2272:
1852:
1550:
1122:(Kawamata log terminal) singularities; in characteristic zero, these are rational singularities and hence are Cohen–Macaulay, One successful analog of rational singularities in positive characteristic is the notion of
2105:
2503:
2073:
311:
245:
1228:
455:
570:
2868:
2796:
1656:
3410:
Matsumura (1989), Theorem 23.5.; NB: although the reference is somehow vague on whether a ring there is assumed to be local or not, the proof there does not need the ring to be local.
1452:
are Cohen–Macaulay if and only if they have no embedded primes. The singularities present in Cohen–Macaulay curves can be classified completely by looking at the plane curve case.
1388:
1427:
636:
607:
519:
1597:
1977:
1940:
728:
658:
3058:) (the coordinate ring of the union of a line and a plane) is reduced, but not equidimensional, and hence not Cohen–Macaulay. Taking the quotient by the non-zero-divisor
175:
1709:
1450:
1351:
1736:
1683:
1398:
Cohen–Macaulay curves are a special case of Cohen–Macaulay schemes, but are useful for compactifying moduli spaces of curves where the boundary of the smooth locus
930:
857:
1875:
1756:
1617:
1325:
786:
690:
478:
387:
363:
343:
2952:
generated by a number of elements equal to its height is unmixed. A Noetherian ring is Cohen–Macaulay if and only if the unmixedness theorem holds for it.
1072:=0, is a 1-dimensional domain which is Gorenstein, and hence Cohen–Macaulay, but not regular. This ring can also be described as the coordinate ring of the
1764:
3097:) (the coordinate ring of the union of two planes meeting in a point) is reduced and equidimensional, but not Cohen–Macaulay. To prove that, one can use
3559:
1466:
3718:
3118:
1460:
Using the criterion, there are easy examples of non-Cohen–Macaulay curves from constructing curves with embedded points. For example, the scheme
55:
is Cohen–Macaulay exactly when it is a finitely generated free module over a regular local subring. Cohen–Macaulay rings play a central role in
2955:
The unmixed theorem applies in particular to the zero ideal (an ideal generated by zero elements) and thus it says a Cohen–Macaulay ring is an
3255:
3978:
576:
unless we define zero modules as Cohen-Macaulay. So we define zero modules as Cohen-Macaulay modules in this definition.) Now, to define
2267:{\displaystyle {\frac {\mathbb {C} }{(y-x^{2})}}\otimes _{\mathbb {C} }{\frac {\mathbb {C} }{(y)}}\cong {\frac {\mathbb {C} }{(x^{2})}}}
1429:
is of Cohen–Macaulay curves. There is a useful criterion for deciding whether or not curves are Cohen–Macaulay. Schemes of dimension
3177:
for
Gorenstein or Cohen–Macaulay schemes retain some of the simplicity of what happens for regular schemes or smooth varieties.
2447:
2013:
460:
The above definition was for a
Noetherian local rings. But we can expand the definition for a more general Noetherian ring: If
250:
3889:
3816:
3730:
3693:
3652:
3537:
184:
1162:
2878:
1293:
400:
527:
2818:
2746:
3859:
3775:
3479:
2799:
314:
3710:
3669:
1626:
2999:) (the coordinate ring of a line with an embedded point) is not Cohen–Macaulay. This follows, for example, by
3632:
3617:
2959:; in fact, in the strong sense: there is no embedded component and each component has the same codimension.
3998:
2967:
953:
1356:
3993:
3612:
3607:
2326:
1401:
3881:
3843:
3800:
3759:
3677:
3529:
3174:
2683:
2322:
2084:
896:
112:
2079:
In general, that multiplicity is given as a length essentially characterizes Cohen–Macaulay ring; see
612:
583:
495:
2687:
2598:
2536:-module. Again, it follows that this freeness is independent of the choice of the polynomial subring
2302:
2087:, on the other hand, roughly characterizes a regular local ring as a local ring of multiplicity one.
1988:
1234:
144:
94:
1558:
3833:
3000:
1949:
1912:
1119:
1065:
937:
83:
63:
3248:
711:
641:
2376:
2293:
2277:
which is Cohen–Macaulay of length two, hence the intersection multiplicity is two, as expected.
1053:²) has dimension 0 and hence is Cohen–Macaulay, but it is not reduced and therefore not regular.
3602:
Cohen's paper was written when "local ring" meant what is now called a "Noetherian local ring".
979:
945:
3344:
smoothness here is somehow extraneous and is used in part to make sense of a proper component.
3301:
3973:
1658:, a curve with an embedded point can be constructed using the same technique: find the ideal
1115:
1034:
154:
1688:
1432:
1330:
3933:
3921:
3899:
3869:
3826:
3785:
3740:
3703:
3662:
3598:
3547:
3508:
3310:
2956:
2738:
1714:
1661:
1305:
1107:
990:
908:
2549:
A Noetherian local ring is Cohen–Macaulay if and only if its completion is Cohen–Macaulay.
8:
3837:
3122:
2594:
1886:
789:
573:
318:
71:
56:
3925:
3314:
2422:
have the same degree. It is striking that this property is independent of the choice of
801:
3911:
3586:
3496:
3166:
2963:
2934:
1860:
1741:
1602:
1310:
1134:
1073:
998:
903:
736:
705:
675:
463:
372:
348:
328:
118:
48:
40:
3885:
3855:
3812:
3792:
3771:
3747:
3726:
3689:
3648:
3578:
3533:
3328:
2099:
with a line tangent to it, the local ring at the intersection point is isomorphic to
1146:
796:
1847:{\displaystyle X={\text{Proj}}\left({\frac {\mathbb {C} }{I_{C}\cdot I_{x}}}\right)}
3847:
3804:
3763:
3681:
3640:
3568:
3488:
3318:
3146:
3134:
3098:
2911:
2686:. There is also a structure theorem for Cohen–Macaulay rings of codimension 2, the
2568:
2287:
1906:
36:
3296:
3929:
3895:
3865:
3822:
3781:
3736:
3722:
3699:
3658:
3636:
3594:
3543:
3504:
3158:
2731:
2670:). In geometric terms, this holds for a local ring of a subscheme of codimension
2396:
2369:
2285:
There is a remarkable characterization of Cohen–Macaulay rings, sometimes called
1266:
949:
892:
878:
731:
322:
134:
106:
3751:
3624:
3474:
3150:
2882:
2434:
2384:
860:
708:. This leads to various examples of Cohen–Macaulay rings, such as the integers
75:
59:: they form a very broad class, and yet they are well understood in many ways.
44:
3851:
3644:
1545:{\displaystyle X={\text{Spec}}\left({\frac {\mathbb {C} }{(x^{2},xy)}}\right)}
3987:
3808:
3767:
3582:
3332:
3170:
2372:
1980:
1942:, that is, an irreducible component of expected dimension. If the local ring
1111:
867:
522:
21:
3272:
3946:
874:
3951:
3685:
3963:
3554:
3162:
2354:
2342:
2318:
1619:-axis with an embedded point at the origin, which can be thought of as a
1103:=0, is a 1-dimensional domain which is Cohen–Macaulay but not Gorenstein.
885:
131:
28:
89:
For
Noetherian local rings, there is the following chain of inclusions.
3968:
3590:
3500:
3323:
137:
52:
17:
3169:
scheme is
Gorenstein. Thus the statements of duality theorems such as
2429:
Finally, there is a version of
Miracle Flatness for graded rings. Let
3557:(1946), "On the structure and ideal theory of complete local rings",
3573:
3492:
3157:, is represented by a single sheaf. The stronger property of being
2096:
3916:
3109:
is a Cohen–Macaulay local ring of dimension at least 2, then Spec
3906:
Schwede, Karl; Tucker, Kevin (2012), "A survey of test ideals",
863:, for example a smooth variety over a field, is Cohen–Macaulay.
3880:, Cambridge Studies in Advanced Mathematics (2nd ed.),
3133:
One meaning of the Cohen–Macaulay condition can be seen in
700:
Noetherian rings of the following types are Cohen–Macaulay.
3721:. 3. Folge., vol. 2 (2nd ed.), Berlin, New York:
3528:, Cambridge Studies in Advanced Mathematics, vol. 39,
2280:
2498:{\displaystyle R=K\oplus R_{1}\oplus R_{2}\oplus \cdots .}
2068:{\displaystyle i(Z,V\cdot W,X)=\operatorname {length} (A)}
86:. All Cohen–Macaulay rings have the unmixedness property.
3629:
Commutative
Algebra with a View toward Algebraic Geometry
2608:
is a quotient of a Cohen–Macaulay ring, then the locus {
306:{\displaystyle \mathrm {depth} (M)\leq \mathrm {dim} (M)}
3249:"Compactifying Locally Cohen–Macaulay Projective Curves"
1110:
over a field of characteristic zero are Cohen–Macaulay.
580:
Cohen-Macaulay modules for these rings, we require that
2095:
For a simple example, if we take the intersection of a
2644:) be a Noetherian local ring of embedding codimension
2821:
2749:
2450:
2108:
2016:
1952:
1915:
1863:
1767:
1744:
1717:
1691:
1664:
1629:
1605:
1561:
1469:
1435:
1404:
1359:
1333:
1313:
1261:. It follows, for example, that the affine cone Spec
1165:
911:
804:
739:
714:
678:
644:
615:
586:
530:
498:
466:
403:
375:
351:
331:
253:
240:{\displaystyle \mathrm {depth} (M)=\mathrm {dim} (M)}
187:
157:
3141:
is Cohen–Macaulay if the "dualizing complex", which
3015:, with degree 1 over points of the affine line Spec
1885:
Cohen–Macaulay schemes have a special relation with
1223:{\displaystyle R=\bigoplus _{j\geq 0}H^{0}(X,L^{j})}
3297:"Curve singularities of finite Cohen–Macaulay type"
2690:: they are all determinantal rings, defined by the
2556:is a Cohen–Macaulay ring, then the polynomial ring
450:{\displaystyle \mathrm {dim} (M)=\mathrm {dim} (R)}
2862:
2790:
2497:
2266:
2067:
1971:
1934:
1869:
1846:
1750:
1730:
1703:
1677:
1650:
1611:
1591:
1544:
1444:
1421:
1382:
1345:
1319:
1222:
1033:is Cohen−Macaulay. Similarly, coordinate rings of
924:
851:
780:
722:
684:
652:
630:
601:
565:{\displaystyle \mathrm {m} \in \mathrm {Supp} (M)}
564:
513:
472:
449:
381:
357:
337:
325:of a certain kind of modules). On the other hand,
305:
239:
169:
3560:Transactions of the American Mathematical Society
3219:Kollár & Mori (1998), Theorems 5.20 and 5.22.
2885:for rings that generalize this characterization.)
2863:{\displaystyle \operatorname {length} (R/Q)=e(Q)}
2791:{\displaystyle \operatorname {length} (R/Q)=e(Q)}
393:Cohen-Macaulay module is a Cohen-Macaulay module
3985:
3719:Ergebnisse der Mathematik und ihrer Grenzgebiete
2574:in the maximal ideal of a Noetherian local ring
1126:; again, such singularities are Cohen–Macaulay.
2966:(a ring in which the unmixed theorem holds for
2516:(with generators in various degrees) such that
2418:is Cohen–Macaulay if and only if all fibers of
2625:is Cohen–Macaulay } is an open subset of Spec
3910:, Berlin: Walter de Gruyter, pp. 39–99,
3905:
2593:The quotient of a Cohen–Macaulay ring by any
2364:A geometric reformulation is as follows. Let
2313:. Such a subring exists for any localization
866:Any 0-dimensional ring (or equivalently, any
3605:
2508:There is always a graded polynomial subring
936:is a Cohen–Macaulay algebra over a field of
3947:Examples of Cohen-Macaulay integral domains
3523:
2349:-module; it is also equivalent to say that
3797:Singularities of the Minimal Model Program
3756:Birational Geometry of Algebraic Varieties
3746:
3228:Schwede & Tucker (2012), Appendix C.1.
2889:
2333:is complete and contains a field, or when
480:is a commutative Noetherian ring, then an
82:), who proved the unmixedness theorem for
3915:
3875:
3572:
3322:
2682:is Cohen–Macaulay if and only if it is a
2305:as a module over some regular local ring
2230:
2190:
2165:
2113:
1901:closed subschemes of pure dimension. Let
1787:
1651:{\displaystyle C\subset \mathbb {P} ^{2}}
1638:
1489:
1299:
716:
3832:
3623:
3524:Bruns, Winfried; Herzog, Jürgen (1993),
3273:"Lemma 31.4.4 (0BXG)—The Stacks project"
2281:Miracle flatness or Hironaka's criterion
1623:. Given a smooth projective plane curve
1555:has the decomposition into prime ideals
1393:
1005:is equal to the "expected" codimension (
963:be the quotient of a regular local ring
944:is a finite group (or more generally, a
672:if it is a Cohen-Macaulay module (as an
67:
3839:The Algebraic Theory of Modular Systems
3294:
3128:
2395:. By Noether normalization, there is a
2341:is Cohen–Macaulay if and only if it is
1114:over any field are Cohen–Macaulay. The
1099:, or its localization or completion at
369:if it is a Cohen-Macaulay module as an
3986:
3979:Wiles's proof of Fermat's Last Theorem
3791:
3709:
3668:
3353:
3023:≠ 0, but with degree 2 over the point
1880:
1118:makes prominent use of varieties with
3553:
2925:. (This is stronger than saying that
1909:of the scheme-theoretic intersection
1857:is a curve with an embedded point at
1233:is Cohen–Macaulay if and only if the
959:Any determinantal ring. That is, let
79:
3473:
3419:Matsumura (1989), Theorem 17.3.(ii).
3113:minus its closed point is connected.
2433:be a finitely generated commutative
1383:{\displaystyle {\mathcal {O}}_{X,x}}
3635:, vol. 150, Berlin, New York:
3192:Bruns & Herzog, from def. 2.1.1
3007:is finite over the polynomial ring
1327:is Cohen–Macaulay if at each point
521:is a Cohen-Macaulay module for all
13:
3477:(1964), "On unmixedness theorem",
3246:
2973:
1422:{\displaystyle {\mathcal {M}}_{g}}
1408:
1363:
1280:has dimension at least 2 (because
646:
622:
593:
549:
546:
543:
540:
532:
505:
434:
431:
428:
411:
408:
405:
345:is a module on itself, so we call
290:
287:
284:
267:
264:
261:
258:
255:
224:
221:
218:
201:
198:
195:
192:
189:
14:
4010:
3940:
3908:Progress in Commutative Algebra 2
3374:Eisenbud (1995), Corollary 18.17.
3365:Bruns & Herzog, Theorem A.22.
2721:), the following are equivalent:
2582:is Cohen–Macaulay if and only if
2528:is Cohen–Macaulay if and only if
1304:We say that a locally Noetherian
3952:Examples of Cohen-Macaulay rings
3446:Matsumura (1989), Theorem 17.11.
3437:Matsumura (1989), Exercise 24.2.
3383:Eisenbud (1995), Exercise 18.17.
3295:Wiegand, Roger (December 1991),
2910:is equal to the height of every
877:, for example any 1-dimensional
631:{\displaystyle R_{\mathrm {m} }}
602:{\displaystyle M_{\mathrm {m} }}
514:{\displaystyle M_{\mathrm {m} }}
3674:Introduction to Toric Varieties
3480:American Journal of Mathematics
3467:
3464:Eisenbud (1995), Theorem 18.12.
3458:
3455:Matsumura (1989), Theorem 17.6.
3449:
3440:
3431:
3428:Matsumura (1989), Theorem 17.9.
3422:
3413:
3404:
3401:Matsumura (1989), Theorem 17.7.
3395:
3392:Matsumura (1989), Theorem 17.5.
3386:
3377:
3368:
3359:
3347:
3338:
3261:from the original on 5 Mar 2020
3201:Eisenbud (1995), Theorem 18.18.
2879:Generalized Cohen–Macaulay ring
2080:
1711:and multiply it with the ideal
1455:
1294:Generalized Cohen–Macaulay ring
638:-module for each maximal ideal
3288:
3265:
3240:
3231:
3222:
3213:
3204:
3195:
3186:
2857:
2851:
2842:
2828:
2785:
2779:
2770:
2756:
2258:
2245:
2240:
2234:
2217:
2211:
2206:
2194:
2181:
2169:
2153:
2134:
2129:
2117:
2062:
2056:
2044:
2020:
1809:
1791:
1592:{\displaystyle (x)\cdot (x,y)}
1586:
1574:
1568:
1562:
1532:
1510:
1505:
1493:
1276:has dimension 1, but not when
1217:
1198:
846:
843:
811:
808:
775:
743:
559:
553:
444:
438:
421:
415:
300:
294:
277:
271:
234:
228:
211:
205:
74:for polynomial rings, and for
1:
3633:Graduate Texts in Mathematics
3517:
2944:is said to hold for the ring
2713:For a Noetherian local ring (
2543:
1972:{\displaystyle V\times _{X}W}
1935:{\displaystyle V\times _{X}W}
859:. In geometric terms, every
125:
3876:Matsumura, Hideyuki (1989),
3137:theory. A variety or scheme
3125:need not be Cohen-Macaulay.
2968:integral closure of an ideal
2520:is finitely generated as an
1987:is Cohen-Macaulay, then the
948:whose identity component is
723:{\displaystyle \mathbb {Z} }
653:{\displaystyle \mathrm {m} }
64:Francis Sowerby Macaulay
51:. Under mild assumptions, a
7:
3957:
3613:Encyclopedia of Mathematics
3161:means that this sheaf is a
3066:gives the previous example.
2906:in height if the height of
2800:Hilbert–Samuel multiplicity
2337:is a complete domain. Then
2327:Noether normalization lemma
1153:. Then the section ring of
695:
315:Auslander–Buchsbaum formula
113:complete intersection rings
10:
4015:
3882:Cambridge University Press
3844:Cambridge University Press
3801:Cambridge University Press
3760:Cambridge University Press
3678:Princeton University Press
3530:Cambridge University Press
3175:Grothendieck local duality
3073:is a field, then the ring
3046:is a field, then the ring
2983:is a field, then the ring
2560:and the power series ring
2323:finitely generated algebra
2090:
2085:Multiplicity one criterion
2003:is given as the length of
1599:. Geometrically it is the
1141:≥ 1 over a field, and let
897:complete intersection ring
95:Universally catenary rings
15:
3645:10.1007/978-1-4612-5350-1
2811:For some parameter ideal
2737:(an ideal generated by a
2674:in a regular scheme. For
2301:be a local ring which is
1989:intersection multiplicity
1292:) is not zero). See also
1064:, or its localization or
997:. If the codimension (or
317:for the relation between
84:formal power series rings
3809:10.1017/CBO9781139547895
3768:10.1017/CBO9780511662560
3277:stacks.math.columbia.edu
3180:
1893:be a smooth variety and
1124:F-rational singularities
954:Hochster–Roberts theorem
664:. As in the local case,
16:Not to be confused with
3878:Commutative Ring Theory
3852:10.3792/chmm/1263317740
3165:. In particular, every
2890:The unmixedness theorem
1272:is Cohen–Macaulay when
1095:of the polynomial ring
1060:of the polynomial ring
1035:determinantal varieties
170:{\displaystyle M\neq 0}
3974:Gorenstein local rings
3606:V.I. Danilov (2001) ,
3356:, Proposition 8.2. (b)
2864:
2792:
2499:
2329:; it also exists when
2268:
2069:
1973:
1936:
1871:
1848:
1752:
1732:
1705:
1704:{\displaystyle x\in C}
1679:
1652:
1613:
1593:
1546:
1446:
1445:{\displaystyle \leq 1}
1423:
1384:
1347:
1346:{\displaystyle x\in X}
1321:
1300:Cohen–Macaulay schemes
1249:) is zero for all 1 ≤
1224:
1108:Rational singularities
946:linear algebraic group
926:
895:. In particular, any
853:
782:
724:
686:
654:
632:
603:
566:
515:
474:
451:
383:
359:
339:
307:
241:
171:
3686:10.1515/9781400882526
3608:"Cohen–Macaulay ring"
3237:Kollár (2013), (3.4).
3210:Fulton (1993), p. 89.
3103:connectedness theorem
2898:of a Noetherian ring
2865:
2793:
2688:Hilbert–Burch theorem
2564:] are Cohen–Macaulay.
2500:
2325:over a field, by the
2269:
2070:
1974:
1937:
1872:
1849:
1753:
1733:
1731:{\displaystyle I_{C}}
1706:
1680:
1678:{\displaystyle I_{x}}
1653:
1614:
1594:
1547:
1447:
1424:
1394:Cohen–Macaulay curves
1385:
1348:
1322:
1225:
1116:minimal model program
927:
925:{\displaystyle R^{G}}
854:
783:
725:
687:
655:
633:
604:
572:. (This is a kind of
567:
516:
490:Cohen–Macaulay module
475:
452:
384:
360:
340:
308:
247:(in general we have:
242:
179:Cohen-Macaulay module
172:
3526:Cohen–Macaulay Rings
3129:Grothendieck duality
3123:Cohen-Macaulay rings
2957:equidimensional ring
2819:
2747:
2739:system of parameters
2599:universally catenary
2590:) is Cohen–Macaulay.
2532:is free as a graded
2448:
2391:be the dimension of
2294:Hironaka's criterion
2106:
2014:
1950:
1913:
1861:
1765:
1742:
1715:
1689:
1662:
1627:
1603:
1559:
1467:
1433:
1402:
1357:
1331:
1311:
1257:−1 and all integers
1163:
1041:Some more examples:
909:
802:
737:
712:
692:-module on itself).
676:
642:
613:
584:
528:
496:
464:
401:
373:
349:
329:
251:
185:
155:
101:Cohen–Macaulay rings
3999:Commutative algebra
3926:2011arXiv1104.2000S
3715:Intersection theory
3315:1991ArM....29..339W
3302:Arkiv för Matematik
3039:) has dimension 2).
2942:unmixedness theorem
1887:intersection theory
1881:Intersection theory
1390:is Cohen–Macaulay.
1037:are Cohen-Macaulay.
670:Cohen-Macaulay ring
574:circular definition
367:Cohen-Macaulay ring
119:regular local rings
72:unmixedness theorem
62:They are named for
57:commutative algebra
33:Cohen–Macaulay ring
3994:Algebraic geometry
3324:10.1007/BF02384346
2964:quasi-unmixed ring
2860:
2788:
2727:is Cohen–Macaulay.
2495:
2303:finitely generated
2264:
2065:
1969:
1932:
1867:
1844:
1748:
1728:
1701:
1675:
1648:
1609:
1589:
1542:
1442:
1419:
1380:
1343:
1317:
1220:
1187:
1135:projective variety
1027:determinantal ring
922:
904:ring of invariants
884:Any 2-dimensional
873:Any 1-dimensional
852:{\displaystyle K]}
849:
778:
720:
706:regular local ring
682:
650:
628:
599:
562:
511:
470:
447:
379:
355:
335:
303:
237:
167:
145:finitely generated
70:), who proved the
49:equidimensionality
3891:978-0-521-36764-6
3818:978-1-107-03534-8
3732:978-3-540-62046-4
3695:978-0-691-00049-7
3654:978-0-387-94268-1
3539:978-0-521-41068-7
3027:= 0 (because the
2684:hypersurface ring
2383:(for example, an
2262:
2221:
2157:
1889:. Precisely, let
1870:{\displaystyle x}
1838:
1777:
1751:{\displaystyle C}
1612:{\displaystyle y}
1536:
1479:
1320:{\displaystyle X}
1172:
1147:ample line bundle
971:generated by the
797:power series ring
781:{\displaystyle K}
685:{\displaystyle R}
473:{\displaystyle R}
382:{\displaystyle R}
358:{\displaystyle R}
338:{\displaystyle R}
143:, a finite (i.e.
41:algebro-geometric
39:with some of the
4006:
3936:
3919:
3902:
3872:
3829:
3788:
3743:
3706:
3665:
3620:
3601:
3576:
3550:
3512:
3511:
3471:
3465:
3462:
3456:
3453:
3447:
3444:
3438:
3435:
3429:
3426:
3420:
3417:
3411:
3408:
3402:
3399:
3393:
3390:
3384:
3381:
3375:
3372:
3366:
3363:
3357:
3351:
3345:
3342:
3336:
3335:
3326:
3309:(1–2): 339–357,
3292:
3286:
3285:
3284:
3283:
3269:
3263:
3262:
3260:
3253:
3247:Honsen, Morten,
3244:
3238:
3235:
3229:
3226:
3220:
3217:
3211:
3208:
3202:
3199:
3193:
3190:
3147:derived category
3135:coherent duality
3001:Miracle Flatness
2912:associated prime
2869:
2867:
2866:
2861:
2838:
2797:
2795:
2794:
2789:
2766:
2706:matrix for some
2569:non-zero-divisor
2504:
2502:
2501:
2496:
2485:
2484:
2472:
2471:
2406:to affine space
2288:miracle flatness
2273:
2271:
2270:
2265:
2263:
2261:
2257:
2256:
2243:
2233:
2227:
2222:
2220:
2209:
2193:
2187:
2185:
2184:
2168:
2158:
2156:
2152:
2151:
2132:
2116:
2110:
2074:
2072:
2071:
2066:
1978:
1976:
1975:
1970:
1965:
1964:
1941:
1939:
1938:
1933:
1928:
1927:
1907:proper component
1876:
1874:
1873:
1868:
1853:
1851:
1850:
1845:
1843:
1839:
1837:
1836:
1835:
1823:
1822:
1812:
1790:
1784:
1778:
1775:
1757:
1755:
1754:
1749:
1737:
1735:
1734:
1729:
1727:
1726:
1710:
1708:
1707:
1702:
1684:
1682:
1681:
1676:
1674:
1673:
1657:
1655:
1654:
1649:
1647:
1646:
1641:
1618:
1616:
1615:
1610:
1598:
1596:
1595:
1590:
1551:
1549:
1548:
1543:
1541:
1537:
1535:
1522:
1521:
1508:
1492:
1486:
1480:
1477:
1451:
1449:
1448:
1443:
1428:
1426:
1425:
1420:
1418:
1417:
1412:
1411:
1389:
1387:
1386:
1381:
1379:
1378:
1367:
1366:
1352:
1350:
1349:
1344:
1326:
1324:
1323:
1318:
1229:
1227:
1226:
1221:
1216:
1215:
1197:
1196:
1186:
1029:. In that case,
931:
929:
928:
923:
921:
920:
858:
856:
855:
850:
842:
841:
823:
822:
787:
785:
784:
779:
774:
773:
755:
754:
729:
727:
726:
721:
719:
691:
689:
688:
683:
659:
657:
656:
651:
649:
637:
635:
634:
629:
627:
626:
625:
608:
606:
605:
600:
598:
597:
596:
571:
569:
568:
563:
552:
535:
520:
518:
517:
512:
510:
509:
508:
479:
477:
476:
471:
456:
454:
453:
448:
437:
414:
388:
386:
385:
380:
364:
362:
361:
356:
344:
342:
341:
336:
312:
310:
309:
304:
293:
270:
246:
244:
243:
238:
227:
204:
176:
174:
173:
168:
107:Gorenstein rings
47:, such as local
43:properties of a
37:commutative ring
4014:
4013:
4009:
4008:
4007:
4005:
4004:
4003:
3984:
3983:
3960:
3943:
3892:
3862:
3819:
3778:
3752:Mori, Shigefumi
3733:
3723:Springer-Verlag
3711:Fulton, William
3696:
3670:Fulton, William
3655:
3637:Springer-Verlag
3625:Eisenbud, David
3574:10.2307/1990313
3540:
3520:
3515:
3493:10.2307/2373158
3475:Chow, Wei Liang
3472:
3468:
3463:
3459:
3454:
3450:
3445:
3441:
3436:
3432:
3427:
3423:
3418:
3414:
3409:
3405:
3400:
3396:
3391:
3387:
3382:
3378:
3373:
3369:
3364:
3360:
3352:
3348:
3343:
3339:
3293:
3289:
3281:
3279:
3271:
3270:
3266:
3258:
3251:
3245:
3241:
3236:
3232:
3227:
3223:
3218:
3214:
3209:
3205:
3200:
3196:
3191:
3187:
3183:
3131:
2976:
2974:Counterexamples
2948:if every ideal
2935:equidimensional
2892:
2834:
2820:
2817:
2816:
2762:
2748:
2745:
2744:
2732:parameter ideal
2657:
2648:, meaning that
2624:
2546:
2480:
2476:
2467:
2463:
2449:
2446:
2445:
2397:finite morphism
2283:
2252:
2248:
2244:
2229:
2228:
2226:
2210:
2189:
2188:
2186:
2164:
2163:
2159:
2147:
2143:
2133:
2112:
2111:
2109:
2107:
2104:
2103:
2093:
2015:
2012:
2011:
1960:
1956:
1951:
1948:
1947:
1923:
1919:
1914:
1911:
1910:
1883:
1862:
1859:
1858:
1831:
1827:
1818:
1814:
1813:
1786:
1785:
1783:
1779:
1774:
1766:
1763:
1762:
1743:
1740:
1739:
1722:
1718:
1716:
1713:
1712:
1690:
1687:
1686:
1669:
1665:
1663:
1660:
1659:
1642:
1637:
1636:
1628:
1625:
1624:
1604:
1601:
1600:
1560:
1557:
1556:
1517:
1513:
1509:
1488:
1487:
1485:
1481:
1476:
1468:
1465:
1464:
1458:
1434:
1431:
1430:
1413:
1407:
1406:
1405:
1403:
1400:
1399:
1396:
1368:
1362:
1361:
1360:
1358:
1355:
1354:
1353:the local ring
1332:
1329:
1328:
1312:
1309:
1308:
1302:
1267:abelian variety
1211:
1207:
1192:
1188:
1176:
1164:
1161:
1160:
1112:Toric varieties
993:of elements of
952:). This is the
916:
912:
910:
907:
906:
893:Gorenstein ring
837:
833:
818:
814:
803:
800:
799:
769:
765:
750:
746:
738:
735:
734:
732:polynomial ring
715:
713:
710:
709:
698:
677:
674:
673:
645:
643:
640:
639:
621:
620:
616:
614:
611:
610:
592:
591:
587:
585:
582:
581:
539:
531:
529:
526:
525:
504:
503:
499:
497:
494:
493:
465:
462:
461:
427:
404:
402:
399:
398:
374:
371:
370:
350:
347:
346:
330:
327:
326:
283:
254:
252:
249:
248:
217:
188:
186:
183:
182:
156:
153:
152:
128:
76:Irvin Cohen
25:
12:
11:
5:
4012:
4002:
4001:
3996:
3982:
3981:
3976:
3971:
3966:
3959:
3956:
3955:
3954:
3949:
3942:
3941:External links
3939:
3938:
3937:
3903:
3890:
3873:
3860:
3834:Macaulay, F.S.
3830:
3817:
3789:
3776:
3744:
3731:
3707:
3694:
3666:
3653:
3621:
3603:
3551:
3538:
3519:
3516:
3514:
3513:
3466:
3457:
3448:
3439:
3430:
3421:
3412:
3403:
3394:
3385:
3376:
3367:
3358:
3346:
3337:
3287:
3264:
3239:
3230:
3221:
3212:
3203:
3194:
3184:
3182:
3179:
3130:
3127:
3115:
3114:
3067:
3040:
3031:-vector space
2975:
2972:
2937:; see below.)
2891:
2888:
2887:
2886:
2883:Buchsbaum ring
2874:
2873:
2872:
2871:
2859:
2856:
2853:
2850:
2847:
2844:
2841:
2837:
2833:
2830:
2827:
2824:
2809:
2808:
2807:
2787:
2784:
2781:
2778:
2775:
2772:
2769:
2765:
2761:
2758:
2755:
2752:
2728:
2711:
2698:minors of an (
2653:
2630:
2620:
2602:
2591:
2565:
2550:
2545:
2542:
2524:-module. Then
2506:
2505:
2494:
2491:
2488:
2483:
2479:
2475:
2470:
2466:
2462:
2459:
2456:
2453:
2435:graded algebra
2385:affine variety
2282:
2279:
2275:
2274:
2260:
2255:
2251:
2247:
2242:
2239:
2236:
2232:
2225:
2219:
2216:
2213:
2208:
2205:
2202:
2199:
2196:
2192:
2183:
2180:
2177:
2174:
2171:
2167:
2162:
2155:
2150:
2146:
2142:
2139:
2136:
2131:
2128:
2125:
2122:
2119:
2115:
2092:
2089:
2077:
2076:
2064:
2061:
2058:
2055:
2052:
2049:
2046:
2043:
2040:
2037:
2034:
2031:
2028:
2025:
2022:
2019:
1968:
1963:
1959:
1955:
1931:
1926:
1922:
1918:
1882:
1879:
1866:
1855:
1854:
1842:
1834:
1830:
1826:
1821:
1817:
1811:
1808:
1805:
1802:
1799:
1796:
1793:
1789:
1782:
1773:
1770:
1747:
1725:
1721:
1700:
1697:
1694:
1685:of a point in
1672:
1668:
1645:
1640:
1635:
1632:
1608:
1588:
1585:
1582:
1579:
1576:
1573:
1570:
1567:
1564:
1553:
1552:
1540:
1534:
1531:
1528:
1525:
1520:
1516:
1512:
1507:
1504:
1501:
1498:
1495:
1491:
1484:
1475:
1472:
1457:
1454:
1441:
1438:
1416:
1410:
1395:
1392:
1377:
1374:
1371:
1365:
1342:
1339:
1336:
1316:
1301:
1298:
1231:
1230:
1219:
1214:
1210:
1206:
1203:
1200:
1195:
1191:
1185:
1182:
1179:
1175:
1171:
1168:
1105:
1104:
1089:
1054:
1039:
1038:
957:
938:characteristic
919:
915:
900:
889:
882:
871:
864:
861:regular scheme
848:
845:
840:
836:
832:
829:
826:
821:
817:
813:
810:
807:
777:
772:
768:
764:
761:
758:
753:
749:
745:
742:
718:
697:
694:
681:
648:
624:
619:
609:to be such an
595:
590:
561:
558:
555:
551:
548:
545:
542:
538:
534:
523:maximal ideals
507:
502:
469:
446:
443:
440:
436:
433:
430:
426:
423:
420:
417:
413:
410:
407:
378:
354:
334:
302:
299:
296:
292:
289:
286:
282:
279:
276:
273:
269:
266:
263:
260:
257:
236:
233:
230:
226:
223:
220:
216:
213:
210:
207:
203:
200:
197:
194:
191:
166:
163:
160:
127:
124:
123:
122:
45:smooth variety
9:
6:
4:
3:
2:
4011:
4000:
3997:
3995:
3992:
3991:
3989:
3980:
3977:
3975:
3972:
3970:
3967:
3965:
3962:
3961:
3953:
3950:
3948:
3945:
3944:
3935:
3931:
3927:
3923:
3918:
3913:
3909:
3904:
3901:
3897:
3893:
3887:
3883:
3879:
3874:
3871:
3867:
3863:
3861:1-4297-0441-1
3857:
3853:
3849:
3845:
3841:
3840:
3835:
3831:
3828:
3824:
3820:
3814:
3810:
3806:
3802:
3798:
3794:
3793:Kollár, János
3790:
3787:
3783:
3779:
3777:0-521-63277-3
3773:
3769:
3765:
3761:
3757:
3753:
3749:
3748:Kollár, János
3745:
3742:
3738:
3734:
3728:
3724:
3720:
3716:
3712:
3708:
3705:
3701:
3697:
3691:
3687:
3683:
3679:
3675:
3671:
3667:
3664:
3660:
3656:
3650:
3646:
3642:
3638:
3634:
3630:
3626:
3622:
3619:
3615:
3614:
3609:
3604:
3600:
3596:
3592:
3588:
3584:
3580:
3575:
3570:
3567:(1): 54–106,
3566:
3562:
3561:
3556:
3552:
3549:
3545:
3541:
3535:
3531:
3527:
3522:
3521:
3510:
3506:
3502:
3498:
3494:
3490:
3486:
3482:
3481:
3476:
3470:
3461:
3452:
3443:
3434:
3425:
3416:
3407:
3398:
3389:
3380:
3371:
3362:
3355:
3350:
3341:
3334:
3330:
3325:
3320:
3316:
3312:
3308:
3304:
3303:
3298:
3291:
3278:
3274:
3268:
3257:
3250:
3243:
3234:
3225:
3216:
3207:
3198:
3189:
3185:
3178:
3176:
3172:
3171:Serre duality
3168:
3164:
3160:
3156:
3152:
3148:
3144:
3140:
3136:
3126:
3124:
3120:
3119:Segre product
3112:
3108:
3104:
3100:
3096:
3092:
3088:
3084:
3080:
3076:
3072:
3068:
3065:
3061:
3057:
3053:
3049:
3045:
3041:
3038:
3034:
3030:
3026:
3022:
3018:
3014:
3010:
3006:
3002:
2998:
2994:
2990:
2986:
2982:
2978:
2977:
2971:
2969:
2965:
2960:
2958:
2953:
2951:
2947:
2943:
2938:
2936:
2932:
2928:
2924:
2920:
2916:
2913:
2909:
2905:
2901:
2897:
2884:
2880:
2876:
2875:
2854:
2848:
2845:
2839:
2835:
2831:
2825:
2822:
2814:
2810:
2805:
2801:
2798: := the
2782:
2776:
2773:
2767:
2763:
2759:
2753:
2750:
2743:
2742:
2740:
2736:
2733:
2729:
2726:
2723:
2722:
2720:
2716:
2712:
2709:
2705:
2701:
2697:
2693:
2689:
2685:
2681:
2677:
2673:
2669:
2665:
2661:
2656:
2651:
2647:
2643:
2639:
2635:
2631:
2628:
2623:
2619:
2615:
2611:
2607:
2603:
2600:
2596:
2592:
2589:
2585:
2581:
2577:
2573:
2570:
2566:
2563:
2559:
2555:
2551:
2548:
2547:
2541:
2539:
2535:
2531:
2527:
2523:
2519:
2515:
2511:
2492:
2489:
2486:
2481:
2477:
2473:
2468:
2464:
2460:
2457:
2454:
2451:
2444:
2443:
2442:
2440:
2437:over a field
2436:
2432:
2427:
2425:
2421:
2417:
2413:
2409:
2405:
2401:
2398:
2394:
2390:
2386:
2382:
2379:over a field
2378:
2374:
2373:affine scheme
2371:
2367:
2362:
2360:
2356:
2352:
2348:
2344:
2340:
2336:
2332:
2328:
2324:
2320:
2316:
2312:
2309:contained in
2308:
2304:
2300:
2296:
2295:
2290:
2289:
2278:
2253:
2249:
2237:
2223:
2214:
2203:
2200:
2197:
2178:
2175:
2172:
2160:
2148:
2144:
2140:
2137:
2126:
2123:
2120:
2102:
2101:
2100:
2098:
2088:
2086:
2082:
2059:
2053:
2050:
2047:
2041:
2038:
2035:
2032:
2029:
2026:
2023:
2017:
2010:
2009:
2008:
2006:
2002:
1998:
1994:
1990:
1986:
1982:
1981:generic point
1966:
1961:
1957:
1953:
1945:
1929:
1924:
1920:
1916:
1908:
1904:
1900:
1896:
1892:
1888:
1878:
1864:
1840:
1832:
1828:
1824:
1819:
1815:
1806:
1803:
1800:
1797:
1794:
1780:
1771:
1768:
1761:
1760:
1759:
1745:
1723:
1719:
1698:
1695:
1692:
1670:
1666:
1643:
1633:
1630:
1622:
1606:
1583:
1580:
1577:
1571:
1565:
1538:
1529:
1526:
1523:
1518:
1514:
1502:
1499:
1496:
1482:
1473:
1470:
1463:
1462:
1461:
1453:
1439:
1436:
1414:
1391:
1375:
1372:
1369:
1340:
1337:
1334:
1314:
1307:
1297:
1295:
1291:
1287:
1283:
1279:
1275:
1271:
1268:
1264:
1260:
1256:
1252:
1248:
1244:
1240:
1236:
1212:
1208:
1204:
1201:
1193:
1189:
1183:
1180:
1177:
1173:
1169:
1166:
1159:
1158:
1157:
1156:
1152:
1148:
1144:
1140:
1137:of dimension
1136:
1132:
1127:
1125:
1121:
1117:
1113:
1109:
1102:
1098:
1094:
1090:
1087:
1083:
1079:
1075:
1071:
1067:
1063:
1059:
1055:
1052:
1048:
1044:
1043:
1042:
1036:
1032:
1028:
1024:
1020:
1016:
1012:
1008:
1004:
1000:
996:
992:
989:
985:
981:
978:
974:
970:
967:by the ideal
966:
962:
958:
955:
951:
947:
943:
939:
935:
917:
913:
905:
901:
898:
894:
890:
887:
883:
880:
876:
872:
869:
868:Artinian ring
865:
862:
838:
834:
830:
827:
824:
819:
815:
805:
798:
794:
791:
770:
766:
762:
759:
756:
751:
747:
740:
733:
707:
703:
702:
701:
693:
679:
671:
667:
663:
617:
588:
579:
575:
556:
536:
524:
500:
491:
487:
483:
467:
458:
441:
424:
418:
396:
392:
376:
368:
352:
332:
324:
320:
316:
297:
280:
274:
231:
214:
208:
180:
164:
161:
158:
150:
146:
142:
139:
136:
133:
121:
120:
115:
114:
109:
108:
103:
102:
97:
96:
92:
91:
90:
87:
85:
81:
77:
73:
69:
65:
60:
58:
54:
50:
46:
42:
38:
34:
30:
23:
22:Cohen algebra
19:
3907:
3877:
3838:
3796:
3755:
3714:
3673:
3628:
3611:
3564:
3558:
3555:Cohen, I. S.
3525:
3484:
3478:
3469:
3460:
3451:
3442:
3433:
3424:
3415:
3406:
3397:
3388:
3379:
3370:
3361:
3349:
3340:
3306:
3300:
3290:
3280:, retrieved
3276:
3267:
3242:
3233:
3224:
3215:
3206:
3197:
3188:
3154:
3145:lies in the
3142:
3138:
3132:
3116:
3110:
3106:
3102:
3094:
3090:
3086:
3082:
3078:
3074:
3070:
3063:
3059:
3055:
3051:
3047:
3043:
3036:
3032:
3028:
3024:
3020:
3016:
3012:
3008:
3004:
2996:
2992:
2988:
2984:
2980:
2961:
2954:
2949:
2945:
2941:
2939:
2930:
2926:
2922:
2918:
2914:
2907:
2903:
2899:
2895:
2893:
2812:
2803:
2734:
2724:
2718:
2714:
2707:
2703:
2699:
2695:
2691:
2679:
2675:
2671:
2667:
2663:
2659:
2654:
2649:
2645:
2641:
2637:
2633:
2626:
2621:
2617:
2613:
2609:
2605:
2587:
2583:
2579:
2575:
2571:
2561:
2557:
2553:
2537:
2533:
2529:
2525:
2521:
2517:
2513:
2509:
2507:
2438:
2430:
2428:
2423:
2419:
2415:
2411:
2407:
2403:
2399:
2392:
2388:
2380:
2365:
2363:
2358:
2350:
2346:
2338:
2334:
2330:
2314:
2310:
2306:
2298:
2292:
2286:
2284:
2276:
2094:
2078:
2004:
2000:
1996:
1992:
1984:
1943:
1902:
1898:
1894:
1890:
1884:
1856:
1620:
1554:
1459:
1456:Non-examples
1397:
1303:
1289:
1285:
1281:
1277:
1273:
1269:
1262:
1258:
1254:
1250:
1246:
1242:
1238:
1232:
1154:
1150:
1142:
1138:
1130:
1128:
1123:
1106:
1100:
1096:
1092:
1091:The subring
1085:
1081:
1077:
1076:cubic curve
1069:
1061:
1057:
1056:The subring
1050:
1046:
1040:
1030:
1026:
1025:is called a
1022:
1018:
1014:
1010:
1006:
1002:
994:
987:
983:
976:
972:
968:
964:
960:
941:
933:
875:reduced ring
792:
699:
669:
665:
661:
577:
489:
485:
481:
459:
394:
390:
366:
178:
148:
140:
129:
117:
111:
105:
100:
99:
93:
88:
61:
32:
26:
3969:Local rings
3964:Ring theory
3487:: 799–822,
3354:Fulton 1998
3163:line bundle
2881:as well as
2377:finite type
2319:prime ideal
2081:#Properties
886:normal ring
389:-module. A
132:commutative
29:mathematics
3988:Categories
3518:References
3282:2020-03-05
3159:Gorenstein
3099:Hartshorne
2962:See also:
2902:is called
2730:For every
2544:Properties
1235:cohomology
1066:completion
488:is called
397:such that
138:local ring
135:Noetherian
126:Definition
53:local ring
18:Cohen ring
3917:1104.2000
3618:EMS Press
3583:0002-9947
3333:0004-2080
2894:An ideal
2826:
2754:
2490:⋯
2487:⊕
2474:⊕
2461:⊕
2370:connected
2361:-module.
2224:≅
2161:⊗
2141:−
2054:
2033:⋅
1958:×
1921:×
1825:⋅
1696:∈
1634:⊂
1621:fat point
1572:⋅
1437:≤
1338:∈
1181:≥
1174:⨁
1045:The ring
950:reductive
940:zero and
828:…
760:…
537:∈
281:≤
162:≠
3958:See also
3836:(1916),
3795:(2013),
3754:(1998),
3713:(1998),
3672:(1993),
3627:(1995),
3256:archived
3143:a priori
2666:) − dim(
2097:parabola
1265:over an
1074:cuspidal
982:of some
696:Examples
484:-module
151:-module
3934:2932591
3922:Bibcode
3900:0879273
3870:1281612
3827:3057950
3786:1658959
3741:1644323
3704:1234037
3663:1322960
3599:0016094
3591:1990313
3548:1251956
3509:0171804
3501:2373158
3311:Bibcode
3167:regular
3151:sheaves
3121:of two
2904:unmixed
2612:∈ Spec
2414:. Then
2387:). Let
2091:Example
1979:at the
1758:. Then
795:, or a
788:over a
730:, or a
578:maximal
391:maximal
78: (
66: (
3932:
3898:
3888:
3868:
3858:
3825:
3815:
3784:
3774:
3739:
3729:
3702:
3692:
3661:
3651:
3597:
3589:
3581:
3546:
3536:
3507:
3499:
3331:
2823:length
2751:length
2702:+1) ×
2567:For a
2357:as an
2345:as an
2297:. Let
2051:length
1999:along
1306:scheme
1237:group
1145:be an
999:height
991:matrix
980:minors
879:domain
313:, see
130:For a
3912:arXiv
3587:JSTOR
3497:JSTOR
3259:(PDF)
3252:(PDF)
3181:Notes
3105:: if
3019:with
2877:(See
2652:= dim
2632:Let (
2595:ideal
2410:over
2402:from
2368:be a
2321:of a
2317:at a
1905:be a
1133:be a
1084:over
1021:+1),
1001:) of
932:when
790:field
668:is a
319:depth
177:is a
35:is a
3886:ISBN
3856:ISBN
3813:ISBN
3772:ISBN
3727:ISBN
3690:ISBN
3649:ISBN
3579:ISSN
3534:ISBN
3329:ISSN
3117:The
2940:The
2678:=1,
2355:free
2343:flat
1995:and
1776:Proj
1478:Spec
1129:Let
1013:+1)(
902:The
891:Any
704:Any
321:and
80:1946
68:1916
31:, a
3848:doi
3805:doi
3764:doi
3682:doi
3641:doi
3569:doi
3489:doi
3319:doi
3173:or
3153:on
3149:of
3101:'s
3069:If
3042:If
2979:If
2970:).
2933:is
2917:of
2802:of
2741:),
2604:If
2597:is
2552:If
2375:of
2353:is
2291:or
1991:of
1983:of
1946:of
1738:of
1149:on
1120:klt
1068:at
660:of
492:if
323:dim
181:if
98:⊃
27:In
20:or
3990::
3930:MR
3928:,
3920:,
3896:MR
3894:,
3884:,
3866:MR
3864:,
3854:,
3846:,
3842:,
3823:MR
3821:,
3811:,
3803:,
3799:,
3782:MR
3780:,
3770:,
3762:,
3758:,
3750:;
3737:MR
3735:,
3725:,
3717:,
3700:MR
3698:,
3688:,
3680:,
3676:,
3659:MR
3657:,
3647:,
3639:,
3631:,
3616:,
3610:,
3595:MR
3593:,
3585:,
3577:,
3565:59
3563:,
3544:MR
3542:,
3532:,
3505:MR
3503:,
3495:,
3485:86
3483:,
3327:,
3317:,
3307:29
3305:,
3299:,
3275:,
3254:,
3095:xz
3091:xy
3087:wz
3083:wy
3081:/(
3077:=
3056:xz
3052:xy
3050:/(
3035:/(
3011:=
3003::
2997:xy
2991:/(
2987:=
2815:,
2717:,
2694:×
2640:,
2636:,
2616:|
2586:/(
2578:,
2540:.
2512:⊂
2441:,
2426:.
2083:.
2007::
1897:,
1877:.
1296:.
1288:,
1253:≤
1245:,
1080:=
1049:/(
986:×
975:×
870:).
457:.
365:a
147:)
116:⊃
110:⊃
104:⊃
3924::
3914::
3850::
3807::
3766::
3684::
3643::
3571::
3491::
3321::
3313::
3155:X
3139:X
3111:R
3107:R
3093:,
3089:,
3085:,
3079:K
3075:R
3071:K
3064:z
3062:−
3060:x
3054:,
3048:K
3044:K
3037:x
3033:K
3029:K
3025:y
3021:y
3017:A
3013:K
3009:A
3005:R
2995:,
2993:x
2989:K
2985:R
2981:K
2950:I
2946:A
2931:I
2929:/
2927:A
2923:I
2921:/
2919:A
2915:P
2908:I
2900:A
2896:I
2870:.
2858:)
2855:Q
2852:(
2849:e
2846:=
2843:)
2840:Q
2836:/
2832:R
2829:(
2813:Q
2806:.
2804:Q
2786:)
2783:Q
2780:(
2777:e
2774:=
2771:)
2768:Q
2764:/
2760:R
2757:(
2735:Q
2725:R
2719:m
2715:R
2710:.
2708:r
2704:r
2700:r
2696:r
2692:r
2680:R
2676:c
2672:c
2668:R
2664:m
2662:/
2660:m
2658:(
2655:k
2650:c
2646:c
2642:k
2638:m
2634:R
2629:.
2627:R
2622:p
2618:R
2614:R
2610:p
2606:R
2601:.
2588:u
2584:R
2580:R
2576:R
2572:u
2562:R
2558:R
2554:R
2538:A
2534:A
2530:R
2526:R
2522:A
2518:R
2514:R
2510:A
2493:.
2482:2
2478:R
2469:1
2465:R
2458:K
2455:=
2452:R
2439:K
2431:R
2424:f
2420:f
2416:X
2412:K
2408:A
2404:X
2400:f
2393:X
2389:n
2381:K
2366:X
2359:A
2351:R
2347:A
2339:R
2335:R
2331:R
2315:R
2311:R
2307:A
2299:R
2259:)
2254:2
2250:x
2246:(
2241:]
2238:x
2235:[
2231:C
2218:)
2215:y
2212:(
2207:]
2204:y
2201:,
2198:x
2195:[
2191:C
2182:]
2179:y
2176:,
2173:x
2170:[
2166:C
2154:)
2149:2
2145:x
2138:y
2135:(
2130:]
2127:y
2124:,
2121:x
2118:[
2114:C
2075:.
2063:)
2060:A
2057:(
2048:=
2045:)
2042:X
2039:,
2036:W
2030:V
2027:,
2024:Z
2021:(
2018:i
2005:A
2001:Z
1997:W
1993:V
1985:Z
1967:W
1962:X
1954:V
1944:A
1930:W
1925:X
1917:V
1903:Z
1899:W
1895:V
1891:X
1865:x
1841:)
1833:x
1829:I
1820:C
1816:I
1810:]
1807:z
1804:,
1801:y
1798:,
1795:x
1792:[
1788:C
1781:(
1772:=
1769:X
1746:C
1724:C
1720:I
1699:C
1693:x
1671:x
1667:I
1644:2
1639:P
1631:C
1607:y
1587:)
1584:y
1581:,
1578:x
1575:(
1569:)
1566:x
1563:(
1539:)
1533:)
1530:y
1527:x
1524:,
1519:2
1515:x
1511:(
1506:]
1503:y
1500:,
1497:x
1494:[
1490:C
1483:(
1474:=
1471:X
1440:1
1415:g
1409:M
1376:x
1373:,
1370:X
1364:O
1341:X
1335:x
1315:X
1290:O
1286:X
1284:(
1282:H
1278:X
1274:X
1270:X
1263:R
1259:j
1255:n
1251:i
1247:L
1243:X
1241:(
1239:H
1218:)
1213:j
1209:L
1205:,
1202:X
1199:(
1194:0
1190:H
1184:0
1178:j
1170:=
1167:R
1155:L
1151:X
1143:L
1139:n
1131:X
1101:t
1097:K
1093:K
1088:.
1086:K
1082:x
1078:y
1070:t
1062:K
1058:K
1051:x
1047:K
1031:R
1023:R
1019:r
1017:−
1015:q
1011:r
1009:−
1007:p
1003:I
995:S
988:q
984:p
977:r
973:r
969:I
965:S
961:R
956:.
942:G
934:R
918:G
914:R
899:.
888:.
881:.
847:]
844:]
839:n
835:x
831:,
825:,
820:1
816:x
812:[
809:[
806:K
793:K
776:]
771:n
767:x
763:,
757:,
752:1
748:x
744:[
741:K
717:Z
680:R
666:R
662:R
647:m
623:m
618:R
594:m
589:M
560:)
557:M
554:(
550:p
547:p
544:u
541:S
533:m
506:m
501:M
486:M
482:R
468:R
445:)
442:R
439:(
435:m
432:i
429:d
425:=
422:)
419:M
416:(
412:m
409:i
406:d
395:M
377:R
353:R
333:R
301:)
298:M
295:(
291:m
288:i
285:d
278:)
275:M
272:(
268:h
265:t
262:p
259:e
256:d
235:)
232:M
229:(
225:m
222:i
219:d
215:=
212:)
209:M
206:(
202:h
199:t
196:p
193:e
190:d
165:0
159:M
149:R
141:R
24:.
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