3437:
3994:
3274:
2205:
725:
3837:
4196:
1455:
413:
2285:
3698:
2603:
2067:
1930:
4055:
530:
832:
3010:
2948:
3603:
As noted earlier, sets of germs may have algebraic structures such as being rings. In many situations, rings of germs are not arbitrary rings but instead have quite specific properties.
2524:
3856:
894:
1701:
Interpreting germs through sheaves also gives a general explanation for the presence of algebraic structures on sets of germs. The reason is that formation of stalks preserves finite
2843:
2781:
1541:
1502:
3141:
2742:
3067:
1668:
1577:
1634:
1287:
930:
619:
336:
2648:
1819:
1696:
1246:
1199:
1376:
3432:{\displaystyle {_{n}{\mathcal {C}}^{0}},{_{n}{\mathcal {C}}^{k}},{_{n}{\mathcal {C}}^{\infty }},{_{n}{\mathcal {C}}^{\omega }},{_{n}{\mathcal {O}}},{_{n}{\mathfrak {V}}}}
2375:
138:
257:
3267:
2313:
1098:
4498:
4451:
2348:
3507:
3235:
2467:
2418:
2105:
1853:
4471:
4424:
3477:
3457:
3205:
3185:
3165:
3103:
3034:
2972:
2910:
2888:
2866:
2805:
2708:
2688:
2668:
2091:
1987:
1967:
1780:
1760:
186:
166:
657:
3729:
353:(be it on maps or sets), and the equivalence classes are called germs (map-germs, or set-germs accordingly). The equivalence relation is usually written
4087:
1381:
3723:
would imply that a smooth function whose Taylor series vanished would be the zero function. But this is false, as can be seen by considering
3847:
3534:, and indeed the Taylor series of a germ (of a differentiable function) is defined: you only need local information to compute derivatives.
3630:-times differentiable, smooth, or analytic functions on a real manifold (when such functions are defined); for holomorphic functions on a
35:
of that object and others of the same kind that captures their shared local properties. In particular, the objects in question are mostly
3634:; and for regular functions on an algebraic variety. The property that rings of germs are local rings is axiomatized by the theory of
359:
2223:
3656:
2533:
2005:
1868:
59:); it is however necessary that the space on/in which the object is defined is a topological space, in order that the word
4541:
Mozyrska, Dorota; Bartosiewicz, Zbigniew (2006). "Systems of germs and theorems of zeros in infinite-dimensional spaces".
4013:
456:
47:. In specific implementations of this idea, the functions or subsets in question will have some property, such as being
3109:
For germs of sets and varieties, the notation is not so well established: some notations found in literature include:
2744:
As a consequence, germs, constituting stalks of sheaves of various kind of functions, borrow this scheme of notation:
4397:
4381:
4319:
4295:
790:
3989:{\displaystyle \cdots \subsetneq (x^{-j+1}f(x))\subsetneq (x^{-j}f(x))\subsetneq (x^{-j-1}f(x))\subsetneq \cdots .}
2979:
2917:
3642:
2495:
1210:
837:
4524:
3538:
2812:
2750:
1507:
1468:
633:
4519:
3843:
3115:
2713:
193:
3041:
1642:
1550:
4374:
3720:
3561:
1606:
1259:
903:
565:
300:
3641:
The types of local rings that arise, however, depend closely on the theory under consideration. The
3753:
2629:
1800:
1677:
1458:
1227:
1157:
3208:
2421:
1990:
1355:
1139:
one. It therefore makes little or no sense to talk of a convergent sequence of germs. However, if
4390:
Varietà differenziabili e coomologia di De Rham (Differentiable manifolds and De Rham cohomology)
2353:
2094:
1702:
105:
218:
36:
3240:
4575:
3588:
3573:
3565:
2298:
1342:
1077:
2200:{\displaystyle C^{\infty }(X,Y)=\bigcap \nolimits _{k}C^{k}(X,Y)\subseteq {\mbox{Hom}}(X,Y)}
4476:
4429:
3635:
3592:
2433:
2326:
1322:
1298:
350:
4514:
8:
4246:
3486:
3214:
2606:
2446:
2397:
1937:
1832:
1222:
1025:
720:{\displaystyle S\sim _{x}T\Longleftrightarrow \mathbf {1} _{S}\sim _{x}\mathbf {1} _{T}.}
78:
56:
1782:
have additional structure, it is possible to define subsets of the set of all maps from
4542:
4456:
4409:
4231:
3550:
3546:
3462:
3442:
3190:
3170:
3150:
3088:
3082:
3019:
2957:
2895:
2873:
2851:
2790:
2693:
2673:
2653:
2076:
1972:
1952:
1856:
1765:
1745:
1698:. This equivalence relation is an abstraction of the germ equivalence described above.
1113:
644:
171:
151:
16:
Equivalence class of objects sharing local properties at a point in a topological space
3832:{\displaystyle f(x)={\begin{cases}e^{-1/x^{2}},&x\neq 0,\\0,&x=0.\end{cases}}}
3626:. This is the case, for example, for continuous functions on a topological space; for
4570:
4393:
4377:
4334:
4315:
4291:
3577:
2478:
2470:
2320:
2292:
1306:
1152:
1136:
1132:
994:
965:
940:
145:
48:
40:
32:
28:
4554:
4279:
4226:
3631:
3074:
2474:
1334:
4191:{\displaystyle f=|f|^{1/2}\cdot {\big (}\operatorname {sgn} (f)|f|^{1/2}{\big )},}
81:
metaphor, as a germ is (locally) the "heart" of a function, as it is for a grain.
4366:
4287:
4241:
3646:
3557:
2212:
52:
55:, but in general this is not needed (the functions in question need not even be
4530:
3584:
3525:
2846:
2316:
1710:
628:
is then intended as an entire equivalence class of maps, using the same letter
4564:
4346:
4251:
3705:
3531:
2623:
1637:
1249:
1202:
1105:
3711:
of this ring consists of all germs that vanish at the origin, and the power
4236:
3650:
3569:
3480:
2429:
1450:{\displaystyle \mathrm {res} _{VU}:{\mathcal {F}}(U)\to {\mathcal {F}}(V),}
1314:
1109:
1002:
4270:
Tu, L. W. (2007). An introduction to manifolds. New York: Springer. p. 11.
4214:
3542:
2610:
739:, and in particular they don't need to have the same domain. However, if
73:
20:
3704:. This ring is local but not Noetherian. To see why, observe that the
1221:
The idea of germs is behind the definition of sheaves and presheaves. A
4534:
4338:
3623:
3719:− 1 derivatives vanish. If this ring were Noetherian, then the
4547:
1001:, etc. Similarly for subsets: if one representative of a germ is an
1795:
1791:
1148:
1005:
then so are all representatives, at least on some neighbourhood of
1209:
of map(-germs)) do have topologies as they can be identified with
2425:
4333:
2952:
space of germs of infinitely differentiable ("smooth") functions
3850:, but there is an infinite ascending chain of principal ideals
2216:
2098:
1994:
44:
3591:. A tangent vector can be viewed as a point-derivation on the
3237:), it can be dropped in each of the above symbols: also, when
408:{\displaystyle f\sim _{x}g\quad {\text{or}}\quad S\sim _{x}T.}
3528:
of a function at a point can be studied by analyzing its germ
3610:
is a space of some sort. It is often the case that, at each
993:, then they share all local properties, such as continuity,
3825:
2280:{\displaystyle C^{\omega }(X,Y)\subseteq {\mbox{Hom}}(X,Y)}
4213:, whence the conclusion. This is related to the setup of
3700:
be the ring of germs at the origin of smooth functions on
3693:{\displaystyle {\mathcal {C}}_{0}^{\infty }(\mathbf {R} )}
3645:
implies that rings of germs of holomorphic functions are
3572:, and thus the set of germs can be considered to be the
3269:, a subscript before the symbol can be added. As example
2598:{\displaystyle \{g:\exists x\forall y>x\,f(y)=g(y)\}}
2256:
2176:
2038:
1901:
4479:
4459:
4432:
4412:
4090:
4016:
3859:
3732:
3659:
3489:
3465:
3445:
3277:
3243:
3217:
3193:
3173:
3153:
3118:
3091:
3044:
3022:
2982:
2960:
2920:
2898:
2876:
2854:
2815:
2793:
2753:
2716:
2696:
2676:
2656:
2632:
2536:
2498:
2449:
2436:
between them can be defined, and therefore spaces of
2400:
2356:
2329:
2301:
2226:
2108:
2079:
2062:{\displaystyle C^{k}(X,Y)\subseteq {\mbox{Hom}}(X,Y)}
2008:
1975:
1955:
1925:{\displaystyle C^{0}(X,Y)\subseteq {\mbox{Hom}}(X,Y)}
1871:
1835:
1803:
1768:
1748:
1680:
1645:
1609:
1553:
1510:
1471:
1384:
1358:
1262:
1230:
1160:
1080:
906:
840:
793:
660:
568:
459:
362:
303:
221:
174:
154:
108:
4540:
4050:{\displaystyle {\mathcal {C}}_{0}^{0}(\mathbf {R} )}
4309:
1028:, then it makes sense to multiply germs: to define
4492:
4465:
4445:
4418:
4190:
4057:of germs at the origin of continuous functions on
4049:
3988:
3831:
3692:
3537:Germs are useful in determining the properties of
3501:
3471:
3451:
3431:
3261:
3229:
3199:
3179:
3159:
3135:
3097:
3061:
3028:
3004:
2966:
2942:
2904:
2882:
2860:
2837:
2799:
2775:
2736:
2702:
2682:
2662:
2642:
2597:
2518:
2461:
2412:
2369:
2342:
2307:
2279:
2199:
2085:
2061:
1981:
1961:
1924:
1847:
1813:
1774:
1754:
1690:
1662:
1628:
1571:
1535:
1496:
1449:
1370:
1281:
1240:
1193:
1092:
932:. This is particularly relevant in two settings:
924:
888:
826:
719:
613:
525:{\displaystyle _{x}=\{g:X\to Y\mid g\sim _{x}f\}.}
524:
407:
330:
251:
180:
160:
132:
1016:are inherited by the set of germs with values in
4562:
4387:
4278:
4343:Analytic Functions of Several Complex Variables
1297:. Typical examples of abelian groups here are:
827:{\displaystyle S\cap U=T\cap U\neq \emptyset ,}
4453:(Germs of differentiable functions at a point
4406:Germi di funzioni differenziabili in un punto
4180:
4130:
4061:even has the property that its maximal ideal
3848:ascending chain condition on principal ideals
3518:The key word in the applications of germs is
2481:) functions between them can be defined, and
4357:The Elementary Properties of the Local Rings
3583:Germs can also be used in the definition of
3005:{\displaystyle {\mathcal {C}}_{x}^{\omega }}
2943:{\displaystyle {\mathcal {C}}_{x}^{\infty }}
2592:
2537:
2526:at positive infinity (or simply the germ of
639:Notice that two sets are germ-equivalent at
516:
479:
4302:, chapter I, paragraph 6, subparagraph 10 "
3556:When the topological spaces considered are
2519:{\displaystyle f:\mathbb {R} \rightarrow Y}
4553:A research preprint dealing with germs of
4546:
4205:| vanishes at the origin, this expresses
3439:are the spaces of germs shown above when
2564:
2506:
889:{\displaystyle f|_{S\cap V}=g|_{T\cap V}}
4314:(2nd ed.). North-Holland Elsevier.
4201:where sgn is the sign function. Since |
3846:. This is because all UFDs satisfy the
1216:
997:etc., so it makes sense to talk about a
3649:. It can also be shown that these are
3598:
3071:space of germs of holomorphic functions
4563:
4513:Chirka, Evgeniǐ Mikhaǐlovich (2001) ,
4512:
4312:Analysis on Real and Complex Manifolds
2838:{\displaystyle {\mathcal {C}}_{x}^{k}}
2785:space of germs of continuous functions
2776:{\displaystyle {\mathcal {C}}_{x}^{0}}
1536:{\displaystyle g\in {\mathcal {F}}(V)}
1497:{\displaystyle f\in {\mathcal {F}}(U)}
4353:Local Rings of Holomorphic Functions
3715:consists of those germs whose first
3618:, the ring of germs of functions at
3545:: they are one of the main tools in
3145:space of germs of analytic varieties
3014:space of germs of analytic functions
1636:). The equivalence classes form the
282:, then they define the same germ at
84:
4557:in an infinite dimensional setting.
4209:as the product of two functions in
3423:
3136:{\displaystyle {\mathfrak {V}}_{x}}
3122:
3079:space of germs of regular functions
2737:{\displaystyle {\mathcal {F}}_{x}.}
2138:
1012:Algebraic structures on the target
976:
735:Maps need not be defined on all of
89:
13:
4020:
3999:The inclusions are strict because
3674:
3663:
3402:
3375:
3353:
3347:
3319:
3291:
3062:{\displaystyle {\mathcal {O}}_{x}}
3048:
2986:
2935:
2924:
2819:
2757:
2720:
2635:
2552:
2546:
2379:germs of (finitely) differentiable
2362:
2114:
1806:
1683:
1663:{\displaystyle {\mathcal {F}}_{x}}
1649:
1612:
1572:{\displaystyle W\subseteq U\cap V}
1519:
1480:
1430:
1411:
1393:
1390:
1387:
1265:
1233:
818:
341:It is straightforward to see that
286:if there is again a neighbourhood
14:
4587:
4506:
1629:{\displaystyle {\mathcal {F}}(W)}
1282:{\displaystyle {\mathcal {F}}(U)}
925:{\displaystyle x\in V\subseteq U}
896:, for some smaller neighbourhood
730:
614:{\displaystyle f:(X,x)\to (Y,y).}
4040:
3683:
1378:then there is a restriction map
1211:finite-dimensional vector spaces
704:
682:
331:{\displaystyle S\cap U=T\cap U.}
3643:Weierstrass preparation theorem
3530:. They are a generalization of
3513:
2890:-times-differentiable functions
1116:, then so is the set of germs.
1020:. For instance, if the target
999:differentiable or analytic germ
385:
379:
4284:General Topology. Chapters 1-4
4264:
4160:
4151:
4147:
4141:
4107:
4098:
4044:
4036:
3974:
3971:
3965:
3940:
3934:
3931:
3925:
3906:
3900:
3897:
3891:
3866:
3742:
3736:
3687:
3679:
3187:is fixed and known (e.g. when
2643:{\displaystyle {\mathcal {F}}}
2589:
2583:
2574:
2568:
2510:
2438:germs of holomorphic functions
2274:
2262:
2249:
2237:
2194:
2182:
2169:
2157:
2131:
2119:
2056:
2044:
2031:
2019:
1919:
1907:
1894:
1882:
1814:{\displaystyle {\mathcal {F}}}
1691:{\displaystyle {\mathcal {F}}}
1623:
1617:
1530:
1524:
1491:
1485:
1441:
1435:
1425:
1422:
1416:
1276:
1270:
1241:{\displaystyle {\mathcal {F}}}
1194:{\displaystyle J_{x}^{k}(X,Y)}
1188:
1176:
870:
846:
677:
605:
593:
590:
587:
575:
491:
467:
460:
246:
240:
231:
225:
124:
1:
4326:, chapter 2, paragraph 2.1, "
4257:
1942:germs of continuous functions
1070:of the pointwise product map
1039:, first take representatives
4310:Raghavan Narasimhan (1973).
3541:near chosen points of their
1823:some notable examples follow
1547:if there is a neighbourhood
1371:{\displaystyle V\subseteq U}
1047:, defined on neighbourhoods
960:, so is not even defined at
7:
4520:Encyclopedia of Mathematics
4355:", especially paragraph A "
4220:
3844:unique factorization domain
2617:
2605:. These germs are used in
2370:{\displaystyle C^{\infty }}
1737:
956:has a pole of some sort at
10:
4592:
4375:Cambridge University Press
3721:Krull intersection theorem
3653:. On the other hand, let
3562:complex analytic varieties
2483:germs of regular functions
624:When using this notation,
133:{\displaystyle f,g:X\to Y}
4388:Giuseppe Tallini (1973).
4371:Geometric Differentiation
3568:on them can be viewed as
2319:for infinity; this is an
1821:and corresponding germs:
1721:-algebra, then any stalk
1465:, one says that elements
1337:), constant functions on
1256:assigns an abelian group
1055:respectively, and define
968:, which would be defined
436:. Similarly, the germ at
252:{\displaystyle f(u)=g(u)}
77:in a continuation of the
71:The name is derived from
4531:Germ of smooth functions
4003:is in the maximal ideal
3842:This ring is also not a
3595:of germs at that point.
3262:{\displaystyle \dim X=n}
3209:topological vector space
2095:differentiable functions
1991:differentiable structure
1459:compatibility conditions
1100:). In the same way, if
645:characteristic functions
215:are equal; meaning that
203:such that restricted to
188:define the same germ at
2710:is commonly denoted by
2650:on a topological space
2308:{\displaystyle \omega }
1705:. This implies that if
1252:on a topological space
1131:does not have a useful
1093:{\displaystyle U\cap V}
989:are germ equivalent at
783:are germ equivalent at
767:are germ equivalent at
647:are germ-equivalent at
278:are any two subsets of
98:of a topological space
66:
4494:
4467:
4447:
4420:
4392:. Edizioni Cremonese.
4286:(paperback ed.).
4192:
4051:
3990:
3833:
3694:
3503:
3473:
3453:
3433:
3263:
3231:
3201:
3181:
3161:
3137:
3099:
3063:
3030:
3006:
2968:
2944:
2906:
2884:
2862:
2839:
2801:
2777:
2738:
2704:
2684:
2664:
2644:
2599:
2520:
2463:
2414:
2377:), and then spaces of
2371:
2344:
2309:
2281:
2201:
2087:
2063:
1983:
1963:
1926:
1849:
1815:
1790:or more generally sub-
1776:
1756:
1692:
1664:
1630:
1573:
1537:
1498:
1451:
1372:
1343:differential operators
1283:
1242:
1195:
1094:
926:
890:
828:
721:
615:
526:
409:
343:defining the same germ
332:
253:
182:
162:
134:
4495:
4493:{\displaystyle V_{n}}
4468:
4448:
4446:{\displaystyle V_{n}}
4421:
4193:
4052:
3991:
3834:
3695:
3636:locally ringed spaces
3589:differential geometry
3574:analytic continuation
3566:holomorphic functions
3504:
3474:
3454:
3434:
3264:
3232:
3202:
3182:
3162:
3138:
3100:
3064:
3031:
3007:
2969:
2945:
2907:
2885:
2863:
2840:
2802:
2778:
2739:
2705:
2685:
2665:
2645:
2600:
2521:
2464:
2434:holomorphic functions
2430:complex vector spaces
2415:
2372:
2345:
2343:{\displaystyle C^{k}}
2310:
2282:
2202:
2088:
2064:
1984:
1964:
1927:
1850:
1816:
1777:
1757:
1693:
1665:
1631:
1574:
1538:
1499:
1452:
1373:
1323:holomorphic functions
1299:real-valued functions
1284:
1243:
1217:Relation with sheaves
1196:
1151:, then the spaces of
1095:
1074:(which is defined on
927:
891:
829:
722:
643:if and only if their
616:
527:
410:
333:
254:
183:
163:
135:
27:of an object in/on a
4477:
4457:
4430:
4410:
4088:
4014:
3857:
3730:
3657:
3599:Algebraic properties
3487:
3463:
3443:
3275:
3241:
3215:
3191:
3171:
3151:
3116:
3089:
3042:
3020:
2980:
2958:
2918:
2896:
2874:
2852:
2813:
2791:
2751:
2714:
2694:
2674:
2654:
2630:
2534:
2496:
2447:
2398:
2354:
2327:
2299:
2224:
2106:
2093:-times continuously
2077:
2006:
1973:
1953:
1938:continuous functions
1869:
1833:
1801:
1766:
1746:
1678:
1643:
1607:
1603:) (both elements of
1551:
1508:
1469:
1382:
1356:
1260:
1228:
1158:
1119:The set of germs at
1078:
904:
838:
791:
658:
566:
543:that maps the point
457:
360:
351:equivalence relation
301:
219:
172:
152:
106:
4359:" and paragraph E "
4035:
3678:
3502:{\displaystyle x=0}
3230:{\displaystyle x=0}
3001:
2939:
2834:
2772:
2607:asymptotic analysis
2471:algebraic structure
2462:{\displaystyle X,Y}
2440:can be constructed.
2424:(for instance, are
2413:{\displaystyle X,Y}
2389:can be constructed.
1848:{\displaystyle X,Y}
1457:satisfying certain
1175:
964:, as for example a
430:is usually denoted
426:, then its germ at
4555:analytic varieties
4490:
4463:
4443:
4416:
4361:Germs of Varieties
4232:Catastrophe theory
4215:almost ring theory
4188:
4081:can be written as
4047:
4017:
3986:
3829:
3824:
3690:
3660:
3560:or more generally
3551:catastrophe theory
3547:singularity theory
3499:
3469:
3449:
3429:
3259:
3227:
3197:
3177:
3157:
3133:
3095:
3083:algebraic geometry
3059:
3026:
3002:
2983:
2964:
2940:
2921:
2902:
2880:
2870:space of germs of
2858:
2835:
2816:
2797:
2773:
2754:
2734:
2700:
2680:
2660:
2640:
2595:
2516:
2459:
2410:
2387:analytic functions
2367:
2340:
2323:, by analogy with
2305:
2293:analytic functions
2277:
2260:
2197:
2180:
2083:
2059:
2042:
1979:
1959:
1922:
1905:
1857:topological spaces
1845:
1811:
1772:
1752:
1688:
1660:
1626:
1569:
1543:are equivalent at
1533:
1494:
1447:
1368:
1307:differential forms
1279:
1238:
1191:
1161:
1090:
1066:to be the germ at
922:
886:
834:and then moreover
824:
755:, both subsets of
717:
611:
522:
405:
328:
249:
178:
158:
130:
63:has some meaning.
23:, the notion of a
4466:{\displaystyle P}
4419:{\displaystyle P}
4404:, paragraph 31, "
4335:Robert C. Gunning
4328:Basic Definitions
3578:analytic function
3539:dynamical systems
3472:{\displaystyle n}
3452:{\displaystyle X}
3200:{\displaystyle X}
3180:{\displaystyle x}
3167:. When the point
3160:{\displaystyle x}
3098:{\displaystyle x}
3029:{\displaystyle x}
2967:{\displaystyle x}
2905:{\displaystyle x}
2883:{\displaystyle k}
2861:{\displaystyle k}
2800:{\displaystyle x}
2703:{\displaystyle X}
2683:{\displaystyle x}
2663:{\displaystyle X}
2489:) can be defined.
2422:complex structure
2321:abuse of notation
2259:
2179:
2086:{\displaystyle k}
2041:
1982:{\displaystyle Y}
1962:{\displaystyle X}
1904:
1775:{\displaystyle Y}
1755:{\displaystyle X}
1289:to each open set
1135:, except for the
995:differentiability
966:rational function
383:
181:{\displaystyle g}
161:{\displaystyle f}
85:Formal definition
33:equivalence class
29:topological space
4583:
4552:
4550:
4527:
4499:
4497:
4496:
4491:
4489:
4488:
4472:
4470:
4469:
4464:
4452:
4450:
4449:
4444:
4442:
4441:
4425:
4423:
4422:
4417:
4403:
4350:
4325:
4304:Germs at a point
4301:
4280:Nicolas Bourbaki
4271:
4268:
4227:Analytic variety
4197:
4195:
4194:
4189:
4184:
4183:
4177:
4176:
4172:
4163:
4154:
4134:
4133:
4124:
4123:
4119:
4110:
4101:
4056:
4054:
4053:
4048:
4043:
4034:
4029:
4024:
4023:
3995:
3993:
3992:
3987:
3961:
3960:
3921:
3920:
3887:
3886:
3838:
3836:
3835:
3830:
3828:
3827:
3783:
3782:
3781:
3780:
3771:
3699:
3697:
3696:
3691:
3686:
3677:
3672:
3667:
3666:
3647:Noetherian rings
3632:complex manifold
3558:Riemann surfaces
3526:local properties
3508:
3506:
3505:
3500:
3478:
3476:
3475:
3470:
3458:
3456:
3455:
3450:
3438:
3436:
3435:
3430:
3428:
3427:
3426:
3420:
3419:
3407:
3406:
3405:
3399:
3398:
3386:
3385:
3384:
3379:
3378:
3371:
3370:
3358:
3357:
3356:
3351:
3350:
3343:
3342:
3330:
3329:
3328:
3323:
3322:
3315:
3314:
3302:
3301:
3300:
3295:
3294:
3287:
3286:
3268:
3266:
3265:
3260:
3236:
3234:
3233:
3228:
3206:
3204:
3203:
3198:
3186:
3184:
3183:
3178:
3166:
3164:
3163:
3158:
3142:
3140:
3139:
3134:
3132:
3131:
3126:
3125:
3104:
3102:
3101:
3096:
3075:complex geometry
3068:
3066:
3065:
3060:
3058:
3057:
3052:
3051:
3035:
3033:
3032:
3027:
3011:
3009:
3008:
3003:
3000:
2995:
2990:
2989:
2973:
2971:
2970:
2965:
2949:
2947:
2946:
2941:
2938:
2933:
2928:
2927:
2911:
2909:
2908:
2903:
2889:
2887:
2886:
2881:
2867:
2865:
2864:
2859:
2844:
2842:
2841:
2836:
2833:
2828:
2823:
2822:
2806:
2804:
2803:
2798:
2782:
2780:
2779:
2774:
2771:
2766:
2761:
2760:
2743:
2741:
2740:
2735:
2730:
2729:
2724:
2723:
2709:
2707:
2706:
2701:
2689:
2687:
2686:
2681:
2669:
2667:
2666:
2661:
2649:
2647:
2646:
2641:
2639:
2638:
2604:
2602:
2601:
2596:
2529:
2525:
2523:
2522:
2517:
2509:
2468:
2466:
2465:
2460:
2419:
2417:
2416:
2411:
2376:
2374:
2373:
2368:
2366:
2365:
2349:
2347:
2346:
2341:
2339:
2338:
2314:
2312:
2311:
2306:
2295:can be defined (
2286:
2284:
2283:
2278:
2261:
2257:
2236:
2235:
2213:smooth functions
2206:
2204:
2203:
2198:
2181:
2177:
2156:
2155:
2146:
2145:
2118:
2117:
2092:
2090:
2089:
2084:
2068:
2066:
2065:
2060:
2043:
2039:
2018:
2017:
1988:
1986:
1985:
1980:
1968:
1966:
1965:
1960:
1931:
1929:
1928:
1923:
1906:
1902:
1881:
1880:
1854:
1852:
1851:
1846:
1820:
1818:
1817:
1812:
1810:
1809:
1781:
1779:
1778:
1773:
1761:
1759:
1758:
1753:
1697:
1695:
1694:
1689:
1687:
1686:
1674:of the presheaf
1669:
1667:
1666:
1661:
1659:
1658:
1653:
1652:
1635:
1633:
1632:
1627:
1616:
1615:
1578:
1576:
1575:
1570:
1542:
1540:
1539:
1534:
1523:
1522:
1503:
1501:
1500:
1495:
1484:
1483:
1456:
1454:
1453:
1448:
1434:
1433:
1415:
1414:
1405:
1404:
1396:
1377:
1375:
1374:
1369:
1335:complex manifold
1288:
1286:
1285:
1280:
1269:
1268:
1247:
1245:
1244:
1239:
1237:
1236:
1200:
1198:
1197:
1192:
1174:
1169:
1099:
1097:
1096:
1091:
977:Basic properties
939:is defined on a
931:
929:
928:
923:
895:
893:
892:
887:
885:
884:
873:
861:
860:
849:
833:
831:
830:
825:
726:
724:
723:
718:
713:
712:
707:
701:
700:
691:
690:
685:
673:
672:
620:
618:
617:
612:
531:
529:
528:
523:
512:
511:
475:
474:
414:
412:
411:
406:
398:
397:
384:
381:
375:
374:
337:
335:
334:
329:
258:
256:
255:
250:
187:
185:
184:
179:
167:
165:
164:
159:
139:
137:
136:
131:
90:Basic definition
4591:
4590:
4586:
4585:
4584:
4582:
4581:
4580:
4561:
4560:
4509:
4502:" (in Italian).
4484:
4480:
4478:
4475:
4474:
4458:
4455:
4454:
4437:
4433:
4431:
4428:
4427:
4411:
4408:
4407:
4400:
4367:Ian R. Porteous
4322:
4298:
4288:Springer-Verlag
4275:
4274:
4269:
4265:
4260:
4242:Riemann surface
4223:
4179:
4178:
4168:
4164:
4159:
4158:
4150:
4129:
4128:
4115:
4111:
4106:
4105:
4097:
4089:
4086:
4085:
4039:
4030:
4025:
4019:
4018:
4015:
4012:
4011:
3947:
3943:
3913:
3909:
3873:
3869:
3858:
3855:
3854:
3823:
3822:
3811:
3802:
3801:
3787:
3776:
3772:
3767:
3760:
3756:
3749:
3748:
3731:
3728:
3727:
3682:
3673:
3668:
3662:
3661:
3658:
3655:
3654:
3601:
3585:tangent vectors
3516:
3488:
3485:
3484:
3464:
3461:
3460:
3444:
3441:
3440:
3422:
3421:
3415:
3412:
3411:
3401:
3400:
3394:
3391:
3390:
3380:
3374:
3373:
3372:
3366:
3363:
3362:
3352:
3346:
3345:
3344:
3338:
3335:
3334:
3324:
3318:
3317:
3316:
3310:
3307:
3306:
3296:
3290:
3289:
3288:
3282:
3279:
3278:
3276:
3273:
3272:
3242:
3239:
3238:
3216:
3213:
3212:
3192:
3189:
3188:
3172:
3169:
3168:
3152:
3149:
3148:
3127:
3121:
3120:
3119:
3117:
3114:
3113:
3090:
3087:
3086:
3053:
3047:
3046:
3045:
3043:
3040:
3039:
3021:
3018:
3017:
2996:
2991:
2985:
2984:
2981:
2978:
2977:
2959:
2956:
2955:
2934:
2929:
2923:
2922:
2919:
2916:
2915:
2897:
2894:
2893:
2875:
2872:
2871:
2853:
2850:
2849:
2829:
2824:
2818:
2817:
2814:
2811:
2810:
2792:
2789:
2788:
2767:
2762:
2756:
2755:
2752:
2749:
2748:
2725:
2719:
2718:
2717:
2715:
2712:
2711:
2695:
2692:
2691:
2675:
2672:
2671:
2655:
2652:
2651:
2634:
2633:
2631:
2628:
2627:
2620:
2535:
2532:
2531:
2527:
2505:
2497:
2494:
2493:
2448:
2445:
2444:
2399:
2396:
2395:
2361:
2357:
2355:
2352:
2351:
2334:
2330:
2328:
2325:
2324:
2300:
2297:
2296:
2255:
2231:
2227:
2225:
2222:
2221:
2175:
2151:
2147:
2141:
2137:
2113:
2109:
2107:
2104:
2103:
2078:
2075:
2074:
2037:
2013:
2009:
2007:
2004:
2003:
1974:
1971:
1970:
1954:
1951:
1950:
1900:
1876:
1872:
1870:
1867:
1866:
1834:
1831:
1830:
1805:
1804:
1802:
1799:
1798:
1767:
1764:
1763:
1747:
1744:
1743:
1740:
1729:
1682:
1681:
1679:
1676:
1675:
1654:
1648:
1647:
1646:
1644:
1641:
1640:
1611:
1610:
1608:
1605:
1604:
1598:
1588:
1552:
1549:
1548:
1518:
1517:
1509:
1506:
1505:
1479:
1478:
1470:
1467:
1466:
1429:
1428:
1410:
1409:
1397:
1386:
1385:
1383:
1380:
1379:
1357:
1354:
1353:
1264:
1263:
1261:
1258:
1257:
1232:
1231:
1229:
1226:
1225:
1219:
1170:
1165:
1159:
1156:
1155:
1079:
1076:
1075:
1065:
1060:
1038:
1033:
979:
905:
902:
901:
874:
869:
868:
850:
845:
844:
839:
836:
835:
792:
789:
788:
733:
708:
703:
702:
696:
692:
686:
681:
680:
668:
664:
659:
656:
655:
567:
564:
563:
507:
503:
470:
466:
458:
455:
454:
449:
435:
393:
389:
380:
370:
366:
361:
358:
357:
302:
299:
298:
220:
217:
216:
173:
170:
169:
153:
150:
149:
107:
104:
103:
102:, and two maps
92:
87:
69:
17:
12:
11:
5:
4589:
4579:
4578:
4573:
4559:
4558:
4538:
4528:
4508:
4507:External links
4505:
4504:
4503:
4487:
4483:
4462:
4440:
4436:
4415:
4398:
4385:
4364:
4331:
4320:
4307:
4296:
4273:
4272:
4262:
4261:
4259:
4256:
4255:
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4249:
4244:
4239:
4234:
4229:
4222:
4219:
4199:
4198:
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4175:
4171:
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4162:
4157:
4153:
4149:
4146:
4143:
4140:
4137:
4132:
4127:
4122:
4118:
4114:
4109:
4104:
4100:
4096:
4093:
4046:
4042:
4038:
4033:
4028:
4022:
3997:
3996:
3985:
3982:
3979:
3976:
3973:
3970:
3967:
3964:
3959:
3956:
3953:
3950:
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3942:
3939:
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3933:
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3927:
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3800:
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3689:
3685:
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3676:
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3665:
3600:
3597:
3515:
3512:
3511:
3510:
3498:
3495:
3492:
3468:
3448:
3425:
3418:
3414:
3410:
3404:
3397:
3393:
3389:
3383:
3377:
3369:
3365:
3361:
3355:
3349:
3341:
3337:
3333:
3327:
3321:
3313:
3309:
3305:
3299:
3293:
3285:
3281:
3270:
3258:
3255:
3252:
3249:
3246:
3226:
3223:
3220:
3196:
3176:
3156:
3130:
3124:
3107:
3106:
3094:
3056:
3050:
3037:
3025:
2999:
2994:
2988:
2975:
2963:
2937:
2932:
2926:
2913:
2901:
2879:
2857:
2847:natural number
2832:
2827:
2821:
2808:
2796:
2770:
2765:
2759:
2733:
2728:
2722:
2699:
2679:
2659:
2637:
2619:
2616:
2615:
2614:
2594:
2591:
2588:
2585:
2582:
2579:
2576:
2573:
2570:
2567:
2563:
2560:
2557:
2554:
2551:
2548:
2545:
2542:
2539:
2515:
2512:
2508:
2504:
2501:
2490:
2485:(and likewise
2458:
2455:
2452:
2441:
2409:
2406:
2403:
2391:
2390:
2364:
2360:
2337:
2333:
2304:
2289:
2288:
2287:
2276:
2273:
2270:
2267:
2264:
2254:
2251:
2248:
2245:
2242:
2239:
2234:
2230:
2209:
2208:
2207:
2196:
2193:
2190:
2187:
2184:
2174:
2171:
2168:
2165:
2162:
2159:
2154:
2150:
2144:
2140:
2136:
2133:
2130:
2127:
2124:
2121:
2116:
2112:
2082:
2071:
2070:
2069:
2058:
2055:
2052:
2049:
2046:
2036:
2033:
2030:
2027:
2024:
2021:
2016:
2012:
1998:
1997:
1978:
1958:
1946:
1945:
1934:
1933:
1932:
1921:
1918:
1915:
1912:
1909:
1899:
1896:
1893:
1890:
1887:
1884:
1879:
1875:
1861:
1860:
1844:
1841:
1838:
1808:
1771:
1751:
1739:
1736:
1725:
1711:Lawvere theory
1685:
1657:
1651:
1625:
1622:
1619:
1614:
1594:
1584:
1568:
1565:
1562:
1559:
1556:
1532:
1529:
1526:
1521:
1516:
1513:
1493:
1490:
1487:
1482:
1477:
1474:
1461:. For a fixed
1446:
1443:
1440:
1437:
1432:
1427:
1424:
1421:
1418:
1413:
1408:
1403:
1400:
1395:
1392:
1389:
1367:
1364:
1361:
1278:
1275:
1272:
1267:
1250:abelian groups
1235:
1218:
1215:
1201:(finite order
1190:
1187:
1184:
1181:
1178:
1173:
1168:
1164:
1089:
1086:
1083:
1061:
1056:
1034:
1029:
978:
975:
974:
973:
951:
921:
918:
915:
912:
909:
883:
880:
877:
872:
867:
864:
859:
856:
853:
848:
843:
823:
820:
817:
814:
811:
808:
805:
802:
799:
796:
732:
731:More generally
729:
728:
727:
716:
711:
706:
699:
695:
689:
684:
679:
676:
671:
667:
663:
634:representative
622:
621:
610:
607:
604:
601:
598:
595:
592:
589:
586:
583:
580:
577:
574:
571:
559:is denoted as
535:A map germ at
533:
532:
521:
518:
515:
510:
506:
502:
499:
496:
493:
490:
487:
484:
481:
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445:
431:
416:
415:
404:
401:
396:
392:
388:
378:
373:
369:
365:
339:
338:
327:
324:
321:
318:
315:
312:
309:
306:
270:Similarly, if
248:
245:
242:
239:
236:
233:
230:
227:
224:
192:if there is a
177:
157:
129:
126:
123:
120:
117:
114:
111:
94:Given a point
91:
88:
86:
83:
68:
65:
15:
9:
6:
4:
3:
2:
4588:
4577:
4574:
4572:
4569:
4568:
4566:
4556:
4549:
4544:
4539:
4536:
4532:
4529:
4526:
4522:
4521:
4516:
4511:
4510:
4501:
4485:
4481:
4460:
4438:
4434:
4413:
4401:
4399:88-7083-413-1
4395:
4391:
4386:
4383:
4382:0-521-00264-8
4379:
4376:
4372:
4368:
4365:
4362:
4358:
4354:
4351:, chapter 2 "
4348:
4347:Prentice-Hall
4344:
4340:
4336:
4332:
4329:
4323:
4321:0-7204-2501-8
4317:
4313:
4308:
4305:
4299:
4297:3-540-64241-2
4293:
4289:
4285:
4281:
4277:
4276:
4267:
4263:
4253:
4250:
4248:
4245:
4243:
4240:
4238:
4235:
4233:
4230:
4228:
4225:
4224:
4218:
4216:
4212:
4208:
4204:
4185:
4173:
4169:
4165:
4155:
4144:
4138:
4135:
4125:
4120:
4116:
4112:
4102:
4094:
4091:
4084:
4083:
4082:
4080:
4076:
4072:
4068:
4064:
4060:
4031:
4026:
4008:
4006:
4002:
3983:
3980:
3977:
3968:
3962:
3957:
3954:
3951:
3948:
3944:
3937:
3928:
3922:
3917:
3914:
3910:
3903:
3894:
3888:
3883:
3880:
3877:
3874:
3870:
3863:
3860:
3853:
3852:
3851:
3849:
3845:
3819:
3816:
3813:
3808:
3805:
3798:
3795:
3792:
3789:
3784:
3777:
3773:
3768:
3764:
3761:
3757:
3750:
3745:
3739:
3733:
3726:
3725:
3724:
3722:
3718:
3714:
3710:
3707:
3706:maximal ideal
3703:
3669:
3652:
3651:regular rings
3648:
3644:
3639:
3637:
3633:
3629:
3625:
3621:
3617:
3613:
3609:
3606:Suppose that
3604:
3596:
3594:
3590:
3586:
3581:
3579:
3575:
3571:
3567:
3563:
3559:
3554:
3552:
3548:
3544:
3540:
3535:
3533:
3532:Taylor series
3529:
3527:
3521:
3496:
3493:
3490:
3482:
3479:-dimensional
3466:
3446:
3416:
3413:
3408:
3395:
3392:
3387:
3381:
3367:
3364:
3359:
3339:
3336:
3331:
3325:
3311:
3308:
3303:
3297:
3283:
3280:
3271:
3256:
3253:
3250:
3247:
3244:
3224:
3221:
3218:
3210:
3194:
3174:
3154:
3146:
3128:
3112:
3111:
3110:
3092:
3084:
3080:
3076:
3072:
3054:
3038:
3023:
3015:
2997:
2992:
2976:
2961:
2953:
2930:
2914:
2899:
2891:
2877:
2855:
2848:
2830:
2825:
2809:
2794:
2786:
2768:
2763:
2747:
2746:
2745:
2731:
2726:
2697:
2677:
2657:
2625:
2612:
2608:
2586:
2580:
2577:
2571:
2565:
2561:
2558:
2555:
2549:
2543:
2540:
2513:
2502:
2499:
2491:
2488:
2484:
2480:
2476:
2472:
2456:
2453:
2450:
2442:
2439:
2435:
2431:
2427:
2423:
2407:
2404:
2401:
2393:
2392:
2388:
2384:
2380:
2358:
2335:
2331:
2322:
2318:
2302:
2294:
2290:
2271:
2268:
2265:
2252:
2246:
2243:
2240:
2232:
2228:
2220:
2219:
2218:
2214:
2210:
2191:
2188:
2185:
2172:
2166:
2163:
2160:
2152:
2148:
2142:
2134:
2128:
2125:
2122:
2110:
2102:
2101:
2100:
2096:
2080:
2072:
2053:
2050:
2047:
2034:
2028:
2025:
2022:
2014:
2010:
2002:
2001:
2000:
1999:
1996:
1992:
1976:
1956:
1948:
1947:
1943:
1939:
1935:
1916:
1913:
1910:
1897:
1891:
1888:
1885:
1877:
1873:
1865:
1864:
1863:
1862:
1858:
1842:
1839:
1836:
1828:
1827:
1826:
1824:
1797:
1793:
1789:
1785:
1769:
1749:
1735:
1733:
1728:
1724:
1720:
1716:
1712:
1708:
1704:
1699:
1673:
1655:
1639:
1620:
1602:
1597:
1592:
1587:
1582:
1566:
1563:
1560:
1557:
1554:
1546:
1527:
1514:
1511:
1488:
1475:
1472:
1464:
1460:
1444:
1438:
1419:
1406:
1401:
1398:
1365:
1362:
1359:
1350:
1348:
1344:
1340:
1336:
1332:
1328:
1324:
1320:
1316:
1315:vector fields
1312:
1308:
1304:
1300:
1296:
1292:
1273:
1255:
1251:
1224:
1214:
1212:
1208:
1204:
1203:Taylor series
1185:
1182:
1179:
1171:
1166:
1162:
1154:
1150:
1146:
1142:
1138:
1134:
1130:
1126:
1123:of maps from
1122:
1117:
1115:
1111:
1107:
1106:abelian group
1103:
1087:
1084:
1081:
1073:
1069:
1064:
1059:
1054:
1050:
1046:
1042:
1037:
1032:
1027:
1023:
1019:
1015:
1010:
1008:
1004:
1000:
996:
992:
988:
984:
972:a subvariety.
971:
967:
963:
959:
955:
952:
949:
945:
942:
938:
935:
934:
933:
919:
916:
913:
910:
907:
899:
881:
878:
875:
865:
862:
857:
854:
851:
841:
821:
815:
812:
809:
806:
803:
800:
797:
794:
786:
782:
778:
774:
770:
766:
762:
758:
754:
750:
746:
742:
738:
714:
709:
697:
693:
687:
674:
669:
665:
661:
654:
653:
652:
650:
646:
642:
637:
635:
631:
627:
608:
602:
599:
596:
584:
581:
578:
572:
569:
562:
561:
560:
558:
554:
551:to the point
550:
546:
542:
538:
519:
513:
508:
504:
500:
497:
494:
488:
485:
482:
476:
471:
463:
453:
452:
451:
448:
443:
439:
434:
429:
425:
421:
402:
399:
394:
390:
386:
376:
371:
367:
363:
356:
355:
354:
352:
348:
344:
325:
322:
319:
316:
313:
310:
307:
304:
297:
296:
295:
293:
289:
285:
281:
277:
273:
268:
266:
262:
243:
237:
234:
228:
222:
214:
210:
206:
202:
198:
195:
194:neighbourhood
191:
175:
155:
147:
143:
127:
121:
118:
115:
112:
109:
101:
97:
82:
80:
76:
75:
64:
62:
58:
54:
50:
46:
42:
38:
34:
30:
26:
22:
4576:Sheaf theory
4548:math/0612355
4518:
4405:
4389:
4370:
4360:
4356:
4352:
4342:
4327:
4311:
4303:
4283:
4266:
4237:Gluing axiom
4210:
4206:
4202:
4200:
4078:
4074:
4073:. Any germ
4070:
4066:
4062:
4058:
4009:
4004:
4000:
3998:
3841:
3716:
3712:
3708:
3701:
3640:
3627:
3619:
3615:
3611:
3607:
3605:
3602:
3582:
3570:power series
3555:
3536:
3523:
3519:
3517:
3514:Applications
3481:vector space
3144:
3108:
3078:
3070:
3013:
2951:
2869:
2784:
2621:
2611:Hardy fields
2486:
2482:
2437:
2386:
2382:
2378:
2315:here is the
1941:
1859:, the subset
1822:
1787:
1783:
1741:
1731:
1726:
1722:
1718:
1714:
1713:and a sheaf
1706:
1700:
1671:
1600:
1595:
1590:
1585:
1580:
1544:
1462:
1351:
1346:
1338:
1330:
1326:
1318:
1310:
1302:
1294:
1290:
1253:
1220:
1206:
1144:
1140:
1128:
1124:
1120:
1118:
1110:vector space
1101:
1071:
1067:
1062:
1057:
1052:
1048:
1044:
1040:
1035:
1030:
1021:
1017:
1013:
1011:
1006:
1003:analytic set
998:
990:
986:
982:
980:
969:
961:
957:
953:
947:
943:
936:
897:
784:
780:
776:
772:
768:
764:
760:
756:
752:
748:
744:
740:
736:
734:
648:
640:
638:
629:
625:
623:
556:
552:
548:
544:
540:
536:
534:
446:
441:
437:
432:
427:
423:
419:
418:Given a map
417:
346:
342:
340:
291:
287:
283:
279:
275:
271:
269:
264:
260:
212:
208:
204:
200:
196:
189:
141:
99:
95:
93:
72:
70:
60:
24:
18:
4373:, page 71,
3564:, germs of
3543:phase space
2670:at a point
2626:of a sheaf
2492:The germ of
1794:of a given
751:has domain
743:has domain
444:is written
74:cereal germ
21:mathematics
4565:Categories
4535:PlanetMath
4339:Hugo Rossi
4258:References
4065:satisfies
3624:local ring
1792:presheaves
1734:-algebra.
1730:is also a
941:subvariety
294:such that
57:continuous
4525:EMS Press
4139:
4126:⋅
4010:The ring
3981:⋯
3978:⊊
3955:−
3949:−
3938:⊊
3915:−
3904:⊊
3875:−
3864:⊊
3861:⋯
3793:≠
3762:−
3675:∞
3382:ω
3354:∞
3248:
2998:ω
2936:∞
2845:for each
2553:∀
2547:∃
2511:→
2363:∞
2303:ω
2253:⊆
2233:ω
2173:⊆
2139:⋂
2115:∞
2035:⊆
1898:⊆
1855:are both
1564:∩
1558:⊆
1515:∈
1476:∈
1426:→
1363:⊆
1149:manifolds
1085:∩
917:⊆
911:∈
879:∩
855:∩
819:∅
816:≠
810:∩
798:∩
775:if first
694:∼
678:⟺
666:∼
591:→
505:∼
498:∣
492:→
440:of a set
391:∼
368:∼
320:∩
308:∩
148:), then
125:→
37:functions
4571:Topology
4341:(1965).
4282:(1989).
4221:See also
3520:locality
2618:Notation
2487:rational
2479:rational
2469:have an
2215:and the
1989:admit a
1949:If both
1940:defines
1796:presheaf
1738:Examples
1583:with res
1223:presheaf
1137:discrete
1133:topology
632:for any
450:. Thus,
259:for all
49:analytic
4369:(2001)
3593:algebra
3143:is the
3069:is the
3012:is the
2950:is the
2868:is the
2783:is the
2475:regular
2473:, then
2426:subsets
2420:have a
2317:ordinal
1593:) = res
759:, then
144:is any
140:(where
45:subsets
4515:"Germ"
4396:
4380:
4318:
4294:
3576:of an
3077:), or
2383:smooth
2217:subset
2099:subset
2097:, the
1995:subset
1993:, the
1703:limits
1329:(when
1104:is an
787:, say
349:is an
53:smooth
43:) and
31:is an
4543:arXiv
4252:Stalk
4247:Sheaf
3622:is a
3459:is a
3207:is a
3085:) at
2624:stalk
2530:) is
2477:(and
1717:is a
1709:is a
1638:stalk
1333:is a
1112:, or
1026:group
1024:is a
950:, and
900:with
636:map.
79:sheaf
61:local
4394:ISBN
4378:ISBN
4337:and
4316:ISBN
4292:ISBN
3549:and
3524:all
3483:and
3211:and
3081:(in
3073:(in
2622:The
2609:and
2559:>
2350:and
1969:and
1762:and
1504:and
1341:and
1153:jets
1147:are
1143:and
1114:ring
1051:and
1043:and
985:and
779:and
763:and
747:and
274:and
211:and
168:and
67:Name
41:maps
39:(or
25:germ
4533:at
4473:of
4426:di
4136:sgn
3587:in
3245:dim
3147:at
3016:at
2954:at
2892:at
2787:at
2690:of
2443:If
2432:),
2428:of
2394:If
2291:of
2258:Hom
2211:of
2178:Hom
2073:of
2040:Hom
1936:of
1903:Hom
1829:If
1786:to
1742:If
1670:at
1579:of
1352:If
1345:on
1325:on
1317:on
1309:on
1301:on
1293:in
1248:of
1205:at
1127:to
981:If
970:off
946:of
771:in
555:in
547:in
539:in
422:on
345:at
290:of
263:in
199:of
146:set
51:or
19:In
4567::
4523:,
4517:,
4363:".
4345:.
4330:".
4306:".
4290:.
4217:.
4077:∈
4069:=
4007:.
3820:0.
3638:.
3614:∈
3580:.
3553:.
3522::
2385:,
2381:,
1825:.
1596:WV
1586:WU
1349:.
1321:,
1313:,
1305:,
1213:.
1108:,
1072:fg
1009:.
651::
382:or
267:.
207:,
4551:.
4545::
4537:.
4500:)
4486:n
4482:V
4461:P
4439:n
4435:V
4414:P
4402:.
4384:.
4349:.
4324:.
4300:.
4211:m
4207:f
4203:f
4186:,
4181:)
4174:2
4170:/
4166:1
4161:|
4156:f
4152:|
4148:)
4145:f
4142:(
4131:(
4121:2
4117:/
4113:1
4108:|
4103:f
4099:|
4095:=
4092:f
4079:m
4075:f
4071:m
4067:m
4063:m
4059:R
4045:)
4041:R
4037:(
4032:0
4027:0
4021:C
4005:m
4001:x
3984:.
3975:)
3972:)
3969:x
3966:(
3963:f
3958:1
3952:j
3945:x
3941:(
3935:)
3932:)
3929:x
3926:(
3923:f
3918:j
3911:x
3907:(
3901:)
3898:)
3895:x
3892:(
3889:f
3884:1
3881:+
3878:j
3871:x
3867:(
3817:=
3814:x
3809:,
3806:0
3799:,
3796:0
3790:x
3785:,
3778:2
3774:x
3769:/
3765:1
3758:e
3751:{
3746:=
3743:)
3740:x
3737:(
3734:f
3717:k
3713:m
3709:m
3702:R
3688:)
3684:R
3680:(
3670:0
3664:C
3628:k
3620:x
3616:X
3612:x
3608:X
3509:.
3497:0
3494:=
3491:x
3467:n
3447:X
3424:V
3417:n
3409:,
3403:O
3396:n
3388:,
3376:C
3368:n
3360:,
3348:C
3340:n
3332:,
3326:k
3320:C
3312:n
3304:,
3298:0
3292:C
3284:n
3257:n
3254:=
3251:X
3225:0
3222:=
3219:x
3195:X
3175:x
3155:x
3129:x
3123:V
3105:.
3093:x
3055:x
3049:O
3036:.
3024:x
2993:x
2987:C
2974:.
2962:x
2931:x
2925:C
2912:.
2900:x
2878:k
2856:k
2831:k
2826:x
2820:C
2807:.
2795:x
2769:0
2764:x
2758:C
2732:.
2727:x
2721:F
2698:X
2678:x
2658:X
2636:F
2613:.
2593:}
2590:)
2587:y
2584:(
2581:g
2578:=
2575:)
2572:y
2569:(
2566:f
2562:x
2556:y
2550:x
2544::
2541:g
2538:{
2528:f
2514:Y
2507:R
2503::
2500:f
2457:Y
2454:,
2451:X
2408:Y
2405:,
2402:X
2359:C
2336:k
2332:C
2275:)
2272:Y
2269:,
2266:X
2263:(
2250:)
2247:Y
2244:,
2241:X
2238:(
2229:C
2195:)
2192:Y
2189:,
2186:X
2183:(
2170:)
2167:Y
2164:,
2161:X
2158:(
2153:k
2149:C
2143:k
2135:=
2132:)
2129:Y
2126:,
2123:X
2120:(
2111:C
2081:k
2057:)
2054:Y
2051:,
2048:X
2045:(
2032:)
2029:Y
2026:,
2023:X
2020:(
2015:k
2011:C
1977:Y
1957:X
1944:.
1920:)
1917:Y
1914:,
1911:X
1908:(
1895:)
1892:Y
1889:,
1886:X
1883:(
1878:0
1874:C
1843:Y
1840:,
1837:X
1807:F
1788:Y
1784:X
1770:Y
1750:X
1732:T
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