10746:
10739:
10712:
10700:
10687:
10651:Group scheme
10646:Hodge theory
10628:Stable curve
10597:on modules.
10572:
10558:(similar to
10549:
10544:
10534:
10530:
10526:
10512:
10506:
10494:
10483:
10468:. Moreover,
10465:
10461:
10458:direct image
10453:
10449:
10445:
10441:
10434:free modules
10429:
10427:
10422:
10413:
10409:
10405:
10401:
10397:
10392:
10388:
10384:
10383:on a scheme
10378:
10374:
10370:
10363:
10352:
10348:
10341:
10337:
10332:
10328:
10323:that form a
10320:
10312:
10308:
10306:
10298:
10288:
10271:
10267:
10258:
10254:
10250:
10246:
10242:
10238:
10234:
10052:
9760:
9650:
9643:
9639:
9631:
9627:
9623:
9619:
9615:
9611:
9607:
9600:
9592:
9588:
9573:Galois group
9348:
9344:
9340:
9333:
9318:
9310:
9306:
9302:
9298:
9294:
9283:
9279:
9272:
9264:
9260:
9253:
9245:
9239:
9235:
9228:
9220:
9216:
9212:
9208:
9204:
9200:
9196:
9192:
9188:
9184:
9180:
9176:
9172:
9168:
9164:
9160:
9156:
9152:
9148:
9144:
9140:
9136:
9132:
9128:
9124:
9120:
9116:
9112:
9108:
9104:
9100:
9096:
9092:
9088:
9084:
9080:
9076:
9071:
9062:
9049:
9045:adding to it
9040:
9000:
8961:ultrafilters
8895:
8888:
8884:
8880:
8876:
8717:
8684:
8680:
8614:
8610:
8534:
8530:
8526:
8474:
8470:
8466:
8339:
8333:
8328:
8252:
8210:
8209:-space over
8206:
8202:
8165:
8161:
8157:
7483:
7431:
7028:with values
6628:
6534:Galois group
5920:
5468:
5087:
4575:
4348:
4047:
4025:finite field
3866:
3455:
3265:
2893:hypersurface
2829:
2645:
2522:. We define
2520:residue ring
2519:
2345:
2291:
2135:
2105:
2003:
1979:
1285:
1282:Affine space
1277:
1263:
1261:
1256:
1217:
1155:
1151:
1147:
1144:
1140:
1136:
1132:
1128:
1088:
1084:
1080:
1074:
1065:
1061:
1057:
1049:
1045:
1041:
1037:
1033:
1029:
1025:
1018:
1014:
1010:
1006:
1002:
998:
994:
990:
986:
982:
978:
974:
967:
963:
959:
957:
952:
944:
940:
936:
932:
928:
924:
920:
916:
912:
908:
904:
900:
896:
888:
884:
880:
876:
872:
870:
861:
849:
847:
842:
838:
834:
830:
826:
822:
818:
814:
810:
806:
802:
798:
794:
792:
787:
775:
771:
767:
763:
761:
756:
752:
748:
744:
741:
737:
733:
729:
725:
721:
718:
715:
711:
701:
691:
687:
683:
679:
672:
664:
657:
651:
649:
647:
642:
638:
633:
629:
617:
615:
610:
606:
602:
597:
593:
589:
584:
580:
576:
571:
567:
562:
558:
554:
550:
546:
542:
530:
528:
517:
515:
510:
506:
502:
497:
493:
489:
481:
478:prime ideals
473:
466:
459:
455:
453:
435:
417:
412:
408:
398:
373:
371:
334:
332:
315:Emmy Noether
310:
307:prime ideals
302:
298:
294:
290:
279:
274:
266:number rings
238:real numbers
231:
219:
215:moduli space
210:
202:
199:
196:
192:
188:
182:
170:prime ideals
163:
147:ringed space
124:
85:
77:
76:
59:
55:
39:
29:
11357:Terence Tao
11086:Harris, Joe
10747:Les schémas
10595:Hom functor
10493:defined an
9825:, which is
9606:defined by
9234:defined by
8378:, called a
7432:The scheme
4110:, and also
2895:subvariety
1980:The scheme
1013:-points of
931:is a field
915:-points of
793:A morphism
784:finite type
714:, a scheme
228:Development
32:mathematics
10964:References
10763:, note 29.
10642:Chow group
10346:associated
9946:transverse
9646:derivative
9644:and first
9052:March 2024
8679:. To show
7474:valuations
7091:, that is
6924:elements,
5878:for those
3124:defines a
2758:, so that
2644:gives the
782:scheme of
525:Definition
351:André Weil
328:Noetherian
305:, and the
234:polynomial
11166:(1997) .
10988:0806.1033
10679:Citations
10560:orbifolds
10387:means an
10112:≅
10021:⊂
9900:−
9849:−
9636:nilpotent
9612:fat point
9532:−
9517:−
9485:⊂
9430:
9424:→
9409:−
9394:−
9362:
9167:= −1 has
8979:≠
8972:∏
8933:∞
8918:∏
8909:
8818:…
8741:→
8659:−
8566:∖
8518:over the
8419:…
8397:
8096:−
8067:−
8054:−
8013:
7928:−
7912:−
7855:−
7843:Δ
7777:−
7679:
7660:
7653:×
7607:
7544:
7495:∈
7409:¯
7183:∈
7148:α
7130:∈
7124:α
6982:∈
6941:
6856:α
6798:α
6780:≅
6486:−
6481:±
6386:. And at
6358:−
6201:∩
6192:−
6112:±
6035:−
6030:±
6016:, namely
5797:−
5759:−
5717:−
5253:⊂
5206:
5138:∈
5049:∖
4960:ρ
4889:…
4789:…
4703:⋯
4647:…
4601:…
4483:
4322:∈
3878:∈
3867:A number
3844:−
3839:ℓ
3828:…
3817:−
3746:ℓ
3726:…
3696:∖
3661:≥
3644:∈
3594:−
3508:∖
3467:∈
3382:∈
3305:
3218:−
3179:
3014:
2973:⊂
2924:⊂
2909:¯
2863:…
2736:↦
2703:≅
2586:→
2427:¯
2398:∈
2379:…
2268:≥
2258:∈
2243:∈
2187:−
2117:∈
2081:∉
2055:∈
2027:∖
1951:⊂
1921:¯
1912:∈
1848:⊂
1818:∈
1742:¯
1733:⊂
1714:¯
1699:¯
1670:⊂
1610:−
1594:…
1578:−
1466:…
1385:…
1320:¯
1106:×
833:) → Spec(
778:means an
622:separated
618:prescheme
442:, it was
394:Matsusaka
382:manifolds
324:dimension
44:structure
11367:Category
11256:(1999).
11128:(1960).
11088:(1998).
11051:(1985),
10601:See also
10593:and the
10456:(by the
10356:-module
10316:-modules
10171:of this
9657:calculus
9326:ramified
9278:, where
6304:, since
6262:ramified
6141:, since
5605:for all
5330:, where
1274:Examples
891:means a
704:category
660:, for a
539:spectrum
464:spectrum
166:spectrum
108:topology
72:integers
58:= 0 and
11323:2223406
11311:Bibcode
11284:1748380
11246:1917232
11215:0456457
11188:0463157
11156:0217083
11114:1730819
11075:0780183
11042:1848254
11007:3082873
10552:descent
10373:= Spec(
10245:= Spec
9652:reduced
8609:; then
8321:is not.
7452:is not
1145:product
1054:functor
1052:) is a
972:algebra
927:. When
899:→ Spec(
893:section
864:) as a
856:in the
772:variety
755:→ Spec(
650:affine
626:Mumford
343:compact
286:in the
11321:
11282:
11272:
11244:
11234:
11213:
11203:
11186:
11176:
11154:
11112:
11102:
11073:
11063:
11040:
11005:
10523:action
10325:module
10231:smooth
10169:length
9591:. Let
9461:degree
9267:)) → A
8514:is an
8249:proper
7958:
7950:
7830:is an
7454:proper
7306:; and
7072:
7058:
6289:
6281:
6126:
6118:
5132:
5124:
4095:
4087:
4023:, the
3655:
3638:
3630:
2518:, the
2252:
2237:
2234:
2226:
2223:
2075:
2072:
2064:
2061:
1838:
1835:
1827:
1824:
1268:limits
1091:, the
852:is an
848:Since
730:scheme
724:(or a
654:-space
551:scheme
436:scheme
335:affine
282:: the
40:scheme
11301:arXiv
10983:arXiv
10853:(PDF)
10509:stack
10377:). A
9211:) of
8875:. If
8716:when
8251:over
7738:. If
7464:with
6234:. At
4672:with
4291:with
2891:is a
1901:with
1255:over
1216:over
1064:over
1036:over
997:over
983:point
977:, an
885:point
879:, an
786:over
774:over
541:Spec(
533:is a
486:sheaf
469:of a
250:field
151:sheaf
149:or a
143:atlas
131:ideal
42:is a
11270:ISBN
11232:ISBN
11201:ISBN
11174:ISBN
11100:ISBN
11061:ISBN
10163:The
9659:and
9427:Spec
9359:Spec
9309:−1)(
9293:Let
8906:Spec
8894:Let
8525:The
8394:Proj
8010:Spec
7676:Spec
7657:Spec
7604:Spec
7541:Spec
7476:.
5203:Spec
4480:Frac
4346:.
3302:Spec
3176:Spec
3011:Spec
2068:with
1831:with
1286:Let
1187:and
1131:and
1083:and
1075:The
993:) →
821:) →
745:over
719:over
549:. A
426:and
392:and
390:Chow
353:and
317:and
244:and
183:The
110:and
74:).
38:, a
11262:doi
11144:doi
11028:doi
10993:doi
10444:of
10369:on
10305:of
10257:of
9634:is
9630:on
9618:is
9219:of
9143:of
9047:.
8846:to
8478:of
8342:= 0
8247:is
7954:mod
6938:deg
6868:of
6680:is
6285:mod
6122:mod
5918:.
5128:for
5080:.
4845:in
4091:mod
3634:for
3484:is
2827:.
2343:.
2230:for
1977:.
1028:of
985:of
951:of
887:of
813:*:
759:).
694:).
609:on
529:An
480:of
396:.)
297:of
252:of
122:.
30:In
11369::
11359:).
11319:MR
11317:,
11309:,
11280:MR
11278:.
11268:.
11242:MR
11240:.
11230:.
11226:.
11211:MR
11209:.
11184:MR
11182:.
11152:MR
11150:.
11142:.
11138:.
11132:.
11124:;
11110:MR
11108:.
11098:.
11094:.
11084:;
11071:MR
11069:,
11038:MR
11036:,
11024:38
11022:,
11003:MR
11001:,
10991:,
10979:55
10977:,
10724:^
10695:".
10661:,
10657:,
10653:,
10644:,
10640:,
10621:,
10617:,
10613:,
10609:,
10489:.
10425:.
10249:/(
10053:x-
9761:x-
9622:/(
9301:=
9238:+
9163:+
8473:=
8469:+
8324:A
8164:,
8147:.
8099:27
7931:27
7900:18
7468:.
7429:.
5459:.
3864:.
3258:.
2288:.
1270:.
1259:.
1150:×
1072:.
1044:↦
955:.
935:,
868:.
841:→
801:→
797::
790:.
736:→
518:k,
422:,
349:,
224:.
217:.
191:→
180:.
161:.
11313::
11303::
11286:.
11264::
11248:.
11217:.
11190:.
11158:.
11146::
11140:4
11116:.
11030::
10995::
10985::
10944:.
10815:.
10545:G
10531:X
10527:G
10466:X
10462:X
10454:Y
10450:X
10446:X
10442:Y
10423:X
10414:X
10410:O
10406:X
10398:X
10393:X
10389:O
10385:X
10375:R
10371:X
10364:M
10360:~
10353:X
10349:O
10342:R
10338:M
10333:X
10329:O
10321:X
10313:X
10309:O
10299:X
10272:Y
10268:y
10265:T
10259:Y
10255:y
10251:x
10247:C
10243:X
10239:Y
10235:Y
10214:k
10194:]
10191:y
10188:,
10185:x
10182:[
10179:k
10151:.
10145:)
10140:2
10136:x
10132:(
10127:]
10124:x
10121:[
10118:k
10106:)
10103:y
10099:,
10094:2
10090:x
10086:(
10081:]
10078:y
10075:,
10072:x
10069:[
10066:k
10039:]
10036:y
10033:,
10030:x
10027:[
10024:k
10018:)
10015:y
10012:,
10009:x
10006:(
9986:)
9983:0
9980:,
9977:0
9974:(
9971:=
9968:)
9965:y
9962:,
9959:x
9956:(
9932:)
9929:y
9925:,
9920:2
9916:x
9912:(
9909:=
9906:)
9903:y
9895:2
9891:x
9887:(
9884:+
9881:)
9878:y
9875:(
9855:)
9852:y
9844:2
9840:x
9836:(
9833:V
9811:2
9807:x
9803:=
9800:y
9780:)
9777:y
9774:(
9771:V
9747:]
9744:y
9741:,
9738:x
9735:[
9732:k
9712:k
9690:2
9685:k
9680:A
9640:X
9632:X
9628:x
9624:x
9620:C
9616:X
9608:x
9601:C
9593:X
9547:.
9543:)
9538:)
9535:5
9529:x
9526:(
9523:)
9520:1
9514:x
9511:(
9508:x
9503:(
9499:)
9496:x
9493:(
9489:C
9482:)
9479:x
9476:(
9472:C
9447:.
9444:)
9441:x
9438:(
9434:C
9420:)
9415:)
9412:5
9406:x
9403:(
9400:)
9397:1
9391:x
9388:(
9385:x
9380:(
9376:)
9373:x
9370:(
9366:C
9349:X
9345:X
9341:x
9334:C
9319:C
9311:x
9307:x
9305:(
9303:x
9299:y
9295:X
9284:x
9282:(
9280:C
9273:C
9265:x
9263:(
9261:C
9254:C
9240:y
9236:x
9229:R
9221:k
9217:E
9213:E
9209:E
9207:(
9205:X
9201:k
9197:X
9193:C
9189:C
9187:(
9185:X
9181:C
9179:(
9177:X
9173:R
9171:(
9169:X
9165:y
9161:x
9157:X
9153:k
9151:(
9149:X
9145:k
9141:E
9137:E
9135:(
9133:X
9129:k
9127:(
9125:X
9121:k
9117:k
9115:(
9113:X
9109:k
9107:(
9105:X
9101:k
9097:k
9093:k
9091:(
9089:X
9085:k
9081:n
9054:)
9050:(
9001:n
8987:k
8982:n
8976:m
8942:)
8938:k
8928:1
8925:=
8922:n
8913:(
8896:k
8889:X
8885:f
8881:f
8877:X
8863:)
8860:X
8857:(
8854:O
8834:]
8829:n
8825:x
8821:,
8815:,
8810:1
8806:x
8802:[
8798:C
8794:=
8791:)
8786:n
8781:A
8776:(
8773:O
8751:n
8746:A
8738:X
8735::
8732:f
8718:n
8702:n
8697:A
8685:X
8681:X
8667:]
8662:1
8655:x
8651:,
8648:x
8645:[
8641:C
8635:c
8632:e
8629:p
8626:S
8615:n
8611:X
8596:C
8575:}
8572:0
8569:{
8561:n
8556:A
8551:=
8548:X
8535:k
8531:k
8522:.
8500:2
8494:Q
8488:P
8475:z
8471:y
8467:x
8452:.
8449:)
8446:f
8443:(
8439:/
8435:]
8430:n
8426:x
8422:,
8416:,
8411:0
8407:x
8403:[
8400:R
8364:n
8359:R
8354:P
8340:f
8334:R
8329:f
8307:n
8302:C
8278:n
8273:P
8267:C
8253:R
8233:n
8228:R
8223:P
8211:R
8207:n
8203:n
8187:n
8182:R
8177:P
8162:n
8158:R
8135:p
8132:,
8129:3
8107:2
8103:p
8073:)
8070:p
8062:3
8058:x
8049:2
8045:y
8041:(
8036:]
8033:y
8030:,
8027:x
8024:[
8020:Z
8007:=
8004:X
7984:p
7964:,
7961:p
7947:0
7944:=
7939:2
7935:c
7923:3
7919:b
7915:4
7909:c
7906:b
7903:a
7897:+
7892:2
7888:b
7882:2
7878:a
7874:+
7871:c
7866:3
7862:a
7858:4
7852:=
7847:f
7818:c
7815:+
7812:x
7809:b
7806:+
7801:2
7797:x
7793:a
7790:+
7785:3
7781:x
7772:2
7768:y
7764:=
7761:)
7758:y
7755:,
7752:x
7749:(
7746:f
7724:p
7719:F
7697:)
7692:p
7687:F
7682:(
7671:)
7667:Z
7663:(
7649:X
7646:=
7641:p
7637:X
7611:Z
7584:)
7581:)
7578:f
7575:(
7571:/
7567:]
7564:y
7561:,
7558:x
7555:[
7551:Z
7547:(
7538:=
7535:X
7515:]
7512:y
7509:,
7506:x
7503:[
7499:Z
7492:f
7440:Y
7415:p
7405:F
7381:)
7378:x
7375:(
7372:f
7352:p
7332:)
7329:)
7326:x
7323:(
7320:f
7317:(
7314:V
7294:p
7291:=
7288:)
7285:x
7282:(
7279:r
7259:)
7256:p
7253:(
7250:V
7230:)
7225:m
7220:(
7217:r
7197:]
7194:x
7191:[
7187:Z
7180:)
7177:x
7174:(
7171:r
7151:)
7145:(
7140:p
7135:F
7127:)
7121:(
7118:r
7115:=
7112:)
7107:m
7102:(
7099:r
7077:m
7068:d
7065:o
7062:m
7055:r
7052:=
7049:)
7044:m
7039:(
7036:r
7016:Y
6996:]
6993:x
6990:[
6986:Z
6979:)
6976:x
6973:(
6970:r
6950:)
6947:f
6944:(
6935:=
6932:d
6910:d
6906:p
6885:)
6882:x
6879:(
6876:f
6853:=
6850:x
6828:p
6823:F
6801:)
6795:(
6790:p
6785:F
6777:)
6774:)
6771:x
6768:(
6765:f
6762:(
6758:/
6754:]
6751:x
6748:[
6743:p
6738:F
6733:=
6728:m
6722:/
6718:]
6715:x
6712:[
6708:Z
6704:=
6701:)
6696:m
6691:(
6688:k
6668:)
6665:)
6662:x
6659:(
6656:f
6653:,
6650:p
6647:(
6644:=
6639:m
6614:)
6611:1
6608:+
6603:2
6599:x
6595:(
6592:V
6572:)
6569:1
6566:+
6561:2
6557:x
6553:,
6550:3
6547:(
6544:V
6518:3
6513:F
6489:1
6478:=
6475:x
6455:)
6452:1
6449:+
6444:2
6440:x
6436:,
6433:3
6430:(
6427:=
6422:m
6400:3
6397:=
6394:p
6374:)
6369:2
6365:)
6361:1
6355:x
6352:(
6349:,
6346:2
6343:(
6340:=
6337:)
6334:1
6331:+
6326:2
6322:x
6318:,
6315:2
6312:(
6292:2
6278:1
6275:=
6272:x
6248:2
6245:=
6242:p
6222:)
6219:2
6216:+
6213:x
6210:,
6207:5
6204:(
6198:)
6195:2
6189:x
6186:,
6183:5
6180:(
6177:=
6174:)
6171:1
6168:+
6163:2
6159:x
6155:,
6152:5
6149:(
6129:5
6115:2
6109:=
6106:x
6086:5
6083:=
6080:p
6060:p
6038:1
6027:=
6024:x
6004:1
6001:+
5996:2
5992:x
5971:x
5951:)
5948:1
5945:+
5940:2
5936:x
5932:(
5929:V
5906:b
5886:p
5866:)
5863:p
5860:(
5857:V
5837:b
5833:/
5829:a
5826:=
5823:x
5803:)
5800:a
5794:x
5791:b
5788:(
5785:V
5765:)
5762:a
5756:x
5753:(
5750:=
5745:p
5723:)
5720:a
5714:x
5711:(
5708:V
5688:a
5685:=
5682:x
5662:x
5642:p
5622:)
5619:x
5616:(
5613:f
5593:)
5590:)
5587:x
5584:(
5581:f
5578:,
5575:p
5572:(
5569:=
5564:m
5542:)
5539:p
5536:(
5533:=
5528:p
5506:)
5503:p
5500:(
5497:V
5477:p
5447:x
5427:p
5407:Y
5387:p
5367:)
5364:x
5361:(
5358:f
5338:p
5318:)
5315:)
5312:x
5309:(
5306:f
5303:,
5300:p
5297:(
5294:=
5289:m
5267:]
5264:x
5261:[
5257:Z
5248:p
5226:)
5223:]
5220:x
5217:[
5213:Z
5209:(
5200:=
5197:Y
5177:]
5174:x
5171:[
5167:Z
5146:}
5142:Z
5135:a
5121:a
5118:{
5115:=
5110:1
5104:Z
5098:A
5068:)
5063:i
5059:n
5055:(
5052:V
5046:Z
5043:=
5038:i
5034:U
5013:Z
4987:i
4983:n
4977:i
4973:a
4969:=
4964:i
4938:Z
4917:)
4914:1
4911:(
4908:=
4905:)
4900:r
4896:n
4892:,
4886:,
4881:1
4877:n
4873:(
4853:Z
4831:p
4825:m
4800:r
4796:n
4792:,
4786:,
4781:1
4777:n
4756:Z
4732:1
4729:=
4724:r
4720:n
4714:r
4710:a
4706:+
4700:+
4695:1
4691:n
4685:1
4681:a
4658:r
4654:a
4650:,
4644:,
4639:1
4635:a
4612:r
4608:n
4604:,
4598:,
4593:1
4589:n
4559:p
4553:m
4530:b
4526:/
4522:a
4501:Q
4497:=
4494:)
4490:Z
4486:(
4477:=
4474:)
4469:0
4463:p
4457:(
4454:k
4434:Z
4414:p
4394:p
4391:=
4388:n
4366:p
4360:m
4332:p
4327:F
4319:)
4314:p
4308:m
4302:(
4299:f
4279:f
4259:n
4237:p
4231:m
4208:n
4187:Z
4183:=
4180:)
4177:0
4174:(
4170:/
4165:Z
4144:n
4141:=
4138:)
4133:0
4127:p
4121:(
4118:n
4098:p
4084:n
4081:=
4078:)
4073:p
4067:m
4061:(
4058:n
4048::
4035:p
4009:p
4004:F
3999:=
3996:)
3993:p
3990:(
3986:/
3981:Z
3977:=
3974:)
3969:p
3963:m
3957:(
3954:k
3932:p
3926:m
3903:Z
3882:Z
3875:n
3852:]
3847:1
3835:p
3831:,
3825:,
3820:1
3812:1
3808:p
3804:[
3800:Z
3796:=
3793:)
3790:U
3787:(
3782:Z
3776:O
3753:}
3742:p
3735:m
3729:,
3723:,
3716:1
3712:p
3705:m
3699:{
3693:Z
3690:=
3687:U
3667:}
3664:0
3658:m
3652:,
3648:Z
3641:n
3622:m
3618:p
3614:n
3608:{
3605:=
3602:]
3597:1
3590:p
3586:[
3582:Z
3578:=
3575:)
3570:p
3566:U
3562:(
3557:Z
3551:O
3528:}
3523:p
3517:m
3511:{
3505:Z
3502:=
3497:p
3493:U
3471:Z
3464:p
3430:)
3427:0
3424:(
3421:=
3416:0
3410:p
3386:Z
3379:p
3359:)
3356:p
3353:(
3350:=
3345:p
3339:m
3316:)
3312:Z
3308:(
3299:=
3296:Z
3275:Z
3246:)
3243:)
3240:1
3237:+
3234:y
3231:(
3226:2
3222:y
3213:2
3209:x
3205:(
3201:/
3197:]
3194:y
3191:,
3188:x
3185:[
3182:k
3173:=
3170:V
3148:2
3143:k
3138:A
3112:)
3109:1
3106:+
3103:y
3100:(
3095:2
3091:y
3087:=
3082:2
3078:x
3057:k
3037:)
3034:)
3031:f
3028:(
3024:/
3020:R
3017:(
3008:=
3005:)
3002:f
2999:(
2996:V
2976:R
2970:)
2967:f
2964:(
2939:n
2934:k
2929:A
2921:)
2918:f
2915:(
2906:V
2879:)
2874:n
2870:x
2866:,
2860:,
2855:1
2851:x
2847:(
2844:f
2841:=
2838:f
2815:)
2812:a
2809:(
2806:r
2786:)
2781:a
2775:m
2769:(
2766:r
2744:i
2740:a
2731:i
2727:x
2706:k
2698:a
2692:m
2685:/
2681:R
2678:=
2675:)
2670:a
2664:m
2658:(
2655:k
2630:a
2624:m
2599:p
2593:/
2589:R
2583:R
2563:r
2543:)
2538:p
2533:(
2530:r
2504:p
2498:/
2494:R
2472:p
2450:X
2424:X
2401:R
2395:)
2390:n
2386:x
2382:,
2376:,
2371:1
2367:x
2363:(
2360:r
2357:=
2354:r
2331:U
2309:X
2303:O
2276:}
2271:0
2263:Z
2255:m
2249:,
2246:R
2240:r
2215:m
2211:f
2207:r
2201:{
2198:=
2195:]
2190:1
2183:f
2179:[
2176:R
2173:=
2170:)
2165:f
2161:U
2157:(
2152:X
2146:O
2120:R
2114:f
2091:}
2086:p
2078:f
2058:X
2050:p
2045:{
2042:=
2039:)
2036:f
2033:(
2030:V
2024:X
2021:=
2016:f
2012:U
1988:X
1963:a
1957:m
1946:p
1918:V
1909:a
1887:a
1881:m
1858:}
1853:q
1843:p
1821:X
1813:q
1808:{
1805:=
1802:)
1797:p
1792:(
1789:V
1767:p
1739:X
1730:)
1725:p
1720:(
1711:V
1705:=
1696:V
1673:R
1665:p
1643:a
1623:)
1618:n
1614:a
1605:n
1601:x
1597:,
1591:,
1586:1
1582:a
1573:1
1569:x
1565:(
1562:=
1557:a
1551:m
1528:)
1525:R
1522:(
1518:c
1515:e
1512:p
1509:S
1505:=
1502:X
1482:]
1477:n
1473:x
1469:,
1463:,
1458:1
1454:x
1450:[
1447:k
1444:=
1441:R
1421:k
1401:)
1396:n
1392:a
1388:,
1382:,
1377:1
1373:a
1369:(
1366:=
1363:a
1341:n
1336:k
1331:A
1326:=
1317:X
1294:k
1264:Z
1257:k
1241:n
1238:+
1235:m
1230:A
1218:k
1202:n
1197:A
1173:m
1168:A
1156:k
1152:Z
1148:X
1141:k
1137:k
1133:Z
1129:X
1115:Z
1110:Y
1102:X
1089:Y
1085:Z
1081:X
1066:R
1062:X
1058:R
1050:S
1048:(
1046:X
1042:S
1038:R
1034:X
1030:k
1026:E
1019:k
1015:X
1011:S
1007:S
1005:(
1003:X
999:R
995:X
991:S
987:X
981:-
979:S
975:S
970:-
968:R
964:R
960:X
953:X
947:-
945:k
941:k
939:(
937:X
933:k
929:R
925:R
921:X
917:X
913:R
909:R
907:(
905:X
901:R
897:X
889:X
883:-
881:R
877:R
873:X
862:Z
850:Z
843:A
839:B
835:B
831:A
827:X
825:(
823:O
819:Y
817:(
815:O
811:f
803:Y
799:X
795:f
788:k
776:k
768:k
764:k
757:R
753:X
749:R
742:X
738:Y
734:X
728:-
726:Y
722:Y
716:X
712:Y
692:R
688:R
684:n
680:k
673:k
665:n
658:k
652:n
643:,
639:U
637:(
634:X
630:O
611:U
603:U
601:(
598:X
594:O
590:U
585:X
581:O
577:X
572:i
568:U
563:i
559:U
555:X
547:R
543:R
511:R
507:U
503:U
501:(
498:X
494:O
490:U
482:R
474:R
467:X
311:k
303:k
299:n
295:k
291:k
280:k
211:Y
203:Y
197:X
193:Y
189:X
60:x
56:x
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.