5500:
5096:
5495:{\displaystyle {\begin{matrix}\operatorname {Spec} \left({\frac {\mathbb {C} }{\left(y-{\frac {\beta }{\alpha }}x\right)}}\right)&\to &\operatorname {Spec} \left({\frac {\mathbb {C} \left}{\left(y-{\frac {b}{a}}x\right)}}\right)&\to &{\underline {\operatorname {Spec} }}_{X}\left({\frac {{\mathcal {O}}_{X}}{\left(ay-bx\right)}}\right)\\\downarrow &&\downarrow &&\downarrow \\\operatorname {Spec} (\mathbb {C} )&\to &\operatorname {Spec} \left(\mathbb {C} \left\right)=U_{a}&\to &\mathbb {P} _{a,b}^{1}\end{matrix}}}
3471:
4616:
5946:
4409:
Global Spec has a universal property similar to the universal property for ordinary Spec. More precisely, just as Spec and the global section functor are contravariant right adjoints between the category of commutative rings and schemes, global Spec and the direct image functor for the structure map
6698:
Further, the geometric structure of the spectrum of the ring (equivalently, the algebraic structure of the module) captures the behavior of the spectrum of the operator, such as algebraic multiplicity and geometric multiplicity. For instance, for the 2×2 identity matrix has corresponding module:
4467:
4963:
2168:
are essentially identical. By studying spectra of polynomial rings instead of algebraic sets with
Zariski topology, one can generalize the concepts of algebraic geometry to non-algebraically closed fields and beyond, eventually arriving at the language of
2554:
6684:
6632:, while finite sets of points correspond to finite-dimensional representations (which are reducible, corresponding geometrically to being a union, and algebraically to not being a prime ideal). The non-maximal ideals then correspond to
1643:
2769:
5587:
1831:
5823:
2367:
4162:
892:
4103:
4258:
2438:
3602:
6287:
272:
3112:
2924:
2041:, then the Zariski topology defined above coincides with the Zariski topology defined on algebraic sets (which has precisely the algebraic subsets as closed sets). Specifically, the maximal ideals in
3859:
2623:
707:
7035:. Indeed, any commutative C*-algebra can be realized as the algebra of scalars of a Hausdorff space in this way, yielding the same correspondence as between a ring and its spectrum. Generalizing to
4400:
5736:
3944:
2214:
4803:
1163:
5818:
5664:
4611:{\displaystyle \operatorname {Hom} _{{\mathcal {O}}_{S}{\text{-alg}}}({\mathcal {A}},\pi _{*}{\mathcal {O}}_{X})\cong \operatorname {Hom} _{{\text{Sch}}/S}(X,\mathbf {Spec} ({\mathcal {A}})),}
3435:
4695:
3717:
1482:
1382:
2075:
577:
4744:
797:
7029:
4883:
3634:
3542:
2162:
2118:
1310:
1248:
1199:
1096:
1005:
519:
479:
431:
352:
218:
215:
90:
6877:
4851:
4348:
6784:
6339:
3825:
3785:
3765:
3745:
1905:
1878:
1854:
1703:
1532:
3357:
3028:
3897:
4439:
4031:
2033:
are closed in the spectrum, while the elements corresponding to subvarieties have a closure consisting of all their points and subvarieties. If one only considers the points of
611:
5059:
4651:
3198:
5622:
5024:
2276:
969:
397:
4878:
3975:
1735:
5088:
4056:
125:
4284:
4188:
2665:
2240:
1064:
919:
6556:
3258:
3152:
7486:
6516:
4402:. That is, as ring homomorphisms induce opposite maps of spectra, the restriction maps of a sheaf of algebras induce the inclusion maps of the spectra that make up the
6934:
1571:
2994:
6398:
3643:, Hochster considers what he calls the patch topology on a prime spectrum. By definition, the patch topology is the smallest topology in which the sets of the forms
3288:
2443:
6479:
6452:
6425:
6117:
638:
171:
6626:
6152:
6086:
5975:, while more general subvarieties correspond to possibly reducible representations that need not be cyclic. Recall that abstractly, the representation theory of a
3670:
1580:
6048:
2670:
6368:
5941:{\displaystyle {\mathcal {I}}=\left(2\times 2{\text{ minors of }}{\begin{pmatrix}a_{0}&\cdots &a_{n}\\x_{0}&\cdots &x_{n}\end{pmatrix}}\right).}
5511:
4459:
3995:
3917:
3805:
3455:
3377:
3323:
3218:
3048:
2968:
2944:
2842:
2643:
1763:
1755:
1683:
1663:
4995:
2281:
4108:
2128:, one additional non-closed point has been introduced, and this point "keeps track" of the corresponding subvariety. One thinks of this point as the
802:
4061:
7552:
7463:
7383:
7354:
7328:
7301:
7267:
4193:
2376:
3547:
6971:
6176:
7538:
6946:
the eigenvalues (with geometric multiplicity) of the operator correspond to the (reduced) points of the variety, with multiplicity;
227:
7067:
3056:
2848:
2625:
looks topologically like the transverse intersection of two complex planes at a point, although typically this is depicted as a
3830:
2562:
643:
6645:
6165:
In the case that the field is algebraically closed (say, the complex numbers), every maximal ideal corresponds to a point in
1710:
1574:
4353:
5669:
3922:
1912:
1706:
2818:
Here are some examples of schemes that are not affine schemes. They are constructed from gluing affine schemes together.
2185:
7694:
7679:
4749:
1101:
7638:
5741:
5627:
3382:
4664:
7544:
7367:
3675:
1948:
1441:
1341:
7450:
7406:
6975:
4958:{\displaystyle {\underline {\operatorname {Spec} }}_{X}({\mathcal {A}}/{\mathcal {I}})\to \mathbb {P} _{a,b}^{1}}
2048:
1908:
7032:
544:
7623:
6996:
4700:
1166:
7014:
3607:
3515:
2135:
2091:
1283:
1221:
1172:
1069:
978:
746:
492:
452:
404:
325:
188:
67:
7040:
6795:
4808:
4289:
6705:
6292:
3810:
3770:
3750:
3730:
2802:
1890:
1863:
1839:
1688:
1517:
7684:
7277:
6651:
5624:
and the origin. This example can be generalized to parameterize the family of lines through the origin of
3505:
3328:
2999:
1261:
710:
3864:
3501:
Some authors (notably M. Hochster) consider topologies on prime spectra other than the
Zariski topology.
7689:
7004:
5956:
4413:
4005:
2011:
585:
6675:, one can consider the vector space with operator as a module over the polynomial ring in one variable
5029:
4624:
3157:
5595:
5000:
2252:
928:
357:
7062:
7052:
4856:
3998:
3949:
2783:
2170:
1857:
1716:
1313:
1023:
1016:
5064:
4037:
106:
5976:
4661:
The relative spec is the correct tool for parameterizing the family of lines through the origin of
1968:
1881:
1251:
1206:
730:
136:
38:
4263:
4167:
2648:
2223:
1040:
897:
7589:
7571:
7525:
7505:
7433:
7415:
7082:
6668:
6525:
3223:
3117:
1932:
1330:
93:
6488:
2549:{\displaystyle \mathbb {C} {\xrightarrow {ev_{(\alpha _{1},\ldots ,\alpha _{n})}}}\mathbb {C} }
1856:
even defines a contravariant functor from the category of commutative rings to the category of
7617:
7548:
7459:
7379:
7350:
7324:
7297:
7263:
7072:
7057:
6892:
4001:
2947:
2370:
1920:
1544:
1539:
97:
7402:"The Patch Topology and the Ultrafilter Topology on the Prime Spectrum of a Commutative Ring"
2973:
1638:{\displaystyle \operatorname {Spec} (f):\operatorname {Spec} (S)\to \operatorname {Spec} (R)}
7581:
7495:
7445:
7425:
7401:
7316:
7289:
7255:
7077:
6967:
6373:
3263:
2823:
1435:
522:
278:
50:
7473:
7393:
7007:(the bounded continuous functions on the space, being analogous to regular functions) is a
6949:
the primary decomposition of the module corresponds to the unreduced points of the variety;
6457:
6430:
6403:
6095:
2797:) A topological space is homeomorphic to the prime spectrum of a commutative ring (i.e., a
2764:{\displaystyle \{(\alpha _{1},0),(0,\alpha _{2}):\alpha _{1},\alpha _{2}\in \mathbb {C} \}}
616:
149:
7469:
7455:
7389:
7375:
7346:
7234:
7000:
6992:
6661:
6657:
6567:
6170:
6126:
6053:
5987:
5582:{\displaystyle \operatorname {Spec} (\mathbb {C} ){\xrightarrow {}}\mathbb {P} _{a,b}^{1}}
3646:
2794:
1826:{\displaystyle {\mathcal {O}}_{f^{-1}({\mathfrak {p}})}\to {\mathcal {O}}_{\mathfrak {p}}}
1511:
1326:
734:
438:
285:
5992:
1860:. In fact it is the universal such functor, and hence can be used to define the functor
7363:
7281:
7251:
6344:
4444:
3980:
3902:
3790:
3440:
3362:
3293:
3203:
3033:
2953:
2929:
2827:
2798:
2779:
2628:
2243:
1740:
1668:
1648:
530:
101:
7500:
7481:
4968:
3470:
7673:
7593:
7087:
6956:
6088:
As the latter formulation makes clear, a polynomial ring is the group algebra over a
2129:
922:
442:
434:
31:
7639:
https://mathoverflow.net/questions/441029/intrinsic-topology-on-the-zariski-spectrum
7529:
7437:
2014:
or a variety if it cannot be written as the union of two proper algebraic subsets).
7338:
6089:
2775:
2362:{\displaystyle \mathbb {A} _{\mathbb {C} }^{n}=\operatorname {Spec} (\mathbb {C} )}
2217:
2078:
1202:
1008:
7585:
5971:, and the spectrum of a ring corresponds to irreducible cyclic representations of
2556:. This fundamental observation allows us to give meaning to other affine schemes.
2806:
2787:
174:
57:
4157:{\displaystyle f:{\underline {\operatorname {Spec} }}_{S}({\mathcal {A}})\to S}
7652:
7601:
7429:
7293:
7031:
of the algebra of scalars, indeed functorially so; this is the content of the
6988:
6883:
6685:
structure theorem for finitely generated modules over a principal ideal domain
5980:
2950:(in fact, we can define projective space for any base scheme). The projective
1952:
1277:
5090:
the fiber can be computed by looking at the composition of pullback diagrams
17:
3604:
satisfy the axioms for closed sets in a topological space. This topology on
7661:
7605:
1169:. As above, this construction extends to a presheaf on all open subsets of
887:{\textstyle \Gamma (U,{\mathcal {O}}_{X})=\varprojlim _{i\in I}R_{f_{i}},}
972:
482:
449:
are precisely the closed points in this topology. By the same reasoning,
182:
7518:
6952:
a diagonalizable (semisimple) operator corresponds to a reduced variety;
5986:
The connection to representation theory is clearer if one considers the
4098:{\displaystyle {\underline {\operatorname {Spec} }}_{S}({\mathcal {A}})}
7509:
7320:
7011:
C*-algebra, with the space being recovered as a topological space from
4253:{\displaystyle f^{-1}(U)\cong \operatorname {Spec} ({\mathcal {A}}(U))}
1535:
726:
3200:
is the affine plane with the origin taken out. The global sections of
6974:
of the operator, and the product of the invariant factors equals the
6955:
a cyclic module (one generator) corresponds to the operator having a
4410:
are contravariant right adjoints between the category of commutative
2433:{\displaystyle (\alpha _{1},\ldots ,\alpha _{n})\in \mathbb {C} ^{n}}
6119:
corresponds to choosing a basis for the vector space. Then an ideal
5537:
3597:{\displaystyle \varphi ^{*}(\operatorname {Spec} B),\varphi :A\to B}
2492:
7576:
7562:
Tarizadeh, Abolfazl (2019). "Flat topology and its dual aspects".
7420:
7374:, Graduate Texts in Mathematics, vol. 197, Berlin, New York:
6282:{\displaystyle (x_{1}-a_{1}),(x_{2}-a_{2}),\ldots ,(x_{n}-a_{n})}
3919:
is clear from the context, then relative Spec may be denoted by
6939:
showing algebraic multiplicity 2 but geometric multiplicity 1.
1438:
in algebraic geometry. Using this definition, we can describe
1011:. Any ringed space isomorphic to one of this form is called an
3465:
1963:
is such an algebraic set, one considers the commutative ring
288:
for the
Zariski topology can be constructed as follows. For
7313:
Commutative Rings whose
Finitely Generated Modules Decompose
7130:
5829:
5758:
5747:
5301:
4921:
4909:
4862:
4814:
4766:
4755:
4594:
4526:
4505:
4479:
4420:
4378:
4359:
4233:
4140:
4087:
4043:
4012:
1810:
1770:
1460:
1360:
821:
669:
592:
267:{\displaystyle \{V_{I}\colon I{\text{ is an ideal of }}R\}.}
112:
3107:{\displaystyle \mathbb {A} _{k}^{2}=\operatorname {Spec} k}
2919:{\displaystyle \mathbb {P} _{k}^{n}=\operatorname {Proj} k}
6162:-module; this generalizes 1-dimensional representations).
4965:
parameterizes the desired family. In fact, the fiber over
6628:
correspond precisely to 1-dimensional representations of
30:
For the concept of ring spectrum in homotopy theory, see
7288:. Encyclopaedia of Mathematical Sciences. Vol. 17.
3854:{\displaystyle {\underline {\operatorname {Spec} }}_{S}}
2667:
are the evaluation morphisms associated with the points
2618:{\displaystyle \operatorname {Spec} (\mathbb {C} /(xy))}
702:{\displaystyle \Gamma (D_{f},{\mathcal {O}}_{X})=R_{f},}
7235:
https://www.math.ias.edu/~lurie/261ynotes/lecture14.pdf
3482:
2946:. This can be easily generalized to any base ring, see
733:. In more detail, the distinguished open subsets are a
6886:, while a non-trivial 2×2 nilpotent matrix has module
5861:
5101:
4395:{\displaystyle {\mathcal {A}}(U)\to {\mathcal {A}}(V)}
1919:; each of these categories is often thought of as the
805:
749:
737:
of the
Zariski topology, so for an arbitrary open set
7017:
6895:
6798:
6708:
6570:
6528:
6491:
6460:
6433:
6406:
6376:
6347:
6295:
6179:
6129:
6098:
6056:
5995:
5826:
5744:
5672:
5630:
5598:
5514:
5099:
5067:
5032:
5003:
4971:
4886:
4859:
4811:
4752:
4703:
4667:
4627:
4470:
4447:
4416:
4356:
4292:
4266:
4196:
4170:
4111:
4064:
4040:
4008:
3983:
3952:
3925:
3905:
3867:
3833:
3813:
3793:
3773:
3753:
3733:
3678:
3649:
3610:
3550:
3518:
3443:
3385:
3365:
3331:
3296:
3266:
3226:
3206:
3160:
3120:
3059:
3036:
3002:
2976:
2956:
2932:
2851:
2830:
2673:
2651:
2631:
2565:
2446:
2379:
2284:
2255:
2226:
2188:
2138:
2094:
2051:
1893:
1866:
1842:
1766:
1743:
1719:
1691:
1671:
1651:
1583:
1547:
1520:
1444:
1344:
1286:
1224:
1175:
1104:
1072:
1043:
981:
931:
900:
646:
619:
588:
547:
495:
455:
407:
360:
328:
230:
191:
152:
109:
70:
7519:"Remarks on spectra, supports, and Hochster duality"
5731:{\displaystyle X=\mathbb {P} _{a_{0},...,a_{n}}^{n}}
3939:{\displaystyle {\underline {\operatorname {Spec} }}}
1384:
more concretely as follows. We say that an element
2209:{\displaystyle \operatorname {Spec} (\mathbb {Z} )}
1951:) that are defined as the common zeros of a set of
7023:
6928:
6871:
6778:
6620:
6550:
6510:
6473:
6446:
6419:
6392:
6362:
6333:
6281:
6146:
6111:
6080:
6042:
5940:
5812:
5730:
5658:
5616:
5581:
5494:
5082:
5053:
5018:
4989:
4957:
4872:
4845:
4798:{\displaystyle {\mathcal {A}}={\mathcal {O}}_{X},}
4797:
4738:
4689:
4645:
4610:
4453:
4433:
4394:
4342:
4278:
4252:
4182:
4156:
4097:
4050:
4025:
3989:
3969:
3938:
3911:
3891:
3853:
3819:
3799:
3779:
3759:
3739:
3711:
3664:
3628:
3596:
3536:
3449:
3429:
3371:
3351:
3317:
3282:
3252:
3212:
3192:
3146:
3106:
3042:
3022:
2988:
2962:
2938:
2918:
2836:
2763:
2659:
2637:
2617:
2548:
2432:
2361:
2270:
2234:
2208:
2156:
2112:
2069:
1899:
1872:
1848:
1825:
1749:
1729:
1697:
1677:
1657:
1637:
1565:
1526:
1476:
1376:
1304:
1242:
1193:
1158:{\displaystyle \Gamma (D_{f},{\tilde {M}})=M_{f},}
1157:
1090:
1058:
999:
963:
913:
886:
791:
701:
632:
605:
571:
513:
473:
425:
391:
346:
266:
209:
165:
119:
84:
7487:Transactions of the American Mathematical Society
5813:{\displaystyle {\mathcal {A}}={\mathcal {O}}_{X}}
5659:{\displaystyle \mathbb {A} _{\mathbb {C} }^{n+1}}
2124:(with Zariski topology): for every subvariety of
3430:{\displaystyle V_{(x)}\cap V_{(y)}=\varnothing }
1705:can be seen as a contravariant functor from the
1019:are obtained by gluing affine schemes together.
7315:. Lecture Notes in Mathematics. Vol. 723.
7177:
7142:
4690:{\displaystyle \mathbb {A} _{\mathbb {C} }^{2}}
2440:can be identified with the evaluation morphism
2025:together with elements for all subvarieties of
925:with respect to the natural ring homomorphisms
6987:The spectrum can be generalized from rings to
6882:showing geometric multiplicity 2 for the zero
2132:for the subvariety. Furthermore, the sheaf on
1434:. Note that this agrees with the notion of a
6158:(cyclic meaning generated by 1 element as an
3712:{\displaystyle \operatorname {Spec} (A)-V(f)}
1477:{\displaystyle \Gamma (U,{\mathcal {O}}_{X})}
1377:{\displaystyle \Gamma (U,{\mathcal {O}}_{X})}
613:is defined on the distinguished open subsets
8:
7482:"Prime Ideal Structure in Commutative Rings"
7211:
2758:
2674:
2182:The spectrum of integers: The affine scheme
2120:as an "enrichment" of the topological space
386:
361:
258:
231:
5505:where the composition of the bottom arrows
3727:There is a relative version of the functor
2645:, since the only well defined morphisms to
2070:{\displaystyle \operatorname {MaxSpec} (R)}
7118:
6656:The term "spectrum" comes from the use in
3462:Non-Zariski topologies on a prime spectrum
1645:(since the preimage of any prime ideal in
572:{\displaystyle X=\operatorname {Spec} (R)}
7575:
7499:
7419:
7189:
7016:
6917:
6908:
6894:
6846:
6811:
6797:
6756:
6721:
6707:
6606:
6587:
6569:
6533:
6527:
6496:
6490:
6465:
6459:
6438:
6432:
6411:
6405:
6400:the covector being given by sending each
6381:
6375:
6346:
6322:
6303:
6294:
6270:
6257:
6232:
6219:
6200:
6187:
6178:
6133:
6128:
6103:
6097:
6055:
6031:
6012:
5994:
5916:
5899:
5885:
5868:
5856:
5851:
5828:
5827:
5825:
5801:
5776:
5763:
5757:
5756:
5746:
5745:
5743:
5722:
5715:
5690:
5685:
5681:
5680:
5671:
5644:
5639:
5638:
5637:
5633:
5632:
5629:
5597:
5573:
5562:
5558:
5557:
5532:
5525:
5524:
5513:
5482:
5471:
5467:
5466:
5452:
5426:
5418:
5417:
5392:
5391:
5306:
5300:
5299:
5295:
5285:
5275:
5243:
5202:
5194:
5193:
5190:
5149:
5118:
5117:
5114:
5100:
5098:
5066:
5031:
5010:
5006:
5005:
5002:
4970:
4949:
4938:
4934:
4933:
4920:
4919:
4914:
4908:
4907:
4898:
4888:
4885:
4861:
4860:
4858:
4813:
4812:
4810:
4771:
4765:
4764:
4754:
4753:
4751:
4739:{\displaystyle X=\mathbb {P} _{a,b}^{1}.}
4727:
4716:
4712:
4711:
4702:
4681:
4676:
4675:
4674:
4670:
4669:
4666:
4626:
4593:
4592:
4575:
4553:
4548:
4547:
4531:
4525:
4524:
4517:
4504:
4503:
4490:
4484:
4478:
4477:
4475:
4469:
4446:
4425:
4419:
4418:
4415:
4377:
4376:
4358:
4357:
4355:
4322:
4297:
4291:
4265:
4232:
4231:
4201:
4195:
4169:
4139:
4138:
4129:
4119:
4110:
4086:
4085:
4076:
4066:
4063:
4042:
4041:
4039:
4017:
4011:
4010:
4007:
3982:
3953:
3951:
3926:
3924:
3904:
3883:
3869:
3866:
3845:
3835:
3832:
3812:
3792:
3772:
3752:
3732:
3677:
3648:
3609:
3555:
3549:
3517:
3442:
3409:
3390:
3384:
3364:
3343:
3338:
3334:
3333:
3330:
3295:
3271:
3265:
3244:
3231:
3225:
3205:
3178:
3165:
3159:
3138:
3125:
3119:
3114:are distinguished open affine subschemes
3071:
3066:
3062:
3061:
3058:
3035:
3014:
3009:
3005:
3004:
3001:
2975:
2955:
2931:
2907:
2888:
2863:
2858:
2854:
2853:
2850:
2829:
2754:
2753:
2744:
2731:
2715:
2684:
2672:
2653:
2652:
2650:
2630:
2595:
2576:
2575:
2564:
2542:
2541:
2527:
2508:
2500:
2487:
2478:
2459:
2448:
2447:
2445:
2424:
2420:
2419:
2406:
2387:
2378:
2347:
2328:
2317:
2316:
2298:
2293:
2292:
2291:
2287:
2286:
2283:
2262:
2258:
2257:
2254:
2228:
2227:
2225:
2199:
2198:
2187:
2164:and the sheaf of polynomial functions on
2137:
2093:
2077:, together with the Zariski topology, is
2050:
1892:
1865:
1841:
1816:
1815:
1809:
1808:
1793:
1792:
1780:
1775:
1769:
1768:
1765:
1742:
1721:
1720:
1718:
1690:
1670:
1650:
1582:
1546:
1519:
1465:
1459:
1458:
1443:
1365:
1359:
1358:
1343:
1285:
1223:
1174:
1146:
1125:
1124:
1115:
1103:
1071:
1045:
1044:
1042:
980:
949:
936:
930:
901:
899:
873:
868:
849:
839:
826:
820:
819:
804:
792:{\textstyle U=\bigcup _{i\in I}D_{f_{i}}}
781:
776:
760:
748:
690:
674:
668:
667:
657:
645:
624:
618:
597:
591:
590:
587:
546:
494:
454:
406:
368:
359:
327:
250:
238:
229:
190:
157:
151:
111:
110:
108:
77:
69:
7154:
7024:{\displaystyle \operatorname {MaxSpec} }
6370:are then parametrized by the dual space
3640:
3629:{\displaystyle \operatorname {Spec} (A)}
3537:{\displaystyle \operatorname {Spec} (A)}
3260:that restrict to the same polynomial on
2220:in the category of affine schemes since
2157:{\displaystyle \operatorname {Spec} (R)}
2113:{\displaystyle \operatorname {Spec} (R)}
2088:One can thus view the topological space
1305:{\displaystyle \operatorname {Spec} (R)}
1243:{\displaystyle \operatorname {Spec} (R)}
1194:{\displaystyle \operatorname {Spec} (R)}
1098:. On the distinguished open subsets set
1091:{\displaystyle \operatorname {Spec} (R)}
1000:{\displaystyle \operatorname {Spec} (R)}
514:{\displaystyle \operatorname {Spec} (R)}
474:{\displaystyle \operatorname {Spec} (R)}
426:{\displaystyle \operatorname {Spec} (R)}
347:{\displaystyle \operatorname {Spec} (R)}
210:{\displaystyle \operatorname {Spec} (R)}
85:{\displaystyle \operatorname {Spec} {R}}
7400:Fontana, Marco; Loper, K. Alan (2008).
7222:
7099:
3424:
2996:is not affine as the global section of
1408:if it can be represented as a fraction
7615:
7341:; O'Shea, Donal; Little, John (1997),
7131:Arkhangel'skii & Pontryagin (1990)
6872:{\displaystyle K/(T-0)\oplus K/(T-0),}
6564:-space, thought of as the max spec of
4846:{\displaystyle {\mathcal {I}}=(ay-bx)}
4343:{\displaystyle f^{-1}(V)\to f^{-1}(U)}
3636:is called the constructible topology.
3053:Affine plane minus the origin. Inside
725:. It can be shown that this defines a
7165:
7106:
6779:{\displaystyle K/(T-1)\oplus K/(T-1)}
6652:Algebra representation § Weights
6334:{\displaystyle (a_{1},\ldots ,a_{n})}
3820:{\displaystyle \operatorname {Spec} }
3780:{\displaystyle \operatorname {Spec} }
3760:{\displaystyle \operatorname {Spec} }
3740:{\displaystyle \operatorname {Spec} }
2246:in the category of commutative rings.
1900:{\displaystyle \operatorname {Spec} }
1873:{\displaystyle \operatorname {Spec} }
1849:{\displaystyle \operatorname {Spec} }
1698:{\displaystyle \operatorname {Spec} }
1527:{\displaystyle \operatorname {Spec} }
399:is a basis for the Zariski topology.
7:
7662:"27.3 Relative spectrum via glueing"
7200:
6522:numbers, or equivalently a covector
5592:gives the line containing the point
3352:{\displaystyle \mathbb {A} _{k}^{2}}
3023:{\displaystyle \mathbb {P} _{k}^{n}}
2021:therefore consists of the points of
1510:It is useful to use the language of
1484:as precisely the set of elements of
7606:"Foundations Of Algebraic Geometry"
7260:Introduction to Commutative Algebra
6691:(as a ring) equals the spectrum of
3892:{\displaystyle \mathbf {Spec} _{S}}
1817:
1794:
1722:
1250:, that is, a prime ideal, then the
1205:. A sheaf of this form is called a
4997:is the line through the origin of
4434:{\displaystyle {\mathcal {O}}_{S}}
4350:is induced by the restriction map
4026:{\displaystyle {\mathcal {O}}_{S}}
1994:is algebraically closed), and the
1931:Following on from the example, in
1927:Motivation from algebraic geometry
1445:
1345:
1105:
806:
647:
606:{\displaystyle {\mathcal {O}}_{X}}
25:
7501:10.1090/S0002-9947-1969-0251026-X
7343:Ideals, Varieties, and Algorithms
5979:is the study of modules over its
5951:Representation theory perspective
5054:{\displaystyle (\alpha ,\beta ).}
4646:{\displaystyle \pi \colon X\to S}
4260:, and such that for open affines
3193:{\displaystyle D_{x}\cup D_{y}=U}
2249:The scheme-theoretic analogue of
305:to be the set of prime ideals of
7039:-commutative C*-algebras yields
6173:(the maximal ideal generated by
5617:{\displaystyle (\alpha ,\beta )}
5019:{\displaystyle \mathbb {A} ^{2}}
4585:
4582:
4579:
4576:
4164:such that for every open affine
3963:
3960:
3957:
3954:
3879:
3876:
3873:
3870:
3469:
2801:) if and only if it is compact,
2271:{\displaystyle \mathbb {C} ^{n}}
2085:also with the Zariski topology.
1490:that are regular at every point
1338:, then we can describe the ring
964:{\displaystyle R_{f}\to R_{fg}.}
729:and therefore that it defines a
392:{\displaystyle \{D_{f}:f\in R\}}
6789:the 2×2 zero matrix has module
6640:Functional analysis perspective
4873:{\displaystyle {\mathcal {A}}.}
4746:Consider the sheaf of algebras
3970:{\displaystyle \mathbf {Spec} }
1730:{\displaystyle {\mathfrak {p}}}
579:with the Zariski topology, the
6905:
6899:
6863:
6851:
6843:
6837:
6828:
6816:
6808:
6802:
6773:
6761:
6753:
6747:
6738:
6726:
6718:
6712:
6646:Spectrum (functional analysis)
6636:-dimensional representations.
6612:
6580:
6539:
6502:
6357:
6351:
6328:
6296:
6276:
6250:
6238:
6212:
6206:
6180:
6154:is a cyclic representation of
6072:
6066:
6037:
6005:
5807:
5769:
5611:
5599:
5550:
5538:
5529:
5521:
5460:
5401:
5396:
5388:
5375:
5369:
5363:
5324:
5312:
5269:
5228:
5216:
5175:
5134:
5122:
5083:{\displaystyle \alpha \neq 0,}
5045:
5033:
4984:
4972:
4929:
4926:
4904:
4840:
4822:
4789:
4777:
4637:
4602:
4599:
4589:
4566:
4537:
4500:
4389:
4383:
4373:
4370:
4364:
4337:
4331:
4315:
4312:
4306:
4247:
4244:
4238:
4228:
4216:
4210:
4148:
4145:
4135:
4092:
4082:
4051:{\displaystyle {\mathcal {A}}}
3706:
3700:
3691:
3685:
3659:
3653:
3623:
3617:
3588:
3573:
3561:
3531:
3525:
3504:First, there is the notion of
3416:
3410:
3397:
3391:
3312:
3300:
3101:
3089:
2913:
2881:
2721:
2702:
2696:
2677:
2612:
2609:
2600:
2592:
2580:
2572:
2533:
2501:
2484:
2452:
2412:
2380:
2356:
2353:
2321:
2313:
2203:
2195:
2151:
2145:
2107:
2101:
2064:
2058:
1804:
1799:
1789:
1711:category of topological spaces
1632:
1626:
1617:
1614:
1608:
1596:
1590:
1557:
1471:
1448:
1371:
1348:
1299:
1293:
1237:
1231:
1188:
1182:
1136:
1130:
1108:
1085:
1079:
1050:
994:
988:
942:
832:
809:
680:
650:
566:
560:
508:
502:
468:
462:
420:
414:
341:
335:
204:
198:
120:{\displaystyle {\mathcal {O}}}
1:
7586:10.1080/00927872.2018.1469637
7178:Atiyah & Macdonald (1969)
7143:Atiyah & Macdonald (1969)
7068:Serre's theorem on affineness
6995:, yielding the notion of the
2037:, i.e. the maximal ideals in
1913:category of commutative rings
1707:category of commutative rings
7660:The Stacks Project authors.
7540:Steps in Commutative Algebra
6959:(a vector whose orbit under
6341:). These representations of
4279:{\displaystyle V\subseteq U}
4183:{\displaystyle U\subseteq S}
3220:are pairs of polynomials on
2786:Hausdorff space (that is, a
2660:{\displaystyle \mathbb {C} }
2235:{\displaystyle \mathbb {Z} }
2010:(an algebraic set is called
1986:correspond to the points of
1713:. Moreover, for every prime
1059:{\displaystyle {\tilde {M}}}
914:{\displaystyle \varprojlim }
277:This topology is called the
64:, and is usually denoted by
27:Set of a ring's prime ideals
6551:{\displaystyle K^{n}\to K.}
6454:. Thus a representation of
4441:-algebras and schemes over
3807:is a scheme, then relative
3290:, which can be shown to be
3253:{\displaystyle D_{x},D_{y}}
3147:{\displaystyle D_{x},D_{y}}
7711:
7545:Cambridge University Press
7212:Fontana & Loper (2008)
7168:, Chapter 4, example 4.4.1
7145:, Ch. 1. Exercise 23. (iv)
6649:
6643:
6511:{\displaystyle K^{n}\to K}
6092:, and writing in terms of
4657:Example of a relative Spec
4653:is a morphism of schemes.
4190:, there is an isomorphism
1949:algebraically closed field
1917:category of affine schemes
1757:descends to homomorphisms
1254:of the structure sheaf at
743:, written as the union of
252: is an ideal of
29:
7564:Communications in Algebra
7537:Sharp, Rodney Y. (2001).
7430:10.1080/00927870802110326
7407:Communications in Algebra
7294:10.1007/978-3-642-61265-7
7133:, example 21, section 2.6
6976:characteristic polynomial
6970:of the module equals the
6289:corresponds to the point
6123:or equivalently a module
219:collection of closed sets
7121:, p. 70, Definition
6997:spectrum of a C*-algebra
6929:{\displaystyle K/T^{2},}
5963:corresponds to a module
5955:From the perspective of
4853:be a sheaf of ideals of
3325:, the global section of
2774:The prime spectrum of a
1566:{\displaystyle f:R\to S}
1167:localization of a module
1037:, we may define a sheaf
971:One may check that this
7372:The geometry of schemes
7311:Brandal, Willy (1979).
7252:Atiyah, Michael Francis
7109:, p. 44, Def. 3.26
7041:noncommutative topology
6687:. Then the spectrum of
6518:) is given by a set of
4880:Then the relative spec
3723:Global or relative Spec
2989:{\displaystyle n\geq 1}
1907:yields a contravariant
96:it is simultaneously a
7654:The Spectrum of a Ring
7517:Kock, Joachim (2007).
7025:
6930:
6873:
6780:
6622:
6552:
6512:
6475:
6448:
6421:
6394:
6393:{\displaystyle V^{*},}
6364:
6335:
6283:
6148:
6113:
6082:
6044:
5942:
5814:
5732:
5660:
5618:
5583:
5496:
5084:
5055:
5020:
4991:
4959:
4874:
4847:
4799:
4740:
4691:
4647:
4612:
4455:
4435:
4396:
4344:
4280:
4254:
4184:
4158:
4099:
4052:
4027:
3991:
3971:
3940:
3913:
3893:
3855:
3821:
3801:
3781:
3761:
3741:
3713:
3666:
3630:
3598:
3538:
3506:constructible topology
3451:
3431:
3373:
3353:
3319:
3284:
3283:{\displaystyle D_{xy}}
3254:
3214:
3194:
3148:
3108:
3044:
3024:
2990:
2964:
2940:
2920:
2838:
2765:
2661:
2639:
2619:
2550:
2434:
2363:
2272:
2236:
2210:
2158:
2114:
2071:
1901:
1874:
1850:
1827:
1751:
1731:
1699:
1679:
1659:
1639:
1567:
1528:
1506:Functorial perspective
1478:
1396:is regular at a point
1378:
1306:
1244:
1195:
1159:
1092:
1060:
1001:
965:
915:
888:
793:
703:
634:
607:
573:
515:
481:is not, in general, a
475:
427:
393:
348:
268:
211:
167:
121:
86:
7480:Hochster, M. (1969).
7278:Arkhangel’skii, A. V.
7026:
6931:
6874:
6781:
6650:Further information:
6623:
6553:
6513:
6476:
6474:{\displaystyle K^{n}}
6449:
6447:{\displaystyle a_{i}}
6427:to the corresponding
6422:
6420:{\displaystyle x_{i}}
6395:
6365:
6336:
6284:
6149:
6114:
6112:{\displaystyle x_{i}}
6083:
6050:or, without a basis,
6045:
5957:representation theory
5943:
5853: minors of
5815:
5733:
5661:
5619:
5584:
5497:
5085:
5056:
5026:containing the point
5021:
4992:
4960:
4875:
4848:
4800:
4741:
4692:
4648:
4613:
4456:
4436:
4397:
4345:
4281:
4255:
4185:
4159:
4100:
4053:
4028:
3992:
3972:
3941:
3914:
3894:
3856:
3822:
3802:
3782:
3762:
3742:
3714:
3667:
3631:
3599:
3539:
3452:
3432:
3374:
3354:
3320:
3285:
3255:
3215:
3195:
3149:
3109:
3045:
3025:
2991:
2965:
2941:
2921:
2839:
2766:
2662:
2640:
2620:
2551:
2435:
2373:perspective, a point
2364:
2273:
2237:
2211:
2159:
2115:
2072:
1902:
1875:
1858:locally ringed spaces
1851:
1836:of local rings. Thus
1828:
1752:
1732:
1700:
1680:
1660:
1640:
1568:
1529:
1479:
1379:
1307:
1245:
1196:
1160:
1093:
1061:
1002:
966:
916:
889:
794:
704:
635:
633:{\displaystyle D_{f}}
608:
574:
529:axiom); it is also a
516:
476:
428:
394:
349:
322:is an open subset of
269:
212:
168:
166:{\displaystyle V_{I}}
122:
87:
7622:: CS1 maint: year (
7454:, Berlin, New York:
7345:, Berlin, New York:
7180:, Ch. 5, Exercise 27
7063:Spectrum of a matrix
7053:Scheme (mathematics)
7033:Banach–Stone theorem
7015:
6893:
6796:
6706:
6621:{\displaystyle R=K,}
6568:
6526:
6489:
6458:
6431:
6404:
6374:
6345:
6293:
6177:
6147:{\displaystyle R/I,}
6127:
6096:
6081:{\displaystyle R=K.}
6054:
5993:
5824:
5742:
5670:
5628:
5596:
5512:
5097:
5065:
5030:
5001:
4969:
4884:
4857:
4809:
4750:
4701:
4665:
4625:
4468:
4445:
4414:
4354:
4290:
4264:
4194:
4168:
4109:
4062:
4058:, there is a scheme
4038:
4006:
3981:
3950:
3923:
3903:
3865:
3831:
3811:
3791:
3771:
3751:
3731:
3676:
3665:{\displaystyle V(I)}
3647:
3608:
3548:
3516:
3441:
3383:
3363:
3329:
3294:
3264:
3224:
3204:
3158:
3118:
3057:
3034:
3000:
2974:
2954:
2930:
2849:
2828:
2784:totally disconnected
2671:
2649:
2629:
2563:
2444:
2377:
2282:
2278:: The affine scheme
2253:
2224:
2186:
2136:
2092:
2049:
1969:polynomial functions
1891:
1864:
1840:
1764:
1741:
1717:
1689:
1669:
1665:is a prime ideal in
1649:
1581:
1545:
1518:
1442:
1342:
1314:locally ringed space
1284:
1222:
1173:
1102:
1070:
1041:
979:
929:
898:
803:
747:
644:
617:
586:
545:
493:
453:
405:
358:
326:
228:
189:
150:
107:
68:
7695:Functional analysis
7680:Commutative algebra
6043:{\displaystyle R=K}
5727:
5655:
5578:
5553:
5487:
4954:
4732:
4686:
3348:
3076:
3019:
2868:
2814:Non-affine examples
2538:
2303:
1882:natural isomorphism
1207:quasicoherent sheaf
537:Sheaves and schemes
437:, but almost never
39:commutative algebra
7651:Kevin R. Coombes:
7451:Algebraic Geometry
7321:10.1007/BFb0069021
7286:General Topology I
7262:. Westview Press.
7083:Primitive spectrum
7021:
7005:algebra of scalars
6972:minimal polynomial
6926:
6869:
6776:
6695:(as an operator).
6669:finite-dimensional
6618:
6548:
6508:
6471:
6444:
6417:
6390:
6360:
6331:
6279:
6144:
6109:
6078:
6040:
5938:
5924:
5810:
5728:
5679:
5656:
5631:
5614:
5579:
5556:
5492:
5490:
5465:
5283:
5080:
5051:
5016:
4987:
4955:
4932:
4896:
4870:
4843:
4795:
4736:
4710:
4687:
4668:
4643:
4608:
4451:
4431:
4392:
4340:
4276:
4250:
4180:
4154:
4127:
4095:
4074:
4048:
4023:
3987:
3967:
3936:
3934:
3909:
3889:
3851:
3843:
3817:
3797:
3777:
3757:
3737:
3709:
3662:
3626:
3594:
3534:
3481:. You can help by
3447:
3427:
3369:
3349:
3332:
3315:
3280:
3250:
3210:
3190:
3144:
3104:
3060:
3040:
3020:
3003:
2986:
2960:
2936:
2916:
2852:
2834:
2761:
2657:
2635:
2615:
2546:
2430:
2359:
2285:
2268:
2232:
2206:
2154:
2110:
2067:
2002:correspond to the
1939:, i.e. subsets of
1933:algebraic geometry
1897:
1870:
1846:
1823:
1747:
1727:
1695:
1675:
1655:
1635:
1563:
1524:
1474:
1374:
1331:field of fractions
1302:
1240:
1201:and satisfies the
1191:
1155:
1088:
1056:
997:
961:
911:
909:
884:
847:
789:
771:
699:
630:
603:
569:
511:
471:
423:
389:
344:
264:
207:
163:
117:
100:equipped with the
94:algebraic geometry
82:
56:is the set of all
7610:math.stanford.edu
7554:978-0-511-62368-4
7465:978-0-387-90244-9
7446:Hartshorne, Robin
7385:978-0-387-98637-1
7356:978-0-387-94680-1
7330:978-3-540-09507-1
7303:978-3-642-64767-3
7282:Pontryagin, L. S.
7269:978-0-201-40751-8
7119:Hartshorne (1977)
7058:Projective scheme
6999:. Notably, for a
6963:spans the space);
6363:{\displaystyle K}
5854:
5554:
5434:
5353:
5276:
5261:
5251:
5210:
5167:
5157:
4889:
4551:
4493:
4454:{\displaystyle S}
4120:
4067:
3990:{\displaystyle S}
3927:
3912:{\displaystyle S}
3836:
3800:{\displaystyle S}
3512:, the subsets of
3499:
3498:
3450:{\displaystyle U}
3379:is not affine as
3372:{\displaystyle U}
3318:{\displaystyle k}
3213:{\displaystyle U}
3043:{\displaystyle k}
2963:{\displaystyle n}
2948:Proj construction
2939:{\displaystyle k}
2837:{\displaystyle n}
2638:{\displaystyle +}
2539:
2371:functor of points
1921:opposite category
1750:{\displaystyle f}
1737:the homomorphism
1678:{\displaystyle R}
1658:{\displaystyle S}
1540:ring homomorphism
1514:and observe that
1280:. Consequently,
1133:
1053:
1022:Similarly, for a
902:
840:
756:
719:by the powers of
253:
173:to be the set of
98:topological space
16:(Redirected from
7702:
7665:
7627:
7621:
7613:
7597:
7579:
7558:
7543:(2nd ed.).
7533:
7523:
7513:
7503:
7476:
7441:
7423:
7414:(8): 2917–2922.
7396:
7359:
7334:
7307:
7273:
7237:
7231:
7225:
7220:
7214:
7209:
7203:
7198:
7192:
7190:Tarizadeh (2019)
7187:
7181:
7175:
7169:
7163:
7157:
7152:
7146:
7140:
7134:
7128:
7122:
7116:
7110:
7104:
7078:Ziegler spectrum
7030:
7028:
7027:
7022:
6968:invariant factor
6942:In more detail:
6935:
6933:
6932:
6927:
6922:
6921:
6912:
6878:
6876:
6875:
6870:
6850:
6815:
6785:
6783:
6782:
6777:
6760:
6725:
6627:
6625:
6624:
6619:
6611:
6610:
6592:
6591:
6560:Thus, points in
6557:
6555:
6554:
6549:
6538:
6537:
6517:
6515:
6514:
6509:
6501:
6500:
6480:
6478:
6477:
6472:
6470:
6469:
6453:
6451:
6450:
6445:
6443:
6442:
6426:
6424:
6423:
6418:
6416:
6415:
6399:
6397:
6396:
6391:
6386:
6385:
6369:
6367:
6366:
6361:
6340:
6338:
6337:
6332:
6327:
6326:
6308:
6307:
6288:
6286:
6285:
6280:
6275:
6274:
6262:
6261:
6237:
6236:
6224:
6223:
6205:
6204:
6192:
6191:
6153:
6151:
6150:
6145:
6137:
6118:
6116:
6115:
6110:
6108:
6107:
6087:
6085:
6084:
6079:
6049:
6047:
6046:
6041:
6036:
6035:
6017:
6016:
5959:, a prime ideal
5947:
5945:
5944:
5939:
5934:
5930:
5929:
5928:
5921:
5920:
5904:
5903:
5890:
5889:
5873:
5872:
5855:
5852:
5833:
5832:
5819:
5817:
5816:
5811:
5806:
5805:
5781:
5780:
5768:
5767:
5762:
5761:
5751:
5750:
5737:
5735:
5734:
5729:
5726:
5721:
5720:
5719:
5695:
5694:
5684:
5665:
5663:
5662:
5657:
5654:
5643:
5642:
5636:
5623:
5621:
5620:
5615:
5588:
5586:
5585:
5580:
5577:
5572:
5561:
5555:
5533:
5528:
5501:
5499:
5498:
5493:
5491:
5486:
5481:
5470:
5457:
5456:
5444:
5440:
5439:
5435:
5427:
5421:
5395:
5373:
5367:
5358:
5354:
5352:
5348:
5327:
5311:
5310:
5305:
5304:
5296:
5290:
5289:
5284:
5266:
5262:
5260:
5256:
5252:
5244:
5231:
5215:
5211:
5203:
5197:
5191:
5172:
5168:
5166:
5162:
5158:
5150:
5137:
5121:
5115:
5089:
5087:
5086:
5081:
5060:
5058:
5057:
5052:
5025:
5023:
5022:
5017:
5015:
5014:
5009:
4996:
4994:
4993:
4990:{\displaystyle }
4988:
4964:
4962:
4961:
4956:
4953:
4948:
4937:
4925:
4924:
4918:
4913:
4912:
4903:
4902:
4897:
4879:
4877:
4876:
4871:
4866:
4865:
4852:
4850:
4849:
4844:
4818:
4817:
4804:
4802:
4801:
4796:
4776:
4775:
4770:
4769:
4759:
4758:
4745:
4743:
4742:
4737:
4731:
4726:
4715:
4696:
4694:
4693:
4688:
4685:
4680:
4679:
4673:
4652:
4650:
4649:
4644:
4617:
4615:
4614:
4609:
4598:
4597:
4588:
4562:
4561:
4557:
4552:
4549:
4536:
4535:
4530:
4529:
4522:
4521:
4509:
4508:
4496:
4495:
4494:
4491:
4489:
4488:
4483:
4482:
4461:. In formulas,
4460:
4458:
4457:
4452:
4440:
4438:
4437:
4432:
4430:
4429:
4424:
4423:
4401:
4399:
4398:
4393:
4382:
4381:
4363:
4362:
4349:
4347:
4346:
4341:
4330:
4329:
4305:
4304:
4286:, the inclusion
4285:
4283:
4282:
4277:
4259:
4257:
4256:
4251:
4237:
4236:
4209:
4208:
4189:
4187:
4186:
4181:
4163:
4161:
4160:
4155:
4144:
4143:
4134:
4133:
4128:
4104:
4102:
4101:
4096:
4091:
4090:
4081:
4080:
4075:
4057:
4055:
4054:
4049:
4047:
4046:
4032:
4030:
4029:
4024:
4022:
4021:
4016:
4015:
3996:
3994:
3993:
3988:
3977:. For a scheme
3976:
3974:
3973:
3968:
3966:
3945:
3943:
3942:
3937:
3935:
3918:
3916:
3915:
3910:
3898:
3896:
3895:
3890:
3888:
3887:
3882:
3860:
3858:
3857:
3852:
3850:
3849:
3844:
3826:
3824:
3823:
3818:
3806:
3804:
3803:
3798:
3786:
3784:
3783:
3778:
3766:
3764:
3763:
3758:
3746:
3744:
3743:
3738:
3718:
3716:
3715:
3710:
3671:
3669:
3668:
3663:
3635:
3633:
3632:
3627:
3603:
3601:
3600:
3595:
3560:
3559:
3543:
3541:
3540:
3535:
3494:
3491:
3473:
3466:
3456:
3454:
3453:
3448:
3436:
3434:
3433:
3428:
3420:
3419:
3401:
3400:
3378:
3376:
3375:
3370:
3358:
3356:
3355:
3350:
3347:
3342:
3337:
3324:
3322:
3321:
3316:
3289:
3287:
3286:
3281:
3279:
3278:
3259:
3257:
3256:
3251:
3249:
3248:
3236:
3235:
3219:
3217:
3216:
3211:
3199:
3197:
3196:
3191:
3183:
3182:
3170:
3169:
3153:
3151:
3150:
3145:
3143:
3142:
3130:
3129:
3113:
3111:
3110:
3105:
3075:
3070:
3065:
3049:
3047:
3046:
3041:
3029:
3027:
3026:
3021:
3018:
3013:
3008:
2995:
2993:
2992:
2987:
2969:
2967:
2966:
2961:
2945:
2943:
2942:
2937:
2925:
2923:
2922:
2917:
2912:
2911:
2893:
2892:
2867:
2862:
2857:
2843:
2841:
2840:
2835:
2770:
2768:
2767:
2762:
2757:
2749:
2748:
2736:
2735:
2720:
2719:
2689:
2688:
2666:
2664:
2663:
2658:
2656:
2644:
2642:
2641:
2636:
2624:
2622:
2621:
2616:
2599:
2579:
2555:
2553:
2552:
2547:
2545:
2540:
2537:
2536:
2532:
2531:
2513:
2512:
2488:
2483:
2482:
2464:
2463:
2451:
2439:
2437:
2436:
2431:
2429:
2428:
2423:
2411:
2410:
2392:
2391:
2368:
2366:
2365:
2360:
2352:
2351:
2333:
2332:
2320:
2302:
2297:
2296:
2290:
2277:
2275:
2274:
2269:
2267:
2266:
2261:
2241:
2239:
2238:
2233:
2231:
2215:
2213:
2212:
2207:
2202:
2163:
2161:
2160:
2155:
2119:
2117:
2116:
2111:
2076:
2074:
2073:
2068:
2029:. The points of
2017:The spectrum of
1906:
1904:
1903:
1898:
1879:
1877:
1876:
1871:
1855:
1853:
1852:
1847:
1832:
1830:
1829:
1824:
1822:
1821:
1820:
1814:
1813:
1803:
1802:
1798:
1797:
1788:
1787:
1774:
1773:
1756:
1754:
1753:
1748:
1736:
1734:
1733:
1728:
1726:
1725:
1704:
1702:
1701:
1696:
1685:). In this way,
1684:
1682:
1681:
1676:
1664:
1662:
1661:
1656:
1644:
1642:
1641:
1636:
1572:
1570:
1569:
1564:
1533:
1531:
1530:
1525:
1501:
1495:
1489:
1483:
1481:
1480:
1475:
1470:
1469:
1464:
1463:
1436:regular function
1433:
1427:
1421:
1407:
1401:
1395:
1389:
1383:
1381:
1380:
1375:
1370:
1369:
1364:
1363:
1337:
1324:
1311:
1309:
1308:
1303:
1276:, and this is a
1275:
1269:
1259:
1249:
1247:
1246:
1241:
1217:
1200:
1198:
1197:
1192:
1164:
1162:
1161:
1156:
1151:
1150:
1135:
1134:
1126:
1120:
1119:
1097:
1095:
1094:
1089:
1065:
1063:
1062:
1057:
1055:
1054:
1046:
1036:
1030:
1006:
1004:
1003:
998:
970:
968:
967:
962:
957:
956:
941:
940:
920:
918:
917:
912:
910:
893:
891:
890:
885:
880:
879:
878:
877:
860:
859:
848:
831:
830:
825:
824:
798:
796:
795:
790:
788:
787:
786:
785:
770:
742:
724:
718:
708:
706:
705:
700:
695:
694:
679:
678:
673:
672:
662:
661:
639:
637:
636:
631:
629:
628:
612:
610:
609:
604:
602:
601:
596:
595:
578:
576:
575:
570:
541:Given the space
525:(satisfies the T
523:Kolmogorov space
520:
518:
517:
512:
480:
478:
477:
472:
432:
430:
429:
424:
398:
396:
395:
390:
373:
372:
353:
351:
350:
345:
279:Zariski topology
273:
271:
270:
265:
254:
251:
243:
242:
217:by defining the
216:
214:
213:
208:
172:
170:
169:
164:
162:
161:
131:Zariski topology
126:
124:
123:
118:
116:
115:
91:
89:
88:
83:
81:
51:commutative ring
21:
7710:
7709:
7705:
7704:
7703:
7701:
7700:
7699:
7670:
7669:
7668:
7659:
7647:
7635:
7633:Further reading
7630:
7614:
7600:
7561:
7555:
7536:
7521:
7516:
7479:
7466:
7456:Springer-Verlag
7444:
7399:
7386:
7376:Springer-Verlag
7364:Eisenbud, David
7362:
7357:
7347:Springer-Verlag
7337:
7331:
7310:
7304:
7284:, eds. (1990).
7276:
7270:
7256:Macdonald, I.G.
7250:
7246:
7241:
7240:
7232:
7228:
7221:
7217:
7210:
7206:
7199:
7195:
7188:
7184:
7176:
7172:
7164:
7160:
7155:Hochster (1969)
7153:
7149:
7141:
7137:
7129:
7125:
7117:
7113:
7105:
7101:
7096:
7049:
7013:
7012:
7001:Hausdorff space
6993:operator theory
6985:
6983:Generalizations
6913:
6891:
6890:
6794:
6793:
6704:
6703:
6662:linear operator
6658:operator theory
6654:
6648:
6642:
6602:
6583:
6566:
6565:
6529:
6524:
6523:
6492:
6487:
6486:
6461:
6456:
6455:
6434:
6429:
6428:
6407:
6402:
6401:
6377:
6372:
6371:
6343:
6342:
6318:
6299:
6291:
6290:
6266:
6253:
6228:
6215:
6196:
6183:
6175:
6174:
6171:Nullstellensatz
6169:-space, by the
6125:
6124:
6099:
6094:
6093:
6052:
6051:
6027:
6008:
5991:
5990:
5988:polynomial ring
5953:
5923:
5922:
5912:
5910:
5905:
5895:
5892:
5891:
5881:
5879:
5874:
5864:
5857:
5841:
5837:
5822:
5821:
5797:
5772:
5755:
5740:
5739:
5711:
5686:
5668:
5667:
5626:
5625:
5594:
5593:
5510:
5509:
5489:
5488:
5463:
5458:
5448:
5422:
5416:
5412:
5404:
5399:
5379:
5378:
5372:
5366:
5360:
5359:
5332:
5328:
5298:
5297:
5291:
5274:
5272:
5267:
5236:
5232:
5198:
5192:
5186:
5178:
5173:
5142:
5138:
5116:
5110:
5095:
5094:
5063:
5062:
5028:
5027:
5004:
4999:
4998:
4967:
4966:
4887:
4882:
4881:
4855:
4854:
4807:
4806:
4763:
4748:
4747:
4699:
4698:
4663:
4662:
4659:
4623:
4622:
4543:
4523:
4513:
4476:
4471:
4466:
4465:
4443:
4442:
4417:
4412:
4411:
4352:
4351:
4318:
4293:
4288:
4287:
4262:
4261:
4197:
4192:
4191:
4166:
4165:
4118:
4107:
4106:
4105:and a morphism
4065:
4060:
4059:
4036:
4035:
4009:
4004:
4003:
3979:
3978:
3948:
3947:
3921:
3920:
3901:
3900:
3868:
3863:
3862:
3834:
3829:
3828:
3809:
3808:
3789:
3788:
3769:
3768:
3749:
3748:
3729:
3728:
3725:
3674:
3673:
3645:
3644:
3641:Hochster (1969)
3606:
3605:
3551:
3546:
3545:
3514:
3513:
3508:: given a ring
3495:
3489:
3486:
3479:needs expansion
3464:
3439:
3438:
3405:
3386:
3381:
3380:
3361:
3360:
3327:
3326:
3292:
3291:
3267:
3262:
3261:
3240:
3227:
3222:
3221:
3202:
3201:
3174:
3161:
3156:
3155:
3134:
3121:
3116:
3115:
3055:
3054:
3032:
3031:
2998:
2997:
2972:
2971:
2952:
2951:
2928:
2927:
2903:
2884:
2847:
2846:
2826:
2825:
2816:
2803:quasi-separated
2782:) is a compact
2740:
2727:
2711:
2680:
2669:
2668:
2647:
2646:
2627:
2626:
2561:
2560:
2523:
2504:
2496:
2474:
2455:
2442:
2441:
2418:
2402:
2383:
2375:
2374:
2343:
2324:
2280:
2279:
2256:
2251:
2250:
2222:
2221:
2184:
2183:
2179:
2134:
2133:
2090:
2089:
2047:
2046:
1929:
1889:
1888:
1862:
1861:
1838:
1837:
1807:
1776:
1767:
1762:
1761:
1739:
1738:
1715:
1714:
1687:
1686:
1667:
1666:
1647:
1646:
1579:
1578:
1543:
1542:
1516:
1515:
1512:category theory
1508:
1497:
1491:
1485:
1457:
1440:
1439:
1429:
1423:
1409:
1403:
1397:
1391:
1385:
1357:
1340:
1339:
1333:
1327:integral domain
1320:
1282:
1281:
1271:
1265:
1255:
1220:
1219:
1213:
1171:
1170:
1142:
1111:
1100:
1099:
1068:
1067:
1039:
1038:
1032:
1026:
977:
976:
975:is a sheaf, so
945:
932:
927:
926:
896:
895:
869:
864:
838:
818:
801:
800:
777:
772:
745:
744:
738:
720:
714:
686:
666:
653:
642:
641:
620:
615:
614:
589:
584:
583:
581:structure sheaf
543:
542:
539:
528:
491:
490:
486:
451:
450:
441:: in fact, the
403:
402:
364:
356:
355:
324:
323:
321:
309:not containing
304:
234:
226:
225:
187:
186:
181:. We can put a
153:
148:
147:
133:
105:
104:
66:
65:
45:(or simply the
35:
28:
23:
22:
15:
12:
11:
5:
7708:
7706:
7698:
7697:
7692:
7687:
7682:
7672:
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7667:
7666:
7657:
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7645:External links
7643:
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7223:Brandal (1979)
7215:
7204:
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7123:
7111:
7098:
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7073:Étale spectrum
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6644:Main article:
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4406:of the sheaf.
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3999:quasi-coherent
3986:
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3933:
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3881:
3878:
3875:
3872:
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3827:is denoted by
3816:
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3767:, or relative
3756:
3747:called global
3736:
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3181:
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3154:. Their union
3141:
3137:
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2799:spectral space
2791:
2780:power set ring
2772:
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2244:initial object
2230:
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2201:
2197:
2194:
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2178:
2175:
2153:
2150:
2147:
2144:
2141:
2109:
2106:
2103:
2100:
2097:
2066:
2063:
2060:
2057:
2054:
1980:maximal ideals
1959:variables. If
1937:algebraic sets
1928:
1925:
1923:of the other.
1896:
1869:
1845:
1834:
1833:
1819:
1812:
1806:
1801:
1796:
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1507:
1504:
1473:
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1462:
1456:
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1447:
1373:
1368:
1362:
1356:
1353:
1350:
1347:
1301:
1298:
1295:
1292:
1289:
1239:
1236:
1233:
1230:
1227:
1218:is a point in
1190:
1187:
1184:
1181:
1178:
1154:
1149:
1145:
1141:
1138:
1132:
1129:
1123:
1118:
1114:
1110:
1107:
1087:
1084:
1081:
1078:
1075:
1052:
1049:
1031:over the ring
996:
993:
990:
987:
984:
960:
955:
952:
948:
944:
939:
935:
908:
905:
883:
876:
872:
867:
863:
858:
855:
852:
846:
843:
837:
834:
829:
823:
817:
814:
811:
808:
784:
780:
775:
769:
766:
763:
759:
755:
752:
698:
693:
689:
685:
682:
677:
671:
665:
660:
656:
652:
649:
627:
623:
600:
594:
568:
565:
562:
559:
556:
553:
550:
538:
535:
531:spectral space
526:
510:
507:
504:
501:
498:
484:
470:
467:
464:
461:
458:
443:maximal ideals
422:
419:
416:
413:
410:
388:
385:
382:
379:
376:
371:
367:
363:
343:
340:
337:
334:
331:
317:
300:
275:
274:
263:
260:
257:
249:
246:
241:
237:
233:
206:
203:
200:
197:
194:
160:
156:
132:
129:
114:
102:sheaf of rings
80:
76:
73:
43:prime spectrum
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7707:
7696:
7693:
7691:
7688:
7686:
7685:Scheme theory
7683:
7681:
7678:
7677:
7675:
7663:
7658:
7656:
7655:
7650:
7649:
7644:
7640:
7637:
7636:
7632:
7625:
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7607:
7603:
7599:
7595:
7591:
7587:
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7511:
7507:
7502:
7497:
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7439:
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7340:
7336:
7332:
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7314:
7309:
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7299:
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7227:
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7151:
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7139:
7136:
7132:
7127:
7124:
7120:
7115:
7112:
7108:
7103:
7100:
7093:
7089:
7088:Stone duality
7086:
7084:
7081:
7079:
7076:
7074:
7071:
7069:
7066:
7064:
7061:
7059:
7056:
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7050:
7046:
7044:
7042:
7038:
7034:
7018:
7010:
7006:
7002:
6998:
6994:
6990:
6982:
6977:
6973:
6969:
6965:
6962:
6958:
6957:cyclic vector
6954:
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6948:
6945:
6944:
6943:
6940:
6923:
6918:
6914:
6909:
6902:
6896:
6889:
6888:
6887:
6885:
6866:
6860:
6857:
6854:
6847:
6840:
6834:
6831:
6825:
6822:
6819:
6812:
6805:
6799:
6792:
6791:
6790:
6770:
6767:
6764:
6757:
6750:
6744:
6741:
6735:
6732:
6729:
6722:
6715:
6709:
6702:
6701:
6700:
6696:
6694:
6690:
6686:
6682:
6678:
6674:
6671:vector space
6670:
6666:
6663:
6659:
6653:
6647:
6639:
6637:
6635:
6631:
6615:
6607:
6603:
6599:
6596:
6593:
6588:
6584:
6577:
6574:
6571:
6563:
6558:
6545:
6542:
6534:
6530:
6521:
6505:
6497:
6493:
6485:-linear maps
6484:
6466:
6462:
6439:
6435:
6412:
6408:
6387:
6382:
6378:
6354:
6348:
6323:
6319:
6315:
6312:
6309:
6304:
6300:
6271:
6267:
6263:
6258:
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6244:
6241:
6233:
6229:
6225:
6220:
6216:
6209:
6201:
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6193:
6188:
6184:
6172:
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6161:
6157:
6141:
6138:
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6122:
6104:
6100:
6091:
6075:
6069:
6063:
6060:
6057:
6032:
6028:
6024:
6021:
6018:
6013:
6009:
6002:
5999:
5996:
5989:
5984:
5982:
5981:group algebra
5978:
5974:
5970:
5966:
5962:
5958:
5950:
5948:
5935:
5931:
5925:
5917:
5913:
5907:
5900:
5896:
5886:
5882:
5876:
5869:
5865:
5858:
5848:
5845:
5842:
5838:
5834:
5802:
5798:
5794:
5791:
5788:
5785:
5782:
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5773:
5764:
5752:
5723:
5716:
5712:
5708:
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5699:
5696:
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5574:
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5534:
5518:
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5508:
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5475:
5472:
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5428:
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5409:
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5321:
5318:
5315:
5307:
5292:
5286:
5280:
5277:
5263:
5257:
5253:
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5245:
5240:
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5233:
5225:
5222:
5219:
5212:
5207:
5204:
5199:
5187:
5183:
5180:
5169:
5163:
5159:
5154:
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5143:
5139:
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5128:
5125:
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5107:
5104:
5093:
5092:
5091:
5077:
5074:
5071:
5068:
5048:
5042:
5039:
5036:
5011:
4981:
4978:
4975:
4950:
4945:
4942:
4939:
4915:
4899:
4893:
4890:
4867:
4837:
4834:
4831:
4828:
4825:
4819:
4792:
4786:
4783:
4780:
4772:
4760:
4733:
4728:
4723:
4720:
4717:
4707:
4704:
4682:
4656:
4654:
4640:
4634:
4631:
4628:
4605:
4572:
4569:
4563:
4558:
4554:
4544:
4540:
4532:
4518:
4514:
4510:
4497:
4485:
4472:
4464:
4463:
4462:
4448:
4426:
4407:
4405:
4386:
4367:
4334:
4326:
4323:
4319:
4309:
4301:
4298:
4294:
4273:
4270:
4267:
4241:
4225:
4222:
4219:
4213:
4205:
4202:
4198:
4177:
4174:
4171:
4151:
4130:
4124:
4121:
4115:
4112:
4077:
4071:
4068:
4034:
4018:
4000:
3984:
3931:
3928:
3906:
3884:
3846:
3840:
3837:
3814:
3794:
3774:
3754:
3734:
3722:
3720:
3703:
3697:
3694:
3688:
3682:
3679:
3656:
3650:
3642:
3637:
3620:
3614:
3611:
3591:
3585:
3582:
3579:
3576:
3570:
3567:
3564:
3556:
3552:
3528:
3522:
3519:
3511:
3507:
3502:
3493:
3484:
3480:
3477:This section
3475:
3472:
3468:
3467:
3461:
3444:
3421:
3413:
3406:
3402:
3394:
3387:
3366:
3344:
3339:
3309:
3306:
3303:
3297:
3275:
3272:
3268:
3245:
3241:
3237:
3232:
3228:
3207:
3187:
3184:
3179:
3175:
3171:
3166:
3162:
3139:
3135:
3131:
3126:
3122:
3098:
3095:
3092:
3086:
3083:
3080:
3077:
3072:
3067:
3052:
3037:
3015:
3010:
2983:
2980:
2977:
2957:
2949:
2933:
2926:over a field
2908:
2904:
2900:
2897:
2894:
2889:
2885:
2878:
2875:
2872:
2869:
2864:
2859:
2845:
2831:
2821:
2820:
2819:
2813:
2808:
2804:
2800:
2796:
2792:
2789:
2785:
2781:
2777:
2773:
2750:
2745:
2741:
2737:
2732:
2728:
2724:
2716:
2712:
2708:
2705:
2699:
2693:
2690:
2685:
2681:
2632:
2606:
2603:
2596:
2589:
2586:
2583:
2569:
2566:
2558:
2528:
2524:
2520:
2517:
2514:
2509:
2505:
2497:
2493:
2489:
2479:
2475:
2471:
2468:
2465:
2460:
2456:
2425:
2415:
2407:
2403:
2399:
2396:
2393:
2388:
2384:
2372:
2348:
2344:
2340:
2337:
2334:
2329:
2325:
2310:
2307:
2304:
2299:
2263:
2248:
2245:
2219:
2192:
2189:
2181:
2180:
2176:
2174:
2172:
2167:
2148:
2142:
2139:
2131:
2130:generic point
2127:
2123:
2104:
2098:
2095:
2086:
2084:
2080:
2061:
2055:
2052:
2044:
2040:
2036:
2032:
2028:
2024:
2020:
2015:
2013:
2009:
2005:
2001:
1997:
1993:
1989:
1985:
1981:
1977:
1973:
1970:
1966:
1962:
1958:
1954:
1950:
1946:
1942:
1938:
1934:
1926:
1924:
1922:
1918:
1914:
1910:
1894:
1885:
1883:
1867:
1859:
1843:
1784:
1781:
1777:
1760:
1759:
1758:
1744:
1712:
1708:
1692:
1672:
1652:
1629:
1623:
1620:
1611:
1605:
1602:
1599:
1593:
1587:
1584:
1576:
1560:
1554:
1551:
1548:
1541:
1537:
1521:
1513:
1505:
1503:
1500:
1494:
1488:
1466:
1454:
1451:
1437:
1432:
1426:
1420:
1416:
1412:
1406:
1400:
1394:
1388:
1366:
1354:
1351:
1336:
1332:
1328:
1323:
1317:
1315:
1296:
1290:
1287:
1279:
1274:
1270:at the ideal
1268:
1263:
1258:
1253:
1234:
1228:
1225:
1216:
1210:
1208:
1204:
1185:
1179:
1176:
1168:
1152:
1147:
1143:
1139:
1127:
1121:
1116:
1112:
1082:
1076:
1073:
1047:
1035:
1029:
1025:
1020:
1018:
1014:
1013:affine scheme
1010:
991:
985:
982:
974:
958:
953:
950:
946:
937:
933:
924:
923:inverse limit
906:
903:
881:
874:
870:
865:
861:
856:
853:
850:
844:
841:
835:
827:
815:
812:
782:
778:
773:
767:
764:
761:
757:
753:
750:
741:
736:
732:
728:
723:
717:
712:
696:
691:
687:
683:
675:
663:
658:
654:
625:
621:
598:
582:
563:
557:
554:
551:
548:
536:
534:
532:
524:
505:
499:
496:
488:
465:
459:
456:
448:
444:
440:
436:
435:compact space
417:
411:
408:
400:
383:
380:
377:
374:
369:
365:
338:
332:
329:
320:
316:
313:. Then each
312:
308:
303:
299:
295:
291:
287:
282:
280:
261:
255:
247:
244:
239:
235:
224:
223:
222:
220:
201:
195:
192:
184:
180:
176:
158:
154:
145:
141:
138:
130:
128:
103:
99:
95:
78:
74:
71:
63:
59:
55:
52:
48:
44:
40:
33:
32:Ring spectrum
19:
18:Relative Spec
7690:Prime ideals
7653:
7609:
7567:
7563:
7539:
7491:
7485:
7449:
7411:
7405:
7371:
7342:
7312:
7285:
7259:
7229:
7218:
7207:
7196:
7185:
7173:
7166:Vakil (n.d.)
7161:
7150:
7138:
7126:
7114:
7107:Sharp (2001)
7102:
7036:
7008:
6986:
6960:
6941:
6938:
6881:
6788:
6697:
6692:
6688:
6683:, as in the
6680:
6676:
6672:
6664:
6655:
6633:
6629:
6561:
6559:
6519:
6482:
6166:
6164:
6159:
6155:
6120:
6090:vector space
5985:
5972:
5968:
5964:
5960:
5954:
5591:
5504:
4660:
4620:
4408:
4403:
3726:
3719:are closed.
3638:
3544:of the form
3509:
3503:
3500:
3487:
3483:adding to it
3478:
2817:
2776:Boolean ring
2218:final object
2165:
2125:
2121:
2087:
2082:
2079:homeomorphic
2042:
2038:
2034:
2030:
2026:
2022:
2018:
2016:
2007:
2004:subvarieties
2003:
1999:
1996:prime ideals
1995:
1991:
1987:
1983:
1979:
1975:
1971:
1964:
1960:
1956:
1944:
1940:
1936:
1935:one studies
1930:
1916:
1911:between the
1887:The functor
1886:
1835:
1509:
1498:
1492:
1486:
1430:
1424:
1418:
1414:
1410:
1404:
1398:
1392:
1386:
1334:
1321:
1318:
1272:
1266:
1262:localization
1256:
1214:
1211:
1203:gluing axiom
1033:
1027:
1021:
1012:
1009:ringed space
921:denotes the
739:
721:
715:
711:localization
580:
540:
521:is always a
446:
401:
318:
314:
310:
306:
301:
297:
293:
289:
283:
276:
178:
175:prime ideals
143:
139:
134:
61:
58:prime ideals
53:
46:
42:
36:
7602:Vakil, Ravi
7570:: 195–205.
7368:Harris, Joe
7201:Kock (2007)
7009:commutative
6989:C*-algebras
5738:by letting
2970:-space for
2824:projective
2795:M. Hochster
2788:Stone space
2559:The cross:
2369:. From the
2012:irreducible
1953:polynomials
1909:equivalence
1260:equals the
640:by setting
489:. However,
177:containing
7674:Categories
7577:1503.04299
7339:Cox, David
7244:References
6884:eigenvalue
6660:. Given a
1575:continuous
1573:induces a
1278:local ring
1165:using the
1015:. General
7594:119574163
7494:: 43–60.
7421:0707.1525
7094:Citations
6966:the last
6858:−
6832:⊕
6823:−
6768:−
6742:⊕
6733:−
6597:…
6540:→
6503:→
6383:∗
6313:…
6264:−
6245:…
6226:−
6194:−
6022:…
5908:⋯
5877:⋯
5846:×
5609:β
5603:α
5548:β
5542:α
5519:
5461:→
5410:
5402:→
5386:
5376:↓
5370:↓
5364:↓
5340:−
5281:_
5270:→
5241:−
5184:
5176:→
5155:α
5152:β
5147:−
5108:
5072:≠
5069:α
5061:Assuming
5043:β
5037:α
4982:β
4976:α
4930:→
4894:_
4832:−
4638:→
4632::
4629:π
4564:
4541:≅
4519:∗
4515:π
4498:
4374:→
4324:−
4316:→
4299:−
4271:⊆
4226:
4220:≅
4203:−
4175:⊆
4149:→
4125:_
4072:_
4033:-algebras
4002:sheaf of
3932:_
3841:_
3695:−
3683:
3615:
3589:→
3580:φ
3568:
3557:∗
3553:φ
3523:
3490:June 2020
3425:∅
3403:∩
3172:∪
3084:
2981:≥
2898:…
2876:
2778:(e.g., a
2751:∈
2742:α
2729:α
2713:α
2682:α
2570:
2525:α
2518:…
2506:α
2469:…
2416:∈
2404:α
2397:…
2385:α
2338:…
2311:
2193:
2143:
2099:
2056:
1990:(because
1805:→
1782:−
1624:
1618:→
1606:
1588:
1558:→
1446:Γ
1346:Γ
1291:
1229:
1180:
1131:~
1106:Γ
1077:
1051:~
986:
943:→
907:←
862:
854:∈
845:←
807:Γ
799:, we set
765:∈
758:⋃
648:Γ
558:
500:
460:
439:Hausdorff
412:
381:∈
333:
296:, define
245::
196:
146:, define
75:
7618:cite web
7604:(n.d.).
7530:54501563
7448:(1977),
7438:17045655
7370:(2000),
7258:(1969).
7047:See also
6634:infinite
5535:→
4805:and let
2490:→
2177:Examples
1915:and the
1538:. Every
973:presheaf
183:topology
135:For any
47:spectrum
7510:1995344
7474:0463157
7394:1730819
7019:MaxSpec
2242:is the
2216:is the
2171:schemes
2053:MaxSpec
2045:, i.e.
1967:of all
1943:(where
1709:to the
1536:functor
1428:not in
1329:, with
1017:schemes
727:B-sheaf
49:) of a
7592:
7551:
7528:
7508:
7472:
7462:
7436:
7392:
7382:
7353:
7327:
7300:
7266:
7003:, the
4621:where
3997:and a
3787:. If
2844:-space
1978:. The
1947:is an
1880:up to
1325:is an
1024:module
894:where
354:, and
221:to be
41:, the
7590:S2CID
7572:arXiv
7526:S2CID
7522:(PDF)
7506:JSTOR
7434:S2CID
7416:arXiv
6667:on a
5977:group
5666:over
4697:over
3899:. If
2807:sober
1534:is a
1422:with
1312:is a
1252:stalk
1007:is a
735:basis
731:sheaf
487:space
433:is a
286:basis
137:ideal
92:; in
7624:link
7549:ISBN
7460:ISBN
7380:ISBN
7351:ISBN
7325:ISBN
7298:ISBN
7264:ISBN
7233:see
5820:and
5516:Spec
5407:Spec
5383:Spec
5278:Spec
5181:Spec
5105:Spec
4891:Spec
4492:-alg
4404:Spec
4223:Spec
4122:Spec
4069:Spec
3929:Spec
3838:Spec
3815:Spec
3775:Spec
3755:Spec
3735:Spec
3680:Spec
3672:and
3612:Spec
3565:Spec
3520:Spec
3081:Spec
2873:Proj
2822:The
2805:and
2567:Spec
2308:Spec
2190:Spec
2140:Spec
2096:Spec
1895:Spec
1868:Spec
1844:Spec
1693:Spec
1621:Spec
1603:Spec
1585:Spec
1577:map
1522:Spec
1288:Spec
1226:Spec
1177:Spec
1074:Spec
983:Spec
709:the
555:Spec
497:Spec
457:Spec
409:Spec
330:Spec
193:Spec
72:Spec
7582:doi
7496:doi
7492:142
7426:doi
7317:doi
7290:doi
7037:non
6991:in
4550:Sch
4545:Hom
4473:Hom
3946:or
3861:or
3639:In
3485:.
3437:in
3030:is
2081:to
2006:of
1998:of
1982:of
1955:in
1496:in
1402:in
1390:in
1319:If
1264:of
1212:If
1066:on
904:lim
842:lim
713:of
445:in
185:on
142:of
60:of
37:In
7676::
7620:}}
7616:{{
7608:.
7588:.
7580:.
7568:47
7566:.
7547:.
7524:.
7504:.
7490:.
7484:.
7470:MR
7468:,
7458:,
7432:.
7424:.
7412:36
7410:.
7404:.
7390:MR
7388:,
7378:,
7366:;
7349:,
7323:.
7296:.
7280:;
7254:;
7043:.
6679:=
6121:I,
5983:.
3359:.
2790:).
2173:.
1974:→
1884:.
1502:.
1413:=
1316:.
1209:.
533:.
292:∈
284:A
281:.
127:.
7664:.
7626:)
7612:.
7596:.
7584::
7574::
7557:.
7532:.
7512:.
7498::
7440:.
7428::
7418::
7333:.
7319::
7306:.
7292::
7272:.
6978:.
6961:T
6924:,
6919:2
6915:T
6910:/
6906:]
6903:T
6900:[
6897:K
6867:,
6864:)
6861:0
6855:T
6852:(
6848:/
6844:]
6841:T
6838:[
6835:K
6829:)
6826:0
6820:T
6817:(
6813:/
6809:]
6806:T
6803:[
6800:K
6774:)
6771:1
6765:T
6762:(
6758:/
6754:]
6751:T
6748:[
6745:K
6739:)
6736:1
6730:T
6727:(
6723:/
6719:]
6716:T
6713:[
6710:K
6693:T
6689:K
6681:K
6677:R
6673:V
6665:T
6630:R
6616:,
6613:]
6608:n
6604:x
6600:,
6594:,
6589:1
6585:x
6581:[
6578:K
6575:=
6572:R
6562:n
6546:.
6543:K
6535:n
6531:K
6520:n
6506:K
6498:n
6494:K
6483:K
6481:(
6467:n
6463:K
6440:i
6436:a
6413:i
6409:x
6388:,
6379:V
6358:]
6355:V
6352:[
6349:K
6329:)
6324:n
6320:a
6316:,
6310:,
6305:1
6301:a
6297:(
6277:)
6272:n
6268:a
6259:n
6255:x
6251:(
6248:,
6242:,
6239:)
6234:2
6230:a
6221:2
6217:x
6213:(
6210:,
6207:)
6202:1
6198:a
6189:1
6185:x
6181:(
6167:n
6160:R
6156:R
6142:,
6139:I
6135:/
6131:R
6105:i
6101:x
6076:.
6073:]
6070:V
6067:[
6064:K
6061:=
6058:R
6038:]
6033:n
6029:x
6025:,
6019:,
6014:1
6010:x
6006:[
6003:K
6000:=
5997:R
5973:R
5969:I
5967:/
5965:R
5961:I
5936:.
5932:)
5926:)
5918:n
5914:x
5901:0
5897:x
5887:n
5883:a
5870:0
5866:a
5859:(
5849:2
5843:2
5839:(
5835:=
5830:I
5808:]
5803:n
5799:x
5795:,
5792:.
5789:.
5786:.
5783:,
5778:0
5774:x
5770:[
5765:X
5759:O
5753:=
5748:A
5724:n
5717:n
5713:a
5709:,
5706:.
5703:.
5700:.
5697:,
5692:0
5688:a
5682:P
5677:=
5674:X
5652:1
5649:+
5646:n
5640:C
5634:A
5612:)
5606:,
5600:(
5575:1
5570:b
5567:,
5564:a
5559:P
5551:]
5545::
5539:[
5530:)
5526:C
5522:(
5484:1
5479:b
5476:,
5473:a
5468:P
5454:a
5450:U
5446:=
5442:)
5437:]
5432:a
5429:b
5424:[
5419:C
5414:(
5397:)
5393:C
5389:(
5356:)
5350:)
5346:x
5343:b
5337:y
5334:a
5330:(
5325:]
5322:y
5319:,
5316:x
5313:[
5308:X
5302:O
5293:(
5287:X
5264:)
5258:)
5254:x
5249:a
5246:b
5238:y
5234:(
5229:]
5226:y
5223:,
5220:x
5217:[
5213:]
5208:a
5205:b
5200:[
5195:C
5188:(
5170:)
5164:)
5160:x
5144:y
5140:(
5135:]
5132:y
5129:,
5126:x
5123:[
5119:C
5112:(
5078:,
5075:0
5049:.
5046:)
5040:,
5034:(
5012:2
5007:A
4985:]
4979::
4973:[
4951:1
4946:b
4943:,
4940:a
4935:P
4927:)
4922:I
4916:/
4910:A
4905:(
4900:X
4868:.
4863:A
4841:)
4838:x
4835:b
4829:y
4826:a
4823:(
4820:=
4815:I
4793:,
4790:]
4787:y
4784:,
4781:x
4778:[
4773:X
4767:O
4761:=
4756:A
4734:.
4729:1
4724:b
4721:,
4718:a
4713:P
4708:=
4705:X
4683:2
4677:C
4671:A
4641:S
4635:X
4606:,
4603:)
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2755:C
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2305:=
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2166:A
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2031:A
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2000:R
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