4804:
8677:
Since the (bounded) derived category is triangulated, there is a
Grothendieck group for derived categories too. This has applications in representation theory for example. For the unbounded category the Grothendieck group however vanishes. For a derived category of some complex finite-dimensional
4630:
4663:
6518:
4890:
5000:
8263:
3930:
1932:
490:
8858:
6324:
4540:
989:
7613:
1770:
6694:
2129:
7322:
3330:
5893:
7375:
7179:
7105:
6926:
1442:
7962:
6986:
6872:
7690:
4799:{\displaystyle \dim _{\mathbb {Q} }(B\otimes _{\mathbb {Z} }\mathbb {Q} )=\dim _{\mathbb {Q} }(A\otimes _{\mathbb {Z} }\mathbb {Q} )+\dim _{\mathbb {Q} }(C\otimes _{\mathbb {Z} }\mathbb {Q} )}
6195:
7416:
7241:
2882:
5254:
7051:
3860:
2634:
6750:. The distinguished sequences are called "exact sequences", hence the name. The precise axioms for this distinguished class do not matter for the construction of the Grothendieck group.
2010:
5291:
1826:
7826:
2179:
8429:
8315:
5189:
4503:
3189:
2688:
8598:(defined via direct sum of finitely generated projective modules) coincide. In fact, both groups are isomorphic to the free abelian group generated by the isomorphism classes of
6100:
5041:
4256:
3761:
4430:
4147:
5941:
4218:
1663:
6403:
5668:
4096:
3224:
2413:
5343:
8517:
8489:
8453:
8339:
7440:
7269:
7129:
6950:
6775:
6732:
6396:
5972:
5730:
5409:
1306:
623:
580:
548:
8087:
1115:
1069:
777:
7984:
6026:
5518:
5063:
4655:
4532:
4278:
3723:
3672:
3491:
3029:
1957:
1626:
1601:
3134:
817:
658:
4819:
7894:
5448:
1023:
850:
731:
8596:
8560:
8383:
6363:
3650:
3559:
3003:
2930:
2585:
2513:
2477:
2308:
2234:
1241:
1484:
and yields a group which satisfies the corresponding universal properties for semigroups, i.e. the "most general and smallest group containing a homomorphic image of
320:
218:
8863:
5784:
8668:
8633:
8113:
7199:
6132:
5363:
5121:
2754:
2364:
291:
250:
7208:
is an exact category if one just uses the standard interpretation of "exact". This gives the notion of a
Grothendieck group in the previous section if one chooses
6834:
5817:
in block triangular form one easily sees that character functions are additive in the above sense. By the universal property this gives us a "universal character"
4901:
4012:
1471:
1351:
3430:
2038:
4391:
4052:
3830:
2825:
8013:
6236:
6156:
6049:
5992:
5804:
5754:
5688:
5629:
5609:
5578:
5558:
5538:
5468:
5384:
5141:
5094:
4309:
6753:
The
Grothendieck group is defined in the same way as before as the abelian group with one generator for each (isomorphism class of) object(s) of the category
4456:
4339:
4173:
3787:
3698:
3614:
3588:
3359:
3081:
3055:
2956:
8124:
8780:
3872:
1838:
89:
336:
8719:
6245:
4625:{\displaystyle 0\to A\otimes _{\mathbb {Z} }\mathbb {Q} \to B\otimes _{\mathbb {Z} }\mathbb {Q} \to C\otimes _{\mathbb {Z} }\mathbb {Q} \to 0}
8749:
858:
7502:
6529:
8709:
7289:
3240:
4509:
4058:
7324:
with the canonical inclusion and projection morphisms. This procedure produces the
Grothendieck group of the commutative monoid
5820:
7327:
7134:
7060:
6881:
1628:. First one observes that the natural numbers (including 0) together with the usual addition indeed form a commutative monoid
7905:
1665:
Now when one uses the
Grothendieck group construction one obtains the formal differences between natural numbers as elements
6959:
6845:
8878:
8837:
6135:
8883:
7630:
7454:. The construction is essentially similar but uses the relations − + = 0 whenever there is a distinguished triangle
8832:
6161:
1679:
8635:
of a ring or a ringed space which is sometimes useful. The category in the case is chosen to be the category of all
7380:
7211:
2837:
1363:
8678:
positively graded algebra there is a subcategory in the unbounded derived category containing the abelian category
5194:
4066:
1516:
6991:
3835:
2590:
2322:
2046:
1357:
1974:
5259:
4810:
4103:
105:
7718:
8391:
8277:
1259:
5150:
4464:
3150:
2649:
2440:
5810:
5733:
2432:
1512:
85:
7490:, two vector spaces are isomorphic if and only if they have the same dimension. Thus, for a vector space
6513:{\displaystyle \cdots \to 0\to 0\to A^{n}\to A^{n+1}\to \cdots \to A^{m-1}\to A^{m}\to 0\to 0\to \cdots }
6058:
5008:
4223:
3728:
8827:
7451:
6878:
Alternatively and equivalently, one can define the
Grothendieck group using a universal property: A map
5588:
4396:
4113:
2336:
2259:
2255:
173:
5898:
4178:
1631:
6326:
onto the ring of Brauer characters. In this way
Grothendieck groups show up in representation theory.
5634:
5103:
The
Grothendieck group satisfies a universal property. One makes a preliminary definition: A function
4072:
3197:
2376:
8714:
6366:
5814:
5300:
3794:
2828:
2641:
2537:
2328:
2263:
1775:
996:
8498:
8470:
8434:
8320:
7421:
7250:
7110:
6931:
6756:
6713:
6376:
5949:
5693:
5390:
1865:
596:
553:
521:
8028:
7484:
6052:
4885:{\displaystyle \operatorname {rank} (A)=\dim _{\mathbb {Q} }(A\otimes _{\mathbb {Z} }\mathbb {Q} )}
2541:
2134:
1074:
1028:
736:
516:
327:
65:
7967:
6000:
5473:
5046:
4638:
4515:
4261:
3706:
3655:
3466:
3012:
1940:
1609:
1584:
8815:
8789:
8775:
8704:
8342:
6699:
In fact the
Grothendieck group was originally introduced for the study of Euler characteristics.
4458:
of the
Grothendieck group. Suppose one has the following short exact sequence of abelian groups:
3094:
2315:
1504:
1491:. This is known as the "group completion of a semigroup" or "group of fractions of a semigroup".
1309:
782:
628:
591:
158:
62:
43:
8385:
is then defined as the Grothendieck group of this exact category and again this gives a functor.
2246:
is defined to be the Grothendieck group of the commutative monoid of all isomorphism classes of
8859:
The Grothendieck Group of Algebraic Vector Bundles; Calculations of Affine and Projective Space
7843:
5414:
1265:
1002:
829:
710:
8888:
8763:
8745:
8565:
8529:
8352:
7283:
6735:
6332:
6198:
3619:
3499:
2972:
2899:
2554:
2482:
2446:
2367:
2325:
2277:
2203:
1959:. Indeed, this is the usual construction to obtain the integers from the natural numbers. See
1532:
1214:
1170:
705:
261:
101:
4995:{\displaystyle =\operatorname {rank} (B)=\operatorname {rank} (A)+\operatorname {rank} (C)=+}
296:
197:
8799:
8770:, (Notes taken by D.W.Anderson, Fall 1964), published in 1967, W.A. Benjamin Inc., New York.
7286:
is also exact if one declares those and only those sequences to be exact that have the form
7205:
6239:
5763:
5584:
2311:
1166:
109:
8811:
8646:
8611:
8098:
7184:
6105:
5348:
5106:
2703:
2342:
267:
223:
8807:
8519:
is an abelian category and a fortiori an exact category so the construction above applies.
7276:
6783:
6370:
3942:
2420:
2185:
1528:
1500:
1447:
1327:
142:
81:
3367:
2015:
8258:{\displaystyle \chi (V^{*})=\sum _{i}(-1)^{i}\dim V=\sum _{i}(-1)^{i}\dim H^{i}(V^{*}).}
4344:
4311:
Indeed, the observation made from the previous paragraph shows that every abelian group
4025:
3803:
2774:
8636:
8456:
7989:
6708:
6707:
A common generalization of these two concepts is given by the Grothendieck group of an
6212:
6141:
6034:
5977:
5789:
5739:
5673:
5614:
5594:
5563:
5543:
5523:
5453:
5369:
5126:
5068:
4283:
3936:
2416:
1604:
1321:
260:. In particular, in the case of a monoid operation denoted multiplicatively that has a
4435:
4318:
4152:
3766:
3677:
3593:
3567:
3338:
3060:
3034:
2935:
8872:
8819:
8683:
8599:
8464:
2548:
2247:
2237:
733:. The two coordinates are meant to represent a positive part and a negative part, so
323:
52:
4432:. Furthermore, the rank of the abelian group satisfies the conditions of the symbol
3925:{\displaystyle 0\to \mathbb {Z} \to \mathbb {Z} \to \mathbb {Z} /n\mathbb {Z} \to 0}
1927:{\displaystyle \forall n\in \mathbb {N} :\qquad {\begin{cases}n:=\\-n:=\end{cases}}}
8272:
7480:
7450:
Generalizing even further it is also possible to define the Grothendieck group for
6202:
6029:
4015:
3493:. The observation from the previous paragraph hence proves the following equation:
2963:
4809:
On the other hand, one also has the following relation; for more information, see
8639:
on the ringed space which reduces to the category of all modules over some ring
4657:-vector spaces, the sequence splits. Therefore, one has the following equation.
4018:
has its symbol equal to 0. This in turn implies that every finite abelian group
1444:. (Here +′ and −′ denote the addition and subtraction in the free abelian group
31:
17:
485:{\displaystyle x=1.x=(0^{-1}.0).x=0^{-1}.(0.x)=0^{-1}.(0.0)=(0^{-1}.0).0=1.0=0}
8850:
5995:
5757:
3006:
113:
2321:
is the Grothendieck group of the monoid consisting of isomorphism classes of
80:. The Grothendieck group construction takes its name from a specific case in
1563:
1555:
1535:
from the category of abelian groups to the category of commutative monoids.
1478:
61:
in the most universal way, in the sense that any abelian group containing a
8845:
6209:
elements. In this case the analogously defined map that associates to each
1971:
Similarly, the Grothendieck group of the multiplicative commutative monoid
8803:
6319:{\displaystyle G_{0}({\overline {\mathbb {F} _{p}}})\to \mathrm {BCh} (G)}
8526:
is a finite-dimensional algebra over some field, the Grothendieck groups
3439:-vector spaces have the same dimension. Also, any two finite-dimensional
2240:
2197:
1960:
1354:
691:
678:
being the "most general" abelian group containing a homomorphic image of
550:
satisfying the following universal property: for any monoid homomorphism
93:
8491:
being the category of finitely generated modules over a noetherian ring
1477:.) This construction has the advantage that it can be performed for any
1199:
is compatible with our equivalence relation, one obtains an addition on
1579:
1578:
The easiest example of a Grothendieck group is the construction of the
1508:
8562:(defined via short exact sequences of finitely generated modules) and
3451:
of same dimension are isomorphic to each other. In fact, every finite
3194:
Since any short exact sequence of vector spaces splits, it holds that
1191:
to be the set of equivalence classes. Since the addition operation on
984:{\displaystyle (m_{1},m_{2})+(n_{1},n_{2})=(m_{1}+n_{1},m_{2}+n_{2})}
97:
7608:{\displaystyle ={\big }\in K_{0}(\mathrm {Vect} _{\mathrm {fin} }).}
511:
be a commutative monoid. Its Grothendieck group is an abelian group
74:
will also contain a homomorphic image of the Grothendieck group of
8794:
8015:
Finally for a bounded complex of finite-dimensional vector spaces
6689:{\displaystyle =\sum _{i}(-1)^{i}=\sum _{i}(-1)^{i}\in G_{0}(R).}
3335:
The above equality hence satisfies the condition of the symbol
2373:
The two previous examples are related: consider the case where
7317:{\displaystyle A\hookrightarrow A\oplus B\twoheadrightarrow B}
3325:{\displaystyle \dim _{K}(V\oplus W)=\dim _{K}(V)+\dim _{K}(W)}
8674:
a functor, but nevertheless it carries important information.
8504:
8476:
8463:. This includes the special case (if the ringed space is an
8440:
8407:
8326:
8293:
7427:
7400:
7350:
7256:
7217:
7157:
7116:
7083:
6937:
6904:
6762:
6719:
5396:
2759:
This definition implies that for any two finitely generated
2254:
with the monoid operation given by direct sum. This gives a
2262:
to abelian groups. This functor is studied and extended in
1920:
4065:
is isomorphic to a direct sum of a torsion subgroup and a
3832:. The following short exact sequence holds, where the map
2196:
The Grothendieck group is the fundamental construction of
5888:{\displaystyle \chi :G_{0}(R)\to \mathrm {Hom} _{K}(R,K)}
7370:{\displaystyle (\mathrm {Iso} ({\mathcal {A}}),\oplus )}
7174:{\displaystyle \chi :\mathrm {Ob} ({\mathcal {A}})\to X}
7100:{\displaystyle \phi :\mathrm {Ob} ({\mathcal {A}})\to G}
6921:{\displaystyle \chi :\mathrm {Ob} ({\mathcal {A}})\to X}
5123:
from the set of isomorphism classes to an abelian group
4059:
fundamental theorem of finitely generated abelian groups
3226:. In fact, for any two finite-dimensional vector spaces
3140:. Suppose one has the following short exact sequence of
8864:
Grothendieck Group of a Smooth Projective Complex Curve
7957:{\displaystyle K_{0}(\mathrm {Vect} _{\mathrm {fin} })}
6738:
together with a class of distinguished short sequences
6981:{\displaystyle A\hookrightarrow B\twoheadrightarrow C}
6867:{\displaystyle A\hookrightarrow B\twoheadrightarrow C}
4407:
4195:
4124:
153:
always exists; it is called the Grothendieck group of
8649:
8614:
8568:
8532:
8501:
8473:
8437:
8394:
8355:
8323:
8280:
8127:
8101:
8031:
7992:
7970:
7908:
7846:
7721:
7633:
7505:
7424:
7383:
7330:
7292:
7253:
7214:
7187:
7137:
7113:
7063:
6994:
6962:
6934:
6884:
6848:
6786:
6759:
6716:
6532:
6406:
6379:
6335:
6248:
6215:
6164:
6144:
6108:
6061:
6037:
6003:
5980:
5952:
5901:
5823:
5792:
5766:
5742:
5696:
5676:
5637:
5617:
5597:
5566:
5546:
5526:
5476:
5456:
5417:
5411:
to the element representing its isomorphism class in
5393:
5372:
5351:
5303:
5262:
5197:
5153:
5129:
5109:
5071:
5049:
5011:
4904:
4822:
4666:
4641:
4543:
4518:
4467:
4438:
4399:
4347:
4321:
4286:
4264:
4226:
4181:
4155:
4116:
4075:
4054:
by the fundamental theorem of finite abelian groups.
4028:
3945:
3875:
3838:
3806:
3769:
3731:
3709:
3680:
3658:
3622:
3596:
3570:
3502:
3469:
3370:
3341:
3243:
3200:
3153:
3097:
3063:
3037:
3015:
2975:
2938:
2902:
2840:
2777:
2706:
2652:
2593:
2557:
2485:
2449:
2379:
2345:
2280:
2206:
2137:
2049:
2018:
1977:
1943:
1841:
1778:
1682:
1634:
1612:
1587:
1450:
1366:
1330:
1268:
1217:
1077:
1031:
1005:
861:
832:
785:
739:
713:
631:
599:
556:
524:
339:
299:
270:
226:
200:
2184:
which of course can be identified with the positive
8662:
8627:
8590:
8554:
8511:
8483:
8447:
8423:
8377:
8333:
8309:
8257:
8107:
8081:
8007:
7978:
7956:
7888:
7820:
7684:
7607:
7434:
7410:
7369:
7316:
7263:
7235:
7193:
7173:
7123:
7099:
7045:
6980:
6944:
6920:
6866:
6828:
6769:
6726:
6688:
6512:
6390:
6357:
6318:
6230:
6189:
6150:
6126:
6094:
6043:
6020:
5986:
5966:
5935:
5887:
5798:
5778:
5748:
5724:
5682:
5662:
5623:
5603:
5572:
5552:
5532:
5512:
5462:
5442:
5403:
5378:
5357:
5337:
5285:
5248:
5183:
5135:
5115:
5088:
5057:
5035:
4994:
4884:
4798:
4649:
4624:
4526:
4497:
4450:
4424:
4385:
4333:
4303:
4272:
4250:
4212:
4167:
4141:
4090:
4046:
4006:
3924:
3854:
3824:
3781:
3755:
3717:
3692:
3666:
3644:
3608:
3582:
3553:
3485:
3424:
3353:
3324:
3218:
3183:
3128:
3075:
3049:
3023:
2997:
2950:
2924:
2876:
2819:
2748:
2682:
2628:
2579:
2507:
2471:
2407:
2358:
2302:
2228:
2173:
2123:
2032:
2004:
1951:
1926:
1820:
1764:
1657:
1620:
1595:
1465:
1436:
1345:
1300:
1235:
1207:becomes an abelian group. The identity element of
1109:
1063:
1017:
983:
844:
811:
771:
725:
652:
617:
574:
542:
484:
314:
285:
244:
212:
7685:{\displaystyle 0\to k^{l}\to k^{m}\to k^{n}\to 0}
6956:is called "additive" if for every exact sequence
5760:that is given by multiplication with the element
3435:Note that any two isomorphic finite-dimensional
2640:-modules and the following relations: For every
8115:is the standard Euler characteristic defined by
6190:{\displaystyle {\overline {\mathbb {F} _{p}}},}
5631:-algebra, then one can associate the character
5580:is the only group homomorphism that does that.
3725:be the set of integers. The Grothendieck group
1765:{\displaystyle n-m\sim n'-m'\iff n+m'+k=n'+m+k}
1211:is , and the inverse of is . The homomorphism
662:This expresses the fact that any abelian group
8778:(2013), "Completions of Grothendieck groups",
8690:-adic completion of the Grothendieck group of
7479:In the abelian category of finite-dimensional
7446:Grothendieck groups of triangulated categories
7411:{\displaystyle \mathrm {Iso} ({\mathcal {A}})}
7236:{\displaystyle {\mathcal {A}}:=R{\text{-mod}}}
3616:with integer coefficients, which implies that
2877:{\displaystyle 0\to M\to M\oplus N\to N\to 0.}
1437:{\displaystyle \{(x+'y)-'(x+y)\mid x,y\in M\}}
7549:
7520:
5249:{\displaystyle \chi (A)-\chi (B)+\chi (C)=0.}
4635:Since the above is a short exact sequence of
2636:of isomorphism classes of finitely generated
2192:Example: the Grothendieck group of a manifold
2012:(starting at 1) consists of formal fractions
330:with only one element), since one must have
8:
7046:{\displaystyle \chi (A)-\chi (B)+\chi (C)=0}
3855:{\displaystyle \mathbb {Z} \to \mathbb {Z} }
2629:{\displaystyle \{\mid X\in R{\text{-mod}}\}}
2623:
2594:
1431:
1367:
161:and can also be concretely constructed from
27:Abelian group extending a commutative monoid
8781:Bulletin of the London Mathematical Society
8740:Bruns, Winfried; Gubeladze, Joseph (2009).
7418:means the "set" of isomorphism classes in
2528:Another construction that carries the name
2124:{\displaystyle p/q\sim p'/q'\iff pq'r=p'qr}
1473:while + denotes the addition in the monoid
3789:for any finitely generated abelian groups
2587:as the abelian group generated by the set
2145:
2141:
2089:
2085:
2005:{\displaystyle (\mathbb {N} ^{*},\times )}
1786:
1782:
1718:
1714:
120:Grothendieck group of a commutative monoid
8793:
8654:
8648:
8619:
8613:
8573:
8567:
8537:
8531:
8503:
8502:
8500:
8475:
8474:
8472:
8439:
8438:
8436:
8412:
8406:
8405:
8393:
8360:
8354:
8325:
8324:
8322:
8298:
8292:
8291:
8279:
8243:
8230:
8214:
8195:
8173:
8154:
8138:
8126:
8100:
8061:
8039:
8030:
7991:
7972:
7971:
7969:
7938:
7937:
7923:
7913:
7907:
7845:
7778:
7757:
7730:
7720:
7670:
7657:
7644:
7632:
7586:
7585:
7571:
7561:
7548:
7547:
7529:
7519:
7518:
7504:
7426:
7425:
7423:
7399:
7398:
7384:
7382:
7349:
7348:
7334:
7329:
7291:
7255:
7254:
7252:
7228:
7216:
7215:
7213:
7186:
7156:
7155:
7144:
7136:
7115:
7114:
7112:
7082:
7081:
7070:
7062:
6993:
6961:
6936:
6935:
6933:
6903:
6902:
6891:
6883:
6847:
6785:
6761:
6760:
6758:
6718:
6717:
6715:
6668:
6649:
6636:
6623:
6604:
6588:
6575:
6556:
6540:
6531:
6486:
6467:
6442:
6429:
6405:
6383:
6378:
6340:
6334:
6296:
6270:
6266:
6265:
6262:
6253:
6247:
6214:
6173:
6169:
6168:
6165:
6163:
6143:
6107:
6076:
6075:
6066:
6060:
6036:
6005:
6004:
6002:
5979:
5960:
5959:
5951:
5927:
5900:
5864:
5853:
5834:
5822:
5791:
5765:
5741:
5707:
5695:
5675:
5642:
5636:
5616:
5596:
5565:
5545:
5525:
5475:
5455:
5422:
5416:
5395:
5394:
5392:
5371:
5350:
5314:
5302:
5286:{\displaystyle \chi :R{\text{-mod}}\to X}
5272:
5261:
5196:
5152:
5128:
5108:
5076:
5075:
5070:
5051:
5050:
5048:
5026:
5025:
5016:
5010:
4903:
4895:Therefore, the following equation holds:
4875:
4874:
4868:
4867:
4866:
4847:
4846:
4845:
4821:
4789:
4788:
4782:
4781:
4780:
4761:
4760:
4759:
4745:
4744:
4738:
4737:
4736:
4717:
4716:
4715:
4701:
4700:
4694:
4693:
4692:
4673:
4672:
4671:
4665:
4643:
4642:
4640:
4612:
4611:
4605:
4604:
4603:
4589:
4588:
4582:
4581:
4580:
4566:
4565:
4559:
4558:
4557:
4542:
4520:
4519:
4517:
4466:
4437:
4406:
4398:
4376:
4375:
4357:
4353:
4352:
4346:
4320:
4291:
4290:
4285:
4266:
4265:
4263:
4241:
4240:
4231:
4225:
4194:
4180:
4154:
4123:
4115:
4082:
4078:
4077:
4074:
4027:
3991:
3990:
3977:
3976:
3963:
3962:
3954:
3950:
3949:
3944:
3912:
3911:
3903:
3899:
3898:
3891:
3890:
3883:
3882:
3874:
3848:
3847:
3840:
3839:
3837:
3805:
3768:
3763:is an abelian group generated by symbols
3746:
3745:
3736:
3730:
3711:
3710:
3708:
3679:
3660:
3659:
3657:
3627:
3621:
3595:
3569:
3523:
3501:
3474:
3468:
3369:
3340:
3304:
3279:
3248:
3242:
3199:
3152:
3114:
3096:
3062:
3036:
3017:
3016:
3014:
2980:
2974:
2937:
2932:is an abelian group generated by symbols
2907:
2901:
2839:
2776:
2705:
2651:
2618:
2592:
2562:
2556:
2490:
2484:
2454:
2448:
2390:
2378:
2350:
2344:
2335:, with the monoid operation given by the
2285:
2279:
2270:Example: The Grothendieck group of a ring
2211:
2205:
2136:
2072:
2053:
2048:
2022:
2017:
1987:
1983:
1982:
1976:
1945:
1944:
1942:
1860:
1852:
1851:
1840:
1777:
1681:
1639:
1638:
1633:
1614:
1613:
1611:
1589:
1588:
1586:
1562:has the cancellation property, and it is
1449:
1365:
1329:
1267:
1216:
1098:
1085:
1076:
1052:
1039:
1030:
1004:
972:
959:
946:
933:
914:
901:
882:
869:
860:
831:
803:
790:
784:
760:
747:
738:
712:
670:will also contain a homomorphic image of
630:
598:
555:
523:
452:
421:
390:
362:
338:
298:
269:
225:
199:
55:. This abelian group is constructed from
7821:{\displaystyle \left=\left+\left=(l+n).}
6365:the 'universal receiver' of generalized
2896:be a field. Then the Grothendieck group
8732:
6703:Grothendieck groups of exact categories
5583:Examples of additive functions are the
692:K-theory § Grothendieck completion
92:, which resulted in the development of
8424:{\displaystyle (X,{\mathcal {O}}_{X})}
8310:{\displaystyle (X,{\mathcal {O}}_{X})}
7275:was assumed to be artinian (and hence
6734:. Simply put, an exact category is an
5387:and the map that takes each object of
1967:Example: the positive rational numbers
1511:; one thus obtains a functor from the
1250:Alternatively, the Grothendieck group
6051:then this character map even gives a
5184:{\displaystyle 0\to A\to B\to C\to 0}
4498:{\displaystyle 0\to A\to B\to C\to 0}
3184:{\displaystyle 0\to V\to T\to W\to 0}
2683:{\displaystyle 0\to A\to B\to C\to 0}
2551:. Then define the Grothendieck group
1673:and one has the equivalence relation
666:that contains a homomorphic image of
7:
8608:There is another Grothendieck group
7107:is called the Grothendieck group of
3136:, the dimension of the vector space
696:To construct the Grothendieck group
141:is to be constructed by introducing
8720:Atiyah–Hirzebruch spectral sequence
8431:, one can also define the category
7243:the category of finitely generated
6329:This universal property also makes
6095:{\displaystyle G_{0}(\mathbb {C} )}
5036:{\displaystyle G_{0}(\mathbb {Z} )}
4251:{\displaystyle G_{0}(\mathbb {Z} )}
3756:{\displaystyle G_{0}(\mathbb {Z} )}
1519:which sends the commutative monoid
1169:does not hold in all monoids). The
322:the Grothendieck group must be the
157:. It is characterized by a certain
133:, "the most general" abelian group
8722:for computing topological K-theory
7945:
7942:
7939:
7933:
7930:
7927:
7924:
7593:
7590:
7587:
7581:
7578:
7575:
7572:
7391:
7388:
7385:
7341:
7338:
7335:
7148:
7145:
7074:
7071:
7057:together with an additive mapping
6895:
6892:
6303:
6300:
6297:
5860:
5857:
5854:
4425:{\displaystyle r={\mbox{rank}}(A)}
4142:{\displaystyle r={\mbox{rank}}(A)}
2391:
1842:
25:
7271:. This is really abelian because
5936:{\displaystyle \chi ()=\chi _{V}}
4213:{\displaystyle ={\mbox{rank}}(A)}
2519:Grothendieck group and extensions
1963:for a more detailed explanation.
1658:{\displaystyle (\mathbb {N} ,+).}
90:Grothendieck–Riemann–Roch theorem
8686:whose Grothendieck group is the
8317:, one can consider the category
5663:{\displaystyle \chi _{V}:R\to k}
5256:Then, for any additive function
4534:implies the following equation.
4091:{\displaystyle \mathbb {Z} ^{r}}
3219:{\displaystyle T\cong V\oplus W}
2408:{\displaystyle R=C^{\infty }(M)}
7621:Moreover, for an exact sequence
5338:{\displaystyle f:G_{0}(R)\to X}
3031:whose generator is the element
1859:
1821:{\displaystyle k\iff n+m'=n'+m}
252:), then the Grothendieck group
8742:Polytopes, Rings, and K-Theory
8585:
8579:
8549:
8543:
8512:{\displaystyle {\mathcal {A}}}
8484:{\displaystyle {\mathcal {A}}}
8448:{\displaystyle {\mathcal {A}}}
8418:
8395:
8372:
8366:
8334:{\displaystyle {\mathcal {A}}}
8304:
8281:
8249:
8236:
8211:
8201:
8170:
8160:
8144:
8131:
8076:
8070:
8067:
8054:
8045:
8032:
7999:
7993:
7951:
7919:
7880:
7874:
7871:
7865:
7853:
7847:
7812:
7806:
7803:
7791:
7676:
7663:
7650:
7637:
7599:
7567:
7542:
7536:
7512:
7506:
7435:{\displaystyle {\mathcal {A}}}
7405:
7395:
7364:
7355:
7345:
7331:
7308:
7296:
7264:{\displaystyle {\mathcal {A}}}
7165:
7162:
7152:
7124:{\displaystyle {\mathcal {A}}}
7091:
7088:
7078:
7034:
7028:
7019:
7013:
7004:
6998:
6972:
6966:
6945:{\displaystyle {\mathcal {A}}}
6912:
6909:
6899:
6858:
6852:
6817:
6811:
6805:
6799:
6793:
6787:
6770:{\displaystyle {\mathcal {A}}}
6727:{\displaystyle {\mathcal {A}}}
6680:
6674:
6658:
6655:
6642:
6629:
6620:
6610:
6594:
6581:
6572:
6562:
6546:
6533:
6504:
6498:
6492:
6479:
6460:
6454:
6435:
6422:
6416:
6410:
6391:{\displaystyle R{\text{-mod}}}
6352:
6346:
6313:
6307:
6293:
6290:
6287:
6281:
6259:
6242:is also a natural isomorphism
6225:
6219:
6121:
6115:
6089:
6086:
6080:
6072:
6015:
6009:
5967:{\displaystyle k=\mathbb {C} }
5917:
5914:
5908:
5905:
5882:
5870:
5849:
5846:
5840:
5813:and writing the corresponding
5725:{\displaystyle V:\chi _{V}(x)}
5719:
5713:
5654:
5507:
5501:
5492:
5489:
5483:
5480:
5434:
5428:
5404:{\displaystyle {\mathcal {A}}}
5329:
5326:
5320:
5277:
5237:
5231:
5222:
5216:
5207:
5201:
5175:
5169:
5163:
5157:
5080:
5072:
5030:
5022:
4989:
4983:
4977:
4971:
4965:
4959:
4947:
4941:
4929:
4923:
4911:
4905:
4879:
4856:
4835:
4829:
4793:
4770:
4749:
4726:
4705:
4682:
4616:
4593:
4570:
4547:
4489:
4483:
4477:
4471:
4445:
4439:
4419:
4413:
4380:
4372:
4363:
4348:
4328:
4322:
4295:
4287:
4245:
4237:
4220:. Then the Grothendieck group
4207:
4201:
4188:
4182:
4162:
4156:
4136:
4130:
4098:for some non-negative integer
4035:
4029:
3995:
3987:
3981:
3973:
3967:
3946:
3916:
3895:
3887:
3879:
3844:
3813:
3807:
3776:
3770:
3750:
3742:
3687:
3681:
3639:
3633:
3603:
3597:
3577:
3571:
3548:
3542:
3509:
3503:
3419:
3413:
3407:
3401:
3395:
3383:
3377:
3371:
3348:
3342:
3319:
3313:
3294:
3288:
3269:
3257:
3175:
3169:
3163:
3157:
3104:
3098:
3070:
3064:
3044:
3038:
2992:
2986:
2945:
2939:
2919:
2913:
2868:
2862:
2850:
2844:
2814:
2808:
2802:
2796:
2790:
2778:
2737:
2731:
2725:
2719:
2713:
2707:
2674:
2668:
2662:
2656:
2603:
2597:
2574:
2568:
2502:
2496:
2466:
2460:
2427:. In this case the projective
2402:
2396:
2297:
2291:
2223:
2217:
2142:
2086:
1999:
1978:
1914:
1902:
1886:
1874:
1783:
1715:
1649:
1635:
1515:of commutative monoids to the
1460:
1454:
1410:
1398:
1387:
1370:
1340:
1334:
1295:
1281:
1275:
1269:
1258:can also be constructed using
1227:
1187:) is denoted by . One defines
1104:
1078:
1058:
1032:
978:
926:
920:
894:
888:
862:
766:
740:
618:{\displaystyle g\colon K\to A}
609:
575:{\displaystyle f\colon M\to A}
566:
543:{\displaystyle i\colon M\to K}
534:
464:
445:
439:
433:
411:
402:
374:
355:
1:
8082:{\displaystyle =\chi (V^{*})}
6136:modular representation theory
5520:for every finitely generated
2274:The zeroth algebraic K group
2174:{\displaystyle r\iff pq'=p'q}
1961:"Construction" under Integers
1507:construction gives rise to a
1110:{\displaystyle (n_{1},n_{2})}
1064:{\displaystyle (m_{1},m_{2})}
772:{\displaystyle (m_{1},m_{2})}
7979:{\displaystyle \mathbb {Z} }
6523:one has a canonical element
6276:
6179:
6021:{\displaystyle \mathbb {C} }
5670:to every finite-dimensional
5513:{\displaystyle f()=\chi (V)}
5147:if, for each exact sequence
5058:{\displaystyle \mathbb {Z} }
4650:{\displaystyle \mathbb {Q} }
4527:{\displaystyle \mathbb {Q} }
4273:{\displaystyle \mathbb {Z} }
3718:{\displaystyle \mathbb {Z} }
3667:{\displaystyle \mathbb {Z} }
3590:is generated by the element
3486:{\displaystyle K^{\oplus n}}
3024:{\displaystyle \mathbb {Z} }
2366:is a covariant functor from
1952:{\displaystyle \mathbb {Z} }
1621:{\displaystyle \mathbb {N} }
1596:{\displaystyle \mathbb {Z} }
852:is defined coordinate-wise:
96:. This specific case is the
8833:Encyclopedia of Mathematics
8643:in case of affine schemes.
7279:) in the previous section.
6369:. In particular, for every
5450:Concretely this means that
3793:. One first notes that any
3361:in the Grothendieck group.
3129:{\displaystyle =\dim _{K}V}
2958:for any finite-dimensional
2697:-modules, add the relation
812:{\displaystyle m_{1}-m_{2}}
653:{\displaystyle f=g\circ i.}
129:Given a commutative monoid
8905:
8455:to be the category of all
4512:with the rational numbers
4067:torsion-free abelian group
1937:This defines the integers
1523:to its Grothendieck group
1517:category of abelian groups
689:
7889:{\displaystyle =\dim(V),}
7377:in the first sense (here
7282:On the other hand, every
7181:factors uniquely through
5443:{\displaystyle G_{0}(R).}
5005:Hence one has shown that
3083:for a finite-dimensional
1538:For a commutative monoid
1316:, the Grothendieck group
1301:{\displaystyle (Z(M),+')}
1165:is necessary because the
1018:{\displaystyle M\times M}
845:{\displaystyle M\times M}
726:{\displaystyle M\times M}
8744:. Springer. p. 50.
8591:{\displaystyle K_{0}(R)}
8555:{\displaystyle G_{0}(R)}
8378:{\displaystyle K_{0}(X)}
6839:for each exact sequence
6358:{\displaystyle G_{0}(R)}
5611:is a finite-dimensional
4811:Rank of an abelian group
3645:{\displaystyle G_{0}(K)}
3554:{\displaystyle =\left=n}
2998:{\displaystyle G_{0}(K)}
2925:{\displaystyle G_{0}(K)}
2580:{\displaystyle G_{0}(R)}
2536:be a finite-dimensional
2508:{\displaystyle K_{0}(M)}
2472:{\displaystyle K_{0}(R)}
2303:{\displaystyle K_{0}(R)}
2229:{\displaystyle K_{0}(M)}
1260:generators and relations
1236:{\displaystyle i:M\to K}
700:of a commutative monoid
149:. Such an abelian group
7452:triangulated categories
7131:iff every additive map
6102:and the character ring
5809:By choosing a suitable
5470:satisfies the equation
4341:the same to the symbol
2435:to vector bundles over
315:{\displaystyle x\in M,}
213:{\displaystyle a\neq b}
176:(that is, there exists
8682:of finite-dimensional
8664:
8637:quasi-coherent sheaves
8629:
8592:
8556:
8513:
8485:
8449:
8425:
8379:
8335:
8311:
8259:
8109:
8083:
8009:
7980:
7958:
7890:
7822:
7686:
7609:
7436:
7412:
7371:
7318:
7265:
7237:
7195:
7175:
7125:
7101:
7047:
6982:
6952:into an abelian group
6946:
6922:
6868:
6830:
6771:
6728:
6690:
6514:
6392:
6359:
6320:
6232:
6191:
6152:
6128:
6096:
6045:
6022:
5988:
5968:
5937:
5889:
5800:
5780:
5779:{\displaystyle x\in R}
5750:
5726:
5684:
5664:
5625:
5605:
5574:
5554:
5534:
5514:
5464:
5444:
5405:
5380:
5359:
5339:
5287:
5250:
5185:
5137:
5117:
5090:
5059:
5037:
4996:
4886:
4800:
4651:
4626:
4528:
4499:
4452:
4426:
4387:
4335:
4305:
4274:
4252:
4214:
4169:
4143:
4092:
4061:, every abelian group
4048:
4008:
3926:
3856:
3826:
3783:
3757:
3719:
3694:
3668:
3646:
3610:
3584:
3555:
3487:
3426:
3355:
3326:
3220:
3185:
3130:
3077:
3051:
3025:
2999:
2952:
2926:
2878:
2821:
2750:
2684:
2630:
2581:
2532:is the following: Let
2509:
2473:
2423:on a compact manifold
2409:
2360:
2310:of a (not necessarily
2304:
2230:
2175:
2125:
2040:with the equivalence
2034:
2006:
1953:
1928:
1822:
1766:
1659:
1622:
1597:
1467:
1438:
1347:
1302:
1237:
1111:
1065:
1019:
985:
846:
813:
773:
727:
686:Explicit constructions
654:
619:
576:
544:
486:
316:
287:
246:
214:
86:Alexander Grothendieck
8665:
8663:{\displaystyle G_{0}}
8630:
8628:{\displaystyle G_{0}}
8593:
8557:
8514:
8486:
8450:
8426:
8380:
8336:
8312:
8260:
8110:
8108:{\displaystyle \chi }
8084:
8010:
7981:
7959:
7891:
7823:
7687:
7610:
7437:
7413:
7372:
7319:
7266:
7238:
7196:
7194:{\displaystyle \phi }
7176:
7126:
7102:
7048:
6983:
6947:
6923:
6869:
6831:
6772:
6729:
6691:
6515:
6393:
6367:Euler characteristics
6360:
6321:
6233:
6192:
6153:
6129:
6127:{\displaystyle Ch(G)}
6097:
6046:
6023:
5989:
5969:
5938:
5890:
5801:
5781:
5751:
5732:is defined to be the
5727:
5685:
5665:
5626:
5606:
5589:representation theory
5575:
5555:
5535:
5515:
5465:
5445:
5406:
5381:
5360:
5358:{\displaystyle \chi }
5340:
5288:
5251:
5186:
5138:
5118:
5116:{\displaystyle \chi }
5091:
5060:
5038:
4997:
4887:
4801:
4652:
4627:
4529:
4500:
4453:
4427:
4388:
4336:
4306:
4275:
4253:
4215:
4170:
4144:
4093:
4049:
4009:
3927:
3862:is multiplication by
3857:
3827:
3784:
3758:
3720:
3695:
3669:
3647:
3611:
3585:
3556:
3488:
3427:
3356:
3327:
3234:the following holds:
3221:
3186:
3131:
3078:
3052:
3026:
3000:
2953:
2927:
2879:
2831:short exact sequence
2822:
2751:
2749:{\displaystyle -+=0.}
2685:
2631:
2582:
2547:or more generally an
2510:
2474:
2410:
2361:
2359:{\displaystyle K_{0}}
2305:
2256:contravariant functor
2231:
2176:
2126:
2035:
2007:
1954:
1929:
1823:
1767:
1660:
1623:
1598:
1574:Example: the integers
1468:
1439:
1348:
1312:generated by the set
1303:
1238:
1117:if, for some element
1112:
1066:
1020:
986:
847:
814:
774:
728:
655:
620:
577:
545:
487:
317:
288:
286:{\displaystyle 0.x=0}
247:
245:{\displaystyle ac=bc}
215:
174:cancellation property
8879:Algebraic structures
8846:"Grothendieck group"
8828:"Grothendieck group"
8715:Topological K-theory
8647:
8612:
8566:
8530:
8499:
8471:
8435:
8392:
8353:
8343:locally free sheaves
8321:
8278:
8125:
8099:
8029:
7990:
7986:and is generated by
7968:
7906:
7844:
7719:
7631:
7503:
7422:
7381:
7328:
7290:
7251:
7212:
7185:
7135:
7111:
7061:
6992:
6960:
6932:
6882:
6846:
6829:{\displaystyle -+=0}
6784:
6757:
6714:
6530:
6404:
6377:
6333:
6246:
6213:
6162:
6142:
6106:
6059:
6035:
6001:
5978:
5950:
5899:
5821:
5790:
5764:
5740:
5694:
5674:
5635:
5615:
5595:
5564:
5544:
5524:
5474:
5454:
5415:
5391:
5370:
5349:
5301:
5260:
5195:
5151:
5127:
5107:
5069:
5047:
5009:
4902:
4820:
4664:
4639:
4541:
4516:
4465:
4436:
4397:
4345:
4319:
4284:
4262:
4224:
4179:
4153:
4149:. Define the symbol
4114:
4073:
4057:Observe that by the
4026:
4007:{\displaystyle =-=0}
3943:
3873:
3836:
3804:
3795:finite abelian group
3767:
3729:
3707:
3703:More generally, let
3678:
3656:
3620:
3594:
3568:
3564:Hence, every symbol
3500:
3467:
3368:
3339:
3241:
3198:
3151:
3095:
3061:
3035:
3013:
2973:
2936:
2900:
2838:
2775:
2704:
2650:
2642:short exact sequence
2591:
2555:
2515:are the same group.
2483:
2447:
2377:
2343:
2278:
2264:topological K-theory
2204:
2135:
2047:
2016:
1975:
1941:
1839:
1776:
1680:
1632:
1610:
1603:from the (additive)
1585:
1570:is already a group.
1466:{\displaystyle Z(M)}
1448:
1364:
1346:{\displaystyle Z(M)}
1328:
1266:
1215:
1075:
1029:
1003:
997:equivalence relation
995:Next one defines an
859:
830:
783:
737:
711:
629:
597:
590:, there is a unique
586:to an abelian group
554:
522:
337:
297:
268:
224:
198:
88:in his proof of the
40:group of differences
8884:Homological algebra
8804:10.1112/blms/bds079
8776:Stroppel, Catharina
8388:For a ringed space
7053:; an abelian group
6053:natural isomorphism
5297:group homomorphism
3674:with the generator
3425:{\displaystyle ==+}
3057:. Here, the symbol
2370:to abelian groups.
2033:{\displaystyle p/q}
1499:In the language of
517:monoid homomorphism
145:to all elements of
102:isomorphism classes
8774:Achar, Pramod N.;
8705:Field of fractions
8660:
8625:
8588:
8552:
8522:In the case where
8509:
8481:
8445:
8421:
8375:
8331:
8307:
8255:
8200:
8159:
8105:
8079:
8005:
7976:
7954:
7886:
7818:
7682:
7605:
7432:
7408:
7367:
7314:
7261:
7233:
7191:
7171:
7121:
7097:
7043:
6978:
6942:
6918:
6864:
6826:
6767:
6724:
6686:
6609:
6561:
6510:
6388:
6355:
6316:
6228:
6187:
6148:
6138:of finite groups,
6124:
6092:
6041:
6018:
5984:
5964:
5933:
5885:
5796:
5776:
5746:
5722:
5680:
5660:
5621:
5601:
5585:character function
5570:
5550:
5530:
5510:
5460:
5440:
5401:
5376:
5355:
5335:
5283:
5246:
5181:
5133:
5113:
5099:Universal Property
5086:
5055:
5033:
4992:
4882:
4796:
4647:
4622:
4524:
4495:
4448:
4422:
4411:
4386:{\displaystyle =r}
4383:
4331:
4301:
4270:
4248:
4210:
4199:
4165:
4139:
4128:
4088:
4047:{\displaystyle =0}
4044:
4004:
3922:
3852:
3825:{\displaystyle =0}
3822:
3779:
3753:
3715:
3690:
3664:
3642:
3606:
3580:
3551:
3483:
3422:
3351:
3322:
3216:
3181:
3126:
3073:
3047:
3021:
2995:
2948:
2922:
2874:
2820:{\displaystyle =+}
2817:
2746:
2680:
2626:
2577:
2530:Grothendieck group
2505:
2469:
2441:Serre–Swan theorem
2405:
2356:
2323:finitely generated
2300:
2250:of finite rank on
2226:
2171:
2121:
2030:
2002:
1949:
1924:
1919:
1818:
1762:
1655:
1618:
1593:
1527:. This functor is
1463:
1434:
1343:
1310:free abelian group
1298:
1243:sends the element
1233:
1107:
1061:
1015:
981:
842:
809:
769:
723:
650:
615:
592:group homomorphism
572:
540:
503:Universal property
482:
312:
283:
242:
210:
172:does not have the
159:universal property
116:as its operation.
44:commutative monoid
36:Grothendieck group
8764:Michael F. Atiyah
8751:978-0-387-76355-2
8191:
8150:
8008:{\displaystyle .}
7964:is isomorphic to
7284:additive category
7231:
6777:and one relation
6736:additive category
6600:
6552:
6386:
6279:
6231:{\displaystyle k}
6199:algebraic closure
6182:
6151:{\displaystyle k}
6044:{\displaystyle G}
5987:{\displaystyle R}
5799:{\displaystyle V}
5749:{\displaystyle k}
5683:{\displaystyle R}
5624:{\displaystyle k}
5604:{\displaystyle R}
5573:{\displaystyle f}
5553:{\displaystyle V}
5533:{\displaystyle R}
5463:{\displaystyle f}
5379:{\displaystyle f}
5275:
5136:{\displaystyle X}
5089:{\displaystyle .}
5043:is isomorphic to
4410:
4304:{\displaystyle .}
4258:is isomorphic to
4198:
4127:
3652:is isomorphic to
3463:is isomorphic to
2827:, because of the
2621:
1533:forgetful functor
1171:equivalence class
1071:is equivalent to
706:Cartesian product
137:that arises from
16:(Redirected from
8896:
8855:
8841:
8822:
8797:
8756:
8755:
8737:
8669:
8667:
8666:
8661:
8659:
8658:
8634:
8632:
8631:
8626:
8624:
8623:
8597:
8595:
8594:
8589:
8578:
8577:
8561:
8559:
8558:
8553:
8542:
8541:
8518:
8516:
8515:
8510:
8508:
8507:
8495:. In both cases
8490:
8488:
8487:
8482:
8480:
8479:
8457:coherent sheaves
8454:
8452:
8451:
8446:
8444:
8443:
8430:
8428:
8427:
8422:
8417:
8416:
8411:
8410:
8384:
8382:
8381:
8376:
8365:
8364:
8340:
8338:
8337:
8332:
8330:
8329:
8316:
8314:
8313:
8308:
8303:
8302:
8297:
8296:
8264:
8262:
8261:
8256:
8248:
8247:
8235:
8234:
8219:
8218:
8199:
8178:
8177:
8158:
8143:
8142:
8114:
8112:
8111:
8106:
8088:
8086:
8085:
8080:
8066:
8065:
8044:
8043:
8014:
8012:
8011:
8006:
7985:
7983:
7982:
7977:
7975:
7963:
7961:
7960:
7955:
7950:
7949:
7948:
7936:
7918:
7917:
7895:
7893:
7892:
7887:
7827:
7825:
7824:
7819:
7787:
7783:
7782:
7766:
7762:
7761:
7745:
7741:
7740:
7691:
7689:
7688:
7683:
7675:
7674:
7662:
7661:
7649:
7648:
7614:
7612:
7611:
7606:
7598:
7597:
7596:
7584:
7566:
7565:
7553:
7552:
7546:
7545:
7524:
7523:
7474:Further examples
7441:
7439:
7438:
7433:
7431:
7430:
7417:
7415:
7414:
7409:
7404:
7403:
7394:
7376:
7374:
7373:
7368:
7354:
7353:
7344:
7323:
7321:
7320:
7315:
7270:
7268:
7267:
7262:
7260:
7259:
7242:
7240:
7239:
7234:
7232:
7229:
7221:
7220:
7206:abelian category
7200:
7198:
7197:
7192:
7180:
7178:
7177:
7172:
7161:
7160:
7151:
7130:
7128:
7127:
7122:
7120:
7119:
7106:
7104:
7103:
7098:
7087:
7086:
7077:
7052:
7050:
7049:
7044:
6987:
6985:
6984:
6979:
6951:
6949:
6948:
6943:
6941:
6940:
6927:
6925:
6924:
6919:
6908:
6907:
6898:
6873:
6871:
6870:
6865:
6835:
6833:
6832:
6827:
6776:
6774:
6773:
6768:
6766:
6765:
6733:
6731:
6730:
6725:
6723:
6722:
6695:
6693:
6692:
6687:
6673:
6672:
6654:
6653:
6641:
6640:
6628:
6627:
6608:
6593:
6592:
6580:
6579:
6560:
6545:
6544:
6519:
6517:
6516:
6511:
6491:
6490:
6478:
6477:
6453:
6452:
6434:
6433:
6397:
6395:
6394:
6389:
6387:
6384:
6364:
6362:
6361:
6356:
6345:
6344:
6325:
6323:
6322:
6317:
6306:
6280:
6275:
6274:
6269:
6263:
6258:
6257:
6240:Brauer character
6237:
6235:
6234:
6229:
6196:
6194:
6193:
6188:
6183:
6178:
6177:
6172:
6166:
6157:
6155:
6154:
6149:
6133:
6131:
6130:
6125:
6101:
6099:
6098:
6093:
6079:
6071:
6070:
6050:
6048:
6047:
6042:
6027:
6025:
6024:
6019:
6008:
5993:
5991:
5990:
5985:
5973:
5971:
5970:
5965:
5963:
5942:
5940:
5939:
5934:
5932:
5931:
5894:
5892:
5891:
5886:
5869:
5868:
5863:
5839:
5838:
5805:
5803:
5802:
5797:
5785:
5783:
5782:
5777:
5755:
5753:
5752:
5747:
5731:
5729:
5728:
5723:
5712:
5711:
5689:
5687:
5686:
5681:
5669:
5667:
5666:
5661:
5647:
5646:
5630:
5628:
5627:
5622:
5610:
5608:
5607:
5602:
5579:
5577:
5576:
5571:
5559:
5557:
5556:
5551:
5539:
5537:
5536:
5531:
5519:
5517:
5516:
5511:
5469:
5467:
5466:
5461:
5449:
5447:
5446:
5441:
5427:
5426:
5410:
5408:
5407:
5402:
5400:
5399:
5385:
5383:
5382:
5377:
5365:factors through
5364:
5362:
5361:
5356:
5344:
5342:
5341:
5336:
5319:
5318:
5292:
5290:
5289:
5284:
5276:
5273:
5255:
5253:
5252:
5247:
5190:
5188:
5187:
5182:
5142:
5140:
5139:
5134:
5122:
5120:
5119:
5114:
5095:
5093:
5092:
5087:
5079:
5064:
5062:
5061:
5056:
5054:
5042:
5040:
5039:
5034:
5029:
5021:
5020:
5001:
4999:
4998:
4993:
4891:
4889:
4888:
4883:
4878:
4873:
4872:
4871:
4852:
4851:
4850:
4805:
4803:
4802:
4797:
4792:
4787:
4786:
4785:
4766:
4765:
4764:
4748:
4743:
4742:
4741:
4722:
4721:
4720:
4704:
4699:
4698:
4697:
4678:
4677:
4676:
4656:
4654:
4653:
4648:
4646:
4631:
4629:
4628:
4623:
4615:
4610:
4609:
4608:
4592:
4587:
4586:
4585:
4569:
4564:
4563:
4562:
4533:
4531:
4530:
4525:
4523:
4504:
4502:
4501:
4496:
4457:
4455:
4454:
4451:{\displaystyle }
4449:
4431:
4429:
4428:
4423:
4412:
4408:
4392:
4390:
4389:
4384:
4379:
4362:
4361:
4356:
4340:
4338:
4337:
4334:{\displaystyle }
4332:
4310:
4308:
4307:
4302:
4294:
4279:
4277:
4276:
4271:
4269:
4257:
4255:
4254:
4249:
4244:
4236:
4235:
4219:
4217:
4216:
4211:
4200:
4196:
4174:
4172:
4171:
4168:{\displaystyle }
4166:
4148:
4146:
4145:
4140:
4129:
4125:
4097:
4095:
4094:
4089:
4087:
4086:
4081:
4053:
4051:
4050:
4045:
4013:
4011:
4010:
4005:
3994:
3980:
3966:
3958:
3953:
3931:
3929:
3928:
3923:
3915:
3907:
3902:
3894:
3886:
3861:
3859:
3858:
3853:
3851:
3843:
3831:
3829:
3828:
3823:
3788:
3786:
3785:
3782:{\displaystyle }
3780:
3762:
3760:
3759:
3754:
3749:
3741:
3740:
3724:
3722:
3721:
3716:
3714:
3699:
3697:
3696:
3693:{\displaystyle }
3691:
3673:
3671:
3670:
3665:
3663:
3651:
3649:
3648:
3643:
3632:
3631:
3615:
3613:
3612:
3609:{\displaystyle }
3607:
3589:
3587:
3586:
3583:{\displaystyle }
3581:
3560:
3558:
3557:
3552:
3535:
3531:
3530:
3492:
3490:
3489:
3484:
3482:
3481:
3431:
3429:
3428:
3423:
3360:
3358:
3357:
3354:{\displaystyle }
3352:
3331:
3329:
3328:
3323:
3309:
3308:
3284:
3283:
3253:
3252:
3225:
3223:
3222:
3217:
3190:
3188:
3187:
3182:
3144:-vector spaces.
3135:
3133:
3132:
3127:
3119:
3118:
3082:
3080:
3079:
3076:{\displaystyle }
3074:
3056:
3054:
3053:
3050:{\displaystyle }
3048:
3030:
3028:
3027:
3022:
3020:
3004:
3002:
3001:
2996:
2985:
2984:
2957:
2955:
2954:
2951:{\displaystyle }
2949:
2931:
2929:
2928:
2923:
2912:
2911:
2883:
2881:
2880:
2875:
2826:
2824:
2823:
2818:
2755:
2753:
2752:
2747:
2689:
2687:
2686:
2681:
2635:
2633:
2632:
2627:
2622:
2619:
2586:
2584:
2583:
2578:
2567:
2566:
2514:
2512:
2511:
2506:
2495:
2494:
2478:
2476:
2475:
2470:
2459:
2458:
2421:smooth functions
2414:
2412:
2411:
2406:
2395:
2394:
2365:
2363:
2362:
2357:
2355:
2354:
2309:
2307:
2306:
2301:
2290:
2289:
2235:
2233:
2232:
2227:
2216:
2215:
2186:rational numbers
2180:
2178:
2177:
2172:
2167:
2156:
2130:
2128:
2127:
2122:
2114:
2100:
2084:
2076:
2071:
2057:
2039:
2037:
2036:
2031:
2026:
2011:
2009:
2008:
2003:
1992:
1991:
1986:
1958:
1956:
1955:
1950:
1948:
1933:
1931:
1930:
1925:
1923:
1922:
1855:
1827:
1825:
1824:
1819:
1811:
1800:
1771:
1769:
1768:
1763:
1749:
1732:
1713:
1702:
1664:
1662:
1661:
1656:
1642:
1627:
1625:
1624:
1619:
1617:
1602:
1600:
1599:
1594:
1592:
1490:
1472:
1470:
1469:
1464:
1443:
1441:
1440:
1435:
1397:
1383:
1352:
1350:
1349:
1344:
1307:
1305:
1304:
1299:
1294:
1242:
1240:
1239:
1234:
1173:of the element (
1167:cancellation law
1116:
1114:
1113:
1108:
1103:
1102:
1090:
1089:
1070:
1068:
1067:
1062:
1057:
1056:
1044:
1043:
1024:
1022:
1021:
1016:
990:
988:
987:
982:
977:
976:
964:
963:
951:
950:
938:
937:
919:
918:
906:
905:
887:
886:
874:
873:
851:
849:
848:
843:
818:
816:
815:
810:
808:
807:
795:
794:
778:
776:
775:
770:
765:
764:
752:
751:
732:
730:
729:
724:
704:, one forms the
659:
657:
656:
651:
624:
622:
621:
616:
581:
579:
578:
573:
549:
547:
546:
541:
498:
491:
489:
488:
483:
460:
459:
429:
428:
398:
397:
370:
369:
321:
319:
318:
313:
292:
290:
289:
284:
259:
255:
251:
249:
248:
243:
219:
217:
216:
211:
193:
189:
185:
171:
164:
156:
152:
148:
143:inverse elements
140:
136:
132:
110:abelian category
84:, introduced by
79:
73:
60:
50:
21:
18:Group completion
8904:
8903:
8899:
8898:
8897:
8895:
8894:
8893:
8869:
8868:
8844:
8826:
8773:
8760:
8759:
8752:
8739:
8738:
8734:
8729:
8701:
8650:
8645:
8644:
8615:
8610:
8609:
8569:
8564:
8563:
8533:
8528:
8527:
8497:
8496:
8469:
8468:
8433:
8432:
8404:
8390:
8389:
8356:
8351:
8350:
8319:
8318:
8290:
8276:
8275:
8239:
8226:
8210:
8169:
8134:
8123:
8122:
8097:
8096:
8057:
8035:
8027:
8026:
7988:
7987:
7966:
7965:
7922:
7909:
7904:
7903:
7842:
7841:
7774:
7770:
7753:
7749:
7726:
7722:
7717:
7716:
7666:
7653:
7640:
7629:
7628:
7570:
7557:
7525:
7501:
7500:
7476:
7448:
7420:
7419:
7379:
7378:
7326:
7325:
7288:
7287:
7249:
7248:
7210:
7209:
7183:
7182:
7133:
7132:
7109:
7108:
7059:
7058:
6990:
6989:
6958:
6957:
6930:
6929:
6880:
6879:
6844:
6843:
6782:
6781:
6755:
6754:
6712:
6711:
6705:
6664:
6645:
6632:
6619:
6584:
6571:
6536:
6528:
6527:
6482:
6463:
6438:
6425:
6402:
6401:
6375:
6374:
6371:bounded complex
6336:
6331:
6330:
6264:
6249:
6244:
6243:
6211:
6210:
6167:
6160:
6159:
6158:can be a field
6140:
6139:
6104:
6103:
6062:
6057:
6056:
6033:
6032:
5999:
5998:
5976:
5975:
5948:
5947:
5923:
5897:
5896:
5852:
5830:
5819:
5818:
5788:
5787:
5762:
5761:
5738:
5737:
5703:
5692:
5691:
5672:
5671:
5638:
5633:
5632:
5613:
5612:
5593:
5592:
5562:
5561:
5542:
5541:
5522:
5521:
5472:
5471:
5452:
5451:
5418:
5413:
5412:
5389:
5388:
5368:
5367:
5347:
5346:
5310:
5299:
5298:
5258:
5257:
5193:
5192:
5149:
5148:
5125:
5124:
5105:
5104:
5101:
5067:
5066:
5065:with generator
5045:
5044:
5012:
5007:
5006:
4900:
4899:
4862:
4841:
4818:
4817:
4776:
4755:
4732:
4711:
4688:
4667:
4662:
4661:
4637:
4636:
4599:
4576:
4553:
4539:
4538:
4514:
4513:
4463:
4462:
4434:
4433:
4395:
4394:
4351:
4343:
4342:
4317:
4316:
4315:has its symbol
4282:
4281:
4280:with generator
4260:
4259:
4227:
4222:
4221:
4177:
4176:
4151:
4150:
4112:
4111:
4110:and denoted by
4076:
4071:
4070:
4024:
4023:
3941:
3940:
3871:
3870:
3834:
3833:
3802:
3801:
3800:satisfies that
3765:
3764:
3732:
3727:
3726:
3705:
3704:
3676:
3675:
3654:
3653:
3623:
3618:
3617:
3592:
3591:
3566:
3565:
3519:
3515:
3498:
3497:
3470:
3465:
3464:
3443:-vector spaces
3366:
3365:
3337:
3336:
3300:
3275:
3244:
3239:
3238:
3196:
3195:
3149:
3148:
3110:
3093:
3092:
3059:
3058:
3033:
3032:
3011:
3010:
2976:
2971:
2970:
2934:
2933:
2903:
2898:
2897:
2890:
2836:
2835:
2773:
2772:
2702:
2701:
2648:
2647:
2589:
2588:
2558:
2553:
2552:
2526:
2521:
2486:
2481:
2480:
2450:
2445:
2444:
2415:is the ring of
2386:
2375:
2374:
2346:
2341:
2340:
2281:
2276:
2275:
2272:
2207:
2202:
2201:
2194:
2160:
2149:
2133:
2132:
2107:
2093:
2077:
2064:
2045:
2044:
2014:
2013:
1981:
1973:
1972:
1969:
1939:
1938:
1918:
1917:
1890:
1889:
1861:
1837:
1836:
1804:
1793:
1774:
1773:
1742:
1725:
1706:
1695:
1678:
1677:
1630:
1629:
1608:
1607:
1605:natural numbers
1583:
1582:
1576:
1566:if and only if
1558:if and only if
1501:category theory
1497:
1488:
1446:
1445:
1390:
1376:
1362:
1361:
1326:
1325:
1287:
1264:
1263:
1213:
1212:
1186:
1179:
1156:
1149:
1138:
1131:
1094:
1081:
1073:
1072:
1048:
1035:
1027:
1026:
1001:
1000:
968:
955:
942:
929:
910:
897:
878:
865:
857:
856:
828:
827:
799:
786:
781:
780:
779:corresponds to
756:
743:
735:
734:
709:
708:
694:
688:
627:
626:
595:
594:
552:
551:
520:
519:
505:
496:
448:
417:
386:
358:
335:
334:
295:
294:
266:
265:
257:
256:cannot contain
253:
222:
221:
196:
195:
191:
187:
177:
169:
162:
154:
150:
146:
138:
134:
130:
127:
122:
82:category theory
75:
69:
56:
46:
28:
23:
22:
15:
12:
11:
5:
8902:
8900:
8892:
8891:
8886:
8881:
8871:
8870:
8867:
8866:
8861:
8856:
8842:
8824:
8788:(1): 200–212,
8771:
8758:
8757:
8750:
8731:
8730:
8728:
8725:
8724:
8723:
8717:
8712:
8707:
8700:
8697:
8696:
8695:
8684:graded modules
8675:
8657:
8653:
8622:
8618:
8606:
8587:
8584:
8581:
8576:
8572:
8551:
8548:
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7855:
7852:
7849:
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7814:
7811:
7808:
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7799:
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7760:
7756:
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7495:
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7366:
7363:
7360:
7357:
7352:
7347:
7343:
7340:
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7295:
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7227:
7224:
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7170:
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7164:
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7076:
7073:
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7066:
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7027:
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7021:
7018:
7015:
7012:
7009:
7006:
7003:
7000:
6997:
6977:
6974:
6971:
6968:
6965:
6939:
6917:
6914:
6911:
6906:
6901:
6897:
6894:
6890:
6887:
6876:
6875:
6863:
6860:
6857:
6854:
6851:
6837:
6836:
6825:
6822:
6819:
6816:
6813:
6810:
6807:
6804:
6801:
6798:
6795:
6792:
6789:
6764:
6721:
6709:exact category
6704:
6701:
6697:
6696:
6685:
6682:
6679:
6676:
6671:
6667:
6663:
6660:
6657:
6652:
6648:
6644:
6639:
6635:
6631:
6626:
6622:
6618:
6615:
6612:
6607:
6603:
6599:
6596:
6591:
6587:
6583:
6578:
6574:
6570:
6567:
6564:
6559:
6555:
6551:
6548:
6543:
6539:
6535:
6521:
6520:
6509:
6506:
6503:
6500:
6497:
6494:
6489:
6485:
6481:
6476:
6473:
6470:
6466:
6462:
6459:
6456:
6451:
6448:
6445:
6441:
6437:
6432:
6428:
6424:
6421:
6418:
6415:
6412:
6409:
6382:
6373:of objects in
6354:
6351:
6348:
6343:
6339:
6315:
6312:
6309:
6305:
6302:
6299:
6295:
6292:
6289:
6286:
6283:
6278:
6273:
6268:
6261:
6256:
6252:
6227:
6224:
6221:
6218:
6186:
6181:
6176:
6171:
6147:
6123:
6120:
6117:
6114:
6111:
6091:
6088:
6085:
6082:
6078:
6074:
6069:
6065:
6040:
6017:
6014:
6011:
6007:
5983:
5962:
5958:
5955:
5930:
5926:
5922:
5919:
5916:
5913:
5910:
5907:
5904:
5884:
5881:
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5872:
5867:
5862:
5859:
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5848:
5845:
5842:
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5833:
5829:
5826:
5795:
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5745:
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5710:
5706:
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5699:
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5600:
5569:
5549:
5529:
5509:
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5500:
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5485:
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5430:
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5398:
5375:
5354:
5334:
5331:
5328:
5325:
5322:
5317:
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5282:
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5230:
5227:
5224:
5221:
5218:
5215:
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5132:
5112:
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4707:
4703:
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4691:
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4382:
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4365:
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4355:
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4300:
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4268:
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4243:
4239:
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4230:
4209:
4206:
4203:
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4187:
4184:
4164:
4161:
4158:
4138:
4135:
4132:
4122:
4119:
4085:
4080:
4069:isomorphic to
4043:
4040:
4037:
4034:
4031:
4003:
4000:
3997:
3993:
3989:
3986:
3983:
3979:
3975:
3972:
3969:
3965:
3961:
3957:
3952:
3948:
3937:exact sequence
3933:
3932:
3921:
3918:
3914:
3910:
3906:
3901:
3897:
3893:
3889:
3885:
3881:
3878:
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3821:
3818:
3815:
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3809:
3778:
3775:
3772:
3752:
3748:
3744:
3739:
3735:
3713:
3689:
3686:
3683:
3662:
3641:
3638:
3635:
3630:
3626:
3605:
3602:
3599:
3579:
3576:
3573:
3562:
3561:
3550:
3547:
3544:
3541:
3538:
3534:
3529:
3526:
3522:
3518:
3514:
3511:
3508:
3505:
3480:
3477:
3473:
3459:-vector space
3433:
3432:
3421:
3418:
3415:
3412:
3409:
3406:
3403:
3400:
3397:
3394:
3391:
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3333:
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3159:
3156:
3125:
3122:
3117:
3113:
3109:
3106:
3103:
3100:
3091:is defined as
3087:-vector space
3072:
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3066:
3046:
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3040:
3019:
2994:
2991:
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2385:
2382:
2353:
2349:
2299:
2296:
2293:
2288:
2284:
2271:
2268:
2248:vector bundles
2225:
2222:
2219:
2214:
2210:
2193:
2190:
2182:
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2170:
2166:
2163:
2159:
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2001:
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1336:
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1286:
1283:
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1277:
1274:
1271:
1262:: denoting by
1232:
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1220:
1184:
1177:
1154:
1147:
1136:
1129:
1106:
1101:
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1060:
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1034:
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992:
980:
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962:
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949:
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932:
928:
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922:
917:
913:
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877:
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864:
841:
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835:
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802:
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763:
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750:
746:
742:
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687:
684:
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643:
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637:
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611:
608:
605:
602:
571:
568:
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562:
559:
539:
536:
533:
530:
527:
504:
501:
493:
492:
481:
478:
475:
472:
469:
466:
463:
458:
455:
451:
447:
444:
441:
438:
435:
432:
427:
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420:
416:
413:
410:
407:
404:
401:
396:
393:
389:
385:
382:
379:
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368:
365:
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357:
354:
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348:
345:
342:
311:
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282:
279:
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273:
241:
238:
235:
232:
229:
209:
206:
203:
126:
123:
121:
118:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8901:
8890:
8887:
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8877:
8876:
8874:
8865:
8862:
8860:
8857:
8853:
8852:
8847:
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8839:
8835:
8834:
8829:
8825:
8821:
8817:
8813:
8809:
8805:
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8796:
8791:
8787:
8783:
8782:
8777:
8772:
8769:
8765:
8762:
8761:
8753:
8747:
8743:
8736:
8733:
8726:
8721:
8718:
8716:
8713:
8711:
8708:
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8698:
8693:
8689:
8685:
8681:
8676:
8673:
8655:
8651:
8642:
8638:
8620:
8616:
8607:
8604:
8601:
8582:
8574:
8570:
8546:
8538:
8534:
8525:
8521:
8494:
8466:
8465:affine scheme
8462:
8458:
8413:
8401:
8398:
8387:
8369:
8361:
8357:
8348:
8344:
8299:
8287:
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8220:
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8174:
8166:
8163:
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8119:
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8025:
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8018:
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7883:
7877:
7868:
7862:
7859:
7856:
7850:
7840:
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7833:
7832:
7815:
7809:
7800:
7797:
7794:
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7784:
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7771:
7767:
7763:
7758:
7754:
7750:
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7737:
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7715:
7714:
7713:
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7708:
7704:
7700:
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7667:
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7645:
7641:
7634:
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7602:
7562:
7558:
7554:
7539:
7533:
7530:
7526:
7515:
7509:
7499:
7498:
7497:
7496:
7493:
7489:
7486:
7482:
7481:vector spaces
7478:
7477:
7473:
7471:
7469:
7465:
7461:
7457:
7453:
7445:
7443:
7361:
7358:
7311:
7305:
7302:
7299:
7293:
7285:
7280:
7278:
7274:
7246:
7225:
7222:
7207:
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7188:
7168:
7141:
7138:
7094:
7067:
7064:
7056:
7040:
7037:
7031:
7025:
7022:
7016:
7010:
7007:
7001:
6995:
6975:
6969:
6963:
6955:
6915:
6888:
6885:
6861:
6855:
6849:
6842:
6841:
6840:
6823:
6820:
6814:
6808:
6802:
6796:
6790:
6780:
6779:
6778:
6751:
6749:
6745:
6741:
6737:
6710:
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6700:
6683:
6677:
6669:
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6661:
6650:
6646:
6637:
6633:
6624:
6616:
6613:
6605:
6601:
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6589:
6585:
6576:
6568:
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6557:
6553:
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6541:
6537:
6526:
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6507:
6501:
6495:
6487:
6483:
6474:
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6464:
6457:
6449:
6446:
6443:
6439:
6430:
6426:
6419:
6413:
6407:
6400:
6399:
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6380:
6372:
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6341:
6337:
6327:
6310:
6284:
6271:
6254:
6250:
6241:
6222:
6216:
6208:
6204:
6200:
6184:
6174:
6145:
6137:
6118:
6112:
6109:
6083:
6067:
6063:
6054:
6038:
6031:
6012:
5997:
5981:
5956:
5953:
5944:
5928:
5924:
5920:
5911:
5902:
5879:
5876:
5873:
5865:
5843:
5835:
5831:
5827:
5824:
5816:
5812:
5807:
5793:
5773:
5770:
5767:
5759:
5743:
5735:
5716:
5708:
5704:
5700:
5697:
5677:
5657:
5651:
5648:
5643:
5639:
5618:
5598:
5590:
5586:
5581:
5567:
5547:
5527:
5504:
5498:
5495:
5486:
5477:
5457:
5437:
5431:
5423:
5419:
5386:
5373:
5352:
5332:
5323:
5315:
5311:
5307:
5304:
5296:
5293:, there is a
5280:
5269:
5266:
5263:
5243:
5240:
5234:
5228:
5225:
5219:
5213:
5210:
5204:
5198:
5178:
5172:
5166:
5160:
5154:
5146:
5130:
5110:
5098:
5096:
5083:
5017:
5013:
4986:
4980:
4974:
4968:
4962:
4956:
4953:
4950:
4944:
4938:
4935:
4932:
4926:
4920:
4917:
4914:
4908:
4898:
4897:
4896:
4863:
4859:
4853:
4842:
4838:
4832:
4826:
4823:
4816:
4815:
4814:
4812:
4777:
4773:
4767:
4756:
4752:
4733:
4729:
4723:
4712:
4708:
4689:
4685:
4679:
4668:
4660:
4659:
4658:
4619:
4600:
4596:
4577:
4573:
4554:
4550:
4544:
4537:
4536:
4535:
4511:
4492:
4486:
4480:
4474:
4468:
4461:
4460:
4459:
4442:
4416:
4403:
4400:
4369:
4366:
4358:
4325:
4314:
4298:
4232:
4228:
4204:
4191:
4185:
4159:
4133:
4120:
4117:
4109:
4105:
4102:, called the
4101:
4083:
4068:
4064:
4060:
4055:
4041:
4038:
4032:
4021:
4017:
4001:
3998:
3984:
3970:
3959:
3955:
3939:implies that
3938:
3919:
3908:
3904:
3876:
3869:
3868:
3867:
3865:
3819:
3816:
3810:
3799:
3796:
3792:
3773:
3737:
3733:
3701:
3684:
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3628:
3624:
3600:
3574:
3545:
3539:
3536:
3532:
3527:
3524:
3520:
3516:
3512:
3506:
3496:
3495:
3494:
3478:
3475:
3471:
3462:
3458:
3455:-dimensional
3454:
3450:
3446:
3442:
3438:
3416:
3410:
3404:
3398:
3392:
3389:
3386:
3380:
3374:
3364:
3363:
3362:
3345:
3316:
3310:
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3297:
3291:
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3276:
3272:
3266:
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3260:
3254:
3249:
3245:
3237:
3236:
3235:
3233:
3229:
3213:
3210:
3207:
3204:
3201:
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3166:
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3154:
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3145:
3143:
3139:
3123:
3120:
3115:
3111:
3107:
3101:
3090:
3086:
3067:
3041:
3008:
2989:
2981:
2977:
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2961:
2942:
2916:
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2895:
2887:
2871:
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2853:
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2841:
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2833:
2832:
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2811:
2805:
2799:
2793:
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2766:
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2740:
2734:
2728:
2722:
2716:
2710:
2700:
2699:
2698:
2696:
2677:
2671:
2665:
2659:
2653:
2646:
2645:
2644:
2643:
2639:
2615:
2612:
2609:
2606:
2600:
2571:
2563:
2559:
2550:
2549:artinian ring
2546:
2543:
2539:
2535:
2531:
2523:
2518:
2516:
2499:
2491:
2487:
2463:
2455:
2451:
2442:
2438:
2434:
2431:-modules are
2430:
2426:
2422:
2418:
2399:
2387:
2383:
2380:
2371:
2369:
2351:
2347:
2338:
2334:
2330:
2327:
2324:
2320:
2317:
2313:
2294:
2286:
2282:
2269:
2267:
2265:
2261:
2257:
2253:
2249:
2245:
2242:
2239:
2220:
2212:
2208:
2199:
2191:
2189:
2187:
2168:
2164:
2161:
2157:
2153:
2150:
2146:
2138:
2118:
2115:
2111:
2108:
2104:
2101:
2097:
2094:
2090:
2081:
2078:
2073:
2068:
2065:
2061:
2058:
2054:
2050:
2043:
2042:
2041:
2027:
2023:
2019:
1996:
1993:
1988:
1966:
1964:
1962:
1911:
1908:
1905:
1899:
1896:
1893:
1883:
1880:
1877:
1871:
1868:
1862:
1856:
1848:
1845:
1835:
1834:
1833:
1815:
1812:
1808:
1805:
1801:
1797:
1794:
1790:
1787:
1779:
1759:
1756:
1753:
1750:
1746:
1743:
1739:
1736:
1733:
1729:
1726:
1722:
1719:
1710:
1707:
1703:
1699:
1696:
1692:
1689:
1686:
1683:
1676:
1675:
1674:
1672:
1668:
1652:
1646:
1643:
1606:
1581:
1573:
1571:
1569:
1565:
1561:
1557:
1553:
1549:
1546: :
1545:
1541:
1536:
1534:
1530:
1526:
1522:
1518:
1514:
1510:
1506:
1502:
1494:
1492:
1487:
1483:
1480:
1476:
1457:
1451:
1428:
1425:
1422:
1419:
1416:
1413:
1407:
1404:
1401:
1394:
1391:
1384:
1380:
1377:
1373:
1359:
1356:
1337:
1331:
1323:
1319:
1315:
1311:
1291:
1288:
1284:
1278:
1272:
1261:
1257:
1253:
1248:
1246:
1230:
1224:
1221:
1218:
1210:
1206:
1202:
1198:
1194:
1190:
1183:
1176:
1172:
1168:
1164:
1161:(the element
1160:
1153:
1146:
1142:
1135:
1128:
1124:
1120:
1099:
1095:
1091:
1086:
1082:
1053:
1049:
1045:
1040:
1036:
1012:
1009:
1006:
998:
973:
969:
965:
960:
956:
952:
947:
943:
939:
934:
930:
923:
915:
911:
907:
902:
898:
891:
883:
879:
875:
870:
866:
855:
854:
853:
839:
836:
833:
824:
822:
804:
800:
796:
791:
787:
761:
757:
753:
748:
744:
720:
717:
714:
707:
703:
699:
693:
685:
683:
681:
677:
673:
669:
665:
660:
647:
644:
641:
638:
635:
632:
612:
606:
603:
600:
593:
589:
585:
569:
563:
560:
557:
537:
531:
528:
525:
518:
514:
510:
502:
500:
479:
476:
473:
470:
467:
461:
456:
453:
449:
442:
436:
430:
425:
422:
418:
414:
408:
405:
399:
394:
391:
387:
383:
380:
377:
371:
366:
363:
359:
352:
349:
346:
343:
340:
333:
332:
331:
329:
325:
324:trivial group
309:
306:
303:
300:
280:
277:
274:
271:
263:
239:
236:
233:
230:
227:
207:
204:
201:
184:
180:
175:
166:
160:
144:
124:
119:
117:
115:
111:
107:
103:
99:
95:
91:
87:
83:
78:
72:
67:
64:
59:
54:
53:abelian group
51:is a certain
49:
45:
41:
37:
33:
19:
8849:
8831:
8785:
8779:
8767:
8741:
8735:
8710:Localization
8691:
8687:
8679:
8671:
8640:
8602:
8523:
8492:
8460:
8346:
8273:ringed space
8016:
7706:
7702:
7698:
7491:
7487:
7467:
7463:
7459:
7455:
7449:
7281:
7272:
7247:-modules as
7244:
7203:
7054:
6953:
6877:
6838:
6752:
6747:
6743:
6739:
6706:
6698:
6522:
6328:
6238:-module its
6206:
6203:finite field
6030:finite group
5945:
5808:
5582:
5366:
5294:
5144:
5102:
5004:
4894:
4808:
4634:
4507:
4312:
4107:
4099:
4062:
4056:
4019:
4016:cyclic group
3934:
3863:
3797:
3790:
3702:
3563:
3460:
3456:
3452:
3448:
3444:
3440:
3436:
3434:
3334:
3231:
3227:
3193:
3141:
3137:
3088:
3084:
2966:
2964:vector space
2959:
2893:
2891:
2768:
2764:
2760:
2758:
2694:
2692:
2637:
2544:
2533:
2529:
2527:
2436:
2428:
2424:
2372:
2332:
2318:
2273:
2251:
2243:
2200:. The group
2195:
2183:
1970:
1936:
1831:
1670:
1666:
1577:
1567:
1559:
1551:
1547:
1543:
1539:
1537:
1529:left adjoint
1524:
1520:
1498:
1485:
1481:
1474:
1317:
1313:
1255:
1251:
1249:
1244:
1208:
1204:
1200:
1196:
1192:
1188:
1181:
1174:
1162:
1158:
1151:
1144:
1140:
1133:
1126:
1122:
1118:
1025:, such that
994:
826:Addition on
825:
820:
701:
697:
695:
679:
675:
671:
667:
663:
661:
587:
583:
512:
508:
506:
494:
262:zero element
182:
178:
167:
128:
76:
70:
57:
47:
39:
35:
29:
4014:, so every
2969:. In fact,
2312:commutative
1832:Now define
264:satisfying
112:, with the
63:homomorphic
32:mathematics
8873:Categories
8851:PlanetMath
8727:References
7277:noetherian
5996:group ring
5895:such that
5758:linear map
5345:such that
5191:, one has
5143:is called
4022:satisfies
3007:isomorphic
2540:over some
2524:Definition
2337:direct sum
2326:projective
1542:, the map
1495:Properties
690:See also:
625:such that
495:for every
293:for every
194:such that
125:Motivation
114:direct sum
8838:EMS Press
8820:260493607
8795:1105.2715
8605:-modules.
8245:∗
8224:
8205:−
8193:∑
8183:
8164:−
8152:∑
8140:∗
8129:χ
8103:χ
8063:∗
8052:χ
8041:∗
8019: *,
7863:
7677:→
7664:→
7651:→
7638:→
7555:∈
7534:
7362:⊕
7309:↠
7303:⊕
7297:↪
7189:ϕ
7166:→
7139:χ
7092:→
7065:ϕ
7026:χ
7011:χ
7008:−
6996:χ
6973:↠
6967:↪
6913:→
6886:χ
6859:↠
6853:↪
6797:−
6662:∈
6651:∗
6614:−
6602:∑
6566:−
6554:∑
6542:∗
6508:⋯
6505:→
6499:→
6493:→
6480:→
6472:−
6461:→
6458:⋯
6455:→
6436:→
6423:→
6417:→
6411:→
6408:⋯
6294:→
6277:¯
6180:¯
6134:. In the
5925:χ
5903:χ
5850:→
5825:χ
5771:∈
5705:χ
5655:→
5640:χ
5499:χ
5353:χ
5330:→
5278:→
5264:χ
5229:χ
5214:χ
5211:−
5199:χ
5176:→
5170:→
5164:→
5158:→
5111:χ
4957:
4939:
4921:
4864:⊗
4854:
4827:
4778:⊗
4768:
4734:⊗
4724:
4690:⊗
4680:
4617:→
4601:⊗
4594:→
4578:⊗
4571:→
4555:⊗
4548:→
4510:tensoring
4490:→
4484:→
4478:→
4472:→
3985:−
3917:→
3896:→
3888:→
3880:→
3845:→
3525:⊕
3476:⊕
3390:⊕
3311:
3286:
3264:⊕
3255:
3211:⊕
3205:≅
3176:→
3170:→
3164:→
3158:→
3121:
2869:→
2863:→
2857:⊕
2851:→
2845:→
2785:⊕
2763:-modules
2717:−
2675:→
2669:→
2663:→
2657:→
2613:∈
2607:∣
2392:∞
2260:manifolds
2143:⟺
2131:for some
2087:⟺
2062:∼
1997:×
1989:∗
1909:−
1894:−
1881:−
1849:∈
1843:∀
1784:⟺
1772:for some
1716:⟺
1704:−
1693:∼
1687:−
1564:bijective
1556:injective
1505:universal
1479:semigroup
1426:∈
1414:∣
1392:−
1358:generated
1228:→
1010:×
837:×
797:−
718:×
642:∘
610:→
604::
567:→
561::
535:→
529::
454:−
423:−
392:−
364:−
304:∈
205:≠
8889:K-theory
8768:K-Theory
8699:See also
6988:one has
5815:matrices
5690:-module
5540:-module
5145:additive
2888:Examples
2443:). Thus
2439:(by the
2419:-valued
2241:manifold
2198:K-theory
2165:′
2154:′
2112:′
2098:′
2082:′
2069:′
1809:′
1798:′
1747:′
1730:′
1711:′
1700:′
1580:integers
1513:category
1395:′
1381:′
1355:subgroup
1322:quotient
1292:′
94:K-theory
8840:, 2001
8812:3033967
8341:of all
7483:over a
6201:of the
5994:is the
5736:of the
2538:algebra
2417:complex
2339:. Then
2329:modules
2238:compact
1531:to the
1509:functor
1353:by the
1320:is the
515:with a
106:objects
42:, of a
8818:
8810:
8748:
8600:simple
8271:For a
8095:where
7204:Every
5295:unique
4393:where
1503:, any
1203:, and
108:of an
98:monoid
34:, the
8816:S2CID
8790:arXiv
8467:) of
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6928:from
6205:with
6028:of a
5811:basis
5734:trace
5591:: If
5587:from
4508:Then
2829:split
2542:field
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2236:of a
1247:to .
582:from
328:group
66:image
38:, or
8746:ISBN
7902:and
7834:Thus
7709:, so
7230:-mod
6385:-mod
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4954:rank
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2767:and
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220:and
186:and
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5915:]
5912:V
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5906:(
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5877:,
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5496:=
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5490:]
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5478:f
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5220:B
5217:(
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5081:]
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5073:[
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5031:)
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4990:]
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4981:+
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4972:[
4969:=
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4963:C
4960:(
4951:+
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4933:=
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4927:B
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4915:=
4912:]
4909:B
4906:[
4880:)
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4683:(
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4364:]
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4313:A
4299:.
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4192:=
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4121:=
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4036:]
4033:G
4030:[
4020:G
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3996:]
3992:Z
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3982:]
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3974:[
3971:=
3968:]
3964:Z
3960:n
3956:/
3951:Z
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3920:0
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3814:]
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3738:0
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3578:]
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3513:=
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3507:V
3504:[
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3417:W
3414:[
3411:+
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3402:[
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3372:[
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916:2
912:n
908:,
903:1
899:n
895:(
892:+
889:)
884:2
880:m
876:,
871:1
867:m
863:(
840:M
834:M
821:K
805:2
801:m
792:1
788:m
767:)
762:2
758:m
754:,
749:1
745:m
741:(
721:M
715:M
702:M
698:K
680:M
676:K
672:K
668:M
664:A
648:.
645:i
639:g
636:=
633:f
613:A
607:K
601:g
588:A
584:M
570:A
564:M
558:f
538:K
532:M
526:i
513:K
509:M
497:x
480:0
477:=
471:=
465:)
457:1
450:0
446:(
443:=
440:)
434:(
431:.
426:1
419:0
415:=
412:)
409:x
403:(
400:.
395:1
388:0
384:=
381:x
378:.
375:)
367:1
360:0
356:(
353:=
350:x
344:=
341:x
326:(
310:,
307:M
301:x
281:0
278:=
275:x
258:M
254:K
240:c
237:b
234:=
231:c
228:a
208:b
202:a
192:M
188:c
183:b
179:a
170:M
163:M
155:M
151:K
147:M
139:M
135:K
131:M
77:M
71:M
58:M
48:M
20:)
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