1387:
6306:
5849:
7482:
In some areas of module theory, a flat resolution must satisfy the additional requirement that each map is a flat pre-cover of the kernel of the map to the right. For projective resolutions, this condition is almost invisible: a projective pre-cover is simply an
524:
7535:
Since flat covers exist for all modules over all rings, minimal flat resolutions can take the place of minimal projective resolutions in many circumstances. The measurement of the departure of flat resolutions from projective resolutions is called
3057:
5730:
7849:
6727:
7564:, so theorems about projective modules, even if proved constructively, do not necessarily apply to free modules. In contrast, no choice is needed to prove that free modules are flat, so theorems about flat modules can still apply.
6180:
3662:
4043:
if taking the tensor product with a sequence produces an exact sequence if and only if the original sequence is exact. Although the concept is defined for modules over a non-necessary commutative ring, it is used mainly for
4693:
7532:, p. 196). The conjecture turned out to be true, resolved positively and proved simultaneously by L. Bican, R. El Bashir and E. Enochs. This was preceded by important contributions by P. Eklof, J. Trlifaj and J. Xu.
7487:
from a projective module. These ideas are inspired from
Auslander's work in approximations. These ideas are also familiar from the more common notion of minimal projective resolutions, where each map is required to be a
6022:
441:
247:
2954:
5078:
1367:
that are flat and not projective. However, finitely generated flat modules are all projective over the rings that are most commonly considered. Moreover, a finitely generated module is flat if and only it is
3428:
7127:
4630:
333:
1016:
5936:
5328:
8086:
6553:
6420:
2345:
7727:
891:
8904:
7982:
3589:
1959:
1908:
1560:
5388:
2222:
109:
7412:
5154:
5118:
673:
4400:
7492:
of the kernel of the map to the right. However, projective covers need not exist in general, so minimal projective resolutions are only of limited use over rings like the integers.
7334:
5447:
8752:
4481:
3552:
1631:
283:
7457:
3998:
7888:
5544:
4828:
446:
2019:
5655:
3256:
3227:
3170:
3141:
6049:
3286:
3197:
6155:
6131:
6073:
4554:
4507:
4344:
2887:
2863:
2827:
169:
4081:
2696:
1336:
1057:
932:
2615:
817:
4029:
3704:
3480:
2590:
2288:
2266:
2244:
5876:
2967:
1855:
8907:, Exposé VIII – this is the main reference (but it depends on a result from Giraud (1964), which replaced (in much more general form) the unpublished Exposé VII of SGA1)
8494:
6656:
4911:
1667:
775:
742:
709:
5718:
4031:(see also below for the geometric meaning). Such flat extensions can be used to yield examples of flat modules that are not free and do not result from a localization.
1168:
7642:
3348:
1820:
1753:
1513:
1481:
355:
8112:
7914:
7232:
6446:
2554:
1331:
1245:
7752:
7258:
6103:
5275:
5195:
3854:
2446:
1788:
7199:
7157:
6630:
5619:
5478:
4985:
3736:
2372:
1118:
1091:
6674:
3885:
2062:
1696:
3808:
2155:
2111:
1586:
7669:
4789:
4742:
4530:
4294:
4228:
2783:
2664:
8152:
8132:
8030:
8010:
7934:
7747:
7477:
7374:
7354:
7298:
7278:
7046:
7014:
6994:
6974:
6954:
6930:
6910:
6890:
6870:
6850:
6830:
6810:
6790:
6770:
6750:
6596:
6576:
6486:
6466:
6361:
6341:
6175:
5956:
5675:
5588:
5568:
5498:
5215:
5025:
5005:
4951:
4931:
4868:
4848:
4762:
4713:
4574:
4442:
4422:
4364:
4317:
4271:
4248:
4205:
4185:
4165:
4145:
4125:
4105:
3925:
3905:
3828:
3110:
3090:
2756:
2736:
2716:
2637:
2518:
2498:
2175:
1386:
1293:
1265:
1216:
1188:
7500:
While projective covers for modules do not always exist, it was speculated that for general rings, every module would have a flat cover, that is, every module
6301:{\displaystyle \operatorname {length} _{S}(S/{\mathfrak {q}}S)=\operatorname {length} _{S}(S/{\mathfrak {m}}S)\operatorname {length} _{R}(R/{\mathfrak {q}}).}
1719:(that is the quotient of a finitely generated free module by a finitely generated submodule) that is flat is always projective. This can be proven by taking
5844:{\displaystyle 0\to A{\overset {f}{\to }}B{\overset {\delta ^{0}}{\to }}B\otimes _{A}B{\overset {\delta ^{1}}{\to }}B\otimes _{A}B\otimes _{A}B\to \cdots }
3597:
2068:
is the ideal of the sequences with a finite number of nonzero terms, is thus flat and finitely generated (only one generator), but it is not projective.
4635:
3448:
Thus, properties of flat (or faithfully flat) ring homomorphisms extends naturally to geometric properties of flat morphisms in algebraic geometry.
1637:
Every projective module is flat. This can be proven from the above characterizations of flatness and projectivity in terms of linear maps by taking
1972:, every finitely generated flat module is projective, since every finitely generated module is finitely presented. The same result is true over an
8391:
4048:. So, this is the only case that is considered here, even if some results can be generalized to the case of modules over a non-commutaive ring.
5966:
Here is one characterization of a faithfully flat homomorphism for a not-necessarily-flat homomorphism. Given an injective local homomorphism
8866:
8726:
8646:
8560:
1348:
Flatness is related to various other module properties, such as being free, projective, or torsion-free. In particular, every flat module is
5969:
399:
181:
2892:
5033:
7356:
does not admit a finite flat resolution, then by convention the flat dimension is said to be infinite. As an example, consider a module
7159:
are all flat modules. Any free or projective resolution is necessarily a flat resolution. Flat resolutions can be used to compute the
5450:
3928:
2838:
2521:
3364:
8816:
7058:
3291:
This property is fundamental in commutative algebra and algebraic geometry, since it reduces the study of flatness to the case of
8544:
8922:
8669:
4045:
4579:
288:
5677:
at all its prime ideals is a faithfully flat module that is not an algebra, except if there are finitely many prime ideals.
5156:(this is a special case of the fact that a faithfully flat quasi-compact morphism of schemes has this property.). See also
8579:
8540:
5881:
5284:
943:
8035:
6502:
6369:
2305:
8970:
7674:
7560:, where projective modules are less useful. For example, that all free modules are projective is equivalent to the full
5681:
The two last examples are implicitly behind the wide use of localization in commutative algebra and algebraic geometry.
8975:
825:
7939:
1335:
8858:
3743:
3557:
2386:
1917:
1860:
1716:
1701:
1518:
1364:
73:
7583:
7557:
5338:
2667:
2195:
1373:
358:
7379:
5127:
5091:
4369:
614:
7307:
5396:
519:{\displaystyle 0\rightarrow K\otimes _{R}M\rightarrow L\otimes _{R}M\rightarrow J\otimes _{R}M\rightarrow 0}
4447:
3485:
1592:
255:
8980:
7417:
7049:
3934:
1414:
47:
7854:
5503:
4794:
6106:
2478:
2300:
1989:
5631:
3232:
3203:
3146:
3117:
6027:
3264:
3175:
8384:
6666:
6136:
6112:
6054:
4535:
4488:
4325:
3351:
3312:
2868:
2844:
2808:
1124:
elements. This allows rewriting the previous characterization in terms of homomorphisms, as follows.
538:
389:
142:
55:
4054:
3052:{\displaystyle M_{\mathfrak {p}}=(R\setminus {\mathfrak {p}})^{-1}M=R_{\mathfrak {p}}\otimes _{R}M.}
2672:
1021:
896:
6662:
2595:
2563:
2189:
1705:
1400:
1349:
780:
527:
137:
51:
4003:
3670:
3454:
2573:
2271:
2249:
2227:
1369:
8686:
8614:
8513:
7593:
5854:
5591:
4716:
1825:
1709:
62:
7544:) and in more recent works focussing on flat resolutions such as Enochs and Jenda (
6635:
4881:
2029:. It is a commutative ring with addition and multiplication defined componentwise. This ring is
1712:. This makes the concept of flatness useful mainly for modules that are not finitely generated.
1640:
747:
714:
681:
7844:{\displaystyle M\otimes _{R}S\to M\otimes _{R}(S\otimes S)\simeq (M\otimes _{R}S)\otimes _{R}S}
5688:
2460:-module for the module structure induced by the homomorphism. For example, the polynomial ring
1138:
8941:
8913:
8884:
8862:
8812:
8806:
8774:
8722:
8642:
8598:
8556:
8434:
7615:
5121:
4084:
3739:
3321:
2567:
2420:
2022:
1793:
1726:
1486:
1454:
1441:
1353:
340:
100:
43:
8091:
7893:
7204:
6722:{\displaystyle 0\to A{\overset {f}{\longrightarrow }}B{\overset {g}{\longrightarrow }}C\to 0}
6425:
2526:
1298:
1221:
8931:
8917:
8830:
8761:
8714:
8678:
8634:
8633:, de Gruyter Expositions in Mathematics, vol. 30, Berlin: Walter de Gruyter & Co.,
8588:
8548:
8503:
8480:
8466:
8426:
7573:
7489:
7237:
6082:
5278:
5245:
5229:
5174:
5085:
3833:
2802:
2759:
2425:
1758:
1425:
581:
Flatness can also be characterized by the following equational condition, which means that
8953:
8896:
8826:
8790:
8736:
8698:
8656:
8610:
8570:
8525:
8446:
7177:
7135:
6601:
5597:
5456:
4956:
3709:
2350:
1096:
1069:
8949:
8892:
8834:
8822:
8786:
8732:
8710:
8694:
8652:
8606:
8566:
8521:
8442:
7561:
6320:
5721:
5236:
5225:
3859:
2640:
2126:
2036:
2030:
1973:
1969:
1672:
586:
7280:
admits a finite flat resolution, the minimal length among all finite flat resolutions of
3752:
2132:
2088:
1565:
8875:
Richman, Fred (1997), "Flat dimension, constructivity, and the
Hilbert syzygy theorem",
7651:
4771:
4724:
4512:
4276:
4210:
2765:
2646:
8782:
8626:
8532:
8454:
8137:
8117:
8015:
7995:
7919:
7732:
7462:
7359:
7339:
7301:
7283:
7263:
7031:
6999:
6979:
6959:
6939:
6933:
6915:
6895:
6875:
6855:
6835:
6815:
6795:
6775:
6755:
6735:
6581:
6561:
6471:
6451:
6346:
6326:
6160:
5941:
5660:
5573:
5553:
5483:
5391:
5200:
5010:
4990:
4936:
4916:
4853:
4833:
4747:
4698:
4559:
4427:
4407:
4349:
4302:
4256:
4233:
4190:
4170:
4150:
4130:
4110:
4090:
3910:
3890:
3813:
3296:
3095:
3075:
2741:
2721:
2701:
2622:
2503:
2483:
2392:
2160:
1278:
1250:
1201:
1173:
377:
93:
85:
8811:. Cambridge Studies in Advanced Mathematics. Vol. 8. Cambridge University Press.
17:
8964:
8842:
8743:
8618:
8402:
7578:
6076:
5158:
3657:{\displaystyle \pi :\operatorname {Spec} (R)\to \operatorname {Spec} (\mathbb {C} ).}
3355:
3308:
3259:
2408:
1986:
A finitely generated flat module that is not projective can be built as follows. Let
1418:
570:
385:
8747:
4765:
2786:
2378:
2192:. Thus a module that contains nonzero torsion elements is not flat. In particular
1704:
flat modules are projective under mild conditions that are generally satisfied in
5878:
are the alternating sums of the maps obtained by inserting 1 in each spot; e.g.,
1755:
in the above characterization of flatness in terms of linear maps. The condition
8663:
Kunz, Ernst (1969), "Characterizations of regular local rings of characteristic
8577:
Enochs, Edgar E. (1981), "Injective and flat covers, envelopes and resolvents",
8314:
7484:
7160:
6316:
4688:{\displaystyle f^{*}\colon \operatorname {Spec} (S)\to \operatorname {Spec} (R)}
2830:
2382:
1445:
1377:
1357:
39:
1403:. This results from the above characterization in terms of relations by taking
526:
is also exact. (This is an equivalent definition since the tensor product is a
8718:
8552:
8430:
8406:
5625:
5547:
4875:
3292:
2182:
2118:
1980:
8945:
8888:
8602:
8438:
8417:
Bican, L.; El Bashir, R.; Enochs, E. (2001), "All modules have flat covers",
5030:
The last but one condition can be stated in the following strengthened form:
1424:
An integral domain over which every torsion-free module is flat is called a
134:
2385:
is flat. Conversely, every flat module can be written as a direct limit of
8781:, Die Grundlehren der mathematischen Wissenschaften, Bd. 114, Boston, MA:
8709:, Graduate Texts in Mathematics No. 189, vol. 189, Berlin, New York:
8638:
92:
if taking the tensor product with a sequence produces an exact sequence
8766:
8690:
8593:
8517:
8471:
6492:
In fact, it is enough to check that the first Tor term vanishes, i.e.,
31:
6017:{\displaystyle (R,{\mathfrak {m}})\hookrightarrow (S,{\mathfrak {n}})}
436:{\displaystyle 0\rightarrow K\rightarrow L\rightarrow J\rightarrow 0,}
242:{\displaystyle \varphi \otimes _{R}M:K\otimes _{R}M\to L\otimes _{R}M}
2949:{\displaystyle R_{\mathfrak {p}}=(R\setminus {\mathfrak {p}})^{-1}R,}
8682:
8508:
7459:
indicates that the arrow in the center is an isomorphism, and hence
5073:{\displaystyle \operatorname {Spec} (S)\to \operatorname {Spec} (R)}
1135:
is flat if and only if the following condition holds: for every map
8936:
2395:
of flat modules need not in general be flat. In fact, given a ring
5228:
is faithfully flat. This property is implicitly behind the use of
2411:(that is, every finitely generated ideal is finitely presented).
8455:"Questions de rationalité des diviseurs en géométrie algébrique"
4874:
The second condition implies that a flat local homomorphism of
3423:{\displaystyle {\mathcal {O}}_{Y,f(x)}\to {\mathcal {O}}_{X,x}}
1413:
The converse holds over the integers, and more generally over
7122:{\displaystyle \cdots \to F_{2}\to F_{1}\to F_{0}\to M\to 0,}
337:
For this definition, it is enough to restrict the injections
4878:
is faithfully flat. It follows from the last condition that
3403:
3371:
2121:(that is an element equal to its square). In particular, if
1385:
1334:
5239:
is a faithfully flat extension of its ring of coefficients.
4187:-module flat (or faithfully flat), one says commonly that
3931:(the coefficients generate the unit ideal). An example is
3315:
results immediately from the local property of flatness.
8492:
Chase, Stephen U. (1960), "Direct products of modules",
4625:{\displaystyle {\mathfrak {p}}=f^{-1}({\mathfrak {P}}).}
3810:
be a polynomial ring over a commutative
Noetherian ring
3295:. They are often expressed by saying that flatness is a
2224:
and all fields of positive characteristics are non-flat
8163:
A module isomorphic to a flat module is of course flat.
3069:-module the three following conditions are equivalent:
328:{\displaystyle k\otimes m\mapsto \varphi (k)\otimes m.}
5400:
2309:
946:
828:
617:
8140:
8120:
8094:
8038:
8018:
7998:
7942:
7922:
7896:
7857:
7755:
7735:
7677:
7654:
7618:
7465:
7420:
7382:
7362:
7342:
7310:
7286:
7266:
7240:
7207:
7180:
7138:
7061:
7034:
7002:
6982:
6962:
6942:
6918:
6898:
6878:
6858:
6838:
6818:
6798:
6778:
6758:
6738:
6677:
6638:
6604:
6584:
6564:
6505:
6474:
6454:
6428:
6372:
6349:
6329:
6183:
6163:
6139:
6115:
6085:
6057:
6030:
5972:
5944:
5884:
5857:
5733:
5691:
5663:
5634:
5600:
5576:
5556:
5506:
5486:
5459:
5399:
5341:
5287:
5248:
5203:
5177:
5130:
5094:
5036:
5013:
4993:
4959:
4939:
4919:
4884:
4856:
4836:
4797:
4774:
4750:
4727:
4701:
4638:
4582:
4562:
4538:
4515:
4491:
4450:
4430:
4410:
4372:
4352:
4328:
4305:
4279:
4259:
4236:
4213:
4193:
4173:
4153:
4133:
4113:
4093:
4057:
4006:
3937:
3913:
3893:
3862:
3836:
3816:
3755:
3712:
3673:
3600:
3560:
3488:
3457:
3367:
3324:
3267:
3235:
3206:
3178:
3149:
3120:
3098:
3078:
2970:
2895:
2871:
2847:
2811:
2768:
2744:
2724:
2704:
2675:
2649:
2625:
2598:
2576:
2529:
2506:
2486:
2428:
2353:
2308:
2274:
2252:
2230:
2198:
2163:
2135:
2091:
2039:
1992:
1920:
1863:
1828:
1796:
1761:
1729:
1675:
1643:
1595:
1568:
1562:
In particular, every free module is projective (take
1521:
1489:
1457:
1301:
1281:
1253:
1224:
1204:
1176:
1141:
1099:
1072:
1024:
1011:{\textstyle x_{i}=\sum _{j=1}^{n}a_{i,j}y_{j}\qquad }
899:
783:
750:
717:
684:
449:
402:
343:
291:
258:
184:
145:
8355:
7540:, and is covered in classics such as Mac Lane (
5931:{\displaystyle \delta ^{0}(b)=b\otimes 1-1\otimes b}
5323:{\displaystyle R\hookrightarrow R/\langle p\rangle }
1383:
This is partly summarized in the following graphic.
8081:{\displaystyle \operatorname {Tor} _{n}^{R}(M,X)=0}
7414:. In this situation, the exactness of the sequence
7170:of a finite flat resolution is the first subscript
6548:{\displaystyle \operatorname {Tor} _{1}^{R}(N,M)=0}
6415:{\displaystyle \operatorname {Tor} _{n}^{R}(X,M)=0}
2340:{\displaystyle \textstyle \bigoplus _{i\in I}M_{i}}
133:if the following condition is satisfied: for every
8146:
8126:
8106:
8080:
8024:
8004:
7976:
7928:
7908:
7882:
7843:
7741:
7722:{\displaystyle S=R\otimes _{R}S\to S\otimes _{R}S}
7721:
7663:
7636:
7471:
7451:
7406:
7368:
7348:
7328:
7292:
7272:
7252:
7226:
7193:
7151:
7121:
7040:
7008:
6988:
6968:
6948:
6924:
6904:
6884:
6864:
6844:
6824:
6804:
6784:
6764:
6744:
6721:
6650:
6624:
6590:
6570:
6547:
6480:
6460:
6440:
6414:
6355:
6335:
6300:
6169:
6149:
6125:
6097:
6067:
6043:
6016:
5950:
5930:
5870:
5843:
5712:
5669:
5649:
5613:
5582:
5562:
5538:
5492:
5472:
5441:
5382:
5322:
5269:
5209:
5189:
5148:
5112:
5072:
5019:
4999:
4979:
4945:
4925:
4905:
4862:
4842:
4822:
4783:
4756:
4736:
4707:
4687:
4624:
4568:
4548:
4524:
4501:
4475:
4436:
4416:
4394:
4358:
4338:
4311:
4288:
4265:
4242:
4222:
4199:
4179:
4159:
4139:
4119:
4099:
4075:
4023:
3992:
3919:
3899:
3879:
3848:
3822:
3802:
3730:
3698:
3656:
3583:
3546:
3474:
3422:
3342:
3280:
3250:
3221:
3191:
3164:
3135:
3104:
3084:
3051:
2948:
2881:
2857:
2821:
2777:
2750:
2730:
2710:
2690:
2658:
2631:
2609:
2584:
2548:
2512:
2492:
2440:
2381:of flat is flat. In particular, a direct limit of
2366:
2339:
2282:
2260:
2238:
2216:
2169:
2149:
2105:
2056:
2013:
1953:
1902:
1849:
1814:
1782:
1747:
1690:
1661:
1625:
1580:
1554:
1507:
1475:
1325:
1287:
1259:
1239:
1210:
1182:
1162:
1112:
1085:
1051:
1010:
926:
885:
811:
769:
736:
703:
667:
608:is flat if and only if, for every linear relation
518:
435:
349:
327:
277:
241:
163:
8495:Transactions of the American Mathematical Society
5159:Flat morphism § Properties of flat morphisms
1093:to this module, which maps the standard basis of
886:{\textstyle \sum _{i=1}^{m}r_{i}a_{i,j}=0\qquad }
7504:would be the epimorphic image of a flat module
6311:Homological characterization using Tor functors
5938:. Then (Grothendieck) this complex is exact if
8918:"Géométrie algébrique et géométrie analytique"
7977:{\displaystyle 0\neq M\subset M\otimes _{R}S.}
2738:is flat. It is faithfully flat if and only if
1983:every finitely generated flat module is free.
8753:Bulletin de la Société Mathématique de France
8459:Bulletin de la Société Mathématique de France
3584:{\displaystyle \mathbb {C} \hookrightarrow R}
2081:is an ideal in a Noetherian commutative ring
8:
7528:was explicitly first stated in Enochs (
7508:such that every map from a flat module onto
5317:
5311:
2033:(that is, every module is flat). The module
1066:elements of a module, and a linear map from
110:Géometrie Algébrique et Géométrie Analytique
7545:
1954:{\displaystyle h\circ i=\mathrm {id} _{M},}
1903:{\displaystyle h\circ i\circ f=h\circ g=f.}
1555:{\displaystyle p\circ i=\mathrm {id} _{M}.}
7556:Flat modules have increased importance in
6665:, one can then easily prove facts about a
5720:there is an associated complex called the
5232:for proving results on real vector spaces.
3433:is a flat ring homomorphism for any point
2188:Over an integral domain, a flat module is
1194:-module, and for every finitely generated
565:-modules, and the tensor products are not
8935:
8765:
8592:
8507:
8470:
8331:
8253:
8241:
8207:
8186:, Ch. I, § 2. Proposition 13, Corollary 1
8139:
8119:
8093:
8048:
8043:
8037:
8017:
7997:
7962:
7941:
7921:
7895:
7871:
7856:
7832:
7816:
7782:
7763:
7754:
7734:
7710:
7691:
7676:
7653:
7617:
7464:
7431:
7419:
7381:
7361:
7341:
7309:
7285:
7265:
7239:
7212:
7206:
7185:
7179:
7143:
7137:
7098:
7085:
7072:
7060:
7033:
7001:
6981:
6961:
6941:
6917:
6912:need not be flat in general. However, if
6897:
6877:
6857:
6837:
6817:
6797:
6777:
6757:
6737:
6700:
6687:
6676:
6637:
6614:
6603:
6583:
6563:
6515:
6510:
6504:
6473:
6453:
6427:
6382:
6377:
6371:
6348:
6328:
6315:Flatness may also be expressed using the
6286:
6285:
6280:
6265:
6249:
6248:
6243:
6228:
6209:
6208:
6203:
6188:
6182:
6162:
6141:
6140:
6138:
6117:
6116:
6114:
6084:
6059:
6058:
6056:
6032:
6031:
6029:
6005:
6004:
5983:
5982:
5971:
5943:
5889:
5883:
5862:
5856:
5826:
5813:
5798:
5789:
5780:
5765:
5756:
5743:
5732:
5690:
5662:
5640:
5639:
5633:
5605:
5599:
5575:
5555:
5530:
5511:
5505:
5485:
5464:
5458:
5426:
5421:
5405:
5398:
5383:{\displaystyle t_{1},\ldots ,t_{k}\in R.}
5365:
5346:
5340:
5306:
5286:
5247:
5221:-module is faithfully flat. For example:
5202:
5176:
5129:
5093:
5035:
5012:
4992:
4969:
4958:
4938:
4918:
4883:
4855:
4835:
4811:
4796:
4773:
4749:
4726:
4700:
4643:
4637:
4610:
4609:
4597:
4584:
4583:
4581:
4561:
4540:
4539:
4537:
4514:
4493:
4492:
4490:
4458:
4449:
4429:
4409:
4374:
4373:
4371:
4351:
4330:
4329:
4327:
4304:
4296:the following conditions are equivalent.
4278:
4258:
4235:
4212:
4192:
4172:
4152:
4132:
4112:
4092:
4056:
4008:
4007:
4005:
3964:
3939:
3938:
3936:
3912:
3892:
3866:
3861:
3835:
3815:
3791:
3772:
3754:
3711:
3678:
3672:
3635:
3634:
3599:
3562:
3561:
3559:
3521:
3496:
3495:
3487:
3459:
3458:
3456:
3408:
3402:
3401:
3376:
3370:
3369:
3366:
3323:
3269:
3268:
3266:
3241:
3240:
3234:
3212:
3211:
3205:
3180:
3179:
3177:
3155:
3154:
3148:
3126:
3125:
3119:
3097:
3077:
3037:
3026:
3025:
3006:
2996:
2995:
2976:
2975:
2969:
2931:
2921:
2920:
2901:
2900:
2894:
2873:
2872:
2870:
2849:
2848:
2846:
2813:
2812:
2810:
2767:
2743:
2723:
2703:
2677:
2676:
2674:
2648:
2624:
2600:
2599:
2597:
2578:
2577:
2575:
2570:only in exceptional cases). For example,
2534:
2528:
2505:
2485:
2427:
2358:
2352:
2330:
2314:
2307:
2276:
2275:
2273:
2254:
2253:
2251:
2232:
2231:
2229:
2217:{\displaystyle \mathbb {Q} /\mathbb {Z} }
2210:
2209:
2204:
2200:
2199:
2197:
2162:
2139:
2134:
2095:
2090:
2043:
2038:
2005:
2004:
2003:
1991:
1942:
1934:
1919:
1862:
1827:
1795:
1760:
1728:
1674:
1642:
1617:
1609:
1594:
1567:
1543:
1535:
1520:
1488:
1456:
1300:
1280:
1252:
1223:
1203:
1175:
1140:
1104:
1098:
1077:
1071:
1023:
1001:
985:
975:
964:
951:
945:
898:
864:
854:
844:
833:
827:
788:
782:
755:
749:
722:
716:
689:
683:
653:
643:
633:
622:
616:
501:
482:
463:
448:
401:
342:
290:
266:
257:
230:
211:
192:
183:
144:
8304:, Exercise (3) after Proposition III.5.2
8265:
7541:
7407:{\displaystyle \operatorname {fd} (M)=0}
5149:{\displaystyle \operatorname {Spec} (S)}
5113:{\displaystyle \operatorname {Spec} (R)}
1390:Module properties in commutative algebra
668:{\textstyle \sum _{i=1}^{m}r_{i}x_{i}=0}
8366:
8195:
8176:
7605:
4395:{\displaystyle {\mathfrak {m}}S\neq S.}
2992:
2917:
2347:of modules is flat if and only if each
8397:from the original on 18 November 2019.
8219:
7529:
7329:{\displaystyle \operatorname {fd} (M)}
6105:is faithfully flat if and only if the
5442:{\displaystyle \textstyle \prod _{i}R}
1790:implies the existence of a linear map
8847:The red book of varieties and schemes
8301:
8230:
4476:{\displaystyle M\otimes _{R}S\neq 0.}
4087:of commutative rings, which gives to
3547:{\displaystyle R=\mathbb {C} /(xy-t)}
2290:is the field of the rational numbers.
1626:{\displaystyle i=p=\mathrm {id} _{M}}
278:{\displaystyle \varphi \otimes _{R}M}
104:
7:
8183:
7452:{\displaystyle 0\to F_{0}\to M\to 0}
3993:{\displaystyle \mathbb {C} /(xy-t),}
2025:whose terms belong to a fixed field
1344:Relations to other module properties
8543:, vol. 150, Berlin, New York:
8343:
8244:, Corollary 1 of Theorem 55, p. 170
7883:{\displaystyle M\to M\otimes _{R}S}
6287:
6250:
6210:
6142:
6118:
6060:
6033:
6006:
5984:
5962:Faithfully flat local homomorphisms
5641:
5539:{\displaystyle t_{1},\ldots ,t_{k}}
4823:{\displaystyle M\to M\otimes _{R}S}
4611:
4585:
4541:
4494:
4375:
4331:
4000:which is flat (and even free) over
3270:
3242:
3213:
3181:
3156:
3127:
3027:
2997:
2977:
2922:
2902:
2874:
2850:
2814:
8877:New Zealand Journal of Mathematics
8356:Bican, El Bashir & Enochs 2001
6598:or, even more restrictively, when
4207:is flat (or faithfully flat) over
3358:if the induced map on local rings
2014:{\displaystyle R=F^{\mathbb {N} }}
1938:
1935:
1613:
1610:
1539:
1536:
27:Algebraic structure in ring theory
25:
8289:
8277:
6658:is any finitely generated ideal.
5650:{\displaystyle R_{\mathfrak {p}}}
3251:{\displaystyle R_{\mathfrak {m}}}
3222:{\displaystyle M_{\mathfrak {m}}}
3165:{\displaystyle R_{\mathfrak {p}}}
3136:{\displaystyle M_{\mathfrak {p}}}
6109:holds for it; that is, for each
6044:{\displaystyle {\mathfrak {m}}S}
3281:{\displaystyle {\mathfrak {m}}.}
3192:{\displaystyle {\mathfrak {p}};}
2592:is flat and not projective over
2403:-modules is flat if and only if
2295:Direct sums, limits and products
2113:is not a flat module, except if
1976:, even if it is not Noetherian.
1339:Factor property of a flat module
533:These definitions apply also if
96:the original sequence is exact.
8670:American Journal of Mathematics
6150:{\displaystyle {\mathfrak {q}}}
6126:{\displaystyle {\mathfrak {m}}}
6068:{\displaystyle {\mathfrak {n}}}
5851:where the coboundary operators
4549:{\displaystyle {\mathfrak {P}}}
4502:{\displaystyle {\mathfrak {p}}}
4339:{\displaystyle {\mathfrak {m}}}
3451:For example, consider the flat
2882:{\displaystyle {\mathfrak {p}}}
2858:{\displaystyle {\mathfrak {p}}}
2822:{\displaystyle {\mathfrak {p}}}
2399:, every direct product of flat
1007:
882:
164:{\displaystyle \varphi :K\to L}
8069:
8057:
7861:
7825:
7806:
7800:
7788:
7772:
7700:
7628:
7443:
7437:
7424:
7395:
7389:
7323:
7317:
7110:
7104:
7091:
7078:
7065:
6713:
6702:
6689:
6681:
6536:
6524:
6403:
6391:
6323:of the tensor product. A left
6292:
6274:
6258:
6237:
6218:
6197:
6089:
6011:
5995:
5992:
5989:
5973:
5901:
5895:
5835:
5791:
5758:
5745:
5737:
5701:
5685:For a given ring homomorphism
5435:
5414:
5303:
5297:
5291:
5264:
5258:
5181:
5143:
5137:
5107:
5101:
5067:
5061:
5052:
5049:
5043:
4801:
4682:
4676:
4667:
4664:
4658:
4616:
4606:
4250:is flat (or faithfully flat).
4076:{\displaystyle f\colon R\to S}
4067:
4018:
4012:
3984:
3969:
3961:
3943:
3797:
3765:
3693:
3687:
3648:
3645:
3639:
3631:
3622:
3619:
3613:
3575:
3572:
3566:
3541:
3526:
3518:
3500:
3469:
3463:
3397:
3392:
3386:
3334:
3172:-module for every prime ideal
3003:
2986:
2928:
2911:
2691:{\displaystyle {\widehat {R}}}
2432:
1806:
1771:
1765:
1499:
1467:
1311:
1305:
1151:
1052:{\displaystyle i=1,\ldots ,m.}
927:{\displaystyle j=1,\ldots ,n,}
593:stem from linear relations in
569:-modules in general, but only
510:
491:
472:
453:
424:
418:
412:
406:
313:
307:
301:
220:
155:
1:
8923:Annales de l'Institut Fourier
8707:Lectures on modules and rings
8580:Israel Journal of Mathematics
8541:Graduate Texts in Mathematics
7890:is injective. Conversely, if
2610:{\displaystyle \mathbb {Z} .}
2268:is the ring of integers, and
1190:is a finitely generated free
812:{\displaystyle a_{i,j}\in R,}
8805:Matsumura, Hideyuki (1986).
8797:Matsumura, Hideyuki (1970),
8631:Relative homological algebra
8292:, Exposé VIII., Corollay 4.3
7538:relative homological algebra
4024:{\displaystyle \mathbb {C} }
3699:{\displaystyle \pi ^{-1}(t)}
3475:{\displaystyle \mathbb {C} }
2585:{\displaystyle \mathbb {Q} }
2283:{\displaystyle \mathbb {Q} }
2261:{\displaystyle \mathbb {Z} }
2239:{\displaystyle \mathbb {Z} }
1914:is surjective, one has thus
7644:is faithfully flat. For an
7552:In constructive mathematics
5871:{\displaystyle \delta ^{n}}
5007:is a Noetherian ring, then
3554:(see below). The inclusion
2865:is, as usual, denoted with
1850:{\displaystyle i\circ f=g,}
1432:Free and projective modules
1062:It is equivalent to define
99:Flatness was introduced by
8997:
8859:Cambridge University Press
7516:, and any endomorphism of
6651:{\displaystyle I\subset R}
4906:{\displaystyle I=IS\cap R}
3744:deformation to normal cone
3591:induces the flat morphism
1662:{\displaystyle g=i\circ f}
1444:if and only if there is a
1365:finitely generated modules
770:{\displaystyle y_{j}\in M}
737:{\displaystyle x_{i}\in M}
704:{\displaystyle r_{i}\in R}
8853:Northcott, D. G. (1984),
8719:10.1007/978-1-4612-0525-8
8553:10.1007/978-1-4612-5350-1
8431:10.1017/S0024609301008104
7749:as a pure subring and so
7524:is an automorphism. This
5713:{\displaystyle f:A\to B,}
3706:is the curve of equation
3303:Flat morphisms of schemes
1717:finitely presented module
1163:{\displaystyle f:F\to M,}
359:finitely generated ideals
252:is also injective, where
8748:"Autour de la platitude"
8453:Cartier, Pierre (1958).
7637:{\displaystyle f:R\to S}
7584:von Neumann regular ring
7558:constructive mathematics
6661:Using the Tor functor's
5480:is faithfully flat over
4632:In other words, the map
3343:{\displaystyle f:X\to Y}
1815:{\displaystyle i:M\to G}
1748:{\displaystyle K=\ker f}
1508:{\displaystyle p:G\to M}
1476:{\displaystyle i:M\to G}
388:; that is if, for every
350:{\displaystyle \varphi }
8808:Commutative ring theory
8705:Lam, Tsit-Yuen (1999),
8419:Bull. London Math. Soc.
8107:{\displaystyle n\geq 1}
8032:is flat if and only if
7909:{\displaystyle M\neq 0}
7227:{\displaystyle F_{i}=0}
6496:is flat if and only if
6441:{\displaystyle n\geq 1}
6363:is flat if and only if
5084:, which means that the
4830:is injective for every
4532:there is a prime ideal
4322:For each maximal ideal
3856:a nonzerodivisor. Then
3667:Each (geometric) fiber
2549:{\displaystyle S^{-1}R}
1415:principal ideal domains
1326:{\displaystyle g(K)=0:}
1240:{\displaystyle \ker f,}
744:, there exist elements
549:-module; in this case,
8407:"Noncommutative Rings"
8334:, Ch. 8, Exercise 22.1
8148:
8128:
8108:
8082:
8026:
8006:
7978:
7930:
7910:
7884:
7845:
7743:
7723:
7665:
7638:
7473:
7453:
7408:
7370:
7350:
7330:
7294:
7274:
7254:
7253:{\displaystyle i>n}
7228:
7195:
7153:
7123:
7042:
7010:
6990:
6970:
6950:
6926:
6906:
6886:
6866:
6846:
6826:
6806:
6786:
6766:
6746:
6723:
6652:
6626:
6592:
6572:
6549:
6482:
6462:
6442:
6416:
6357:
6337:
6302:
6171:
6151:
6127:
6099:
6098:{\displaystyle S\to B}
6069:
6045:
6018:
5952:
5932:
5872:
5845:
5714:
5671:
5651:
5615:
5584:
5564:
5540:
5494:
5474:
5443:
5384:
5324:
5271:
5270:{\displaystyle p\in R}
5211:
5191:
5190:{\displaystyle R\to S}
5150:
5114:
5074:
5021:
5001:
4981:
4947:
4927:
4907:
4864:
4844:
4824:
4785:
4758:
4738:
4709:
4689:
4626:
4570:
4550:
4526:
4503:
4485:For every prime ideal
4477:
4438:
4418:
4396:
4360:
4340:
4313:
4290:
4267:
4244:
4224:
4201:
4181:
4161:
4141:
4121:
4101:
4077:
4025:
3994:
3921:
3901:
3881:
3850:
3849:{\displaystyle f\in S}
3824:
3804:
3732:
3700:
3658:
3585:
3548:
3476:
3424:
3344:
3282:
3252:
3223:
3193:
3166:
3137:
3106:
3086:
3053:
2950:
2889:as an index. That is,
2883:
2859:
2823:
2779:
2752:
2732:
2712:
2692:
2660:
2633:
2611:
2586:
2550:
2514:
2500:of a commutative ring
2494:
2442:
2441:{\displaystyle R\to S}
2368:
2341:
2284:
2262:
2240:
2218:
2171:
2151:
2107:
2058:
2015:
1955:
1904:
1851:
1816:
1784:
1783:{\displaystyle g(K)=0}
1749:
1692:
1663:
1627:
1582:
1556:
1509:
1477:
1391:
1340:
1327:
1289:
1267:factors through a map
1261:
1241:
1212:
1184:
1164:
1114:
1087:
1053:
1012:
980:
928:
887:
849:
813:
771:
738:
705:
669:
638:
520:
437:
351:
329:
285:is the map induced by
279:
243:
165:
48:principal ideal domain
18:Faithfully flat module
8639:10.1515/9783110803662
8627:Jenda, Overtoun M. G.
8149:
8129:
8109:
8083:
8027:
8007:
7979:
7931:
7911:
7885:
7851:is injective. Hence,
7846:
7744:
7724:
7666:
7639:
7526:flat cover conjecture
7474:
7454:
7409:
7371:
7351:
7331:
7295:
7275:
7255:
7229:
7196:
7194:{\displaystyle F_{n}}
7154:
7152:{\displaystyle F_{i}}
7124:
7043:
7011:
6991:
6971:
6951:
6927:
6907:
6887:
6867:
6847:
6832:are flat, then so is
6827:
6807:
6787:
6772:are flat, then so is
6767:
6747:
6724:
6653:
6627:
6625:{\displaystyle N=R/I}
6593:
6573:
6550:
6483:
6463:
6443:
6417:
6358:
6338:
6321:left derived functors
6303:
6172:
6152:
6128:
6107:theorem of transition
6100:
6070:
6046:
6019:
5953:
5933:
5873:
5846:
5715:
5672:
5652:
5628:of the localizations
5616:
5614:{\displaystyle t_{i}}
5585:
5565:
5541:
5495:
5475:
5473:{\displaystyle t_{i}}
5444:
5385:
5325:
5272:
5212:
5192:
5151:
5115:
5075:
5022:
5002:
4987:). In particular, if
4982:
4980:{\displaystyle M=R/I}
4948:
4928:
4908:
4865:
4845:
4825:
4786:
4759:
4739:
4710:
4690:
4627:
4571:
4551:
4527:
4504:
4478:
4439:
4419:
4397:
4361:
4341:
4314:
4291:
4268:
4245:
4225:
4202:
4182:
4162:
4142:
4122:
4107:the structures of an
4102:
4078:
4026:
3995:
3922:
3902:
3882:
3851:
3825:
3805:
3733:
3731:{\displaystyle xy=t.}
3701:
3659:
3586:
3549:
3477:
3425:
3345:
3283:
3253:
3224:
3194:
3167:
3138:
3107:
3087:
3054:
2951:
2884:
2860:
2824:
2780:
2753:
2733:
2713:
2693:
2661:
2634:
2612:
2587:
2551:
2515:
2495:
2479:multiplicative subset
2443:
2369:
2367:{\displaystyle M_{i}}
2342:
2285:
2263:
2241:
2219:
2172:
2152:
2108:
2059:
2016:
1956:
1905:
1852:
1817:
1785:
1750:
1693:
1664:
1628:
1583:
1557:
1510:
1478:
1399:Every flat module is
1389:
1338:
1328:
1290:
1262:
1242:
1213:
1185:
1165:
1115:
1113:{\displaystyle R^{n}}
1088:
1086:{\displaystyle R^{n}}
1054:
1013:
960:
929:
888:
829:
814:
772:
739:
706:
670:
618:
521:
438:
357:to the inclusions of
352:
330:
280:
244:
166:
101:Jean-Pierre Serre
8385:"Deformation theory"
8138:
8118:
8092:
8036:
8016:
7996:
7940:
7920:
7894:
7855:
7753:
7733:
7675:
7652:
7616:
7463:
7418:
7380:
7360:
7340:
7308:
7284:
7264:
7238:
7205:
7178:
7136:
7059:
7032:
7000:
6980:
6960:
6940:
6916:
6896:
6876:
6856:
6836:
6816:
6796:
6776:
6756:
6736:
6675:
6667:short exact sequence
6663:long exact sequences
6636:
6602:
6582:
6562:
6503:
6472:
6452:
6426:
6370:
6347:
6327:
6181:
6161:
6137:
6113:
6083:
6055:
6028:
5970:
5958:is faithfully flat.
5942:
5882:
5855:
5731:
5689:
5661:
5632:
5598:
5574:
5554:
5504:
5484:
5457:
5397:
5339:
5285:
5246:
5201:
5175:
5171:A ring homomorphism
5128:
5092:
5034:
5027:is also Noetherian.
5011:
4991:
4957:
4937:
4917:
4882:
4854:
4834:
4795:
4772:
4748:
4725:
4699:
4636:
4580:
4560:
4536:
4513:
4489:
4448:
4428:
4408:
4370:
4350:
4326:
4303:
4277:
4257:
4234:
4211:
4191:
4171:
4151:
4131:
4111:
4091:
4055:
4046:commutative algebras
4004:
3935:
3911:
3891:
3880:{\displaystyle S/fS}
3860:
3834:
3814:
3753:
3710:
3671:
3598:
3558:
3486:
3455:
3365:
3322:
3307:The definition of a
3265:
3233:
3204:
3176:
3147:
3118:
3096:
3076:
2968:
2893:
2869:
2845:
2809:
2766:
2758:is contained in the
2742:
2722:
2702:
2673:
2647:
2623:
2596:
2574:
2527:
2504:
2484:
2426:
2415:Flat ring extensions
2351:
2306:
2272:
2250:
2228:
2196:
2161:
2133:
2089:
2057:{\displaystyle R/I,}
2037:
1990:
1918:
1861:
1826:
1794:
1759:
1727:
1691:{\displaystyle h=p.}
1673:
1641:
1593:
1566:
1519:
1487:
1455:
1451:and two linear maps
1395:Torsion-free modules
1299:
1279:
1251:
1222:
1202:
1174:
1139:
1097:
1070:
1022:
944:
897:
826:
781:
748:
715:
682:
615:
539:non-commutative ring
447:
400:
390:short exact sequence
341:
289:
256:
182:
143:
52:torsion-free modules
8971:Homological algebra
8855:Multilinear algebra
8799:Commutative algebra
8537:Commutative algebra
8485:Commutative Algebra
8053:
7992:Similarly, a right
7586:– rings over which
6520:
6387:
6079:, the homomorphism
5434:
5330:is faithfully flat.
4319:is faithfully flat.
3803:{\displaystyle S=R}
2150:{\displaystyle R/I}
2117:is generated by an
2106:{\displaystyle R/I}
1706:commutative algebra
1581:{\displaystyle G=M}
1356:is flat, and every
528:right exact functor
8976:Algebraic geometry
8914:Serre, Jean-Pierre
8775:Mac Lane, Saunders
8767:10.24033/bsmf.1675
8625:Enochs, Edgar E.;
8594:10.1007/BF02760849
8472:10.24033/bsmf.1503
8144:
8124:
8104:
8078:
8039:
8022:
8002:
7974:
7926:
7906:
7880:
7841:
7739:
7719:
7664:{\displaystyle M,}
7661:
7634:
7594:Normally flat ring
7469:
7449:
7404:
7366:
7346:
7326:
7290:
7270:
7250:
7224:
7191:
7149:
7119:
7038:
7006:
6986:
6966:
6946:
6922:
6902:
6882:
6862:
6842:
6822:
6802:
6782:
6762:
6742:
6719:
6648:
6622:
6588:
6568:
6545:
6506:
6478:
6458:
6438:
6412:
6373:
6353:
6333:
6298:
6167:
6147:
6123:
6095:
6065:
6041:
6014:
5948:
5928:
5868:
5841:
5710:
5667:
5647:
5611:
5592:linear combination
5580:
5560:
5536:
5490:
5470:
5439:
5438:
5417:
5410:
5380:
5320:
5267:
5217:is a nonzero free
5207:
5187:
5146:
5110:
5070:
5017:
4997:
4977:
4943:
4923:
4903:
4860:
4840:
4820:
4784:{\displaystyle S;}
4781:
4754:
4744:is injective, and
4737:{\displaystyle f,}
4734:
4705:
4685:
4622:
4566:
4546:
4525:{\displaystyle R,}
4522:
4499:
4473:
4434:
4414:
4392:
4356:
4336:
4309:
4289:{\displaystyle R,}
4286:
4263:
4240:
4223:{\displaystyle R,}
4220:
4197:
4177:
4157:
4137:
4117:
4097:
4073:
4021:
3990:
3917:
3897:
3877:
3846:
3820:
3800:
3728:
3696:
3654:
3581:
3544:
3472:
3420:
3340:
3278:
3258:-module for every
3248:
3219:
3189:
3162:
3133:
3102:
3082:
3049:
2946:
2879:
2855:
2819:
2778:{\displaystyle A.}
2775:
2748:
2728:
2708:
2688:
2659:{\displaystyle R,}
2656:
2629:
2607:
2582:
2546:
2510:
2490:
2438:
2387:finitely-generated
2364:
2337:
2336:
2325:
2280:
2258:
2236:
2214:
2167:
2147:
2103:
2054:
2023:infinite sequences
2021:be the set of the
2011:
1951:
1900:
1847:
1812:
1780:
1745:
1710:algebraic geometry
1702:finitely generated
1688:
1659:
1623:
1578:
1552:
1505:
1473:
1392:
1380:are free modules.
1372:, meaning all the
1341:
1323:
1285:
1257:
1237:
1208:
1180:
1160:
1110:
1083:
1049:
1008:
924:
883:
809:
767:
734:
701:
665:
516:
433:
347:
325:
275:
239:
175:-modules, the map
161:
44:projective modules
8868:978-0-521-26269-9
8728:978-0-387-98428-5
8648:978-3-11-016633-0
8562:978-0-387-94268-1
8481:Bourbaki, Nicolas
8315:"Amitsur Complex"
8198:, Lemme 5, p. 249
8147:{\displaystyle X}
8127:{\displaystyle R}
8025:{\displaystyle M}
8005:{\displaystyle R}
7929:{\displaystyle R}
7916:is a module over
7742:{\displaystyle S}
7590:modules are flat.
7472:{\displaystyle M}
7369:{\displaystyle M}
7349:{\displaystyle M}
7293:{\displaystyle M}
7273:{\displaystyle M}
7041:{\displaystyle M}
7009:{\displaystyle C}
6989:{\displaystyle A}
6969:{\displaystyle B}
6949:{\displaystyle B}
6925:{\displaystyle A}
6905:{\displaystyle C}
6885:{\displaystyle B}
6865:{\displaystyle A}
6845:{\displaystyle A}
6825:{\displaystyle C}
6805:{\displaystyle B}
6785:{\displaystyle B}
6765:{\displaystyle C}
6745:{\displaystyle A}
6708:
6695:
6591:{\displaystyle N}
6571:{\displaystyle R}
6481:{\displaystyle X}
6461:{\displaystyle R}
6356:{\displaystyle M}
6336:{\displaystyle R}
6170:{\displaystyle R}
5951:{\displaystyle f}
5804:
5771:
5751:
5670:{\displaystyle R}
5583:{\displaystyle 1}
5563:{\displaystyle R}
5493:{\displaystyle R}
5401:
5210:{\displaystyle S}
5122:quotient topology
5020:{\displaystyle R}
5000:{\displaystyle S}
4946:{\displaystyle R}
4926:{\displaystyle I}
4863:{\displaystyle M}
4843:{\displaystyle R}
4757:{\displaystyle R}
4708:{\displaystyle f}
4569:{\displaystyle S}
4437:{\displaystyle R}
4417:{\displaystyle M}
4359:{\displaystyle R}
4312:{\displaystyle S}
4266:{\displaystyle S}
4243:{\displaystyle f}
4200:{\displaystyle S}
4180:{\displaystyle R}
4160:{\displaystyle S}
4140:{\displaystyle R}
4120:{\displaystyle R}
4100:{\displaystyle S}
4085:ring homomorphism
4051:In this section,
4035:Faithful flatness
3920:{\displaystyle f}
3900:{\displaystyle R}
3823:{\displaystyle R}
3740:flat degeneration
3105:{\displaystyle R}
3085:{\displaystyle M}
2797:In this section,
2751:{\displaystyle I}
2731:{\displaystyle I}
2711:{\displaystyle R}
2685:
2643:commutative ring
2639:is an ideal of a
2632:{\displaystyle I}
2513:{\displaystyle R}
2493:{\displaystyle S}
2421:ring homomorphism
2310:
2170:{\displaystyle I}
1354:projective module
1288:{\displaystyle G}
1260:{\displaystyle f}
1211:{\displaystyle K}
1183:{\displaystyle F}
577:Characterizations
368:Equivalently, an
16:(Redirected from
8988:
8956:
8939:
8909:
8899:
8871:
8849:
8838:
8801:
8793:
8770:
8769:
8739:
8701:
8659:
8621:
8596:
8573:
8528:
8511:
8488:
8476:
8474:
8449:
8413:
8411:
8398:
8396:
8389:
8369:
8364:
8358:
8353:
8347:
8341:
8335:
8329:
8323:
8322:
8311:
8305:
8299:
8293:
8287:
8281:
8275:
8269:
8263:
8257:
8251:
8245:
8239:
8233:
8228:
8222:
8217:
8211:
8205:
8199:
8193:
8187:
8181:
8164:
8161:
8155:
8153:
8151:
8150:
8145:
8133:
8131:
8130:
8125:
8113:
8111:
8110:
8105:
8087:
8085:
8084:
8079:
8052:
8047:
8031:
8029:
8028:
8023:
8011:
8009:
8008:
8003:
7990:
7984:
7983:
7981:
7980:
7975:
7967:
7966:
7935:
7933:
7932:
7927:
7915:
7913:
7912:
7907:
7889:
7887:
7886:
7881:
7876:
7875:
7850:
7848:
7847:
7842:
7837:
7836:
7821:
7820:
7787:
7786:
7768:
7767:
7748:
7746:
7745:
7740:
7728:
7726:
7725:
7720:
7715:
7714:
7696:
7695:
7670:
7668:
7667:
7662:
7647:
7643:
7641:
7640:
7635:
7610:
7574:Generic flatness
7512:factors through
7490:projective cover
7479:itself is flat.
7478:
7476:
7475:
7470:
7458:
7456:
7455:
7450:
7436:
7435:
7413:
7411:
7410:
7405:
7375:
7373:
7372:
7367:
7355:
7353:
7352:
7347:
7335:
7333:
7332:
7327:
7299:
7297:
7296:
7291:
7279:
7277:
7276:
7271:
7259:
7257:
7256:
7251:
7233:
7231:
7230:
7225:
7217:
7216:
7200:
7198:
7197:
7192:
7190:
7189:
7158:
7156:
7155:
7150:
7148:
7147:
7128:
7126:
7125:
7120:
7103:
7102:
7090:
7089:
7077:
7076:
7047:
7045:
7044:
7039:
7020:Flat resolutions
7015:
7013:
7012:
7007:
6995:
6993:
6992:
6987:
6975:
6973:
6972:
6967:
6955:
6953:
6952:
6947:
6931:
6929:
6928:
6923:
6911:
6909:
6908:
6903:
6891:
6889:
6888:
6883:
6871:
6869:
6868:
6863:
6851:
6849:
6848:
6843:
6831:
6829:
6828:
6823:
6811:
6809:
6808:
6803:
6791:
6789:
6788:
6783:
6771:
6769:
6768:
6763:
6751:
6749:
6748:
6743:
6728:
6726:
6725:
6720:
6709:
6701:
6696:
6688:
6657:
6655:
6654:
6649:
6631:
6629:
6628:
6623:
6618:
6597:
6595:
6594:
6589:
6577:
6575:
6574:
6569:
6554:
6552:
6551:
6546:
6519:
6514:
6487:
6485:
6484:
6479:
6467:
6465:
6464:
6459:
6447:
6445:
6444:
6439:
6421:
6419:
6418:
6413:
6386:
6381:
6362:
6360:
6359:
6354:
6342:
6340:
6339:
6334:
6307:
6305:
6304:
6299:
6291:
6290:
6284:
6270:
6269:
6254:
6253:
6247:
6233:
6232:
6214:
6213:
6207:
6193:
6192:
6176:
6174:
6173:
6168:
6156:
6154:
6153:
6148:
6146:
6145:
6132:
6130:
6129:
6124:
6122:
6121:
6104:
6102:
6101:
6096:
6074:
6072:
6071:
6066:
6064:
6063:
6050:
6048:
6047:
6042:
6037:
6036:
6023:
6021:
6020:
6015:
6010:
6009:
5988:
5987:
5957:
5955:
5954:
5949:
5937:
5935:
5934:
5929:
5894:
5893:
5877:
5875:
5874:
5869:
5867:
5866:
5850:
5848:
5847:
5842:
5831:
5830:
5818:
5817:
5805:
5803:
5802:
5790:
5785:
5784:
5772:
5770:
5769:
5757:
5752:
5744:
5719:
5717:
5716:
5711:
5676:
5674:
5673:
5668:
5656:
5654:
5653:
5648:
5646:
5645:
5644:
5620:
5618:
5617:
5612:
5610:
5609:
5589:
5587:
5586:
5581:
5569:
5567:
5566:
5561:
5545:
5543:
5542:
5537:
5535:
5534:
5516:
5515:
5499:
5497:
5496:
5491:
5479:
5477:
5476:
5471:
5469:
5468:
5448:
5446:
5445:
5440:
5433:
5425:
5409:
5389:
5387:
5386:
5381:
5370:
5369:
5351:
5350:
5329:
5327:
5326:
5321:
5310:
5281:, the inclusion
5279:monic polynomial
5276:
5274:
5273:
5268:
5230:complexification
5220:
5216:
5214:
5213:
5208:
5196:
5194:
5193:
5188:
5155:
5153:
5152:
5147:
5119:
5117:
5116:
5111:
5086:Zariski topology
5079:
5077:
5076:
5071:
5026:
5024:
5023:
5018:
5006:
5004:
5003:
4998:
4986:
4984:
4983:
4978:
4973:
4952:
4950:
4949:
4944:
4932:
4930:
4929:
4924:
4913:for every ideal
4912:
4910:
4909:
4904:
4869:
4867:
4866:
4861:
4849:
4847:
4846:
4841:
4829:
4827:
4826:
4821:
4816:
4815:
4790:
4788:
4787:
4782:
4763:
4761:
4760:
4755:
4743:
4741:
4740:
4735:
4714:
4712:
4711:
4706:
4694:
4692:
4691:
4686:
4648:
4647:
4631:
4629:
4628:
4623:
4615:
4614:
4605:
4604:
4589:
4588:
4575:
4573:
4572:
4567:
4555:
4553:
4552:
4547:
4545:
4544:
4531:
4529:
4528:
4523:
4508:
4506:
4505:
4500:
4498:
4497:
4482:
4480:
4479:
4474:
4463:
4462:
4443:
4441:
4440:
4435:
4423:
4421:
4420:
4415:
4401:
4399:
4398:
4393:
4379:
4378:
4365:
4363:
4362:
4357:
4345:
4343:
4342:
4337:
4335:
4334:
4318:
4316:
4315:
4310:
4295:
4293:
4292:
4287:
4272:
4270:
4269:
4264:
4249:
4247:
4246:
4241:
4229:
4227:
4226:
4221:
4206:
4204:
4203:
4198:
4186:
4184:
4183:
4178:
4166:
4164:
4163:
4158:
4146:
4144:
4143:
4138:
4127:-algebra and an
4126:
4124:
4123:
4118:
4106:
4104:
4103:
4098:
4082:
4080:
4079:
4074:
4030:
4028:
4027:
4022:
4011:
3999:
3997:
3996:
3991:
3968:
3942:
3926:
3924:
3923:
3918:
3906:
3904:
3903:
3898:
3886:
3884:
3883:
3878:
3870:
3855:
3853:
3852:
3847:
3829:
3827:
3826:
3821:
3809:
3807:
3806:
3801:
3796:
3795:
3777:
3776:
3737:
3735:
3734:
3729:
3705:
3703:
3702:
3697:
3686:
3685:
3663:
3661:
3660:
3655:
3638:
3590:
3588:
3587:
3582:
3565:
3553:
3551:
3550:
3545:
3525:
3499:
3481:
3479:
3478:
3473:
3462:
3444:
3438:
3429:
3427:
3426:
3421:
3419:
3418:
3407:
3406:
3396:
3395:
3375:
3374:
3349:
3347:
3346:
3341:
3287:
3285:
3284:
3279:
3274:
3273:
3257:
3255:
3254:
3249:
3247:
3246:
3245:
3228:
3226:
3225:
3220:
3218:
3217:
3216:
3198:
3196:
3195:
3190:
3185:
3184:
3171:
3169:
3168:
3163:
3161:
3160:
3159:
3142:
3140:
3139:
3134:
3132:
3131:
3130:
3111:
3109:
3108:
3103:
3091:
3089:
3088:
3083:
3068:
3064:
3058:
3056:
3055:
3050:
3042:
3041:
3032:
3031:
3030:
3014:
3013:
3001:
3000:
2982:
2981:
2980:
2963:
2959:
2955:
2953:
2952:
2947:
2939:
2938:
2926:
2925:
2907:
2906:
2905:
2888:
2886:
2885:
2880:
2878:
2877:
2864:
2862:
2861:
2856:
2854:
2853:
2836:
2828:
2826:
2825:
2820:
2818:
2817:
2803:commutative ring
2800:
2784:
2782:
2781:
2776:
2760:Jacobson radical
2757:
2755:
2754:
2749:
2737:
2735:
2734:
2729:
2718:with respect to
2717:
2715:
2714:
2709:
2697:
2695:
2694:
2689:
2687:
2686:
2678:
2665:
2663:
2662:
2657:
2638:
2636:
2635:
2630:
2616:
2614:
2613:
2608:
2603:
2591:
2589:
2588:
2583:
2581:
2561:
2555:
2553:
2552:
2547:
2542:
2541:
2519:
2517:
2516:
2511:
2499:
2497:
2496:
2491:
2473:
2469:
2465:
2459:
2455:
2447:
2445:
2444:
2439:
2406:
2402:
2398:
2373:
2371:
2370:
2365:
2363:
2362:
2346:
2344:
2343:
2338:
2335:
2334:
2324:
2289:
2287:
2286:
2281:
2279:
2267:
2265:
2264:
2259:
2257:
2246:-modules, where
2245:
2243:
2242:
2237:
2235:
2223:
2221:
2220:
2215:
2213:
2208:
2203:
2180:
2176:
2174:
2173:
2168:
2157:is flat only if
2156:
2154:
2153:
2148:
2143:
2124:
2116:
2112:
2110:
2109:
2104:
2099:
2084:
2080:
2067:
2063:
2061:
2060:
2055:
2047:
2028:
2020:
2018:
2017:
2012:
2010:
2009:
2008:
1964:
1960:
1958:
1957:
1952:
1947:
1946:
1941:
1913:
1909:
1907:
1906:
1901:
1856:
1854:
1853:
1848:
1821:
1819:
1818:
1813:
1789:
1787:
1786:
1781:
1754:
1752:
1751:
1746:
1722:
1697:
1695:
1694:
1689:
1668:
1666:
1665:
1660:
1634:
1632:
1630:
1629:
1624:
1622:
1621:
1616:
1587:
1585:
1584:
1579:
1561:
1559:
1558:
1553:
1548:
1547:
1542:
1514:
1512:
1511:
1506:
1482:
1480:
1479:
1474:
1450:
1439:
1409:
1360:is projective.
1332:
1330:
1329:
1324:
1294:
1292:
1291:
1286:
1274:
1270:
1266:
1264:
1263:
1258:
1246:
1244:
1243:
1238:
1217:
1215:
1214:
1209:
1197:
1193:
1189:
1187:
1186:
1181:
1169:
1167:
1166:
1161:
1134:
1130:
1123:
1119:
1117:
1116:
1111:
1109:
1108:
1092:
1090:
1089:
1084:
1082:
1081:
1065:
1058:
1056:
1055:
1050:
1017:
1015:
1014:
1009:
1006:
1005:
996:
995:
979:
974:
956:
955:
933:
931:
930:
925:
892:
890:
889:
884:
875:
874:
859:
858:
848:
843:
818:
816:
815:
810:
799:
798:
776:
774:
773:
768:
760:
759:
743:
741:
740:
735:
727:
726:
710:
708:
707:
702:
694:
693:
674:
672:
671:
666:
658:
657:
648:
647:
637:
632:
607:
603:
596:
592:
587:linear relations
584:
568:
564:
560:
556:
552:
548:
544:
536:
525:
523:
522:
517:
506:
505:
487:
486:
468:
467:
442:
440:
439:
434:
395:
383:
375:
371:
364:
356:
354:
353:
348:
334:
332:
331:
326:
284:
282:
281:
276:
271:
270:
248:
246:
245:
240:
235:
234:
216:
215:
197:
196:
174:
170:
168:
167:
162:
128:
124:
21:
8996:
8995:
8991:
8990:
8989:
8987:
8986:
8985:
8961:
8960:
8959:
8912:
8902:
8874:
8869:
8852:
8841:
8819:
8804:
8796:
8773:
8742:
8729:
8711:Springer-Verlag
8704:
8683:10.2307/2373351
8662:
8649:
8624:
8576:
8563:
8545:Springer-Verlag
8533:Eisenbud, David
8531:
8509:10.2307/1993382
8491:
8479:
8452:
8416:
8409:
8401:
8394:
8387:
8382:
8378:
8373:
8372:
8365:
8361:
8354:
8350:
8342:
8338:
8330:
8326:
8313:
8312:
8308:
8300:
8296:
8288:
8284:
8276:
8272:
8264:
8260:
8252:
8248:
8240:
8236:
8229:
8225:
8218:
8214:
8206:
8202:
8194:
8190:
8182:
8178:
8173:
8168:
8167:
8162:
8158:
8136:
8135:
8116:
8115:
8090:
8089:
8034:
8033:
8014:
8013:
7994:
7993:
7991:
7987:
7958:
7938:
7937:
7918:
7917:
7892:
7891:
7867:
7853:
7852:
7828:
7812:
7778:
7759:
7751:
7750:
7731:
7730:
7706:
7687:
7673:
7672:
7650:
7649:
7645:
7614:
7613:
7612:Proof: Suppose
7611:
7607:
7602:
7570:
7562:axiom of choice
7554:
7498:
7461:
7460:
7427:
7416:
7415:
7378:
7377:
7358:
7357:
7338:
7337:
7306:
7305:
7282:
7281:
7262:
7261:
7236:
7235:
7208:
7203:
7202:
7201:is nonzero and
7181:
7176:
7175:
7139:
7134:
7133:
7094:
7081:
7068:
7057:
7056:
7030:
7029:
7026:flat resolution
7022:
6998:
6997:
6978:
6977:
6958:
6957:
6938:
6937:
6914:
6913:
6894:
6893:
6874:
6873:
6854:
6853:
6834:
6833:
6814:
6813:
6794:
6793:
6774:
6773:
6754:
6753:
6734:
6733:
6673:
6672:
6634:
6633:
6600:
6599:
6580:
6579:
6560:
6559:
6501:
6500:
6470:
6469:
6450:
6449:
6424:
6423:
6368:
6367:
6345:
6344:
6325:
6324:
6313:
6261:
6224:
6184:
6179:
6178:
6159:
6158:
6135:
6134:
6133:-primary ideal
6111:
6110:
6081:
6080:
6053:
6052:
6026:
6025:
5968:
5967:
5964:
5940:
5939:
5885:
5880:
5879:
5858:
5853:
5852:
5822:
5809:
5794:
5776:
5761:
5729:
5728:
5722:Amitsur complex
5687:
5686:
5659:
5658:
5635:
5630:
5629:
5601:
5596:
5595:
5572:
5571:
5552:
5551:
5526:
5507:
5502:
5501:
5500:if and only if
5482:
5481:
5460:
5455:
5454:
5395:
5394:
5361:
5342:
5337:
5336:
5283:
5282:
5244:
5243:
5237:polynomial ring
5226:field extension
5218:
5199:
5198:
5173:
5172:
5168:
5126:
5125:
5090:
5089:
5032:
5031:
5009:
5008:
4989:
4988:
4955:
4954:
4935:
4934:
4915:
4914:
4880:
4879:
4852:
4851:
4832:
4831:
4807:
4793:
4792:
4770:
4769:
4746:
4745:
4723:
4722:
4697:
4696:
4639:
4634:
4633:
4593:
4578:
4577:
4558:
4557:
4534:
4533:
4511:
4510:
4487:
4486:
4454:
4446:
4445:
4426:
4425:
4406:
4405:
4368:
4367:
4348:
4347:
4324:
4323:
4301:
4300:
4275:
4274:
4255:
4254:
4232:
4231:
4209:
4208:
4189:
4188:
4169:
4168:
4149:
4148:
4129:
4128:
4109:
4108:
4089:
4088:
4053:
4052:
4041:faithfully flat
4037:
4002:
4001:
3933:
3932:
3909:
3908:
3907:if and only if
3889:
3888:
3858:
3857:
3832:
3831:
3812:
3811:
3787:
3768:
3751:
3750:
3708:
3707:
3674:
3669:
3668:
3596:
3595:
3556:
3555:
3484:
3483:
3453:
3452:
3440:
3434:
3400:
3368:
3363:
3362:
3320:
3319:
3305:
3263:
3262:
3236:
3231:
3230:
3207:
3202:
3201:
3174:
3173:
3150:
3145:
3144:
3121:
3116:
3115:
3094:
3093:
3074:
3073:
3066:
3062:
3033:
3021:
3002:
2971:
2966:
2965:
2961:
2957:
2927:
2896:
2891:
2890:
2867:
2866:
2843:
2842:
2834:
2807:
2806:
2798:
2795:
2764:
2763:
2740:
2739:
2720:
2719:
2700:
2699:
2671:
2670:
2645:
2644:
2621:
2620:
2594:
2593:
2572:
2571:
2557:
2530:
2525:
2524:
2502:
2501:
2482:
2481:
2471:
2470:, for any ring
2467:
2461:
2457:
2453:
2424:
2423:
2417:
2404:
2400:
2396:
2393:Direct products
2354:
2349:
2348:
2326:
2304:
2303:
2297:
2270:
2269:
2248:
2247:
2226:
2225:
2194:
2193:
2178:
2159:
2158:
2131:
2130:
2127:integral domain
2122:
2114:
2087:
2086:
2082:
2078:
2074:
2065:
2035:
2034:
2031:absolutely flat
2026:
1999:
1988:
1987:
1974:integral domain
1970:Noetherian ring
1965:is projective.
1962:
1933:
1916:
1915:
1911:
1859:
1858:
1824:
1823:
1792:
1791:
1757:
1756:
1725:
1724:
1723:surjective and
1720:
1671:
1670:
1639:
1638:
1608:
1591:
1590:
1589:
1564:
1563:
1534:
1517:
1516:
1485:
1484:
1453:
1452:
1448:
1437:
1434:
1404:
1397:
1346:
1297:
1296:
1277:
1276:
1272:
1268:
1249:
1248:
1220:
1219:
1200:
1199:
1195:
1191:
1172:
1171:
1137:
1136:
1132:
1128:
1121:
1100:
1095:
1094:
1073:
1068:
1067:
1063:
1020:
1019:
997:
981:
947:
942:
941:
895:
894:
860:
850:
824:
823:
784:
779:
778:
751:
746:
745:
718:
713:
712:
685:
680:
679:
649:
639:
613:
612:
605:
601:
594:
590:
582:
579:
566:
562:
558:
554:
550:
546:
542:
534:
497:
478:
459:
445:
444:
398:
397:
393:
381:
376:is flat if the
373:
369:
362:
339:
338:
287:
286:
262:
254:
253:
226:
207:
188:
180:
179:
172:
141:
140:
126:
122:
119:
107:) in his paper
90:faithfully flat
86:exact sequences
72:if taking the
28:
23:
22:
15:
12:
11:
5:
8994:
8992:
8984:
8983:
8978:
8973:
8963:
8962:
8958:
8957:
8937:10.5802/aif.59
8910:
8900:
8883:(2): 263–273,
8872:
8867:
8861:, p. 33,
8850:
8843:Mumford, David
8839:
8817:
8802:
8794:
8783:Academic Press
8771:
8744:Lazard, Daniel
8740:
8727:
8702:
8677:(3): 772–784,
8660:
8647:
8622:
8587:(3): 189–209,
8574:
8561:
8529:
8502:(3): 457–473,
8489:
8477:
8450:
8425:(4): 385–390,
8414:
8403:Artin, Michael
8399:
8379:
8377:
8374:
8371:
8370:
8359:
8348:
8336:
8332:Matsumura 1986
8324:
8306:
8294:
8282:
8270:
8268:, Exercise 6.4
8258:
8254:Matsumura 1970
8246:
8242:Matsumura 1970
8234:
8223:
8212:
8210:, Theorem 7.10
8208:Matsumura 1986
8200:
8188:
8175:
8174:
8172:
8169:
8166:
8165:
8156:
8143:
8123:
8103:
8100:
8097:
8077:
8074:
8071:
8068:
8065:
8062:
8059:
8056:
8051:
8046:
8042:
8021:
8001:
7985:
7973:
7970:
7965:
7961:
7957:
7954:
7951:
7948:
7945:
7925:
7905:
7902:
7899:
7879:
7874:
7870:
7866:
7863:
7860:
7840:
7835:
7831:
7827:
7824:
7819:
7815:
7811:
7808:
7805:
7802:
7799:
7796:
7793:
7790:
7785:
7781:
7777:
7774:
7771:
7766:
7762:
7758:
7738:
7718:
7713:
7709:
7705:
7702:
7699:
7694:
7690:
7686:
7683:
7680:
7660:
7657:
7633:
7630:
7627:
7624:
7621:
7604:
7603:
7601:
7598:
7597:
7596:
7591:
7581:
7576:
7569:
7566:
7553:
7550:
7497:
7494:
7468:
7448:
7445:
7442:
7439:
7434:
7430:
7426:
7423:
7403:
7400:
7397:
7394:
7391:
7388:
7385:
7365:
7345:
7325:
7322:
7319:
7316:
7313:
7302:flat dimension
7300:is called its
7289:
7269:
7260:. If a module
7249:
7246:
7243:
7223:
7220:
7215:
7211:
7188:
7184:
7146:
7142:
7130:
7129:
7118:
7115:
7112:
7109:
7106:
7101:
7097:
7093:
7088:
7084:
7080:
7075:
7071:
7067:
7064:
7037:
7021:
7018:
7005:
6985:
6976:is flat, then
6965:
6945:
6921:
6901:
6881:
6861:
6841:
6821:
6801:
6781:
6761:
6741:
6730:
6729:
6718:
6715:
6712:
6707:
6704:
6699:
6694:
6691:
6686:
6683:
6680:
6647:
6644:
6641:
6621:
6617:
6613:
6610:
6607:
6587:
6567:
6556:
6555:
6544:
6541:
6538:
6535:
6532:
6529:
6526:
6523:
6518:
6513:
6509:
6490:
6489:
6477:
6457:
6448:and all right
6437:
6434:
6431:
6411:
6408:
6405:
6402:
6399:
6396:
6393:
6390:
6385:
6380:
6376:
6352:
6332:
6312:
6309:
6297:
6294:
6289:
6283:
6279:
6276:
6273:
6268:
6264:
6260:
6257:
6252:
6246:
6242:
6239:
6236:
6231:
6227:
6223:
6220:
6217:
6212:
6206:
6202:
6199:
6196:
6191:
6187:
6166:
6144:
6120:
6094:
6091:
6088:
6062:
6040:
6035:
6013:
6008:
6003:
6000:
5997:
5994:
5991:
5986:
5981:
5978:
5975:
5963:
5960:
5947:
5927:
5924:
5921:
5918:
5915:
5912:
5909:
5906:
5903:
5900:
5897:
5892:
5888:
5865:
5861:
5840:
5837:
5834:
5829:
5825:
5821:
5816:
5812:
5808:
5801:
5797:
5793:
5788:
5783:
5779:
5775:
5768:
5764:
5760:
5755:
5750:
5747:
5742:
5739:
5736:
5726:
5725:
5709:
5706:
5703:
5700:
5697:
5694:
5679:
5678:
5666:
5643:
5638:
5622:
5608:
5604:
5579:
5559:
5533:
5529:
5525:
5522:
5519:
5514:
5510:
5489:
5467:
5463:
5437:
5432:
5429:
5424:
5420:
5416:
5413:
5408:
5404:
5392:direct product
5379:
5376:
5373:
5368:
5364:
5360:
5357:
5354:
5349:
5345:
5333:
5332:
5331:
5319:
5316:
5313:
5309:
5305:
5302:
5299:
5296:
5293:
5290:
5266:
5263:
5260:
5257:
5254:
5251:
5240:
5233:
5206:
5186:
5183:
5180:
5167:
5164:
5145:
5142:
5139:
5136:
5133:
5109:
5106:
5103:
5100:
5097:
5069:
5066:
5063:
5060:
5057:
5054:
5051:
5048:
5045:
5042:
5039:
5016:
4996:
4976:
4972:
4968:
4965:
4962:
4942:
4922:
4902:
4899:
4896:
4893:
4890:
4887:
4872:
4871:
4859:
4839:
4819:
4814:
4810:
4806:
4803:
4800:
4780:
4777:
4753:
4733:
4730:
4720:
4719:is surjective.
4704:
4684:
4681:
4678:
4675:
4672:
4669:
4666:
4663:
4660:
4657:
4654:
4651:
4646:
4642:
4621:
4618:
4613:
4608:
4603:
4600:
4596:
4592:
4587:
4565:
4543:
4521:
4518:
4496:
4483:
4472:
4469:
4466:
4461:
4457:
4453:
4444:-module, then
4433:
4413:
4402:
4391:
4388:
4385:
4382:
4377:
4355:
4333:
4320:
4308:
4285:
4282:
4262:
4239:
4219:
4216:
4196:
4176:
4156:
4136:
4116:
4096:
4072:
4069:
4066:
4063:
4060:
4036:
4033:
4020:
4017:
4014:
4010:
3989:
3986:
3983:
3980:
3977:
3974:
3971:
3967:
3963:
3960:
3957:
3954:
3951:
3948:
3945:
3941:
3916:
3896:
3876:
3873:
3869:
3865:
3845:
3842:
3839:
3819:
3799:
3794:
3790:
3786:
3783:
3780:
3775:
3771:
3767:
3764:
3761:
3758:
3727:
3724:
3721:
3718:
3715:
3695:
3692:
3689:
3684:
3681:
3677:
3665:
3664:
3653:
3650:
3647:
3644:
3641:
3637:
3633:
3630:
3627:
3624:
3621:
3618:
3615:
3612:
3609:
3606:
3603:
3580:
3577:
3574:
3571:
3568:
3564:
3543:
3540:
3537:
3534:
3531:
3528:
3524:
3520:
3517:
3514:
3511:
3508:
3505:
3502:
3498:
3494:
3491:
3471:
3468:
3465:
3461:
3431:
3430:
3417:
3414:
3411:
3405:
3399:
3394:
3391:
3388:
3385:
3382:
3379:
3373:
3339:
3336:
3333:
3330:
3327:
3304:
3301:
3297:local property
3289:
3288:
3277:
3272:
3244:
3239:
3215:
3210:
3199:
3188:
3183:
3158:
3153:
3129:
3124:
3113:
3101:
3081:
3048:
3045:
3040:
3036:
3029:
3024:
3020:
3017:
3012:
3009:
3005:
2999:
2994:
2991:
2988:
2985:
2979:
2974:
2945:
2942:
2937:
2934:
2930:
2924:
2919:
2916:
2913:
2910:
2904:
2899:
2876:
2852:
2816:
2794:
2793:Local property
2791:
2774:
2771:
2747:
2727:
2707:
2684:
2681:
2655:
2652:
2628:
2606:
2602:
2580:
2545:
2540:
2537:
2533:
2509:
2489:
2437:
2434:
2431:
2416:
2413:
2389:free modules.
2361:
2357:
2333:
2329:
2323:
2320:
2317:
2313:
2296:
2293:
2292:
2291:
2278:
2256:
2234:
2212:
2207:
2202:
2186:
2166:
2146:
2142:
2138:
2102:
2098:
2094:
2073:
2070:
2053:
2050:
2046:
2042:
2007:
2002:
1998:
1995:
1950:
1945:
1940:
1937:
1932:
1929:
1926:
1923:
1899:
1896:
1893:
1890:
1887:
1884:
1881:
1878:
1875:
1872:
1869:
1866:
1846:
1843:
1840:
1837:
1834:
1831:
1811:
1808:
1805:
1802:
1799:
1779:
1776:
1773:
1770:
1767:
1764:
1744:
1741:
1738:
1735:
1732:
1687:
1684:
1681:
1678:
1658:
1655:
1652:
1649:
1646:
1620:
1615:
1612:
1607:
1604:
1601:
1598:
1577:
1574:
1571:
1551:
1546:
1541:
1538:
1533:
1530:
1527:
1524:
1504:
1501:
1498:
1495:
1492:
1472:
1469:
1466:
1463:
1460:
1433:
1430:
1419:Dedekind rings
1396:
1393:
1345:
1342:
1322:
1319:
1316:
1313:
1310:
1307:
1304:
1284:
1256:
1236:
1233:
1230:
1227:
1207:
1179:
1159:
1156:
1153:
1150:
1147:
1144:
1107:
1103:
1080:
1076:
1060:
1059:
1048:
1045:
1042:
1039:
1036:
1033:
1030:
1027:
1004:
1000:
994:
991:
988:
984:
978:
973:
970:
967:
963:
959:
954:
950:
935:
934:
923:
920:
917:
914:
911:
908:
905:
902:
881:
878:
873:
870:
867:
863:
857:
853:
847:
842:
839:
836:
832:
808:
805:
802:
797:
794:
791:
787:
766:
763:
758:
754:
733:
730:
725:
721:
700:
697:
692:
688:
676:
675:
664:
661:
656:
652:
646:
642:
636:
631:
628:
625:
621:
578:
575:
571:abelian groups
561:must be right
515:
512:
509:
504:
500:
496:
493:
490:
485:
481:
477:
474:
471:
466:
462:
458:
455:
452:
432:
429:
426:
423:
420:
417:
414:
411:
408:
405:
378:tensor product
346:
324:
321:
318:
315:
312:
309:
306:
303:
300:
297:
294:
274:
269:
265:
261:
250:
249:
238:
233:
229:
225:
222:
219:
214:
210:
206:
203:
200:
195:
191:
187:
160:
157:
154:
151:
148:
121:A left module
118:
115:
94:if and only if
88:. A module is
74:tensor product
54:. Formally, a
46:, and, over a
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8993:
8982:
8981:Module theory
8979:
8977:
8974:
8972:
8969:
8968:
8966:
8955:
8951:
8947:
8943:
8938:
8933:
8929:
8925:
8924:
8919:
8915:
8911:
8908:
8906:
8901:
8898:
8894:
8890:
8886:
8882:
8878:
8873:
8870:
8864:
8860:
8856:
8851:
8848:
8844:
8840:
8836:
8832:
8828:
8824:
8820:
8818:0-521-36764-6
8814:
8810:
8809:
8803:
8800:
8795:
8792:
8788:
8784:
8780:
8776:
8772:
8768:
8763:
8759:
8755:
8754:
8749:
8745:
8741:
8738:
8734:
8730:
8724:
8720:
8716:
8712:
8708:
8703:
8700:
8696:
8692:
8688:
8684:
8680:
8676:
8672:
8671:
8666:
8661:
8658:
8654:
8650:
8644:
8640:
8636:
8632:
8628:
8623:
8620:
8616:
8612:
8608:
8604:
8600:
8595:
8590:
8586:
8582:
8581:
8575:
8572:
8568:
8564:
8558:
8554:
8550:
8546:
8542:
8538:
8534:
8530:
8527:
8523:
8519:
8515:
8510:
8505:
8501:
8497:
8496:
8490:
8486:
8482:
8478:
8473:
8468:
8464:
8461:(in French).
8460:
8456:
8451:
8448:
8444:
8440:
8436:
8432:
8428:
8424:
8420:
8415:
8408:
8404:
8400:
8393:
8386:
8381:
8380:
8375:
8368:
8363:
8360:
8357:
8352:
8349:
8346:, p. 183
8345:
8340:
8337:
8333:
8328:
8325:
8320:
8316:
8310:
8307:
8303:
8298:
8295:
8291:
8286:
8283:
8279:
8274:
8271:
8267:
8266:Eisenbud 1995
8262:
8259:
8255:
8250:
8247:
8243:
8238:
8235:
8232:
8227:
8224:
8221:
8216:
8213:
8209:
8204:
8201:
8197:
8192:
8189:
8185:
8180:
8177:
8170:
8160:
8157:
8141:
8121:
8114:and all left
8101:
8098:
8095:
8075:
8072:
8066:
8063:
8060:
8054:
8049:
8044:
8040:
8019:
7999:
7989:
7986:
7971:
7968:
7963:
7959:
7955:
7952:
7949:
7946:
7943:
7923:
7903:
7900:
7897:
7877:
7872:
7868:
7864:
7858:
7838:
7833:
7829:
7822:
7817:
7813:
7809:
7803:
7797:
7794:
7791:
7783:
7779:
7775:
7769:
7764:
7760:
7756:
7736:
7716:
7711:
7707:
7703:
7697:
7692:
7688:
7684:
7681:
7678:
7658:
7655:
7631:
7625:
7622:
7619:
7609:
7606:
7599:
7595:
7592:
7589:
7585:
7582:
7580:
7579:Flat morphism
7577:
7575:
7572:
7571:
7567:
7565:
7563:
7559:
7551:
7549:
7547:
7543:
7539:
7533:
7531:
7527:
7523:
7519:
7515:
7511:
7507:
7503:
7495:
7493:
7491:
7486:
7480:
7466:
7446:
7440:
7432:
7428:
7421:
7401:
7398:
7392:
7386:
7383:
7363:
7343:
7320:
7314:
7311:
7303:
7287:
7267:
7247:
7244:
7241:
7221:
7218:
7213:
7209:
7186:
7182:
7173:
7169:
7164:
7162:
7144:
7140:
7116:
7113:
7107:
7099:
7095:
7086:
7082:
7073:
7069:
7062:
7055:
7054:
7053:
7051:
7035:
7027:
7019:
7017:
7003:
6983:
6963:
6943:
6935:
6919:
6899:
6879:
6859:
6839:
6819:
6799:
6779:
6759:
6739:
6716:
6710:
6705:
6697:
6692:
6684:
6678:
6671:
6670:
6669:
6668:
6664:
6659:
6645:
6642:
6639:
6619:
6615:
6611:
6608:
6605:
6585:
6565:
6542:
6539:
6533:
6530:
6527:
6521:
6516:
6511:
6507:
6499:
6498:
6497:
6495:
6475:
6455:
6435:
6432:
6429:
6409:
6406:
6400:
6397:
6394:
6388:
6383:
6378:
6374:
6366:
6365:
6364:
6350:
6330:
6322:
6318:
6310:
6308:
6295:
6281:
6277:
6271:
6266:
6262:
6255:
6244:
6240:
6234:
6229:
6225:
6221:
6215:
6204:
6200:
6194:
6189:
6185:
6164:
6108:
6092:
6086:
6078:
6077:primary ideal
6038:
6001:
5998:
5979:
5976:
5961:
5959:
5945:
5925:
5922:
5919:
5916:
5913:
5910:
5907:
5904:
5898:
5890:
5886:
5863:
5859:
5838:
5832:
5827:
5823:
5819:
5814:
5810:
5806:
5799:
5795:
5786:
5781:
5777:
5773:
5766:
5762:
5753:
5748:
5740:
5734:
5723:
5707:
5704:
5698:
5695:
5692:
5684:
5683:
5682:
5664:
5636:
5627:
5623:
5606:
5602:
5593:
5577:
5570:(that is, if
5557:
5549:
5546:generate the
5531:
5527:
5523:
5520:
5517:
5512:
5508:
5487:
5465:
5461:
5452:
5451:localizations
5430:
5427:
5422:
5418:
5411:
5406:
5402:
5393:
5377:
5374:
5371:
5366:
5362:
5358:
5355:
5352:
5347:
5343:
5334:
5314:
5307:
5300:
5294:
5288:
5280:
5261:
5255:
5252:
5249:
5241:
5238:
5234:
5231:
5227:
5223:
5222:
5204:
5184:
5178:
5170:
5169:
5165:
5163:
5161:
5160:
5140:
5134:
5131:
5123:
5104:
5098:
5095:
5087:
5083:
5064:
5058:
5055:
5046:
5040:
5037:
5028:
5014:
4994:
4974:
4970:
4966:
4963:
4960:
4940:
4920:
4900:
4897:
4894:
4891:
4888:
4885:
4877:
4857:
4837:
4817:
4812:
4808:
4804:
4798:
4778:
4775:
4767:
4751:
4731:
4728:
4721:
4718:
4702:
4679:
4673:
4670:
4661:
4655:
4652:
4649:
4644:
4640:
4619:
4601:
4598:
4594:
4590:
4563:
4519:
4516:
4484:
4470:
4467:
4464:
4459:
4455:
4451:
4431:
4424:is a nonzero
4411:
4403:
4389:
4386:
4383:
4380:
4353:
4321:
4306:
4299:
4298:
4297:
4283:
4280:
4273:is flat over
4260:
4251:
4237:
4217:
4214:
4194:
4174:
4154:
4134:
4114:
4094:
4086:
4070:
4064:
4061:
4058:
4049:
4047:
4042:
4034:
4032:
4015:
3987:
3981:
3978:
3975:
3972:
3965:
3958:
3955:
3952:
3949:
3946:
3930:
3914:
3894:
3887:is flat over
3874:
3871:
3867:
3863:
3843:
3840:
3837:
3817:
3792:
3788:
3784:
3781:
3778:
3773:
3769:
3762:
3759:
3756:
3747:
3745:
3741:
3725:
3722:
3719:
3716:
3713:
3690:
3682:
3679:
3675:
3651:
3642:
3628:
3625:
3616:
3610:
3607:
3604:
3601:
3594:
3593:
3592:
3578:
3569:
3538:
3535:
3532:
3529:
3522:
3515:
3512:
3509:
3506:
3503:
3492:
3489:
3466:
3449:
3446:
3443:
3437:
3415:
3412:
3409:
3389:
3383:
3380:
3377:
3361:
3360:
3359:
3357:
3356:flat morphism
3353:
3337:
3331:
3328:
3325:
3316:
3314:
3310:
3309:flat morphism
3302:
3300:
3298:
3294:
3275:
3261:
3260:maximal ideal
3237:
3208:
3200:
3186:
3151:
3122:
3114:
3099:
3079:
3072:
3071:
3070:
3059:
3046:
3043:
3038:
3034:
3022:
3018:
3015:
3010:
3007:
2989:
2983:
2972:
2943:
2940:
2935:
2932:
2914:
2908:
2897:
2840:
2832:
2804:
2792:
2790:
2788:
2772:
2769:
2761:
2745:
2725:
2705:
2682:
2679:
2669:
2653:
2650:
2642:
2626:
2617:
2604:
2569:
2565:
2560:
2543:
2538:
2535:
2531:
2523:
2507:
2487:
2480:
2475:
2466:is flat over
2464:
2451:
2435:
2429:
2422:
2414:
2412:
2410:
2409:coherent ring
2394:
2390:
2388:
2384:
2380:
2375:
2359:
2355:
2331:
2327:
2321:
2318:
2315:
2311:
2302:
2294:
2205:
2191:
2187:
2184:
2164:
2144:
2140:
2136:
2128:
2120:
2100:
2096:
2092:
2076:
2075:
2071:
2069:
2051:
2048:
2044:
2040:
2032:
2024:
2000:
1996:
1993:
1984:
1982:
1977:
1975:
1971:
1966:
1948:
1943:
1930:
1927:
1924:
1921:
1897:
1894:
1891:
1888:
1885:
1882:
1879:
1876:
1873:
1870:
1867:
1864:
1844:
1841:
1838:
1835:
1832:
1829:
1809:
1803:
1800:
1797:
1777:
1774:
1768:
1762:
1742:
1739:
1736:
1733:
1730:
1718:
1713:
1711:
1707:
1703:
1698:
1685:
1682:
1679:
1676:
1656:
1653:
1650:
1647:
1644:
1635:
1618:
1605:
1602:
1599:
1596:
1575:
1572:
1569:
1549:
1544:
1531:
1528:
1525:
1522:
1502:
1496:
1493:
1490:
1470:
1464:
1461:
1458:
1447:
1443:
1431:
1429:
1427:
1426:PrĂĽfer domain
1422:
1420:
1416:
1411:
1407:
1402:
1394:
1388:
1384:
1381:
1379:
1375:
1374:localizations
1371:
1366:
1361:
1359:
1355:
1351:
1343:
1337:
1333:
1320:
1317:
1314:
1308:
1302:
1282:
1254:
1234:
1231:
1228:
1225:
1205:
1177:
1157:
1154:
1148:
1145:
1142:
1125:
1105:
1101:
1078:
1074:
1046:
1043:
1040:
1037:
1034:
1031:
1028:
1025:
1002:
998:
992:
989:
986:
982:
976:
971:
968:
965:
961:
957:
952:
948:
940:
939:
938:
921:
918:
915:
912:
909:
906:
903:
900:
879:
876:
871:
868:
865:
861:
855:
851:
845:
840:
837:
834:
830:
822:
821:
820:
806:
803:
800:
795:
792:
789:
785:
764:
761:
756:
752:
731:
728:
723:
719:
698:
695:
690:
686:
662:
659:
654:
650:
644:
640:
634:
629:
626:
623:
619:
611:
610:
609:
598:
588:
576:
574:
572:
540:
531:
529:
513:
507:
502:
498:
494:
488:
483:
479:
475:
469:
464:
460:
456:
450:
443:the sequence
430:
427:
421:
415:
409:
403:
391:
387:
386:exact functor
379:
366:
360:
344:
335:
322:
319:
316:
310:
304:
298:
295:
292:
272:
267:
263:
259:
236:
231:
227:
223:
217:
212:
208:
204:
201:
198:
193:
189:
185:
178:
177:
176:
158:
152:
149:
146:
139:
136:
132:
116:
114:
112:
111:
106:
102:
97:
95:
91:
87:
83:
79:
75:
71:
67:
64:
60:
57:
53:
49:
45:
41:
37:
33:
19:
8927:
8921:
8903:
8880:
8876:
8854:
8846:
8807:
8798:
8778:
8757:
8751:
8706:
8674:
8668:
8664:
8630:
8584:
8578:
8536:
8499:
8493:
8484:
8462:
8458:
8422:
8418:
8367:Richman 1997
8362:
8351:
8339:
8327:
8318:
8309:
8297:
8285:
8273:
8261:
8256:, Theorem 56
8249:
8237:
8226:
8215:
8203:
8196:Cartier 1958
8191:
8179:
8159:
7988:
7608:
7587:
7555:
7537:
7534:
7525:
7521:
7517:
7513:
7509:
7505:
7501:
7499:
7481:
7304:and denoted
7171:
7167:
7165:
7131:
7052:of the form
7028:of a module
7025:
7023:
6731:
6660:
6557:
6493:
6491:
6317:Tor functors
6314:
5965:
5727:
5680:
5157:
5124:of that of
5081:
5029:
4873:
4766:pure subring
4252:
4147:-module. If
4050:
4040:
4039:A module is
4038:
3748:
3666:
3450:
3447:
3441:
3435:
3432:
3317:
3306:
3290:
3060:
2839:localization
2796:
2787:Zariski ring
2618:
2558:
2522:localization
2476:
2462:
2449:
2418:
2391:
2383:free modules
2379:direct limit
2376:
2298:
2190:torsion free
2072:Non-examples
1985:
1978:
1967:
1714:
1700:Conversely,
1699:
1636:
1435:
1423:
1412:
1405:
1401:torsion-free
1398:
1382:
1378:prime ideals
1370:locally free
1362:
1350:torsion-free
1347:
1126:
1061:
936:
677:
599:
580:
532:
367:
336:
251:
130:
125:over a ring
120:
108:
98:
89:
81:
77:
69:
65:
58:
40:free modules
36:flat modules
35:
29:
8465:: 177–251.
8319:ncatlab.org
8280:, p. 3
8220:Lazard 1969
7496:Flat covers
7485:epimorphism
7161:Tor functor
6792:. Also, if
4876:local rings
4695:induced by
3318:A morphism
3293:local rings
2831:prime ideal
1446:free module
1358:free module
1198:-submodule
8965:Categories
8835:0603.13001
8760:: 81–128,
8376:References
8302:Artin 1999
8231:Chase 1960
7376:such that
7174:such that
7132:where the
7050:resolution
7016:are flat.
6892:are flat,
6024:such that
5626:direct sum
5548:unit ideal
5197:such that
5082:submersive
4576:such that
4366:, one has
3738:(See also
3229:is a flat
3143:is a flat
3092:is a flat
2801:denotes a
2785:(See also
2668:completion
2641:Noetherian
2568:projective
2556:is a flat
2456:is a flat
2301:direct sum
2183:zero ideal
2181:or is the
2119:idempotent
1981:local ring
1822:such that
1515:such that
1442:projective
1363:There are
1295:such that
1271:to a free
819:such that
545:is a left
138:linear map
117:Definition
84:preserves
8946:0373-0956
8889:1171-6096
8619:120567780
8603:0021-2172
8439:0024-6093
8171:Citations
8134:-modules
8099:≥
8055:
7960:⊗
7953:⊂
7947:≠
7901:≠
7869:⊗
7862:→
7830:⊗
7814:⊗
7804:≃
7795:⊗
7780:⊗
7773:→
7761:⊗
7729:exhibits
7708:⊗
7701:→
7689:⊗
7629:→
7444:→
7438:→
7425:→
7387:
7315:
7111:→
7105:→
7092:→
7079:→
7066:→
7063:⋯
6714:→
6703:⟶
6690:⟶
6682:→
6643:⊂
6522:
6468:-modules
6433:≥
6389:
6272:
6235:
6195:
6090:→
5993:↪
5923:⊗
5917:−
5911:⊗
5887:δ
5860:δ
5839:⋯
5836:→
5824:⊗
5811:⊗
5796:δ
5792:→
5778:⊗
5763:δ
5759:→
5746:→
5738:→
5702:→
5521:…
5428:−
5403:∏
5372:∈
5356:…
5318:⟩
5312:⟨
5292:↪
5253:∈
5182:→
5135:
5099:
5059:
5053:→
5041:
4898:∩
4809:⊗
4802:→
4791:that is,
4674:
4668:→
4656:
4650::
4645:∗
4599:−
4468:≠
4456:⊗
4384:≠
4230:and that
4068:→
4062::
3979:−
3929:primitive
3841:∈
3782:…
3680:−
3676:π
3629:
3623:→
3611:
3602:π
3576:↪
3536:−
3482:-algebra
3398:→
3335:→
3035:⊗
3008:−
2993:∖
2964:-module,
2933:−
2918:∖
2683:^
2536:−
2433:→
2374:is flat.
2319:∈
2312:⨁
1925:∘
1886:∘
1874:∘
1868:∘
1857:and thus
1833:∘
1807:→
1740:
1654:∘
1526:∘
1500:→
1468:→
1436:A module
1229:
1152:→
1038:…
962:∑
913:…
831:∑
801:∈
762:∈
729:∈
696:∈
620:∑
511:→
499:⊗
492:→
480:⊗
473:→
461:⊗
454:→
425:→
419:→
413:→
407:→
396:-modules
345:φ
317:⊗
305:φ
302:↦
296:⊗
264:⊗
260:φ
228:⊗
221:→
209:⊗
190:⊗
186:φ
171:of right
156:→
147:φ
135:injective
8930:: 1–42,
8916:(1956),
8779:Homology
8777:(1963),
8746:(1969),
8629:(2000),
8535:(1995),
8405:(1999).
8392:Archived
8344:Lam 1999
8184:Bourbaki
8088:for all
8012:-module
7671:the map
7648:-module
7568:See also
6578:-module
6558:for any
6422:for all
6343:-module
5166:Examples
4850:-module
3112:-module;
2956:and, if
2477:For any
1352:, every
1275:-module
1247:the map
1131:-module
604:-module
372:-module
38:include
8954:0082175
8897:1601663
8827:0879273
8791:0156879
8737:1653294
8699:0252389
8691:2373351
8657:1753146
8611:0636889
8571:1322960
8526:0120260
8518:1993382
8447:1832549
8383:Artin.
7936:, then
5594:of the
5453:at the
5449:of the
5120:is the
4717:spectra
4715:on the
3352:schemes
3313:schemes
2566:(it is
2564:algebra
2177:equals
2085:, then
1968:Over a
1120:to the
600:A left
103: (
61:over a
32:algebra
8952:
8944:
8895:
8887:
8865:
8833:
8825:
8815:
8789:
8735:
8725:
8697:
8689:
8655:
8645:
8617:
8609:
8601:
8569:
8559:
8524:
8516:
8445:
8437:
7168:length
6319:, the
6263:length
6226:length
6186:length
6051:is an
5224:Every
4953:(take
3065:is an
2960:is an
2837:, the
2520:, the
2125:is an
2064:where
1170:where
541:, and
384:is an
56:module
8905:SGA 1
8687:JSTOR
8615:S2CID
8514:JSTOR
8410:(PDF)
8395:(PDF)
8388:(PDF)
8290:SGA I
8278:Artin
7600:Notes
7520:over
7336:. If
7048:is a
6852:. If
5590:is a
5277:is a
4764:is a
4167:is a
4083:is a
3354:is a
2829:is a
2805:. If
2407:is a
1979:On a
937:and
777:and
678:with
537:is a
380:with
361:into
80:with
76:over
8942:ISSN
8885:ISSN
8863:ISBN
8813:ISBN
8723:ISBN
8643:ISBN
8599:ISSN
8557:ISBN
8435:ISSN
7546:2000
7542:1963
7530:1981
7245:>
7234:for
7166:The
6996:and
6956:and
6934:pure
6872:and
6812:and
6752:and
6632:and
5624:The
5390:The
5335:Let
5132:Spec
5096:Spec
5056:Spec
5038:Spec
4671:Spec
4653:Spec
3830:and
3749:Let
3742:and
3626:Spec
3608:Spec
2666:the
2452:if
2450:flat
1961:and
1708:and
1669:and
1588:and
1483:and
1417:and
1127:An
1018:for
893:for
711:and
557:and
131:flat
105:1956
70:flat
63:ring
8932:doi
8831:Zbl
8762:doi
8715:doi
8679:doi
8667:",
8635:doi
8589:doi
8549:doi
8504:doi
8467:doi
8427:doi
8041:Tor
7588:all
7548:).
6936:in
6932:is
6732:If
6508:Tor
6375:Tor
6157:of
5657:of
5550:of
5242:If
5088:of
5080:is
4933:of
4768:of
4556:of
4509:of
4404:If
4346:of
4253:If
3927:is
3746:.)
3445:.
3439:in
3350:of
3311:of
3061:If
2841:at
2833:of
2789:.)
2762:of
2698:of
2619:If
2474:.
2448:is
2077:If
1910:As
1737:ker
1440:is
1408:= 1
1376:at
1226:ker
1218:of
597:.
589:in
530:.)
392:of
129:is
68:is
30:In
8967::
8950:MR
8948:,
8940:,
8926:,
8920:,
8893:MR
8891:,
8881:26
8879:,
8857:,
8845:,
8829:.
8823:MR
8821:.
8787:MR
8785:,
8758:97
8756:,
8750:,
8733:MR
8731:,
8721:,
8713:,
8695:MR
8693:,
8685:,
8675:91
8673:,
8653:MR
8651:,
8641:,
8613:,
8607:MR
8605:,
8597:,
8585:39
8583:,
8567:MR
8565:,
8555:,
8547:,
8539:,
8522:MR
8520:,
8512:,
8500:97
8498:,
8483:.
8463:86
8457:.
8443:MR
8441:,
8433:,
8423:33
8421:,
8390:.
8317:.
7384:fd
7312:fd
7163:.
7024:A
6488:).
6177:,
5621:).
5235:A
5162:.
4471:0.
3299:.
2419:A
2377:A
2299:A
2129:,
1715:A
1633:).
1428:.
1421:.
1410:.
573:.
553:,
365:.
113:.
50:,
42:,
34:,
8934::
8928:6
8837:.
8764::
8717::
8681::
8665:p
8637::
8591::
8551::
8506::
8487:.
8475:.
8469::
8429::
8412:.
8321:.
8154:.
8142:X
8122:R
8102:1
8096:n
8076:0
8073:=
8070:)
8067:X
8064:,
8061:M
8058:(
8050:R
8045:n
8020:M
8000:R
7972:.
7969:S
7964:R
7956:M
7950:M
7944:0
7924:R
7904:0
7898:M
7878:S
7873:R
7865:M
7859:M
7839:S
7834:R
7826:)
7823:S
7818:R
7810:M
7807:(
7801:)
7798:S
7792:S
7789:(
7784:R
7776:M
7770:S
7765:R
7757:M
7737:S
7717:S
7712:R
7704:S
7698:S
7693:R
7685:R
7682:=
7679:S
7659:,
7656:M
7646:R
7632:S
7626:R
7623::
7620:f
7522:M
7518:F
7514:F
7510:M
7506:F
7502:M
7467:M
7447:0
7441:M
7433:0
7429:F
7422:0
7402:0
7399:=
7396:)
7393:M
7390:(
7364:M
7344:M
7324:)
7321:M
7318:(
7288:M
7268:M
7248:n
7242:i
7222:0
7219:=
7214:i
7210:F
7187:n
7183:F
7172:n
7145:i
7141:F
7117:,
7114:0
7108:M
7100:0
7096:F
7087:1
7083:F
7074:2
7070:F
7036:M
7004:C
6984:A
6964:B
6944:B
6920:A
6900:C
6880:B
6860:A
6840:A
6820:C
6800:B
6780:B
6760:C
6740:A
6717:0
6711:C
6706:g
6698:B
6693:f
6685:A
6679:0
6646:R
6640:I
6620:I
6616:/
6612:R
6609:=
6606:N
6586:N
6566:R
6543:0
6540:=
6537:)
6534:M
6531:,
6528:N
6525:(
6517:R
6512:1
6494:M
6476:X
6456:R
6436:1
6430:n
6410:0
6407:=
6404:)
6401:M
6398:,
6395:X
6392:(
6384:R
6379:n
6351:M
6331:R
6296:.
6293:)
6288:q
6282:/
6278:R
6275:(
6267:R
6259:)
6256:S
6251:m
6245:/
6241:S
6238:(
6230:S
6222:=
6219:)
6216:S
6211:q
6205:/
6201:S
6198:(
6190:S
6165:R
6143:q
6119:m
6093:B
6087:S
6075:-
6061:n
6039:S
6034:m
6012:)
6007:n
6002:,
5999:S
5996:(
5990:)
5985:m
5980:,
5977:R
5974:(
5946:f
5926:b
5920:1
5914:1
5908:b
5905:=
5902:)
5899:b
5896:(
5891:0
5864:n
5833:B
5828:A
5820:B
5815:A
5807:B
5800:1
5787:B
5782:A
5774:B
5767:0
5754:B
5749:f
5741:A
5735:0
5724::
5708:,
5705:B
5699:A
5696::
5693:f
5665:R
5642:p
5637:R
5607:i
5603:t
5578:1
5558:R
5532:k
5528:t
5524:,
5518:,
5513:1
5509:t
5488:R
5466:i
5462:t
5436:]
5431:1
5423:i
5419:t
5415:[
5412:R
5407:i
5378:.
5375:R
5367:k
5363:t
5359:,
5353:,
5348:1
5344:t
5315:p
5308:/
5304:]
5301:t
5298:[
5295:R
5289:R
5265:]
5262:x
5259:[
5256:R
5250:p
5219:R
5205:S
5185:S
5179:R
5144:)
5141:S
5138:(
5108:)
5105:R
5102:(
5068:)
5065:R
5062:(
5050:)
5047:S
5044:(
5015:R
4995:S
4975:I
4971:/
4967:R
4964:=
4961:M
4941:R
4921:I
4901:R
4895:S
4892:I
4889:=
4886:I
4870:.
4858:M
4838:R
4818:S
4813:R
4805:M
4799:M
4779:;
4776:S
4752:R
4732:,
4729:f
4703:f
4683:)
4680:R
4677:(
4665:)
4662:S
4659:(
4641:f
4620:.
4617:)
4612:P
4607:(
4602:1
4595:f
4591:=
4586:p
4564:S
4542:P
4520:,
4517:R
4495:p
4465:S
4460:R
4452:M
4432:R
4412:M
4390:.
4387:S
4381:S
4376:m
4354:R
4332:m
4307:S
4284:,
4281:R
4261:S
4238:f
4218:,
4215:R
4195:S
4175:R
4155:S
4135:R
4115:R
4095:S
4071:S
4065:R
4059:f
4019:]
4016:t
4013:[
4009:C
3988:,
3985:)
3982:t
3976:y
3973:x
3970:(
3966:/
3962:]
3959:y
3956:,
3953:x
3950:,
3947:t
3944:[
3940:C
3915:f
3895:R
3875:S
3872:f
3868:/
3864:S
3844:S
3838:f
3818:R
3798:]
3793:r
3789:x
3785:,
3779:,
3774:1
3770:x
3766:[
3763:R
3760:=
3757:S
3726:.
3723:t
3720:=
3717:y
3714:x
3694:)
3691:t
3688:(
3683:1
3652:.
3649:)
3646:]
3643:t
3640:[
3636:C
3632:(
3620:)
3617:R
3614:(
3605::
3579:R
3573:]
3570:t
3567:[
3563:C
3542:)
3539:t
3533:y
3530:x
3527:(
3523:/
3519:]
3516:y
3513:,
3510:x
3507:,
3504:t
3501:[
3497:C
3493:=
3490:R
3470:]
3467:t
3464:[
3460:C
3442:X
3436:x
3416:x
3413:,
3410:X
3404:O
3393:)
3390:x
3387:(
3384:f
3381:,
3378:Y
3372:O
3338:Y
3332:X
3329::
3326:f
3276:.
3271:m
3243:m
3238:R
3214:m
3209:M
3187:;
3182:p
3157:p
3152:R
3128:p
3123:M
3100:R
3080:M
3067:R
3063:M
3047:.
3044:M
3039:R
3028:p
3023:R
3019:=
3016:M
3011:1
3004:)
2998:p
2990:R
2987:(
2984:=
2978:p
2973:M
2962:R
2958:M
2944:,
2941:R
2936:1
2929:)
2923:p
2915:R
2912:(
2909:=
2903:p
2898:R
2875:p
2851:p
2835:R
2815:p
2799:R
2773:.
2770:A
2746:I
2726:I
2706:R
2680:R
2654:,
2651:R
2627:I
2605:.
2601:Z
2579:Q
2562:-
2559:R
2544:R
2539:1
2532:S
2508:R
2488:S
2472:R
2468:R
2463:R
2458:R
2454:S
2436:S
2430:R
2405:R
2401:R
2397:R
2360:i
2356:M
2332:i
2328:M
2322:I
2316:i
2277:Q
2255:Z
2233:Z
2211:Z
2206:/
2201:Q
2185:.
2179:R
2165:I
2145:I
2141:/
2137:R
2123:R
2115:I
2101:I
2097:/
2093:R
2083:R
2079:I
2066:I
2052:,
2049:I
2045:/
2041:R
2027:F
2006:N
2001:F
1997:=
1994:R
1963:M
1949:,
1944:M
1939:d
1936:i
1931:=
1928:i
1922:h
1912:f
1898:.
1895:f
1892:=
1889:g
1883:h
1880:=
1877:f
1871:i
1865:h
1845:,
1842:g
1839:=
1836:f
1830:i
1810:G
1804:M
1801::
1798:i
1778:0
1775:=
1772:)
1769:K
1766:(
1763:g
1743:f
1734:=
1731:K
1721:f
1686:.
1683:p
1680:=
1677:h
1657:f
1651:i
1648:=
1645:g
1619:M
1614:d
1611:i
1606:=
1603:p
1600:=
1597:i
1576:M
1573:=
1570:G
1550:.
1545:M
1540:d
1537:i
1532:=
1529:i
1523:p
1503:M
1497:G
1494::
1491:p
1471:G
1465:M
1462::
1459:i
1449:G
1438:M
1406:m
1321::
1318:0
1315:=
1312:)
1309:K
1306:(
1303:g
1283:G
1273:R
1269:g
1255:f
1235:,
1232:f
1206:K
1196:R
1192:R
1178:F
1158:,
1155:M
1149:F
1146::
1143:f
1133:M
1129:R
1122:n
1106:n
1102:R
1079:n
1075:R
1064:n
1047:.
1044:m
1041:,
1035:,
1032:1
1029:=
1026:i
1003:j
999:y
993:j
990:,
987:i
983:a
977:n
972:1
969:=
966:j
958:=
953:i
949:x
922:,
919:n
916:,
910:,
907:1
904:=
901:j
880:0
877:=
872:j
869:,
866:i
862:a
856:i
852:r
846:m
841:1
838:=
835:i
807:,
804:R
796:j
793:,
790:i
786:a
765:M
757:j
753:y
732:M
724:i
720:x
699:R
691:i
687:r
663:0
660:=
655:i
651:x
645:i
641:r
635:m
630:1
627:=
624:i
606:M
602:R
595:R
591:M
585:-
583:R
567:R
563:R
559:J
555:L
551:K
547:R
543:M
535:R
514:0
508:M
503:R
495:J
489:M
484:R
476:L
470:M
465:R
457:K
451:0
431:,
428:0
422:J
416:L
410:K
404:0
394:R
382:M
374:M
370:R
363:R
323:.
320:m
314:)
311:k
308:(
299:m
293:k
273:M
268:R
237:M
232:R
224:L
218:M
213:R
205:K
202::
199:M
194:R
173:R
159:L
153:K
150::
127:R
123:M
82:M
78:R
66:R
59:M
20:)
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