9022:
5452:
3439:
10574:). 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
509:
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
310:
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
10473:
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
9446:
2275:
3666:
6800:
1857:
2090:
7696:
3752:
10543:. 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
9546:
10488:
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
7963:
4501:
5145:
8075:
7150:
4010:
1744:
9343:
8942:
7078:
1622:
602:. 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.
10150:
2705:
9052:
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.
3527:
3245:
4333:
1964:
7583:
4097:
2940:
8574:
5225:
3315:
9248:, 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(
5266:
3851:
249:. 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
7611:
2127:
8833:
3532:
8501:
6221:
4731:
1342:
7817:
7416:
5067:
3036:
8666:
6672:
4187:
2400:
1773:
8279:
8451:
6454:
134:
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
9931:
6949:
6373:
4916:
6667:
6128:
5592:
5317:
4143:
2878:
1672:
8752:
3429:
3358:
1400:
2785:
2600:
1995:
5764:
10490:
9691:
8365:
8234:
8188:
7196:
6995:
6291:
4988:
3149:
1527:
5541:
4832:
4801:
4659:
4613:
4560:
4367:
4238:
3933:
2631:
1888:
7620:
7229:
6490:
6039:
2745:
2542:
2310:
1242:
7514:
3882:
3471:
3386:
2505:
361:
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
9455:
8703:
8308:
7725:
6829:
6519:
3111:
1203:
1174:
7826:
2473:
1923:
1768:
10038:
9548:
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
8986:
404:. 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.)
1114:
8596:
6571:
5176:
4938:
3275:
2975:
9985:
10536:. 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.
9854:
6613:
5950:
2429:
6855:
8108:
3671:
9812:
7331:
6003:
5802:
7293:
5722:
2119:
7988:
6911:
5836:
9779:
8862:
7380:
7258:
6884:
5865:
5621:
5505:
5366:
2814:
8134:
6399:
6247:
6085:
5687:
4529:
4393:
1481:
10213:
10193:
9746:
9711:
8202:
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
7983:
7439:
7351:
7015:
6059:
5970:
5905:
5885:
5661:
5641:
5476:
5446:
5426:
5406:
5386:
5337:
5012:
4852:
4755:
4433:
4413:
4278:
4258:
4207:
4034:
3902:
3056:
2562:
2449:
2330:
1987:
1642:
1420:
1293:
423:
relating number theory and algebraic geometry, further extended the objects of algebraic geometry, for example by generalizing the base rings allowed. The word
4438:
510:
a projective variety. Applying
Grothendieck's theory to schemes over the integers and other number fields led to powerful new perspectives in number theory.
5080:
613:
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
8526:, 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.
354:
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
10047:
10570:
or "spectral algebraic geometry", the structure sheaf is replaced by a homotopical analog of a sheaf of commutative rings (for example, a sheaf of
7083:
759:
with certain properties. There are different conventions about exactly which schemes should be called varieties. One standard choice is that a
3938:
1677:
11007:
10681:
75:
11123:
8890:
7020:
1532:
108:
3114:
2639:
8948:
of the positive integers (with the discrete topology). In fact, the prime ideals of this ring are in one-to-one correspondence with the
435:
who suggested to Serre the possibility of using the spectrum of an arbitrary commutative ring as a foundation for algebraic geometry.
3476:
173:
10551:
in topology), for which the stabilizer groups are finite, and algebraic spaces, for which the stabilizer groups are trivial. The
8378:
3154:
4283:
1928:
9549:
9441:{\displaystyle \operatorname {Spec} \mathbf {C} (x)\left({\sqrt {x(x-1)(x-5)}}\right)\to \operatorname {Spec} \mathbf {C} (x).}
7519:
4042:
2887:
634:
which are not necessarily functions, can neverthess be patched together from their restrictions in the same way as functions.
11262:
11224:
11166:
11092:
11053:
9627:
but not zero. To indicate the meaning of this scheme: two regular functions on the affine line have the same restriction to
8532:
8529:
A simple reason to go beyond affine schemes is that an open subset of an affine scheme need not be affine. For example, let
3435:. Closed sets are finite sets, and open sets are their complements, the cofinite sets; any infinite set of points is dense.
451:(SGA), bringing to a conclusion a generation of experimental suggestions and partial developments. Grothendieck defined the
10429:
of a smooth variety over a field. However, coherent sheaves are richer; for example, a vector bundle on a closed subscheme
9004:, this is an example of a non-Noetherian quasi-compact scheme with infinitely many irreducible components. (By contrast, a
5181:
3280:
389:
2270:{\displaystyle {\mathcal {O}}_{X}(U_{f})=R=\{{\tfrac {r}{f^{m}}}\ \ {\text{for}}\ \ r\in R,\ m\in \mathbb {Z} _{\geq 0}\}}
11005:(2001), "A mad day's work: from Grothendieck to Connes and Kontsevich. The evolution of concepts of space and symmetry",
9279:
in one variable. To see why it is useful to have an actual "generic point" in the scheme, consider the following example.
5230:
3661:{\displaystyle {\mathcal {O}}_{Z}(U_{p})=\mathbb {Z} =\{{\tfrac {n}{p^{m}}}\ {\text{for}}\ n\in \mathbb {Z} ,\ m\geq 0\}}
3757:
11193:
7588:
166:
10263:. This again indicates that non-reduced subschemes have geometric meaning, related to derivatives and tangent vectors.
8945:
8757:
6795:{\displaystyle k({\mathfrak {m}})=\mathbb {Z} /{\mathfrak {m}}=\mathbb {F} _{p}/(f(x))\cong \mathbb {F} _{p}(\alpha )}
1852:{\displaystyle V({\mathfrak {p}})=\{{\mathfrak {q}}\in X\ \ {\text{with}}\ \ {\mathfrak {p}}\subset {\mathfrak {q}}\}}
8470:
6133:
4664:
1298:
202:. In many cases, the family of all varieties of a given type can itself be viewed as a variety or scheme, known as a
7730:
7385:
5017:
11084:
9061:
Here are some of the ways in which schemes go beyond older notions of algebraic varieties, and their significance.
8609:
4148:
2980:
2338:
8250:
10962:
10571:
6404:
335:
9092:
is algebraically closed (for example the complex numbers), then one can base algebraic geometry on sets such as
908:. In examples, this definition reconstructs the old notion of the set of solutions of the defining equations of
396:
of an algebraic variety. What is true for the generic point is true for "most" points of the variety. In Weil's
11002:
9859:
7461:
overcomes this obstacle by compactifying affine arithmetic schemes, adding points at infinity corresponding to
6916:
6296:
4857:
2085:{\displaystyle U_{f}=X\smallsetminus V(f)=\{{\mathfrak {p}}\in X\ \ {\text{with}}\ \ f\notin {\mathfrak {p}}\}}
768:
428:
346:
applied commutative algebra as a new foundation for algebraic geometry in the richer setting of projective (or
6621:
6090:
5546:
5271:
4102:
2822:
1647:
11285:
8716:
3391:
3320:
1347:
846:
818:). In the case of affine schemes, this construction gives a one-to-one correspondence between morphisms Spec(
400:(1946), generic points are constructed by taking points in a very large algebraically closed field, called a
2750:
2567:
126:
consisting of closed points which correspond to geometric points, together with non-closed points which are
10334:
10218:
For a more elaborate example, one can describe all the zero-dimensional closed subschemes of degree 2 in a
9934:
9033:
7691:{\displaystyle X_{p}=X\times _{\operatorname {Spec} (\mathbb {Z} )}\operatorname {Spec} (\mathbb {F} _{p})}
5727:
4734:
264:
165:. The coordinate ring of an affine scheme is the ring itself, and the coordinate rings of open subsets are
11247:
The Red Book of
Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians
9662:
8336:
8205:
8159:
7155:
6954:
6256:
4943:
3120:
1486:
772:
9449:
5510:
4806:
4760:
4618:
4572:
4534:
4341:
4212:
3907:
2605:
1862:
118:
of regular algebraic functions defined on it (or on its subsets), and each subvariety corresponds to the
10838:
7201:
6459:
6008:
4260:
as a kind of "regular function" on the closed points, a very special type among the arbitrary functions
2710:
2514:
2284:
1212:
11074:
10656:
10567:
10562:
Another type of generalization is to enrich the structure sheaf, bringing algebraic geometry closer to
10511:
7476:
3859:
3448:
3363:
2478:
1081:
427:
was first used in the 1956 Chevalley
Seminar, in which Chevalley pursued Zariski's ideas. According to
246:
51:= 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any
40:
11344:- the comment section contains some interesting discussion on scheme theory (including the posts from
11326:
9569:
8679:
8284:
7701:
6805:
6495:
3061:
1179:
1150:
11156:
10544:
10462:
10407:
2454:
1893:
1749:
881:
210:
9990:
319:, he proved that this definition satisfies many of the intuitive properties of geometric dimension.
11341:
8955:
8245:
1086:
1065:
250:
10510:
attached to each point, which is viewed as the automorphism group of that point. For example, any
8579:
6528:
5150:
4921:
3258:
2948:
11216:
11078:
9940:
8368:
3747:{\displaystyle U=Z\smallsetminus \{{\mathfrak {m}}_{p_{1}},\ldots ,{\mathfrak {m}}_{p_{\ell }}\}}
1256:
502:) are the affine schemes; a general scheme is then obtained by "gluing together" affine schemes.
10242:) as in the previous paragraph. Subschemes of the latter type are determined by a complex point
9817:
6576:
5913:
2405:
11361:
11110:
10647:
8314:
6834:
70:
32:
10960:
Arapura, Donu (2011), "Frobenius amplitude, ultraproducts, and vanishing on singular spaces",
8080:
7446:
114:
Schemes elaborate the fundamental idea that an algebraic variety is best analyzed through the
10552:
10540:
9784:
8997:
7450:
7298:
5975:
5769:
1529:
is a topological space with the
Zariski topology, whose closed points are the maximal ideals
7263:
5692:
2098:
505:
Much of algebraic geometry focuses on projective or quasi-projective varieties over a field
377:
of a curve over any field. (Later, Jacobians were shown to be projective varieties by Weil,
11311:
11299:
11272:
11234:
11203:
11176:
11144:
11102:
11063:
11030:
10995:
10368:
10313:
7454:
6889:
5807:
523:
327:
230:
11280:
Vistoli, Angelo (2005), "Grothendieck topologies, fibered categories and descent theory",
9755:
8838:
7356:
7234:
6860:
5841:
5597:
5481:
5342:
2790:
8:
10621:
10556:
10529:
10497:
10153:
10044:
axis tangent direction (the common tangent of the two curves) and having coordinate ring:
9656:
9541:{\displaystyle \mathbf {C} (x)\subset \mathbf {C} (x)\left({\sqrt {x(x-1)(x-5)}}\right).}
8113:
7462:
6378:
6226:
6064:
5666:
4506:
4372:
1425:
960:
696:
474:
238:
139:
119:
100:
92:
11303:
10626:
7958:{\displaystyle \Delta _{f}=-4a^{3}c+a^{2}b^{2}+18abc-4b^{3}-27c^{2}=0\ {\text{mod}}\ p,}
153:
Formally, a scheme is a ringed space covered by affine schemes. An affine scheme is the
11289:
11042:
10971:
10734:
10461:
has good properties for coherent (and quasi-coherent) sheaves. The resulting theory of
10198:
10163:
10157:
9716:
9696:
9108:), consider polynomial mappings between different sets of this type, and so on. But if
8710:
7968:
7614:
7424:
7336:
7000:
6250:
6044:
5955:
5890:
5870:
5646:
5626:
5461:
5431:
5411:
5391:
5371:
5322:
4997:
4991:
4837:
4740:
4496:{\displaystyle k({\mathfrak {p}}_{0})=\operatorname {Frac} (\mathbb {Z} )=\mathbb {Q} }
4418:
4398:
4263:
4243:
4192:
4019:
3887:
3041:
2547:
2434:
2315:
1972:
1627:
1405:
1278:
527:
452:
382:
154:
80:
24:
5140:{\displaystyle \mathbb {A} _{\mathbb {Z} }^{1}=\{a\ {\text{for}}\ a\in \mathbb {Z} \}}
432:
11258:
11220:
11189:
11162:
11088:
11049:
9553:
9314:
9276:
9005:
8372:
4566:
1058:
416:
195:
143:
123:
79:(EGA); one of its aims was developing the formalism needed to solve deep problems of
36:
11114:
11037:
10611:
10475:
5368:
is a non-constant polynomial with no integer factor and which is irreducible modulo
834:. In this sense, scheme theory completely subsumes the theory of commutative rings.
11250:
11152:
11132:
11016:
10981:
10458:
10296:
10291:
8606:≥ 2. (However, the affine line minus the origin is isomorphic to the affine scheme
6615:
is a kind of fusion of two Galois-symmetric horizonal lines, a curve of degree 2.
610:
459:
420:
412:
408:
374:
366:
351:
347:
326:
algebraic varieties. However, many arguments in algebraic geometry work better for
258:
234:
131:
84:
52:
11021:
10755:
322:
Noether and Krull's commutative algebra can be viewed as an algebraic approach to
237:) that algebraic geometry over the real numbers is simplified by working over the
11307:
11268:
11230:
11199:
11172:
11140:
11118:
11098:
11059:
11026:
10991:
10860:
10651:
10643:
10607:
10603:
10599:
10575:
10563:
10507:
10502:
10493:, gives simple conditions for a functor to be represented by an algebraic space.
10484:
9565:
8993:
8508:
8237:
7458:
6521:; since we cannot distinguish between these values (they are symmetric under the
5408:
as two-dimensional, with a "characteristic direction" measured by the coordinate
2943:
1251:
Since the category of schemes has fiber products and also a terminal object Spec(
1011:
854:
692:
473:
with a natural topology (known as the
Zariski topology), but augmented it with a
355:
316:
312:
276:
147:
115:
11249:. Lecture Notes in Mathematics. Vol. 1358 (2nd ed.). Springer-Verlag.
11070:
10896:
10661:
10579:
10524:
10426:
10385:-module that is the sheaf associated to a module on each affine open subset of
10279:
10273:
9640:
9557:
8504:
7820:
1644:. The scheme also contains a non-closed point for each non-maximal prime ideal
1006:. (This generalizes the old observation that given some equations over a field
937:
842:
650:
378:
307:
267:
suggests an approach to algebraic geometry over any algebraically closed field
242:
88:
8988:
corresponding to the principal ultrafilter associated to the positive integer
8070:{\displaystyle X=\operatorname {Spec} {\frac {\mathbb {Z} }{(y^{2}-x^{3}-p)}}}
315:
of a commutative ring in terms of prime ideals and, at least when the ring is
198:, it can be useful to consider families of algebraic surfaces over any scheme
11355:
11242:
10986:
10595:
10479:
10283:
10251:
10219:
9649:
9001:
7442:
2312:
which gives the usual ring of rational functions regular on a given open set
614:
393:
343:
331:
272:
162:
127:
104:
10699:
8522:) is the scheme defined by starting with two copies of the affine line over
4415:
is equal to zero in the residue field. The field of "rational functions" on
339:
10927:
10798:
10639:
10634:
10616:
10446:
9561:
6522:
4013:
2881:
695:, with morphisms defined as morphisms of locally ringed spaces. (See also:
369:), by gluing affine varieties along open subsets, on the model of abstract
303:
203:
135:
56:
10819:
7145:{\displaystyle r({\mathfrak {m}})=r(\alpha )\in \mathbb {F} _{p}(\alpha )}
494:), which may be thought of as the coordinate ring of regular functions on
11345:
10583:
10578:
in homological algebra yield higher information about operations such as
10422:
8949:
5178:, the polynomials with integer coefficients. The corresponding scheme is
466:
295:
254:
226:
158:
20:
10807:
9021:
4005:{\displaystyle k({\mathfrak {m}}_{p})=\mathbb {Z} /(p)=\mathbb {F} _{p}}
1739:{\displaystyle {\bar {V}}={\bar {V}}({\mathfrak {p}})\subset {\bar {X}}}
122:
of functions which vanish on the subvariety. Intuitively, a scheme is a
11136:
10884:
10630:
9634:
1989:
has a basis of open subsets given by the complements of hypersurfaces,
222:
209:
For some of the detailed definitions in the theory of schemes, see the
10848:
10539:
Grothendieck originally introduced stacks as a tool for the theory of
9572:
treats the fundamental group and the Galois group on the same footing.
8937:{\textstyle \operatorname {Spec} \left(\prod _{n=1}^{\infty }k\right)}
8876:
is not surjective and hence not an isomorphism. Therefore, the scheme
176:
is that much of algebraic geometry should be developed for a morphism
11294:
10687:
10226:. Such a subscheme consists of either two distinct complex points of
9624:
7073:{\displaystyle r({\mathfrak {m}})=r\ \mathrm {mod} \ {\mathfrak {m}}}
3432:
1617:{\displaystyle {\mathfrak {m}}_{a}=(x_{1}-a_{1},\ldots ,x_{n}-a_{n})}
10716:
10714:
8713:
in complex analysis, though easier to prove. That is, the inclusion
2121:. This set is endowed with its coordinate ring of regular functions
11333:
11254:
10548:
10145:{\displaystyle {\frac {k}{(x^{2},\,y)}}\cong {\frac {k}{(x^{2})}}.}
9645:
2700:{\displaystyle k({\mathfrak {m}}_{a})=R/{\mathfrak {m}}_{a}\cong k}
373:
in topology. He needed this generality for his construction of the
370:
96:
10976:
10421:, which are the sheaves that locally come from finitely generated
194:), rather than for an individual scheme. For example, in studying
10936:
10908:
10767:
10711:
10555:
says that an algebraic stack with finite stabilizer groups has a
1042:
60:
11342:
https://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/
9336:→ A over the generic point of A is exactly the generic point of
1049:-algebras to sets. It is an important observation that a scheme
1010:, one can consider the set of solutions of the equations in any
564:(as a locally ringed space) is an affine scheme. In particular,
8244:; this is an algebro-geometric version of compactness. Indeed,
229:. By the 19th century, it became clear (notably in the work of
9856:. Their scheme-theoretic intersection is defined by the ideal
8944:
is an affine scheme whose underlying topological space is the
11119:"Éléments de géométrie algébrique: I. Le langage des schémas"
9302:−5) over the complex numbers. This is a closed subscheme of A
3522:{\displaystyle U_{p}=Z\smallsetminus \{{\mathfrak {m}}_{p}\}}
221:
The origins of algebraic geometry mostly lie in the study of
103:. Scheme theory also unifies algebraic geometry with much of
10872:
10779:
3445:
The basis open set corresponding to the irreducible element
8672:
is not affine, one computes that every regular function on
3240:{\displaystyle V=\operatorname {Spec} k/(x^{2}-y^{2}(y+1))}
10465:
is perhaps the main technical tool in algebraic geometry.
9120:) is not rich enough. Indeed, one can study the solutions
4328:{\displaystyle f({\mathfrak {m}}_{p})\in \mathbb {F} _{p}}
3038:, a closed subscheme of affine space. For example, taking
1959:{\displaystyle {\mathfrak {p}}\subset {\mathfrak {m}}_{a}}
1859:, including all the closed points of the subvariety, i.e.
1267:
Here and below, all the rings considered are commutative.
11186:
Basic
Algebraic Geometry 2: Schemes and Complex Manifolds
10743:
10449:
construction). In this way, coherent sheaves on a scheme
7578:{\displaystyle X=\operatorname {Spec} (\mathbb {Z} /(f))}
5451:
4092:{\displaystyle n({\mathfrak {m}}_{p})=n\ {\text{mod}}\ p}
2935:{\displaystyle {\bar {V}}(f)\subset \mathbb {A} _{k}^{n}}
9144:) in any reasonable sense. For example, the plane curve
3438:
675:-space can in fact be defined over any commutative ring
130:
of irreducible subvarieties. The space is covered by an
3277:
can be considered as the coordinate ring of the scheme
302:, known as affine varieties. Motivated by these ideas,
11331:
8958:
8893:
8569:{\displaystyle X=\mathbb {A} ^{n}\smallsetminus \{0\}}
8281:
is a compact space in the classical topology, whereas
7823:, then the fibers over its discriminant locus, where
3600:
2983:
2431:, also defines a function on the points of the scheme
2193:
443:
261:
used to study complex varieties do not seem to apply.
95:, scheme theory allows a systematic use of methods of
11109:
10201:
10166:
10050:
9993:
9943:
9862:
9820:
9787:
9758:
9719:
9699:
9665:
9458:
9346:
8841:
8760:
8719:
8682:
8612:
8582:
8535:
8473:
8381:
8339:
8287:
8253:
8208:
8162:
8116:
8083:
7991:
7971:
7829:
7733:
7704:
7623:
7591:
7522:
7479:
7427:
7388:
7359:
7339:
7301:
7266:
7237:
7204:
7158:
7086:
7023:
7003:
6957:
6919:
6892:
6863:
6837:
6808:
6675:
6624:
6579:
6531:
6498:
6462:
6407:
6381:
6299:
6259:
6229:
6136:
6093:
6067:
6047:
6011:
5978:
5958:
5916:
5893:
5873:
5844:
5810:
5772:
5730:
5695:
5669:
5649:
5629:
5600:
5549:
5513:
5484:
5464:
5434:
5414:
5394:
5374:
5345:
5325:
5274:
5233:
5220:{\displaystyle Y=\operatorname {Spec} (\mathbb {Z} )}
5184:
5153:
5083:
5020:
5000:
4946:
4924:
4860:
4840:
4809:
4763:
4743:
4667:
4621:
4615:
have no common prime factor, then there are integers
4575:
4562:
corresponding to prime divisors of the denominator.
4537:
4509:
4441:
4421:
4401:
4375:
4344:
4286:
4266:
4246:
4215:
4195:
4151:
4105:
4045:
4022:
3941:
3910:
3890:
3862:
3760:
3674:
3535:
3479:
3451:
3394:
3366:
3323:
3310:{\displaystyle Z=\operatorname {Spec} (\mathbb {Z} )}
3283:
3261:
3157:
3123:
3064:
3044:
2951:
2890:
2825:
2793:
2753:
2713:
2642:
2608:
2570:
2550:
2517:
2481:
2457:
2437:
2408:
2341:
2318:
2287:
2130:
2101:
1998:
1975:
1931:
1896:
1865:
1776:
1752:
1680:
1650:
1630:
1535:
1489:
1428:
1408:
1350:
1301:
1281:
1215:
1182:
1153:
1089:
1674:, whose vanishing defines an irreducible subvariety
1147:-schemes. For example, the product of affine spaces
11183:
10453:include information about all closed subschemes of
9084:) of solutions of the equations in the product set
5268:. The closed points are maximal ideals of the form
5261:{\displaystyle {\mathfrak {p}}\subset \mathbb {Z} }
4435:is the fraction field of the generic residue ring,
1295:be an algebraically closed field. The affine space
257:like the integers, where the tools of topology and
11041:
10207:
10187:
10144:
10032:
9979:
9925:
9848:
9806:
9773:
9740:
9705:
9685:
9540:
9440:
8980:
8936:
8856:
8827:
8746:
8697:
8660:
8590:
8568:
8495:
8445:
8359:
8302:
8273:
8228:
8182:
8128:
8102:
8069:
7977:
7957:
7811:
7719:
7690:
7605:
7577:
7508:
7433:
7410:
7374:
7345:
7325:
7287:
7260:is the vanishing locus of the constant polynomial
7252:
7223:
7190:
7144:
7072:
7009:
6989:
6943:
6905:
6878:
6849:
6823:
6794:
6661:
6607:
6565:
6513:
6484:
6448:
6393:
6367:
6285:
6241:
6215:
6122:
6079:
6053:
6033:
5997:
5964:
5944:
5899:
5879:
5859:
5830:
5796:
5758:
5716:
5681:
5655:
5635:
5615:
5586:
5535:
5499:
5470:
5440:
5420:
5400:
5380:
5360:
5331:
5311:
5260:
5219:
5170:
5139:
5061:
5006:
4982:
4932:
4910:
4846:
4826:
4795:
4749:
4725:
4653:
4607:
4554:
4523:
4495:
4427:
4407:
4387:
4361:
4327:
4272:
4252:
4232:
4201:
4181:
4137:
4091:
4028:
4004:
3927:
3896:
3876:
3846:{\displaystyle {\mathcal {O}}_{Z}(U)=\mathbb {Z} }
3845:
3746:
3660:
3521:
3465:
3423:
3380:
3352:
3309:
3269:
3239:
3143:
3105:
3050:
3030:
2969:
2934:
2872:
2808:
2779:
2739:
2699:
2625:
2594:
2556:
2536:
2499:
2467:
2443:
2423:
2394:
2324:
2304:
2269:
2113:
2084:
1981:
1958:
1917:
1882:
1851:
1762:
1738:
1666:
1636:
1616:
1521:
1475:
1414:
1394:
1336:
1287:
1236:
1197:
1168:
1108:
10278:A central part of scheme theory is the notion of
7698:are then algebraic curves over the finite fields
7606:{\displaystyle \operatorname {Spec} \mathbb {Z} }
392:had often used the somewhat foggy concept of the
11353:
10417:Coherent sheaves include the important class of
9128:) of the given equations in any field extension
8828:{\displaystyle O(\mathbb {A} ^{n})=\mathbb {C} }
8110:. This curve is singular over the prime numbers
5072:
3058:to be the complex or real numbers, the equation
298:correspond to the irreducible algebraic sets in
9008:has only finitely many irreducible components.)
8496:{\displaystyle \mathbb {P} _{\mathbb {Q} }^{2}}
6216:{\displaystyle (5,x^{2}+1)=(5,x-2)\cap (5,x+2)}
4733:. Geometrically, this is a version of the weak
4726:{\displaystyle a_{1}n_{1}+\cdots +a_{r}n_{r}=1}
1337:{\displaystyle {\bar {X}}=\mathbb {A} _{k}^{n}}
11069:
10890:
10825:
10732:
7812:{\displaystyle f(x,y)=y^{2}-x^{3}+ax^{2}+bx+c}
7411:{\displaystyle {\overline {\mathbb {F} }}_{p}}
5062:{\displaystyle U_{i}=Z\smallsetminus V(n_{i})}
3031:{\textstyle V(f)=\operatorname {Spec} (R/(f))}
1746:; the topological closure of the scheme point
282:are in one-to-one correspondence with the set
11008:Bulletin of the American Mathematical Society
9552:. This generalizes to a relation between the
8661:{\displaystyle \mathrm {Spec} \,\mathbb {C} }
4565:This also gives a geometric interpretaton of
4182:{\displaystyle \mathbb {Z} /(0)=\mathbb {Z} }
2395:{\displaystyle r=r(x_{1},\ldots ,x_{n})\in R}
1422:; its coordinate ring is the polynomial ring
142:of rings. The cases of main interest are the
9584:be the closed subscheme of the affine line A
9192:has enough information to determine the set
8563:
8557:
8274:{\displaystyle \mathbb {C} \mathbb {P} ^{n}}
5910:A higher degree "horizontal" subscheme like
5428:, and a "spatial direction" with coordinate
5134:
5106:
3741:
3687:
3655:
3596:
3516:
3499:
3360:, the principal ideals of the prime numbers
2264:
2189:
2079:
2033:
1846:
1796:
9212:. (In particular, the closed subscheme of A
9204:-rational points for every extension field
8446:{\displaystyle \operatorname {Proj} R/(f).}
7353:corresponding to Galois orbits of roots of
7333:contains the points in each characteristic
6449:{\displaystyle {\mathfrak {m}}=(3,x^{2}+1)}
6041:. This behaves differently under different
5663:-coordinate, we have the "horizontal line"
5227:, whose points are all of the prime ideals
39:in several ways, such as taking account of
11151:
10914:
10866:
10854:
10813:
10773:
10496:A further generalization is the idea of a
10406:-module that is the sheaf associated to a
9112:is not algebraically closed, then the set
8320:of positive degree in the polynomial ring
7965:are all singular schemes. For example, if
4209:is determined by its values at the points
686:
157:of a commutative ring; its points are the
11293:
11036:
11020:
10985:
10975:
10902:
10761:
10720:
10705:
10693:
10506:generalize algebraic spaces by having an
10308:, which are sheaves of abelian groups on
10282:, generalizing the notion of (algebraic)
10090:
9926:{\displaystyle (y)+(x^{2}-y)=(x^{2},\,y)}
9916:
9668:
9056:
8952:on the positive integers, with the ideal
8786:
8769:
8734:
8685:
8629:
8627:
8584:
8544:
8482:
8476:
8342:
8290:
8261:
8255:
8211:
8190:can be constructed as a scheme by gluing
8165:
8008:
7707:
7675:
7655:
7599:
7539:
7487:
7393:
7175:
7123:
6974:
6944:{\displaystyle d=\operatorname {deg} (f)}
6811:
6773:
6726:
6696:
6501:
6368:{\displaystyle (2,x^{2}+1)=(2,(x-1)^{2})}
5804:corresponding to the rational coordinate
5478:defines a "vertical line", the subscheme
5245:
5201:
5155:
5130:
5092:
5086:
4926:
4911:{\displaystyle (n_{1},\ldots ,n_{r})=(1)}
4489:
4478:
4315:
4175:
4153:
3992:
3969:
3870:
3788:
3636:
3570:
3459:
3374:
3317:. The Zariski topology has closed points
3300:
3263:
3126:
2917:
2251:
1319:
1218:
1185:
1156:
671:. In the spirit of scheme theory, affine
11213:Algebraic Geometry and Arithmetic Curves
11210:
10040:, but rather a fat point containing the
9631:if and only if they have the same value
9568:). Indeed, Grothendieck's theory of the
9012:
7449:. This is a major obstacle to analyzing
7447:intersect with the expected multiplicity
6997:corresponds to a function on the scheme
6662:{\displaystyle {\mathfrak {m}}=(p,f(x))}
6123:{\displaystyle x=\pm 2\ {\text{mod}}\ 5}
5587:{\displaystyle {\mathfrak {m}}=(p,f(x))}
5312:{\displaystyle {\mathfrak {m}}=(p,f(x))}
4138:{\displaystyle n({\mathfrak {p}}_{0})=n}
3884:corresponds to a function on the scheme
2873:{\displaystyle f=f(x_{1},\ldots ,x_{n})}
1667:{\displaystyle {\mathfrak {p}}\subset R}
1068:always exists. That is, for any schemes
11279:
11241:
11001:
10959:
10942:
10878:
10785:
10749:
9615:); in particular, the regular function
9332:-coordinate. The fiber of the morphism
9136:, but these sets are not determined by
8747:{\displaystyle f:X\to \mathbb {A} ^{n}}
4369:is the vanishing locus of the function
3424:{\displaystyle {\mathfrak {p}}_{0}=(0)}
3353:{\displaystyle {\mathfrak {m}}_{p}=(p)}
3250:
1395:{\displaystyle a=(a_{1},\ldots ,a_{n})}
1344:is the algebraic variety of all points
667:is the spectrum of the polynomial ring
637:A basic example of an affine scheme is
617:'s "Red Book". The sheaf properties of
161:of the ring, and its closed points are
11354:
11327:Can one explain schemes to biologists?
10920:
10791:
10680:Introduction of the first edition of "
9655:Nilpotent elements arise naturally in
9231:= −1 is a nonempty topological space.)
8996:, and in particular, each point is an
7468:
7445:, so that pairs of curves may fail to
2780:{\displaystyle r({\mathfrak {m}}_{a})}
2595:{\displaystyle R\to R/{\mathfrak {p}}}
1624:, the set of polynomials vanishing at
1116:exists in the category of schemes. If
109:Wiles's proof of Fermat's Last Theorem
10739:, Séminaire Henri Cartan, vol. 8
10437:can be viewed as a coherent sheaf on
8139:
5759:{\displaystyle {\mathfrak {p}}=(x-a)}
605:In the early days, this was called a
577:, which assigns to every open subset
11124:Publications Mathématiques de l'IHÉS
10316:over the sheaf of regular functions
10230:, or else a subscheme isomorphic to
9686:{\displaystyle \mathbb {A} _{k}^{2}}
9016:
8360:{\displaystyle \mathbb {P} _{R}^{n}}
8229:{\displaystyle \mathbb {P} _{R}^{n}}
8183:{\displaystyle \mathbb {P} _{R}^{n}}
7191:{\displaystyle r(x)\in \mathbb {Z} }
6990:{\displaystyle r(x)\in \mathbb {Z} }
6286:{\displaystyle x=1\ {\text{mod}}\ 2}
4983:{\displaystyle \rho _{i}=a_{i}n_{i}}
4940:. Indeed, we may consider the terms
4854:, then they generate the unit ideal
3144:{\displaystyle \mathbb {A} _{k}^{2}}
2819:The vanishing locus of a polynomial
1522:{\displaystyle X=\mathrm {Spec} (R)}
826:) of schemes and ring homomorphisms
438:
146:, in which the coordinate rings are
10267:
9603:. The ring of regular functions on
9068:Given some polynomial equations in
8375:, this subscheme can be written as
7213:
7095:
7065:
7032:
6716:
6684:
6627:
6410:
5733:
5552:
5536:{\displaystyle {\mathfrak {p}}=(p)}
5516:
5277:
5236:
4827:{\displaystyle {\mathfrak {m}}_{p}}
4813:
4796:{\displaystyle n_{1},\ldots ,n_{r}}
4654:{\displaystyle a_{1},\ldots ,a_{r}}
4608:{\displaystyle n_{1},\ldots ,n_{r}}
4555:{\displaystyle {\mathfrak {m}}_{p}}
4541:
4451:
4362:{\displaystyle {\mathfrak {m}}_{p}}
4348:
4296:
4233:{\displaystyle {\mathfrak {m}}_{p}}
4219:
4115:
4055:
3951:
3928:{\displaystyle {\mathfrak {m}}_{p}}
3914:
3723:
3693:
3505:
3398:
3327:
2763:
2680:
2652:
2626:{\displaystyle {\mathfrak {m}}_{a}}
2612:
2587:
2526:
2492:
2460:
2074:
2038:
1945:
1934:
1883:{\displaystyle {\mathfrak {m}}_{a}}
1869:
1841:
1831:
1801:
1785:
1755:
1713:
1653:
1539:
849:, the category of schemes has Spec(
798:on the rings of regular functions,
609:, and a scheme was defined to be a
245:, which has the advantage of being
16:Generalization of algebraic variety
13:
10468:
9448:This in turn is equivalent to the
9100:): define the Zariski topology on
8921:
8868:were affine, it would follow that
8623:
8620:
8617:
8614:
8453:For example, the closed subscheme
7831:
7224:{\displaystyle r({\mathfrak {m}})}
7056:
7053:
7050:
6485:{\displaystyle x=\pm {\sqrt {-1}}}
6456:is a prime ideal corresponding to
6034:{\displaystyle x=\pm {\sqrt {-1}}}
5643:points" of the scheme. Fixing the
5450:
5147:is a variety with coordinate ring
3764:
3539:
3437:
2787:corresponds to the original value
2740:{\displaystyle x_{i}\mapsto a_{i}}
2537:{\displaystyle r({\mathfrak {p}})}
2305:{\displaystyle {\mathcal {O}}_{X}}
2291:
2134:
1506:
1503:
1500:
1497:
1237:{\displaystyle \mathbb {A} ^{m+n}}
751:An algebraic variety over a field
546:admitting a covering by open sets
14:
11373:
11319:
10764:, sections VII.4, VIII.2, VIII.3.
10195:-module, i.e. its dimension as a
9638:at the origin. Allowing such non-
9317:double cover of the affine line A
9148:over the real numbers defined by
8676:extends to a regular function on
7509:{\displaystyle f\in \mathbb {Z} }
3877:{\displaystyle n\in \mathbb {Z} }
3466:{\displaystyle p\in \mathbb {Z} }
3381:{\displaystyle p\in \mathbb {Z} }
2500:{\displaystyle R/{\mathfrak {p}}}
449:Séminaire de géométrie algébrique
398:Foundations of Algebraic Geometry
87:(the last of which was proved by
10682:Éléments de géométrie algébrique
10290:, one starts by considering the
9933:. Since the intersection is not
9477:
9460:
9422:
9354:
9020:
8698:{\displaystyle \mathbb {A} ^{n}}
8303:{\displaystyle \mathbb {C} ^{n}}
7720:{\displaystyle \mathbb {F} _{p}}
6824:{\displaystyle \mathbb {F} _{p}}
6514:{\displaystyle \mathbb {F} _{3}}
4803:have no common vanishing points
3106:{\displaystyle x^{2}=y^{2}(y+1)}
1198:{\displaystyle \mathbb {A} ^{n}}
1169:{\displaystyle \mathbb {A} ^{m}}
1128:, their fiber product over Spec(
755:can be defined as a scheme over
445:Éléments de géométrie algébrique
350:) varieties. In particular, the
76:Éléments de géométrie algébrique
10963:Illinois Journal of Mathematics
10831:
9937:, this is not merely the point
9237:The points of the affine line A
8576:, say over the complex numbers
4994:subordinate to the covering of
4395:, the point where the value of
3388:; as well as the generic point
2707:, with the natural isomorphism
2468:{\displaystyle {\mathfrak {p}}}
1918:{\displaystyle a\in {\bar {V}}}
1763:{\displaystyle {\mathfrak {p}}}
1270:
481:he assigned a commutative ring
477:of rings: to every open subset
388:The algebraic geometers of the
334:. From the 1920s to the 1940s,
330:, essentially because they are
11282:Fundamental Algebraic Geometry
10869:, Exercises I.3.6 and III.4.3.
10726:
10674:
10491:Artin representability theorem
10410:on each affine open subset of
10182:
10170:
10133:
10120:
10115:
10109:
10094:
10074:
10069:
10057:
10033:{\displaystyle (x,y)\subset k}
10027:
10015:
10006:
9994:
9974:
9962:
9956:
9944:
9920:
9900:
9894:
9875:
9869:
9863:
9843:
9824:
9768:
9762:
9735:
9723:
9526:
9514:
9511:
9499:
9487:
9481:
9470:
9464:
9432:
9426:
9412:
9403:
9391:
9388:
9376:
9364:
9358:
8981:{\textstyle \prod _{m\neq n}k}
8851:
8845:
8822:
8790:
8779:
8764:
8729:
8655:
8633:
8437:
8431:
8423:
8391:
8326:determines a closed subscheme
8061:
8029:
8024:
8012:
7749:
7737:
7685:
7670:
7659:
7651:
7572:
7569:
7563:
7555:
7543:
7535:
7503:
7491:
7369:
7363:
7320:
7317:
7311:
7305:
7276:
7270:
7247:
7241:
7218:
7208:
7185:
7179:
7168:
7162:
7139:
7133:
7115:
7109:
7100:
7090:
7037:
7027:
6984:
6978:
6967:
6961:
6938:
6932:
6886:; this is a finite field with
6873:
6867:
6789:
6783:
6765:
6762:
6756:
6750:
6742:
6736:
6706:
6700:
6689:
6679:
6656:
6653:
6647:
6635:
6602:
6583:
6573:as two fused points. Overall,
6560:
6535:
6443:
6418:
6362:
6353:
6340:
6331:
6325:
6300:
6210:
6192:
6186:
6168:
6162:
6137:
5939:
5920:
5854:
5848:
5791:
5776:
5753:
5741:
5711:
5699:
5610:
5604:
5581:
5578:
5572:
5560:
5530:
5524:
5494:
5488:
5355:
5349:
5306:
5303:
5297:
5285:
5255:
5249:
5214:
5211:
5205:
5197:
5165:
5159:
5056:
5043:
4905:
4899:
4893:
4861:
4482:
4474:
4462:
4445:
4307:
4290:
4168:
4162:
4126:
4109:
4066:
4049:
3984:
3978:
3962:
3945:
3840:
3792:
3781:
3775:
3590:
3574:
3563:
3550:
3418:
3412:
3347:
3341:
3304:
3296:
3234:
3231:
3219:
3193:
3185:
3173:
3151:, corresponding to the scheme
3100:
3088:
3025:
3022:
3016:
3005:
2993:
2987:
2977:. The corresponding scheme is
2958:
2952:
2909:
2903:
2897:
2867:
2835:
2803:
2797:
2774:
2757:
2724:
2663:
2646:
2574:
2531:
2521:
2415:
2383:
2351:
2183:
2167:
2158:
2145:
2027:
2021:
1909:
1790:
1780:
1730:
1718:
1708:
1702:
1687:
1611:
1553:
1516:
1510:
1470:
1438:
1389:
1357:
1308:
216:
1:
11286:American Mathematical Society
11044:History of Algebraic Geometry
11022:10.1090/S0273-0979-01-00913-2
10952:
10905:, sections VIII.2 and VIII.3.
5073:Affine line over the integers
4569:stating that if the integers
1109:{\displaystyle X\times _{Y}Z}
947:More generally, for a scheme
847:category of commutative rings
513:
11332:The Stacks Project Authors,
11184:Igor R. Shafarevich (2013).
10667:
10566:. In this setting, known as
10559:that is an algebraic space.
10250:together with a line in the
9644:schemes brings the ideas of
9184:− 0.) By contrast, a scheme
8992:. This topological space is
8887:be a field. Then the scheme
8754:induces an isomorphism from
8591:{\displaystyle \mathbb {C} }
7585:has a canonical morphism to
7473:If we consider a polynomial
7397:
7198:is determined by its values
6566:{\displaystyle V(3,x^{2}+1)}
5171:{\displaystyle \mathbb {Z} }
4933:{\displaystyle \mathbb {Z} }
4145:in the generic residue ring
3904:, a function whose value at
3270:{\displaystyle \mathbb {Z} }
2970:{\displaystyle (f)\subset R}
2281:This induces a unique sheaf
2095:for irreducible polynomials
932:) is also called the set of
184:of schemes (called a scheme
35:that enlarges the notion of
7:
10733:Chevalley, C. (1955–1956),
10708:, sections VII.2 and VII.5.
10589:
9980:{\displaystyle (x,y)=(0,0)}
9752:axis, which is the variety
9659:. For example in the plane
9000:. Since affine schemes are
8946:Stone–Čech compactification
5972:-values which are roots of
5838:, which does not intersect
4039:the function is defined by
3433:closure is the whole scheme
2402:, a polynomial function on
1483:. The corresponding scheme
1262:
1076:with morphisms to a scheme
419:, motivated in part by the
10:
11378:
10891:Eisenbud & Harris 1998
10826:Eisenbud & Harris 1998
10657:Moduli of algebraic curves
10568:derived algebraic geometry
10325:. In particular, a module
10271:
9849:{\displaystyle V(x^{2}-y)}
9564:(which classifies certain
8709:≥ 2: this is analogous to
6608:{\displaystyle V(x^{2}+1)}
5945:{\displaystyle V(x^{2}+1)}
4531:has "poles" at the points
3935:lies in the residue field
2475:lies in the quotient ring
2424:{\displaystyle {\bar {X}}}
542:is a locally ringed space
253:, and more generally over
107:, which eventually led to
10463:coherent sheaf cohomology
10408:finitely generated module
10154:intersection multiplicity
9180:) can be identified with
8145:For any commutative ring
8077:then its discriminant is
7382:in the algebraic closure
6850:{\displaystyle x=\alpha }
6492:in an extension field of
4240:only, so we can think of
1124:are schemes over a field
1082:categorical fiber product
596:ring of regular functions
358:of the complex numbers).
265:Hilbert's Nullstellensatz
211:glossary of scheme theory
10929:Stacks Project, Tag 07Y1
10800:Stacks Project, Tag 020D
10518:on an algebraic variety
10393:(on a Noetherian scheme
10329:over a commutative ring
9652:into algebraic geometry.
9599:= 0, sometimes called a
9340:, yielding the morphism
9313:. It can be viewed as a
9076:, one can study the set
8246:complex projective space
8103:{\displaystyle -27p^{2}}
5766:. We also have the line
3431:, the zero ideal, whose
1066:fiber product of schemes
951:over a commutative ring
864:over a commutative ring
794:of schemes determines a
630:) mean that its elements
534:) of a commutative ring
73:in 1960 in his treatise
11217:Oxford University Press
11111:Grothendieck, Alexandre
11080:The Geometry of Schemes
10572:E-infinity ring spectra
10156:of 2 is defined as the
9807:{\displaystyle y=x^{2}}
9713:, with coordinate ring
9570:étale fundamental group
9452:-2 extension of fields
9172:) not empty. (In fact,
9072:variables over a field
8872:is an isomorphism, but
8369:projective hypersurface
7516:then the affine scheme
7326:{\displaystyle V(f(x))}
6802:, a field extension of
5998:{\displaystyle x^{2}+1}
5797:{\displaystyle V(bx-a)}
5388:. Thus, we may picture
5339:is a prime number, and
4918:in the coordinate ring
4735:Hilbert Nullstellensatz
3529:, with coordinate ring
2942:, corresponding to the
687:The category of schemes
290:-tuples of elements of
10987:10.1215/ijm/1373636688
10648:Linear algebraic group
10545:Deligne–Mumford stacks
10528:, which remembers the
10514:of an algebraic group
10209:
10189:
10146:
10034:
9981:
9927:
9850:
9808:
9775:
9742:
9707:
9687:
9542:
9442:
9057:Motivation for schemes
8982:
8938:
8925:
8858:
8829:
8748:
8699:
8662:
8592:
8570:
8497:
8447:
8361:
8315:homogeneous polynomial
8304:
8275:
8230:
8184:
8130:
8104:
8071:
7985:is a prime number and
7979:
7959:
7813:
7721:
7692:
7607:
7579:
7510:
7435:
7412:
7376:
7347:
7327:
7289:
7288:{\displaystyle r(x)=p}
7254:
7225:
7192:
7146:
7074:
7011:
6991:
6945:
6907:
6880:
6851:
6825:
6796:
6663:
6609:
6567:
6515:
6486:
6450:
6395:
6369:
6287:
6243:
6217:
6124:
6081:
6055:
6035:
5999:
5966:
5946:
5901:
5881:
5861:
5832:
5798:
5760:
5718:
5717:{\displaystyle V(x-a)}
5683:
5657:
5637:
5623:, the "characteristic
5617:
5588:
5537:
5501:
5472:
5455:
5442:
5422:
5402:
5382:
5362:
5333:
5313:
5262:
5221:
5172:
5141:
5063:
5008:
4984:
4934:
4912:
4848:
4828:
4797:
4751:
4727:
4655:
4609:
4556:
4525:
4497:
4429:
4409:
4389:
4363:
4329:
4274:
4254:
4234:
4203:
4183:
4139:
4093:
4030:
4006:
3929:
3898:
3878:
3847:
3748:
3662:
3523:
3467:
3442:
3425:
3382:
3354:
3311:
3271:
3241:
3145:
3107:
3052:
3032:
2971:
2936:
2874:
2810:
2781:
2741:
2701:
2627:
2596:
2564:under the natural map
2558:
2538:
2501:
2469:
2445:
2425:
2396:
2326:
2306:
2279:
2271:
2115:
2114:{\displaystyle f\in R}
2093:
2086:
1983:
1960:
1919:
1884:
1853:
1764:
1740:
1668:
1638:
1618:
1523:
1477:
1416:
1396:
1338:
1289:
1238:
1199:
1170:
1110:
1057:is determined by this
978:means a morphism Spec(
174:relative point of view
71:Alexander Grothendieck
10816:, Proposition II.2.3.
10441:that is zero outside
10210:
10190:
10147:
10035:
9987:defined by the ideal
9982:
9928:
9851:
9809:
9776:
9743:
9708:
9688:
9560:in topology) and the
9543:
9443:
9328:by projecting to the
9013:Examples of morphisms
8998:irreducible component
8983:
8939:
8905:
8859:
8830:
8749:
8700:
8663:
8593:
8571:
8516:line with two origins
8498:
8448:
8362:
8305:
8276:
8231:
8194:+ 1 copies of affine
8185:
8131:
8105:
8072:
7980:
7960:
7814:
7722:
7693:
7608:
7580:
7511:
7451:Diophantine equations
7436:
7413:
7377:
7348:
7328:
7290:
7255:
7226:
7193:
7147:
7075:
7012:
6992:
6946:
6908:
6906:{\displaystyle p^{d}}
6881:
6852:
6826:
6797:
6664:
6618:The residue field at
6610:
6568:
6525:), we should picture
6516:
6487:
6451:
6396:
6370:
6288:
6244:
6218:
6125:
6082:
6056:
6036:
6000:
5967:
5947:
5902:
5882:
5862:
5833:
5831:{\displaystyle x=a/b}
5799:
5761:
5719:
5684:
5658:
5638:
5618:
5589:
5538:
5502:
5473:
5458:A given prime number
5454:
5443:
5423:
5403:
5383:
5363:
5334:
5314:
5263:
5222:
5173:
5142:
5064:
5009:
4990:as forming a kind of
4985:
4935:
4913:
4849:
4829:
4798:
4752:
4728:
4656:
4610:
4557:
4526:
4498:
4430:
4410:
4390:
4364:
4330:
4275:
4255:
4235:
4204:
4184:
4140:
4094:
4031:
4007:
3930:
3899:
3879:
3848:
3749:
3663:
3524:
3468:
3441:
3426:
3383:
3355:
3312:
3272:
3255:The ring of integers
3242:
3146:
3108:
3053:
3033:
2972:
2937:
2875:
2811:
2782:
2742:
2702:
2628:
2597:
2559:
2539:
2502:
2470:
2446:
2426:
2397:
2327:
2307:
2272:
2123:
2116:
2087:
1991:
1984:
1961:
1920:
1885:
1854:
1765:
1741:
1669:
1639:
1619:
1524:
1478:
1417:
1397:
1339:
1290:
1255:), it has all finite
1239:
1200:
1171:
1111:
796:pullback homomorphism
729:of schemes. A scheme
498:. These objects Spec(
336:B. L. van der Waerden
91:). Strongly based on
59:are defined over the
10696:, Chapters IV and V.
10530:stabilizer subgroups
10500:. Crudely speaking,
10425:. An example is the
10369:quasi-coherent sheaf
10199:
10164:
10048:
9991:
9941:
9860:
9818:
9785:
9774:{\displaystyle V(y)}
9756:
9717:
9697:
9663:
9456:
9344:
8956:
8891:
8857:{\displaystyle O(X)}
8839:
8758:
8717:
8680:
8610:
8580:
8533:
8471:
8379:
8337:
8333:in projective space
8285:
8251:
8206:
8160:
8114:
8081:
7989:
7969:
7827:
7731:
7702:
7621:
7589:
7520:
7477:
7425:
7386:
7375:{\displaystyle f(x)}
7357:
7337:
7299:
7264:
7253:{\displaystyle V(p)}
7235:
7202:
7156:
7084:
7021:
7001:
6955:
6917:
6890:
6879:{\displaystyle f(x)}
6861:
6835:
6806:
6673:
6622:
6577:
6529:
6496:
6460:
6405:
6379:
6297:
6257:
6227:
6134:
6091:
6087:, we get two points
6065:
6045:
6009:
5976:
5956:
5914:
5891:
5871:
5860:{\displaystyle V(p)}
5842:
5808:
5770:
5728:
5693:
5667:
5647:
5627:
5616:{\displaystyle f(x)}
5598:
5547:
5511:
5500:{\displaystyle V(p)}
5482:
5462:
5432:
5412:
5392:
5372:
5361:{\displaystyle f(x)}
5343:
5323:
5272:
5231:
5182:
5151:
5081:
5018:
4998:
4944:
4922:
4858:
4838:
4807:
4761:
4741:
4665:
4619:
4573:
4535:
4507:
4439:
4419:
4399:
4373:
4342:
4338:Note that the point
4284:
4264:
4244:
4213:
4193:
4149:
4103:
4043:
4020:
3939:
3908:
3888:
3860:
3758:
3672:
3533:
3477:
3449:
3392:
3364:
3321:
3281:
3259:
3251:Spec of the integers
3155:
3121:
3117:in the affine plane
3062:
3042:
2981:
2949:
2888:
2823:
2809:{\displaystyle r(a)}
2791:
2751:
2711:
2640:
2606:
2568:
2548:
2515:
2479:
2455:
2435:
2406:
2339:
2316:
2285:
2128:
2099:
1996:
1973:
1929:
1894:
1863:
1774:
1750:
1678:
1648:
1628:
1533:
1487:
1426:
1406:
1402:with coordinates in
1348:
1299:
1279:
1213:
1180:
1151:
1132:) may be called the
1087:
955:and any commutative
524:locally ringed space
447:(EGA) and the later
328:projective varieties
247:algebraically closed
231:Jean-Victor Poncelet
11304:2004math.....12512V
11188:. Springer-Verlag.
11161:. Springer-Verlag.
10857:, Example II.4.0.1.
10828:, Proposition VI-2.
10622:Birational geometry
10557:coarse moduli space
9781:, and the parabola
9682:
9657:intersection theory
9286:be the plane curve
8492:
8356:
8225:
8179:
8149:and natural number
8129:{\displaystyle 3,p}
7469:Arithmetic surfaces
6394:{\displaystyle p=3}
6242:{\displaystyle p=2}
6080:{\displaystyle p=5}
5724:of the prime ideal
5682:{\displaystyle x=a}
5507:of the prime ideal
5102:
4757:: if the functions
4524:{\displaystyle a/b}
4388:{\displaystyle n=p}
4016:of integers modulo
3839:
3812:
3668:. For the open set
3140:
2931:
1476:{\displaystyle R=k}
1333:
1143:in the category of
736:a commutative ring
721:) means a morphism
697:morphism of schemes
581:a commutative ring
568:comes with a sheaf
225:equations over the
101:homological algebra
93:commutative algebra
11335:The Stacks Project
11288:, pp. 1–104,
11284:, Providence, RI:
11158:Algebraic Geometry
11137:10.1007/bf02684778
10532:for the action of
10205:
10185:
10142:
10030:
9977:
9923:
9846:
9804:
9771:
9738:
9703:
9683:
9666:
9578:Nilpotent elements
9556:(which classifies
9538:
9438:
9277:rational functions
9275:) is the field of
9032:. You can help by
8978:
8974:
8934:
8854:
8825:
8744:
8695:
8658:
8602:is not affine for
8588:
8566:
8493:
8474:
8443:
8371:. In terms of the
8357:
8340:
8300:
8271:
8226:
8209:
8180:
8163:
8140:Non-affine schemes
8126:
8100:
8067:
7975:
7955:
7809:
7717:
7688:
7615:arithmetic surface
7603:
7575:
7506:
7431:
7408:
7372:
7343:
7323:
7285:
7250:
7231:at closed points;
7221:
7188:
7142:
7070:
7007:
6987:
6941:
6903:
6876:
6847:
6821:
6792:
6659:
6605:
6563:
6511:
6482:
6446:
6391:
6365:
6283:
6239:
6213:
6120:
6077:
6051:
6031:
5995:
5962:
5942:
5897:
5877:
5857:
5828:
5794:
5756:
5714:
5679:
5653:
5633:
5613:
5584:
5533:
5497:
5468:
5456:
5438:
5418:
5398:
5378:
5358:
5329:
5309:
5258:
5217:
5168:
5137:
5084:
5059:
5004:
4992:partition of unity
4980:
4930:
4908:
4844:
4824:
4793:
4747:
4723:
4651:
4605:
4552:
4521:
4493:
4425:
4405:
4385:
4359:
4325:
4270:
4250:
4230:
4199:
4179:
4135:
4089:
4026:
4002:
3925:
3894:
3874:
3843:
3822:
3795:
3744:
3658:
3616:
3519:
3463:
3443:
3421:
3378:
3350:
3307:
3267:
3237:
3141:
3124:
3103:
3048:
3028:
2967:
2932:
2915:
2870:
2806:
2777:
2737:
2697:
2623:
2602:. A maximal ideal
2592:
2554:
2534:
2497:
2465:
2441:
2421:
2392:
2335:Each ring element
2322:
2302:
2267:
2209:
2111:
2082:
1979:
1956:
1925:, or equivalently
1915:
1880:
1849:
1760:
1736:
1664:
1634:
1614:
1519:
1473:
1412:
1392:
1334:
1317:
1285:
1234:
1195:
1166:
1106:
769:integral separated
656:. By definition, A
526:isomorphic to the
196:algebraic surfaces
167:rings of fractions
144:Noetherian schemes
81:algebraic geometry
69:was introduced by
25:algebraic geometry
11264:978-3-540-63293-1
11226:978-0-19-850284-5
11211:Qing Liu (2002).
11168:978-0-387-90244-9
11153:Hartshorne, Robin
11094:978-0-387-98637-1
11055:978-0-534-03723-9
10945:, Definition 4.6.
10839:"Elliptic curves"
10553:Keel–Mori theorem
10208:{\displaystyle k}
10188:{\displaystyle k}
10137:
10098:
9741:{\displaystyle k}
9706:{\displaystyle k}
9554:fundamental group
9529:
9406:
9066:Field extensions.
9050:
9049:
9006:Noetherian scheme
8959:
8373:Proj construction
8065:
7978:{\displaystyle p}
7948:
7944:
7940:
7613:and is called an
7434:{\displaystyle Y}
7400:
7346:{\displaystyle p}
7062:
7048:
7010:{\displaystyle Y}
6831:adjoining a root
6480:
6279:
6275:
6271:
6116:
6112:
6108:
6061:-coordinates. At
6054:{\displaystyle p}
6029:
5965:{\displaystyle x}
5900:{\displaystyle b}
5880:{\displaystyle p}
5656:{\displaystyle x}
5636:{\displaystyle p}
5471:{\displaystyle p}
5441:{\displaystyle x}
5421:{\displaystyle p}
5401:{\displaystyle Y}
5381:{\displaystyle p}
5332:{\displaystyle p}
5122:
5118:
5114:
5077:The affine space
5014:by the open sets
5007:{\displaystyle Z}
4847:{\displaystyle Z}
4750:{\displaystyle Z}
4428:{\displaystyle Z}
4408:{\displaystyle p}
4273:{\displaystyle f}
4253:{\displaystyle n}
4202:{\displaystyle n}
4085:
4081:
4077:
4029:{\displaystyle p}
3897:{\displaystyle Z}
3645:
3628:
3624:
3620:
3615:
3115:nodal cubic curve
3051:{\displaystyle k}
2900:
2557:{\displaystyle r}
2444:{\displaystyle X}
2418:
2325:{\displaystyle U}
2242:
2227:
2224:
2220:
2216:
2213:
2208:
2065:
2062:
2058:
2054:
2051:
1982:{\displaystyle X}
1912:
1828:
1825:
1821:
1817:
1814:
1770:is the subscheme
1733:
1705:
1690:
1637:{\displaystyle a}
1415:{\displaystyle k}
1311:
1288:{\displaystyle k}
1059:functor of points
1045:from commutative
1029:, the assignment
998:) for the set of
900:) for the set of
740:means a morphism
555:, such that each
439:Origin of schemes
417:Jean-Pierre Serre
365:(not embedded in
124:topological space
37:algebraic variety
11369:
11338:
11314:
11297:
11276:
11238:
11207:
11180:
11148:
11106:
11066:
11047:
11033:
11024:
10998:
10989:
10979:
10970:(4): 1367–1384,
10946:
10940:
10934:
10932:
10924:
10918:
10912:
10906:
10900:
10894:
10893:, Example II-10.
10888:
10882:
10876:
10870:
10864:
10858:
10852:
10846:
10845:
10843:
10835:
10829:
10823:
10817:
10811:
10805:
10803:
10795:
10789:
10783:
10777:
10771:
10765:
10759:
10753:
10747:
10741:
10740:
10730:
10724:
10723:, section VII.4.
10718:
10709:
10703:
10697:
10691:
10685:
10678:
10627:Étale cohomology
10576:derived functors
10503:algebraic stacks
10459:sheaf cohomology
10357:
10356:
10355:
10350:
10292:abelian category
10280:coherent sheaves
10268:Coherent sheaves
10222:complex variety
10214:
10212:
10211:
10206:
10194:
10192:
10191:
10186:
10151:
10149:
10148:
10143:
10138:
10136:
10132:
10131:
10118:
10104:
10099:
10097:
10086:
10085:
10072:
10052:
10039:
10037:
10036:
10031:
9986:
9984:
9983:
9978:
9932:
9930:
9929:
9924:
9912:
9911:
9887:
9886:
9855:
9853:
9852:
9847:
9836:
9835:
9813:
9811:
9810:
9805:
9803:
9802:
9780:
9778:
9777:
9772:
9747:
9745:
9744:
9739:
9712:
9710:
9709:
9704:
9692:
9690:
9689:
9684:
9681:
9676:
9671:
9594:
9593:
9566:field extensions
9547:
9545:
9544:
9539:
9534:
9530:
9495:
9480:
9463:
9447:
9445:
9444:
9439:
9425:
9411:
9407:
9372:
9357:
9327:
9326:
9312:
9311:
9266:
9265:
9247:
9246:
9222:
9221:
9045:
9042:
9024:
9017:
8994:zero-dimensional
8987:
8985:
8984:
8979:
8973:
8943:
8941:
8940:
8935:
8933:
8929:
8924:
8919:
8863:
8861:
8860:
8855:
8834:
8832:
8831:
8826:
8821:
8820:
8802:
8801:
8789:
8778:
8777:
8772:
8753:
8751:
8750:
8745:
8743:
8742:
8737:
8704:
8702:
8701:
8696:
8694:
8693:
8688:
8667:
8665:
8664:
8659:
8654:
8653:
8632:
8626:
8597:
8595:
8594:
8589:
8587:
8575:
8573:
8572:
8567:
8553:
8552:
8547:
8509:rational numbers
8502:
8500:
8499:
8494:
8491:
8486:
8485:
8479:
8466:
8452:
8450:
8449:
8444:
8430:
8422:
8421:
8403:
8402:
8366:
8364:
8363:
8358:
8355:
8350:
8345:
8332:
8325:
8309:
8307:
8306:
8301:
8299:
8298:
8293:
8280:
8278:
8277:
8272:
8270:
8269:
8264:
8258:
8235:
8233:
8232:
8227:
8224:
8219:
8214:
8189:
8187:
8186:
8181:
8178:
8173:
8168:
8155:projective space
8135:
8133:
8132:
8127:
8109:
8107:
8106:
8101:
8099:
8098:
8076:
8074:
8073:
8068:
8066:
8064:
8054:
8053:
8041:
8040:
8027:
8011:
8005:
7984:
7982:
7981:
7976:
7964:
7962:
7961:
7956:
7946:
7945:
7942:
7938:
7931:
7930:
7915:
7914:
7884:
7883:
7874:
7873:
7858:
7857:
7839:
7838:
7818:
7816:
7815:
7810:
7793:
7792:
7777:
7776:
7764:
7763:
7726:
7724:
7723:
7718:
7716:
7715:
7710:
7697:
7695:
7694:
7689:
7684:
7683:
7678:
7663:
7662:
7658:
7633:
7632:
7612:
7610:
7609:
7604:
7602:
7584:
7582:
7581:
7576:
7562:
7542:
7515:
7513:
7512:
7507:
7490:
7440:
7438:
7437:
7432:
7417:
7415:
7414:
7409:
7407:
7406:
7401:
7396:
7391:
7381:
7379:
7378:
7373:
7352:
7350:
7349:
7344:
7332:
7330:
7329:
7324:
7294:
7292:
7291:
7286:
7259:
7257:
7256:
7251:
7230:
7228:
7227:
7222:
7217:
7216:
7197:
7195:
7194:
7189:
7178:
7151:
7149:
7148:
7143:
7132:
7131:
7126:
7099:
7098:
7079:
7077:
7076:
7071:
7069:
7068:
7060:
7059:
7046:
7036:
7035:
7016:
7014:
7013:
7008:
6996:
6994:
6993:
6988:
6977:
6950:
6948:
6947:
6942:
6912:
6910:
6909:
6904:
6902:
6901:
6885:
6883:
6882:
6877:
6856:
6854:
6853:
6848:
6830:
6828:
6827:
6822:
6820:
6819:
6814:
6801:
6799:
6798:
6793:
6782:
6781:
6776:
6749:
6735:
6734:
6729:
6720:
6719:
6713:
6699:
6688:
6687:
6668:
6666:
6665:
6660:
6631:
6630:
6614:
6612:
6611:
6606:
6595:
6594:
6572:
6570:
6569:
6564:
6553:
6552:
6520:
6518:
6517:
6512:
6510:
6509:
6504:
6491:
6489:
6488:
6483:
6481:
6473:
6455:
6453:
6452:
6447:
6436:
6435:
6414:
6413:
6400:
6398:
6397:
6392:
6374:
6372:
6371:
6366:
6361:
6360:
6318:
6317:
6292:
6290:
6289:
6284:
6277:
6276:
6273:
6269:
6248:
6246:
6245:
6240:
6222:
6220:
6219:
6214:
6155:
6154:
6129:
6127:
6126:
6121:
6114:
6113:
6110:
6106:
6086:
6084:
6083:
6078:
6060:
6058:
6057:
6052:
6040:
6038:
6037:
6032:
6030:
6022:
6004:
6002:
6001:
5996:
5988:
5987:
5971:
5969:
5968:
5963:
5951:
5949:
5948:
5943:
5932:
5931:
5906:
5904:
5903:
5898:
5886:
5884:
5883:
5878:
5866:
5864:
5863:
5858:
5837:
5835:
5834:
5829:
5824:
5803:
5801:
5800:
5795:
5765:
5763:
5762:
5757:
5737:
5736:
5723:
5721:
5720:
5715:
5689:, the subscheme
5688:
5686:
5685:
5680:
5662:
5660:
5659:
5654:
5642:
5640:
5639:
5634:
5622:
5620:
5619:
5614:
5593:
5591:
5590:
5585:
5556:
5555:
5543:: this contains
5542:
5540:
5539:
5534:
5520:
5519:
5506:
5504:
5503:
5498:
5477:
5475:
5474:
5469:
5447:
5445:
5444:
5439:
5427:
5425:
5424:
5419:
5407:
5405:
5404:
5399:
5387:
5385:
5384:
5379:
5367:
5365:
5364:
5359:
5338:
5336:
5335:
5330:
5318:
5316:
5315:
5310:
5281:
5280:
5267:
5265:
5264:
5259:
5248:
5240:
5239:
5226:
5224:
5223:
5218:
5204:
5177:
5175:
5174:
5169:
5158:
5146:
5144:
5143:
5138:
5133:
5120:
5119:
5116:
5112:
5101:
5096:
5095:
5089:
5068:
5066:
5065:
5060:
5055:
5054:
5030:
5029:
5013:
5011:
5010:
5005:
4989:
4987:
4986:
4981:
4979:
4978:
4969:
4968:
4956:
4955:
4939:
4937:
4936:
4931:
4929:
4917:
4915:
4914:
4909:
4892:
4891:
4873:
4872:
4853:
4851:
4850:
4845:
4833:
4831:
4830:
4825:
4823:
4822:
4817:
4816:
4802:
4800:
4799:
4794:
4792:
4791:
4773:
4772:
4756:
4754:
4753:
4748:
4732:
4730:
4729:
4724:
4716:
4715:
4706:
4705:
4687:
4686:
4677:
4676:
4660:
4658:
4657:
4652:
4650:
4649:
4631:
4630:
4614:
4612:
4611:
4606:
4604:
4603:
4585:
4584:
4561:
4559:
4558:
4553:
4551:
4550:
4545:
4544:
4530:
4528:
4527:
4522:
4517:
4502:
4500:
4499:
4494:
4492:
4481:
4461:
4460:
4455:
4454:
4434:
4432:
4431:
4426:
4414:
4412:
4411:
4406:
4394:
4392:
4391:
4386:
4368:
4366:
4365:
4360:
4358:
4357:
4352:
4351:
4334:
4332:
4331:
4326:
4324:
4323:
4318:
4306:
4305:
4300:
4299:
4279:
4277:
4276:
4271:
4259:
4257:
4256:
4251:
4239:
4237:
4236:
4231:
4229:
4228:
4223:
4222:
4208:
4206:
4205:
4200:
4188:
4186:
4185:
4180:
4178:
4161:
4156:
4144:
4142:
4141:
4136:
4125:
4124:
4119:
4118:
4098:
4096:
4095:
4090:
4083:
4082:
4079:
4075:
4065:
4064:
4059:
4058:
4035:
4033:
4032:
4027:
4011:
4009:
4008:
4003:
4001:
4000:
3995:
3977:
3972:
3961:
3960:
3955:
3954:
3934:
3932:
3931:
3926:
3924:
3923:
3918:
3917:
3903:
3901:
3900:
3895:
3883:
3881:
3880:
3875:
3873:
3852:
3850:
3849:
3844:
3838:
3830:
3811:
3803:
3791:
3774:
3773:
3768:
3767:
3753:
3751:
3750:
3745:
3740:
3739:
3738:
3737:
3727:
3726:
3710:
3709:
3708:
3707:
3697:
3696:
3667:
3665:
3664:
3659:
3643:
3639:
3626:
3625:
3622:
3618:
3617:
3614:
3613:
3601:
3589:
3588:
3573:
3562:
3561:
3549:
3548:
3543:
3542:
3528:
3526:
3525:
3520:
3515:
3514:
3509:
3508:
3489:
3488:
3472:
3470:
3469:
3464:
3462:
3430:
3428:
3427:
3422:
3408:
3407:
3402:
3401:
3387:
3385:
3384:
3379:
3377:
3359:
3357:
3356:
3351:
3337:
3336:
3331:
3330:
3316:
3314:
3313:
3308:
3303:
3276:
3274:
3273:
3268:
3266:
3246:
3244:
3243:
3238:
3218:
3217:
3205:
3204:
3192:
3150:
3148:
3147:
3142:
3139:
3134:
3129:
3112:
3110:
3109:
3104:
3087:
3086:
3074:
3073:
3057:
3055:
3054:
3049:
3037:
3035:
3034:
3029:
3015:
2976:
2974:
2973:
2968:
2941:
2939:
2938:
2933:
2930:
2925:
2920:
2902:
2901:
2893:
2879:
2877:
2876:
2871:
2866:
2865:
2847:
2846:
2815:
2813:
2812:
2807:
2786:
2784:
2783:
2778:
2773:
2772:
2767:
2766:
2746:
2744:
2743:
2738:
2736:
2735:
2723:
2722:
2706:
2704:
2703:
2698:
2690:
2689:
2684:
2683:
2676:
2662:
2661:
2656:
2655:
2632:
2630:
2629:
2624:
2622:
2621:
2616:
2615:
2601:
2599:
2598:
2593:
2591:
2590:
2584:
2563:
2561:
2560:
2555:
2544:as the image of
2543:
2541:
2540:
2535:
2530:
2529:
2506:
2504:
2503:
2498:
2496:
2495:
2489:
2474:
2472:
2471:
2466:
2464:
2463:
2450:
2448:
2447:
2442:
2430:
2428:
2427:
2422:
2420:
2419:
2411:
2401:
2399:
2398:
2393:
2382:
2381:
2363:
2362:
2331:
2329:
2328:
2323:
2311:
2309:
2308:
2303:
2301:
2300:
2295:
2294:
2276:
2274:
2273:
2268:
2263:
2262:
2254:
2240:
2225:
2222:
2221:
2218:
2214:
2211:
2210:
2207:
2206:
2194:
2182:
2181:
2157:
2156:
2144:
2143:
2138:
2137:
2120:
2118:
2117:
2112:
2091:
2089:
2088:
2083:
2078:
2077:
2063:
2060:
2059:
2056:
2052:
2049:
2042:
2041:
2008:
2007:
1988:
1986:
1985:
1980:
1965:
1963:
1962:
1957:
1955:
1954:
1949:
1948:
1938:
1937:
1924:
1922:
1921:
1916:
1914:
1913:
1905:
1889:
1887:
1886:
1881:
1879:
1878:
1873:
1872:
1858:
1856:
1855:
1850:
1845:
1844:
1835:
1834:
1826:
1823:
1822:
1819:
1815:
1812:
1805:
1804:
1789:
1788:
1769:
1767:
1766:
1761:
1759:
1758:
1745:
1743:
1742:
1737:
1735:
1734:
1726:
1717:
1716:
1707:
1706:
1698:
1692:
1691:
1683:
1673:
1671:
1670:
1665:
1657:
1656:
1643:
1641:
1640:
1635:
1623:
1621:
1620:
1615:
1610:
1609:
1597:
1596:
1578:
1577:
1565:
1564:
1549:
1548:
1543:
1542:
1528:
1526:
1525:
1520:
1509:
1482:
1480:
1479:
1474:
1469:
1468:
1450:
1449:
1421:
1419:
1418:
1413:
1401:
1399:
1398:
1393:
1388:
1387:
1369:
1368:
1343:
1341:
1340:
1335:
1332:
1327:
1322:
1313:
1312:
1304:
1294:
1292:
1291:
1286:
1243:
1241:
1240:
1235:
1233:
1232:
1221:
1209:is affine space
1204:
1202:
1201:
1196:
1194:
1193:
1188:
1175:
1173:
1172:
1167:
1165:
1164:
1159:
1115:
1113:
1112:
1107:
1102:
1101:
1021:.) For a scheme
884:of the morphism
699:.) For a scheme
666:
665:
465:as the space of
460:commutative ring
421:Weil conjectures
413:Masayoshi Nagata
409:Claude Chevalley
402:universal domain
375:Jacobian variety
367:projective space
363:abstract variety
352:Zariski topology
348:quasi-projective
259:complex analysis
235:Bernhard Riemann
148:Noetherian rings
85:Weil conjectures
53:commutative ring
11377:
11376:
11372:
11371:
11370:
11368:
11367:
11366:
11352:
11351:
11325:David Mumford,
11322:
11317:
11265:
11227:
11196:
11169:
11115:Dieudonné, Jean
11095:
11085:Springer-Verlag
11071:Eisenbud, David
11056:
11038:Dieudonné, Jean
11003:Cartier, Pierre
10955:
10950:
10949:
10941:
10937:
10926:
10925:
10921:
10915:Hartshorne 1997
10913:
10909:
10901:
10897:
10889:
10885:
10877:
10873:
10867:Hartshorne 1997
10865:
10861:
10855:Hartshorne 1997
10853:
10849:
10841:
10837:
10836:
10832:
10824:
10820:
10814:Hartshorne 1997
10812:
10808:
10797:
10796:
10792:
10784:
10780:
10776:, section II.2.
10774:Hartshorne 1997
10772:
10768:
10760:
10756:
10748:
10744:
10731:
10727:
10719:
10712:
10704:
10700:
10692:
10688:
10679:
10675:
10670:
10652:Reductive group
10644:Abelian variety
10608:Finite morphism
10604:Proper morphism
10600:Smooth morphism
10592:
10564:homotopy theory
10508:algebraic group
10485:algebraic space
10471:
10469:Generalizations
10405:
10384:
10351:
10348:
10347:
10346:
10344:
10324:
10304:
10286:. For a scheme
10276:
10270:
10259:
10200:
10197:
10196:
10165:
10162:
10161:
10127:
10123:
10119:
10105:
10103:
10081:
10077:
10073:
10053:
10051:
10049:
10046:
10045:
9992:
9989:
9988:
9942:
9939:
9938:
9907:
9903:
9882:
9878:
9861:
9858:
9857:
9831:
9827:
9819:
9816:
9815:
9798:
9794:
9786:
9783:
9782:
9757:
9754:
9753:
9748:, consider the
9718:
9715:
9714:
9698:
9695:
9694:
9677:
9672:
9667:
9664:
9661:
9660:
9592:
9587:
9586:
9585:
9558:covering spaces
9550:function fields
9494:
9490:
9476:
9459:
9457:
9454:
9453:
9421:
9371:
9367:
9353:
9345:
9342:
9341:
9325:
9320:
9319:
9318:
9310:
9305:
9304:
9303:
9264:
9259:
9258:
9257:
9245:
9240:
9239:
9238:
9220:
9215:
9214:
9213:
9088:. If the field
9059:
9046:
9040:
9037:
9030:needs expansion
9015:
8963:
8957:
8954:
8953:
8920:
8909:
8904:
8900:
8892:
8889:
8888:
8840:
8837:
8836:
8816:
8812:
8797:
8793:
8785:
8773:
8768:
8767:
8759:
8756:
8755:
8738:
8733:
8732:
8718:
8715:
8714:
8711:Hartogs's lemma
8689:
8684:
8683:
8681:
8678:
8677:
8646:
8642:
8628:
8613:
8611:
8608:
8607:
8583:
8581:
8578:
8577:
8548:
8543:
8542:
8534:
8531:
8530:
8487:
8481:
8480:
8475:
8472:
8469:
8468:
8454:
8426:
8417:
8413:
8398:
8394:
8380:
8377:
8376:
8351:
8346:
8341:
8338:
8335:
8334:
8327:
8321:
8294:
8289:
8288:
8286:
8283:
8282:
8265:
8260:
8259:
8254:
8252:
8249:
8248:
8220:
8215:
8210:
8207:
8204:
8203:
8174:
8169:
8164:
8161:
8158:
8157:
8142:
8115:
8112:
8111:
8094:
8090:
8082:
8079:
8078:
8049:
8045:
8036:
8032:
8028:
8007:
8006:
8004:
7990:
7987:
7986:
7970:
7967:
7966:
7941:
7926:
7922:
7910:
7906:
7879:
7875:
7869:
7865:
7853:
7849:
7834:
7830:
7828:
7825:
7824:
7788:
7784:
7772:
7768:
7759:
7755:
7732:
7729:
7728:
7711:
7706:
7705:
7703:
7700:
7699:
7679:
7674:
7673:
7654:
7644:
7640:
7628:
7624:
7622:
7619:
7618:
7598:
7590:
7587:
7586:
7558:
7538:
7521:
7518:
7517:
7486:
7478:
7475:
7474:
7471:
7459:Arakelov theory
7455:geometric tools
7426:
7423:
7422:
7402:
7392:
7390:
7389:
7387:
7384:
7383:
7358:
7355:
7354:
7338:
7335:
7334:
7300:
7297:
7296:
7265:
7262:
7261:
7236:
7233:
7232:
7212:
7211:
7203:
7200:
7199:
7174:
7157:
7154:
7153:
7127:
7122:
7121:
7094:
7093:
7085:
7082:
7081:
7064:
7063:
7049:
7031:
7030:
7022:
7019:
7018:
7002:
6999:
6998:
6973:
6956:
6953:
6952:
6951:. A polynomial
6918:
6915:
6914:
6897:
6893:
6891:
6888:
6887:
6862:
6859:
6858:
6836:
6833:
6832:
6815:
6810:
6809:
6807:
6804:
6803:
6777:
6772:
6771:
6745:
6730:
6725:
6724:
6715:
6714:
6709:
6695:
6683:
6682:
6674:
6671:
6670:
6626:
6625:
6623:
6620:
6619:
6590:
6586:
6578:
6575:
6574:
6548:
6544:
6530:
6527:
6526:
6505:
6500:
6499:
6497:
6494:
6493:
6472:
6461:
6458:
6457:
6431:
6427:
6409:
6408:
6406:
6403:
6402:
6380:
6377:
6376:
6356:
6352:
6313:
6309:
6298:
6295:
6294:
6272:
6258:
6255:
6254:
6228:
6225:
6224:
6150:
6146:
6135:
6132:
6131:
6109:
6092:
6089:
6088:
6066:
6063:
6062:
6046:
6043:
6042:
6021:
6010:
6007:
6006:
5983:
5979:
5977:
5974:
5973:
5957:
5954:
5953:
5952:corresponds to
5927:
5923:
5915:
5912:
5911:
5892:
5889:
5888:
5872:
5869:
5868:
5843:
5840:
5839:
5820:
5809:
5806:
5805:
5771:
5768:
5767:
5732:
5731:
5729:
5726:
5725:
5694:
5691:
5690:
5668:
5665:
5664:
5648:
5645:
5644:
5628:
5625:
5624:
5599:
5596:
5595:
5551:
5550:
5548:
5545:
5544:
5515:
5514:
5512:
5509:
5508:
5483:
5480:
5479:
5463:
5460:
5459:
5433:
5430:
5429:
5413:
5410:
5409:
5393:
5390:
5389:
5373:
5370:
5369:
5344:
5341:
5340:
5324:
5321:
5320:
5276:
5275:
5273:
5270:
5269:
5244:
5235:
5234:
5232:
5229:
5228:
5200:
5183:
5180:
5179:
5154:
5152:
5149:
5148:
5129:
5115:
5097:
5091:
5090:
5085:
5082:
5079:
5078:
5075:
5050:
5046:
5025:
5021:
5019:
5016:
5015:
4999:
4996:
4995:
4974:
4970:
4964:
4960:
4951:
4947:
4945:
4942:
4941:
4925:
4923:
4920:
4919:
4887:
4883:
4868:
4864:
4859:
4856:
4855:
4839:
4836:
4835:
4818:
4812:
4811:
4810:
4808:
4805:
4804:
4787:
4783:
4768:
4764:
4762:
4759:
4758:
4742:
4739:
4738:
4737:for the scheme
4711:
4707:
4701:
4697:
4682:
4678:
4672:
4668:
4666:
4663:
4662:
4645:
4641:
4626:
4622:
4620:
4617:
4616:
4599:
4595:
4580:
4576:
4574:
4571:
4570:
4546:
4540:
4539:
4538:
4536:
4533:
4532:
4513:
4508:
4505:
4504:
4488:
4477:
4456:
4450:
4449:
4448:
4440:
4437:
4436:
4420:
4417:
4416:
4400:
4397:
4396:
4374:
4371:
4370:
4353:
4347:
4346:
4345:
4343:
4340:
4339:
4319:
4314:
4313:
4301:
4295:
4294:
4293:
4285:
4282:
4281:
4265:
4262:
4261:
4245:
4242:
4241:
4224:
4218:
4217:
4216:
4214:
4211:
4210:
4194:
4191:
4190:
4189:. The function
4174:
4157:
4152:
4150:
4147:
4146:
4120:
4114:
4113:
4112:
4104:
4101:
4100:
4078:
4060:
4054:
4053:
4052:
4044:
4041:
4040:
4021:
4018:
4017:
3996:
3991:
3990:
3973:
3968:
3956:
3950:
3949:
3948:
3940:
3937:
3936:
3919:
3913:
3912:
3911:
3909:
3906:
3905:
3889:
3886:
3885:
3869:
3861:
3858:
3857:
3831:
3826:
3804:
3799:
3787:
3769:
3763:
3762:
3761:
3759:
3756:
3755:
3754:, this induces
3733:
3729:
3728:
3722:
3721:
3720:
3703:
3699:
3698:
3692:
3691:
3690:
3673:
3670:
3669:
3635:
3621:
3609:
3605:
3599:
3581:
3577:
3569:
3557:
3553:
3544:
3538:
3537:
3536:
3534:
3531:
3530:
3510:
3504:
3503:
3502:
3484:
3480:
3478:
3475:
3474:
3458:
3450:
3447:
3446:
3403:
3397:
3396:
3395:
3393:
3390:
3389:
3373:
3365:
3362:
3361:
3332:
3326:
3325:
3324:
3322:
3319:
3318:
3299:
3282:
3279:
3278:
3262:
3260:
3257:
3256:
3253:
3213:
3209:
3200:
3196:
3188:
3156:
3153:
3152:
3135:
3130:
3125:
3122:
3119:
3118:
3082:
3078:
3069:
3065:
3063:
3060:
3059:
3043:
3040:
3039:
3011:
2982:
2979:
2978:
2950:
2947:
2946:
2944:principal ideal
2926:
2921:
2916:
2892:
2891:
2889:
2886:
2885:
2861:
2857:
2842:
2838:
2824:
2821:
2820:
2792:
2789:
2788:
2768:
2762:
2761:
2760:
2752:
2749:
2748:
2731:
2727:
2718:
2714:
2712:
2709:
2708:
2685:
2679:
2678:
2677:
2672:
2657:
2651:
2650:
2649:
2641:
2638:
2637:
2617:
2611:
2610:
2609:
2607:
2604:
2603:
2586:
2585:
2580:
2569:
2566:
2565:
2549:
2546:
2545:
2525:
2524:
2516:
2513:
2512:
2491:
2490:
2485:
2480:
2477:
2476:
2459:
2458:
2456:
2453:
2452:
2451:whose value at
2436:
2433:
2432:
2410:
2409:
2407:
2404:
2403:
2377:
2373:
2358:
2354:
2340:
2337:
2336:
2317:
2314:
2313:
2296:
2290:
2289:
2288:
2286:
2283:
2282:
2255:
2250:
2249:
2217:
2202:
2198:
2192:
2174:
2170:
2152:
2148:
2139:
2133:
2132:
2131:
2129:
2126:
2125:
2100:
2097:
2096:
2073:
2072:
2055:
2037:
2036:
2003:
1999:
1997:
1994:
1993:
1974:
1971:
1970:
1950:
1944:
1943:
1942:
1933:
1932:
1930:
1927:
1926:
1904:
1903:
1895:
1892:
1891:
1874:
1868:
1867:
1866:
1864:
1861:
1860:
1840:
1839:
1830:
1829:
1818:
1800:
1799:
1784:
1783:
1775:
1772:
1771:
1754:
1753:
1751:
1748:
1747:
1725:
1724:
1712:
1711:
1697:
1696:
1682:
1681:
1679:
1676:
1675:
1652:
1651:
1649:
1646:
1645:
1629:
1626:
1625:
1605:
1601:
1592:
1588:
1573:
1569:
1560:
1556:
1544:
1538:
1537:
1536:
1534:
1531:
1530:
1496:
1488:
1485:
1484:
1464:
1460:
1445:
1441:
1427:
1424:
1423:
1407:
1404:
1403:
1383:
1379:
1364:
1360:
1349:
1346:
1345:
1328:
1323:
1318:
1303:
1302:
1300:
1297:
1296:
1280:
1277:
1276:
1273:
1265:
1222:
1217:
1216:
1214:
1211:
1210:
1189:
1184:
1183:
1181:
1178:
1177:
1160:
1155:
1154:
1152:
1149:
1148:
1097:
1093:
1088:
1085:
1084:
1012:field extension
938:rational points
912:with values in
855:terminal object
691:Schemes form a
689:
679:, meaning Spec(
664:
659:
658:
657:
625:
589:
576:
563:
554:
516:
489:
441:
433:André Martineau
356:metric topology
277:polynomial ring
243:complex numbers
219:
116:coordinate ring
43:(the equations
23:, specifically
17:
12:
11:
5:
11375:
11365:
11364:
11350:
11349:
11339:
11329:
11321:
11320:External links
11318:
11316:
11315:
11277:
11263:
11255:10.1007/b62130
11243:Mumford, David
11239:
11225:
11208:
11195:978-3642380099
11194:
11181:
11167:
11149:
11107:
11093:
11067:
11054:
11034:
11015:(4): 389–408,
10999:
10956:
10954:
10951:
10948:
10947:
10935:
10919:
10917:, Chapter III.
10907:
10903:Dieudonné 1985
10895:
10883:
10871:
10859:
10847:
10830:
10818:
10806:
10790:
10778:
10766:
10762:Dieudonné 1985
10754:
10742:
10725:
10721:Dieudonné 1985
10710:
10706:Dieudonné 1985
10698:
10694:Dieudonné 1985
10686:
10672:
10671:
10669:
10666:
10665:
10664:
10662:Gluing schemes
10659:
10654:
10637:
10624:
10619:
10614:
10612:Étale morphism
10591:
10588:
10580:tensor product
10525:quotient stack
10476:étale topology
10470:
10467:
10427:tangent bundle
10419:vector bundles
10401:
10391:coherent sheaf
10380:
10340:
10333:determines an
10320:
10300:
10284:vector bundles
10274:Coherent sheaf
10272:Main article:
10269:
10266:
10265:
10264:
10255:
10216:
10215:-vector space.
10204:
10184:
10181:
10178:
10175:
10172:
10169:
10141:
10135:
10130:
10126:
10122:
10117:
10114:
10111:
10108:
10102:
10096:
10093:
10089:
10084:
10080:
10076:
10071:
10068:
10065:
10062:
10059:
10056:
10029:
10026:
10023:
10020:
10017:
10014:
10011:
10008:
10005:
10002:
9999:
9996:
9976:
9973:
9970:
9967:
9964:
9961:
9958:
9955:
9952:
9949:
9946:
9922:
9919:
9915:
9910:
9906:
9902:
9899:
9896:
9893:
9890:
9885:
9881:
9877:
9874:
9871:
9868:
9865:
9845:
9842:
9839:
9834:
9830:
9826:
9823:
9801:
9797:
9793:
9790:
9770:
9767:
9764:
9761:
9737:
9734:
9731:
9728:
9725:
9722:
9702:
9680:
9675:
9670:
9653:
9650:infinitesimals
9588:
9574:
9573:
9537:
9533:
9528:
9525:
9522:
9519:
9516:
9513:
9510:
9507:
9504:
9501:
9498:
9493:
9489:
9486:
9483:
9479:
9475:
9472:
9469:
9466:
9462:
9437:
9434:
9431:
9428:
9424:
9420:
9417:
9414:
9410:
9405:
9402:
9399:
9396:
9393:
9390:
9387:
9384:
9381:
9378:
9375:
9370:
9366:
9363:
9360:
9356:
9352:
9349:
9321:
9306:
9280:
9260:
9241:
9235:Generic point.
9232:
9216:
9058:
9055:
9048:
9047:
9027:
9025:
9014:
9011:
9010:
9009:
8977:
8972:
8969:
8966:
8962:
8932:
8928:
8923:
8918:
8915:
8912:
8908:
8903:
8899:
8896:
8881:
8880:is not affine.
8853:
8850:
8847:
8844:
8824:
8819:
8815:
8811:
8808:
8805:
8800:
8796:
8792:
8788:
8784:
8781:
8776:
8771:
8766:
8763:
8741:
8736:
8731:
8728:
8725:
8722:
8692:
8687:
8657:
8652:
8649:
8645:
8641:
8638:
8635:
8631:
8625:
8622:
8619:
8616:
8586:
8565:
8562:
8559:
8556:
8551:
8546:
8541:
8538:
8527:
8518:(over a field
8512:
8505:elliptic curve
8490:
8484:
8478:
8442:
8439:
8436:
8433:
8429:
8425:
8420:
8416:
8412:
8409:
8406:
8401:
8397:
8393:
8390:
8387:
8384:
8354:
8349:
8344:
8311:
8297:
8292:
8268:
8263:
8257:
8223:
8218:
8213:
8177:
8172:
8167:
8141:
8138:
8125:
8122:
8119:
8097:
8093:
8089:
8086:
8063:
8060:
8057:
8052:
8048:
8044:
8039:
8035:
8031:
8026:
8023:
8020:
8017:
8014:
8010:
8003:
8000:
7997:
7994:
7974:
7954:
7951:
7937:
7934:
7929:
7925:
7921:
7918:
7913:
7909:
7905:
7902:
7899:
7896:
7893:
7890:
7887:
7882:
7878:
7872:
7868:
7864:
7861:
7856:
7852:
7848:
7845:
7842:
7837:
7833:
7821:elliptic curve
7808:
7805:
7802:
7799:
7796:
7791:
7787:
7783:
7780:
7775:
7771:
7767:
7762:
7758:
7754:
7751:
7748:
7745:
7742:
7739:
7736:
7714:
7709:
7687:
7682:
7677:
7672:
7669:
7666:
7661:
7657:
7653:
7650:
7647:
7643:
7639:
7636:
7631:
7627:
7601:
7597:
7594:
7574:
7571:
7568:
7565:
7561:
7557:
7554:
7551:
7548:
7545:
7541:
7537:
7534:
7531:
7528:
7525:
7505:
7502:
7499:
7496:
7493:
7489:
7485:
7482:
7470:
7467:
7430:
7405:
7399:
7395:
7371:
7368:
7365:
7362:
7342:
7322:
7319:
7316:
7313:
7310:
7307:
7304:
7284:
7281:
7278:
7275:
7272:
7269:
7249:
7246:
7243:
7240:
7220:
7215:
7210:
7207:
7187:
7184:
7181:
7177:
7173:
7170:
7167:
7164:
7161:
7141:
7138:
7135:
7130:
7125:
7120:
7117:
7114:
7111:
7108:
7105:
7102:
7097:
7092:
7089:
7067:
7058:
7055:
7052:
7045:
7042:
7039:
7034:
7029:
7026:
7006:
6986:
6983:
6980:
6976:
6972:
6969:
6966:
6963:
6960:
6940:
6937:
6934:
6931:
6928:
6925:
6922:
6900:
6896:
6875:
6872:
6869:
6866:
6846:
6843:
6840:
6818:
6813:
6791:
6788:
6785:
6780:
6775:
6770:
6767:
6764:
6761:
6758:
6755:
6752:
6748:
6744:
6741:
6738:
6733:
6728:
6723:
6718:
6712:
6708:
6705:
6702:
6698:
6694:
6691:
6686:
6681:
6678:
6658:
6655:
6652:
6649:
6646:
6643:
6640:
6637:
6634:
6629:
6604:
6601:
6598:
6593:
6589:
6585:
6582:
6562:
6559:
6556:
6551:
6547:
6543:
6540:
6537:
6534:
6508:
6503:
6479:
6476:
6471:
6468:
6465:
6445:
6442:
6439:
6434:
6430:
6426:
6423:
6420:
6417:
6412:
6401:, we get that
6390:
6387:
6384:
6364:
6359:
6355:
6351:
6348:
6345:
6342:
6339:
6336:
6333:
6330:
6327:
6324:
6321:
6316:
6312:
6308:
6305:
6302:
6282:
6268:
6265:
6262:
6238:
6235:
6232:
6212:
6209:
6206:
6203:
6200:
6197:
6194:
6191:
6188:
6185:
6182:
6179:
6176:
6173:
6170:
6167:
6164:
6161:
6158:
6153:
6149:
6145:
6142:
6139:
6119:
6105:
6102:
6099:
6096:
6076:
6073:
6070:
6050:
6028:
6025:
6020:
6017:
6014:
5994:
5991:
5986:
5982:
5961:
5941:
5938:
5935:
5930:
5926:
5922:
5919:
5896:
5876:
5856:
5853:
5850:
5847:
5827:
5823:
5819:
5816:
5813:
5793:
5790:
5787:
5784:
5781:
5778:
5775:
5755:
5752:
5749:
5746:
5743:
5740:
5735:
5713:
5710:
5707:
5704:
5701:
5698:
5678:
5675:
5672:
5652:
5632:
5612:
5609:
5606:
5603:
5583:
5580:
5577:
5574:
5571:
5568:
5565:
5562:
5559:
5554:
5532:
5529:
5526:
5523:
5518:
5496:
5493:
5490:
5487:
5467:
5437:
5417:
5397:
5377:
5357:
5354:
5351:
5348:
5328:
5308:
5305:
5302:
5299:
5296:
5293:
5290:
5287:
5284:
5279:
5257:
5254:
5251:
5247:
5243:
5238:
5216:
5213:
5210:
5207:
5203:
5199:
5196:
5193:
5190:
5187:
5167:
5164:
5161:
5157:
5136:
5132:
5128:
5125:
5111:
5108:
5105:
5100:
5094:
5088:
5074:
5071:
5058:
5053:
5049:
5045:
5042:
5039:
5036:
5033:
5028:
5024:
5003:
4977:
4973:
4967:
4963:
4959:
4954:
4950:
4928:
4907:
4904:
4901:
4898:
4895:
4890:
4886:
4882:
4879:
4876:
4871:
4867:
4863:
4843:
4821:
4815:
4790:
4786:
4782:
4779:
4776:
4771:
4767:
4746:
4722:
4719:
4714:
4710:
4704:
4700:
4696:
4693:
4690:
4685:
4681:
4675:
4671:
4648:
4644:
4640:
4637:
4634:
4629:
4625:
4602:
4598:
4594:
4591:
4588:
4583:
4579:
4567:Bezout's lemma
4549:
4543:
4520:
4516:
4512:
4491:
4487:
4484:
4480:
4476:
4473:
4470:
4467:
4464:
4459:
4453:
4447:
4444:
4424:
4404:
4384:
4381:
4378:
4356:
4350:
4322:
4317:
4312:
4309:
4304:
4298:
4292:
4289:
4269:
4249:
4227:
4221:
4198:
4177:
4173:
4170:
4167:
4164:
4160:
4155:
4134:
4131:
4128:
4123:
4117:
4111:
4108:
4088:
4074:
4071:
4068:
4063:
4057:
4051:
4048:
4025:
3999:
3994:
3989:
3986:
3983:
3980:
3976:
3971:
3967:
3964:
3959:
3953:
3947:
3944:
3922:
3916:
3893:
3872:
3868:
3865:
3842:
3837:
3834:
3829:
3825:
3821:
3818:
3815:
3810:
3807:
3802:
3798:
3794:
3790:
3786:
3783:
3780:
3777:
3772:
3766:
3743:
3736:
3732:
3725:
3719:
3716:
3713:
3706:
3702:
3695:
3689:
3686:
3683:
3680:
3677:
3657:
3654:
3651:
3648:
3642:
3638:
3634:
3631:
3612:
3608:
3604:
3598:
3595:
3592:
3587:
3584:
3580:
3576:
3572:
3568:
3565:
3560:
3556:
3552:
3547:
3541:
3518:
3513:
3507:
3501:
3498:
3495:
3492:
3487:
3483:
3461:
3457:
3454:
3420:
3417:
3414:
3411:
3406:
3400:
3376:
3372:
3369:
3349:
3346:
3343:
3340:
3335:
3329:
3306:
3302:
3298:
3295:
3292:
3289:
3286:
3265:
3252:
3249:
3236:
3233:
3230:
3227:
3224:
3221:
3216:
3212:
3208:
3203:
3199:
3195:
3191:
3187:
3184:
3181:
3178:
3175:
3172:
3169:
3166:
3163:
3160:
3138:
3133:
3128:
3102:
3099:
3096:
3093:
3090:
3085:
3081:
3077:
3072:
3068:
3047:
3027:
3024:
3021:
3018:
3014:
3010:
3007:
3004:
3001:
2998:
2995:
2992:
2989:
2986:
2966:
2963:
2960:
2957:
2954:
2929:
2924:
2919:
2914:
2911:
2908:
2905:
2899:
2896:
2869:
2864:
2860:
2856:
2853:
2850:
2845:
2841:
2837:
2834:
2831:
2828:
2805:
2802:
2799:
2796:
2776:
2771:
2765:
2759:
2756:
2734:
2730:
2726:
2721:
2717:
2696:
2693:
2688:
2682:
2675:
2671:
2668:
2665:
2660:
2654:
2648:
2645:
2620:
2614:
2589:
2583:
2579:
2576:
2573:
2553:
2533:
2528:
2523:
2520:
2494:
2488:
2484:
2462:
2440:
2417:
2414:
2391:
2388:
2385:
2380:
2376:
2372:
2369:
2366:
2361:
2357:
2353:
2350:
2347:
2344:
2321:
2299:
2293:
2266:
2261:
2258:
2253:
2248:
2245:
2239:
2236:
2233:
2230:
2205:
2201:
2197:
2191:
2188:
2185:
2180:
2177:
2173:
2169:
2166:
2163:
2160:
2155:
2151:
2147:
2142:
2136:
2110:
2107:
2104:
2081:
2076:
2071:
2068:
2048:
2045:
2040:
2035:
2032:
2029:
2026:
2023:
2020:
2017:
2014:
2011:
2006:
2002:
1978:
1953:
1947:
1941:
1936:
1911:
1908:
1902:
1899:
1877:
1871:
1848:
1843:
1838:
1833:
1811:
1808:
1803:
1798:
1795:
1792:
1787:
1782:
1779:
1757:
1732:
1729:
1723:
1720:
1715:
1710:
1704:
1701:
1695:
1689:
1686:
1663:
1660:
1655:
1633:
1613:
1608:
1604:
1600:
1595:
1591:
1587:
1584:
1581:
1576:
1572:
1568:
1563:
1559:
1555:
1552:
1547:
1541:
1518:
1515:
1512:
1508:
1505:
1502:
1499:
1495:
1492:
1472:
1467:
1463:
1459:
1456:
1453:
1448:
1444:
1440:
1437:
1434:
1431:
1411:
1391:
1386:
1382:
1378:
1375:
1372:
1367:
1363:
1359:
1356:
1353:
1331:
1326:
1321:
1316:
1310:
1307:
1284:
1272:
1269:
1264:
1261:
1231:
1228:
1225:
1220:
1192:
1187:
1163:
1158:
1105:
1100:
1096:
1092:
892:). One writes
843:initial object
688:
685:
660:
651:natural number
621:
585:
572:
559:
550:
515:
512:
485:
440:
437:
429:Pierre Cartier
407:In the 1950s,
390:Italian school
308:Wolfgang Krull
273:maximal ideals
251:characteristic
218:
215:
163:maximal ideals
128:generic points
89:Pierre Deligne
83:, such as the
55:(for example,
41:multiplicities
15:
9:
6:
4:
3:
2:
11374:
11363:
11362:Scheme theory
11360:
11359:
11357:
11347:
11343:
11340:
11337:
11336:
11330:
11328:
11324:
11323:
11313:
11309:
11305:
11301:
11296:
11291:
11287:
11283:
11278:
11274:
11270:
11266:
11260:
11256:
11252:
11248:
11244:
11240:
11236:
11232:
11228:
11222:
11218:
11214:
11209:
11205:
11201:
11197:
11191:
11187:
11182:
11178:
11174:
11170:
11164:
11160:
11159:
11154:
11150:
11146:
11142:
11138:
11134:
11130:
11126:
11125:
11120:
11116:
11112:
11108:
11104:
11100:
11096:
11090:
11086:
11082:
11081:
11076:
11072:
11068:
11065:
11061:
11057:
11051:
11048:, Wadsworth,
11046:
11045:
11039:
11035:
11032:
11028:
11023:
11018:
11014:
11010:
11009:
11004:
11000:
10997:
10993:
10988:
10983:
10978:
10973:
10969:
10965:
10964:
10958:
10957:
10944:
10939:
10931:
10930:
10923:
10916:
10911:
10904:
10899:
10892:
10887:
10880:
10875:
10868:
10863:
10856:
10851:
10844:. p. 20.
10840:
10834:
10827:
10822:
10815:
10810:
10802:
10801:
10794:
10788:, Chapter II.
10787:
10782:
10775:
10770:
10763:
10758:
10751:
10746:
10738:
10737:
10729:
10722:
10717:
10715:
10707:
10702:
10695:
10690:
10683:
10677:
10673:
10663:
10660:
10658:
10655:
10653:
10649:
10645:
10641:
10638:
10636:
10632:
10628:
10625:
10623:
10620:
10618:
10615:
10613:
10609:
10605:
10601:
10597:
10596:Flat morphism
10594:
10593:
10587:
10585:
10581:
10577:
10573:
10569:
10565:
10560:
10558:
10554:
10550:
10546:
10542:
10537:
10535:
10531:
10527:
10526:
10522:determines a
10521:
10517:
10513:
10509:
10505:
10504:
10499:
10494:
10492:
10487:
10486:
10481:
10480:Michael Artin
10477:
10466:
10464:
10460:
10456:
10452:
10448:
10444:
10440:
10436:
10432:
10428:
10424:
10420:
10415:
10413:
10409:
10404:
10400:
10397:, say) is an
10396:
10392:
10389:. Finally, a
10388:
10383:
10379:
10375:
10371:
10370:
10365:
10361:
10354:
10343:
10339:
10336:
10332:
10328:
10323:
10319:
10315:
10311:
10307:
10306:
10303:
10299:
10293:
10289:
10285:
10281:
10275:
10262:
10258:
10253:
10252:tangent space
10249:
10245:
10241:
10237:
10233:
10229:
10225:
10221:
10217:
10202:
10179:
10176:
10173:
10167:
10159:
10155:
10139:
10128:
10124:
10112:
10106:
10100:
10091:
10087:
10082:
10078:
10066:
10063:
10060:
10054:
10043:
10024:
10021:
10018:
10012:
10009:
10003:
10000:
9997:
9971:
9968:
9965:
9959:
9953:
9950:
9947:
9936:
9917:
9913:
9908:
9904:
9897:
9891:
9888:
9883:
9879:
9872:
9866:
9840:
9837:
9832:
9828:
9821:
9799:
9795:
9791:
9788:
9765:
9759:
9751:
9732:
9729:
9726:
9720:
9700:
9693:over a field
9678:
9673:
9658:
9654:
9651:
9647:
9643:
9642:
9637:
9636:
9630:
9626:
9622:
9618:
9614:
9610:
9606:
9602:
9598:
9591:
9583:
9579:
9576:
9575:
9571:
9567:
9563:
9559:
9555:
9551:
9535:
9531:
9523:
9520:
9517:
9508:
9505:
9502:
9496:
9491:
9484:
9473:
9467:
9451:
9435:
9429:
9418:
9415:
9408:
9400:
9397:
9394:
9385:
9382:
9379:
9373:
9368:
9361:
9350:
9347:
9339:
9335:
9331:
9324:
9316:
9309:
9301:
9297:
9293:
9289:
9285:
9281:
9278:
9274:
9270:
9263:
9255:
9251:
9244:
9236:
9233:
9230:
9226:
9219:
9211:
9207:
9203:
9199:
9195:
9191:
9188:over a field
9187:
9183:
9179:
9175:
9171:
9167:
9164:) empty, but
9163:
9159:
9155:
9151:
9147:
9143:
9139:
9135:
9131:
9127:
9123:
9119:
9115:
9111:
9107:
9103:
9099:
9095:
9091:
9087:
9083:
9079:
9075:
9071:
9067:
9064:
9063:
9062:
9054:
9044:
9035:
9031:
9028:This section
9026:
9023:
9019:
9018:
9007:
9003:
9002:quasi-compact
8999:
8995:
8991:
8975:
8970:
8967:
8964:
8960:
8951:
8947:
8930:
8926:
8916:
8913:
8910:
8906:
8901:
8897:
8894:
8886:
8882:
8879:
8875:
8871:
8867:
8848:
8842:
8817:
8813:
8809:
8806:
8803:
8798:
8794:
8782:
8774:
8761:
8739:
8726:
8723:
8720:
8712:
8708:
8690:
8675:
8671:
8650:
8647:
8643:
8639:
8636:
8605:
8601:
8560:
8554:
8549:
8539:
8536:
8528:
8525:
8521:
8517:
8513:
8510:
8506:
8488:
8465:
8461:
8457:
8440:
8434:
8427:
8418:
8414:
8410:
8407:
8404:
8399:
8395:
8388:
8385:
8382:
8374:
8370:
8352:
8347:
8330:
8324:
8319:
8316:
8312:
8295:
8266:
8247:
8243:
8239:
8221:
8216:
8201:
8197:
8193:
8175:
8170:
8156:
8152:
8148:
8144:
8143:
8137:
8123:
8120:
8117:
8095:
8091:
8087:
8084:
8058:
8055:
8050:
8046:
8042:
8037:
8033:
8021:
8018:
8015:
8001:
7998:
7995:
7992:
7972:
7952:
7949:
7935:
7932:
7927:
7923:
7919:
7916:
7911:
7907:
7903:
7900:
7897:
7894:
7891:
7888:
7885:
7880:
7876:
7870:
7866:
7862:
7859:
7854:
7850:
7846:
7843:
7840:
7835:
7822:
7806:
7803:
7800:
7797:
7794:
7789:
7785:
7781:
7778:
7773:
7769:
7765:
7760:
7756:
7752:
7746:
7743:
7740:
7734:
7712:
7680:
7667:
7664:
7648:
7645:
7641:
7637:
7634:
7629:
7625:
7617:. The fibers
7616:
7595:
7592:
7566:
7559:
7552:
7549:
7546:
7532:
7529:
7526:
7523:
7500:
7497:
7494:
7483:
7480:
7466:
7464:
7460:
7456:
7452:
7448:
7444:
7428:
7419:
7403:
7366:
7360:
7340:
7314:
7308:
7302:
7282:
7279:
7273:
7267:
7244:
7238:
7205:
7182:
7171:
7165:
7159:
7152:. Again each
7136:
7128:
7118:
7112:
7106:
7103:
7087:
7043:
7040:
7024:
7004:
6981:
6970:
6964:
6958:
6935:
6929:
6926:
6923:
6920:
6898:
6894:
6870:
6864:
6844:
6841:
6838:
6816:
6786:
6778:
6768:
6759:
6753:
6746:
6739:
6731:
6721:
6710:
6703:
6692:
6676:
6650:
6644:
6641:
6638:
6632:
6616:
6599:
6596:
6591:
6587:
6580:
6557:
6554:
6549:
6545:
6541:
6538:
6532:
6524:
6506:
6477:
6474:
6469:
6466:
6463:
6440:
6437:
6432:
6428:
6424:
6421:
6415:
6388:
6385:
6382:
6357:
6349:
6346:
6343:
6337:
6334:
6328:
6322:
6319:
6314:
6310:
6306:
6303:
6280:
6266:
6263:
6260:
6253:double-point
6252:
6249:, we get one
6236:
6233:
6230:
6207:
6204:
6201:
6198:
6195:
6189:
6183:
6180:
6177:
6174:
6171:
6165:
6159:
6156:
6151:
6147:
6143:
6140:
6117:
6103:
6100:
6097:
6094:
6074:
6071:
6068:
6048:
6026:
6023:
6018:
6015:
6012:
5992:
5989:
5984:
5980:
5959:
5936:
5933:
5928:
5924:
5917:
5908:
5894:
5887:which divide
5874:
5851:
5845:
5825:
5821:
5817:
5814:
5811:
5788:
5785:
5782:
5779:
5773:
5750:
5747:
5744:
5738:
5708:
5705:
5702:
5696:
5676:
5673:
5670:
5650:
5630:
5607:
5601:
5575:
5569:
5566:
5563:
5557:
5527:
5521:
5491:
5485:
5465:
5453:
5449:
5435:
5415:
5395:
5375:
5352:
5346:
5326:
5300:
5294:
5291:
5288:
5282:
5252:
5241:
5208:
5194:
5191:
5188:
5185:
5162:
5126:
5123:
5109:
5103:
5098:
5070:
5051:
5047:
5040:
5037:
5034:
5031:
5026:
5022:
5001:
4993:
4975:
4971:
4965:
4961:
4957:
4952:
4948:
4902:
4896:
4888:
4884:
4880:
4877:
4874:
4869:
4865:
4841:
4819:
4788:
4784:
4780:
4777:
4774:
4769:
4765:
4744:
4736:
4720:
4717:
4712:
4708:
4702:
4698:
4694:
4691:
4688:
4683:
4679:
4673:
4669:
4646:
4642:
4638:
4635:
4632:
4627:
4623:
4600:
4596:
4592:
4589:
4586:
4581:
4577:
4568:
4563:
4547:
4518:
4514:
4510:
4503:. A fraction
4485:
4471:
4468:
4465:
4457:
4442:
4422:
4402:
4382:
4379:
4376:
4354:
4336:
4320:
4310:
4302:
4287:
4267:
4247:
4225:
4196:
4171:
4165:
4158:
4132:
4129:
4121:
4106:
4086:
4072:
4069:
4061:
4046:
4038:
4023:
4015:
3997:
3987:
3981:
3974:
3965:
3957:
3942:
3920:
3891:
3866:
3863:
3854:
3835:
3832:
3827:
3823:
3819:
3816:
3813:
3808:
3805:
3800:
3796:
3784:
3778:
3770:
3734:
3730:
3717:
3714:
3711:
3704:
3700:
3684:
3681:
3678:
3675:
3652:
3649:
3646:
3640:
3632:
3629:
3610:
3606:
3602:
3593:
3585:
3582:
3578:
3566:
3558:
3554:
3545:
3511:
3496:
3493:
3490:
3485:
3481:
3455:
3452:
3440:
3436:
3434:
3415:
3409:
3404:
3370:
3367:
3344:
3338:
3333:
3293:
3290:
3287:
3284:
3248:
3228:
3225:
3222:
3214:
3210:
3206:
3201:
3197:
3189:
3182:
3179:
3176:
3170:
3167:
3164:
3161:
3158:
3136:
3131:
3116:
3097:
3094:
3091:
3083:
3079:
3075:
3070:
3066:
3045:
3019:
3012:
3008:
3002:
2999:
2996:
2990:
2984:
2964:
2961:
2955:
2945:
2927:
2922:
2912:
2906:
2894:
2883:
2862:
2858:
2854:
2851:
2848:
2843:
2839:
2832:
2829:
2826:
2817:
2800:
2794:
2769:
2754:
2732:
2728:
2719:
2715:
2694:
2691:
2686:
2673:
2669:
2666:
2658:
2643:
2636:
2635:residue field
2618:
2581:
2577:
2571:
2551:
2518:
2510:
2486:
2482:
2438:
2412:
2389:
2386:
2378:
2374:
2370:
2367:
2364:
2359:
2355:
2348:
2345:
2342:
2333:
2319:
2297:
2278:
2259:
2256:
2246:
2243:
2237:
2234:
2231:
2228:
2203:
2199:
2195:
2186:
2178:
2175:
2171:
2164:
2161:
2153:
2149:
2140:
2122:
2108:
2105:
2102:
2092:
2069:
2066:
2046:
2043:
2030:
2024:
2018:
2015:
2012:
2009:
2004:
2000:
1990:
1976:
1967:
1951:
1939:
1906:
1900:
1897:
1875:
1836:
1809:
1806:
1793:
1777:
1727:
1721:
1699:
1693:
1684:
1661:
1658:
1631:
1606:
1602:
1598:
1593:
1589:
1585:
1582:
1579:
1574:
1570:
1566:
1561:
1557:
1550:
1545:
1513:
1493:
1490:
1465:
1461:
1457:
1454:
1451:
1446:
1442:
1435:
1432:
1429:
1409:
1384:
1380:
1376:
1373:
1370:
1365:
1361:
1354:
1351:
1329:
1324:
1314:
1305:
1282:
1268:
1260:
1258:
1254:
1249:
1247:
1229:
1226:
1223:
1208:
1190:
1161:
1146:
1142:
1138:
1135:
1131:
1127:
1123:
1119:
1103:
1098:
1094:
1090:
1083:
1079:
1075:
1071:
1067:
1062:
1060:
1056:
1052:
1048:
1044:
1040:
1036:
1032:
1028:
1024:
1020:
1016:
1013:
1009:
1005:
1001:
997:
993:
990:. One writes
989:
985:
981:
977:
973:
969:
965:
962:
958:
954:
950:
945:
943:
939:
935:
931:
927:
923:
919:
915:
911:
907:
903:
899:
895:
891:
887:
883:
879:
875:
871:
867:
863:
860:For a scheme
858:
856:
852:
848:
844:
840:
835:
833:
829:
825:
821:
817:
813:
809:
805:
801:
797:
793:
789:
785:
780:
778:
774:
770:
766:
762:
758:
754:
749:
747:
743:
739:
735:
732:
728:
724:
720:
716:
712:
709:
706:
702:
698:
694:
684:
682:
678:
674:
670:
663:
655:
652:
648:
645:over a field
644:
642:
635:
633:
629:
624:
620:
616:
612:
608:
603:
601:
597:
594:) called the
593:
588:
584:
580:
575:
571:
567:
562:
558:
553:
549:
545:
541:
537:
533:
529:
525:
521:
520:affine scheme
511:
508:
503:
501:
497:
493:
488:
484:
480:
476:
472:
468:
464:
461:
457:
454:
450:
446:
436:
434:
430:
426:
422:
418:
414:
410:
405:
403:
399:
395:
394:generic point
391:
386:
384:
380:
376:
372:
368:
364:
359:
357:
353:
349:
345:
344:Oscar Zariski
341:
337:
333:
329:
325:
320:
318:
314:
309:
305:
301:
297:
293:
289:
285:
281:
278:
274:
270:
266:
262:
260:
256:
252:
248:
244:
240:
236:
232:
228:
224:
214:
212:
207:
205:
201:
197:
193:
190:
189:over the base
187:
183:
179:
175:
170:
168:
164:
160:
156:
151:
149:
145:
141:
137:
133:
129:
125:
121:
117:
112:
110:
106:
105:number theory
102:
98:
94:
90:
86:
82:
78:
77:
72:
68:
67:Scheme theory
64:
62:
58:
57:Fermat curves
54:
50:
46:
42:
38:
34:
30:
26:
22:
11334:
11295:math/0412512
11281:
11246:
11212:
11185:
11157:
11128:
11122:
11079:
11043:
11012:
11006:
10967:
10961:
10943:Vistoli 2005
10938:
10928:
10922:
10910:
10898:
10886:
10881:, section 1.
10879:Arapura 2011
10874:
10862:
10850:
10833:
10821:
10809:
10799:
10793:
10786:Mumford 1999
10781:
10769:
10757:
10750:Cartier 2001
10745:
10735:
10728:
10701:
10689:
10676:
10640:Group scheme
10635:Hodge theory
10617:Stable curve
10586:on modules.
10561:
10547:(similar to
10538:
10533:
10523:
10519:
10515:
10501:
10495:
10483:
10472:
10457:. Moreover,
10454:
10450:
10447:direct image
10442:
10438:
10434:
10430:
10423:free modules
10418:
10416:
10411:
10402:
10398:
10394:
10390:
10386:
10381:
10377:
10373:
10372:on a scheme
10367:
10363:
10359:
10352:
10341:
10337:
10330:
10326:
10321:
10317:
10312:that form a
10309:
10301:
10297:
10295:
10287:
10277:
10260:
10256:
10247:
10243:
10239:
10235:
10231:
10227:
10223:
10041:
9749:
9639:
9632:
9628:
9620:
9616:
9612:
9608:
9604:
9600:
9596:
9589:
9581:
9577:
9562:Galois group
9337:
9333:
9329:
9322:
9307:
9299:
9295:
9291:
9287:
9283:
9272:
9268:
9261:
9253:
9249:
9242:
9234:
9228:
9224:
9217:
9209:
9205:
9201:
9197:
9193:
9189:
9185:
9181:
9177:
9173:
9169:
9165:
9161:
9157:
9153:
9149:
9145:
9141:
9137:
9133:
9129:
9125:
9121:
9117:
9113:
9109:
9105:
9101:
9097:
9093:
9089:
9085:
9081:
9077:
9073:
9069:
9065:
9060:
9051:
9038:
9034:adding to it
9029:
8989:
8950:ultrafilters
8884:
8877:
8873:
8869:
8865:
8706:
8673:
8669:
8603:
8599:
8523:
8519:
8515:
8463:
8459:
8455:
8328:
8322:
8317:
8241:
8199:
8198:-space over
8195:
8191:
8154:
8150:
8146:
7472:
7420:
7017:with values
6617:
6523:Galois group
5909:
5457:
5076:
4564:
4337:
4036:
4014:finite field
3855:
3444:
3254:
2882:hypersurface
2818:
2634:
2511:. We define
2509:residue ring
2508:
2334:
2280:
2124:
2094:
1992:
1968:
1274:
1271:Affine space
1266:
1252:
1250:
1245:
1206:
1144:
1140:
1136:
1133:
1129:
1125:
1121:
1117:
1077:
1073:
1069:
1063:
1054:
1050:
1046:
1038:
1034:
1030:
1026:
1022:
1018:
1014:
1007:
1003:
999:
995:
991:
987:
983:
979:
975:
971:
967:
963:
956:
952:
948:
946:
941:
933:
929:
925:
921:
917:
913:
909:
905:
901:
897:
893:
889:
885:
877:
873:
869:
865:
861:
859:
850:
838:
836:
831:
827:
823:
819:
815:
811:
807:
803:
799:
795:
791:
787:
783:
781:
776:
764:
760:
756:
752:
750:
745:
741:
737:
733:
730:
726:
722:
718:
714:
710:
707:
704:
700:
690:
680:
676:
672:
668:
661:
653:
646:
640:
638:
636:
631:
627:
622:
618:
606:
604:
599:
595:
591:
586:
582:
578:
573:
569:
565:
560:
556:
551:
547:
543:
539:
535:
531:
519:
517:
506:
504:
499:
495:
491:
486:
482:
478:
470:
467:prime ideals
462:
455:
448:
444:
442:
424:
406:
401:
397:
387:
362:
360:
323:
321:
304:Emmy Noether
299:
296:prime ideals
291:
287:
283:
279:
268:
263:
255:number rings
227:real numbers
220:
208:
204:moduli space
199:
191:
188:
185:
181:
177:
171:
159:prime ideals
152:
136:ringed space
113:
74:
66:
65:
48:
44:
28:
18:
11346:Terence Tao
11075:Harris, Joe
10736:Les schémas
10584:Hom functor
10482:defined an
9814:, which is
9595:defined by
9223:defined by
8367:, called a
7421:The scheme
4099:, and also
2884:subvariety
1969:The scheme
1002:-points of
920:is a field
904:-points of
782:A morphism
773:finite type
703:, a scheme
217:Development
21:mathematics
10953:References
10752:, note 29.
10631:Chow group
10335:associated
9935:transverse
9635:derivative
9633:and first
9041:March 2024
8668:. To show
7463:valuations
7080:, that is
6913:elements,
5867:for those
3113:defines a
2747:, so that
2633:gives the
771:scheme of
514:Definition
340:André Weil
317:Noetherian
294:, and the
223:polynomial
11155:(1997) .
10977:0806.1033
10668:Citations
10549:orbifolds
10376:means an
10101:≅
10010:⊂
9889:−
9838:−
9625:nilpotent
9601:fat point
9521:−
9506:−
9474:⊂
9419:
9413:→
9398:−
9383:−
9351:
9156:= −1 has
8968:≠
8961:∏
8922:∞
8907:∏
8898:
8807:…
8730:→
8648:−
8555:∖
8507:over the
8408:…
8386:
8085:−
8056:−
8043:−
8002:
7917:−
7901:−
7844:−
7832:Δ
7766:−
7668:
7649:
7642:×
7596:
7533:
7484:∈
7398:¯
7172:∈
7137:α
7119:∈
7113:α
6971:∈
6930:
6845:α
6787:α
6769:≅
6475:−
6470:±
6375:. And at
6347:−
6190:∩
6181:−
6101:±
6024:−
6019:±
6005:, namely
5786:−
5748:−
5706:−
5242:⊂
5195:
5127:∈
5038:∖
4949:ρ
4878:…
4778:…
4692:⋯
4636:…
4590:…
4472:
4311:∈
3867:∈
3856:A number
3833:−
3828:ℓ
3817:…
3806:−
3735:ℓ
3715:…
3685:∖
3650:≥
3633:∈
3583:−
3497:∖
3456:∈
3371:∈
3294:
3207:−
3168:
3003:
2962:⊂
2913:⊂
2898:¯
2852:…
2725:↦
2692:≅
2575:→
2416:¯
2387:∈
2368:…
2257:≥
2247:∈
2232:∈
2176:−
2106:∈
2070:∉
2044:∈
2016:∖
1940:⊂
1910:¯
1901:∈
1837:⊂
1807:∈
1731:¯
1722:⊂
1703:¯
1688:¯
1659:⊂
1599:−
1583:…
1567:−
1455:…
1374:…
1309:¯
1095:×
822:) → Spec(
767:means an
611:separated
607:prescheme
431:, it was
383:Matsusaka
371:manifolds
313:dimension
33:structure
11356:Category
11245:(1999).
11117:(1960).
11077:(1998).
11040:(1985),
10590:See also
10582:and the
10445:(by the
10345:-module
10305:-modules
10160:of this
9646:calculus
9315:ramified
9267:, where
6293:, since
6251:ramified
6130:, since
5594:for all
5319:, where
1263:Examples
880:means a
693:category
649:, for a
528:spectrum
453:spectrum
155:spectrum
97:topology
61:integers
47:= 0 and
11312:2223406
11300:Bibcode
11273:1748380
11235:1917232
11204:0456457
11177:0463157
11145:0217083
11103:1730819
11064:0780183
11031:1848254
10996:3082873
10541:descent
10362:= Spec(
10234:= Spec
9641:reduced
8598:; then
8310:is not.
7441:is not
1134:product
1043:functor
1041:) is a
961:algebra
916:. When
888:→ Spec(
882:section
853:) as a
845:in the
761:variety
744:→ Spec(
639:affine
615:Mumford
332:compact
275:in the
11310:
11271:
11261:
11233:
11223:
11202:
11192:
11175:
11165:
11143:
11101:
11091:
11062:
11052:
11029:
10994:
10512:action
10314:module
10220:smooth
10158:length
9580:. Let
9450:degree
9256:)) → A
8503:is an
8238:proper
7947:
7939:
7819:is an
7443:proper
7295:; and
7061:
7047:
6278:
6270:
6115:
6107:
5121:
5113:
4084:
4076:
4012:, the
3644:
3627:
3619:
2507:, the
2241:
2226:
2223:
2215:
2212:
2064:
2061:
2053:
2050:
1827:
1824:
1816:
1813:
1257:limits
1080:, the
841:is an
837:Since
719:scheme
713:(or a
643:-space
540:scheme
425:scheme
324:affine
271:: the
29:scheme
11290:arXiv
10972:arXiv
10842:(PDF)
10498:stack
10366:). A
9200:) of
8864:. If
8705:when
8240:over
7727:. If
7453:with
6223:. At
4661:with
4280:with
2880:is a
1890:with
1244:over
1205:over
1053:over
1025:over
986:over
972:point
966:, an
874:point
868:, an
775:over
763:over
530:Spec(
522:is a
475:sheaf
458:of a
239:field
140:sheaf
138:or a
132:atlas
120:ideal
31:is a
11259:ISBN
11221:ISBN
11190:ISBN
11163:ISBN
11089:ISBN
11050:ISBN
10152:The
9648:and
9416:Spec
9348:Spec
9298:−1)(
9282:Let
8895:Spec
8883:Let
8514:The
8383:Proj
7999:Spec
7665:Spec
7646:Spec
7593:Spec
7530:Spec
7465:.
5192:Spec
4469:Frac
4335:.
3291:Spec
3165:Spec
3000:Spec
2057:with
1820:with
1275:Let
1176:and
1120:and
1072:and
1064:The
982:) →
810:) →
734:over
708:over
538:. A
415:and
381:and
379:Chow
342:and
306:and
233:and
172:The
99:and
63:).
27:, a
11251:doi
11133:doi
11017:doi
10982:doi
10433:of
10358:on
10294:of
10246:of
9623:is
9619:on
9607:is
9208:of
9132:of
9036:.
8835:to
8467:of
8331:= 0
8236:is
7943:mod
6927:deg
6857:of
6669:is
6274:mod
6111:mod
5907:.
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49:x
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