9485:
9268:
9506:
9474:
9543:
9516:
9496:
6702:
4353:
8213:
5162:
6526:
4774:
199:
into an abelian group is a necessary ingredient for defining K-theory since all definitions start by constructing an abelian monoid from a suitable category and turning it into an abelian group through this universal construction. Given an abelian monoid
113:
in algebraic topology, the reason for this functorial mapping is that it is easier to compute some topological properties from the mapped rings than from the original spaces or schemes. Examples of results gleaned from the K-theory approach include the
7931:
3326:
4977:
3689:
108:
that map from topological spaces or schemes, or to be even more general: any object of a homotopy category to associated rings; these rings reflect some aspects of the structure of the original spaces or schemes. As with functors to
6164:
7196:
4244:
6890:
8424:
4659:
7328:
505:
5594:
5692:. This makes it possible to compute the Grothendieck group on weighted projective spaces since they typically have isolated quotient singularities. In particular, if these singularities have isotropy groups
7202:
singularities, giving techniques for computing the
Grothendieck group of any singular algebraic curve. This is because reduction gives a generically smooth curve, and all singularities are Cohen-Macaulay.
7049:
4866:
2048:
8053:
5990:
5865:
4693:
6945:
4064:. Since a vector bundle over this space is just a finite dimensional vector space, which is a free object in the category of coherent sheaves, hence projective, the monoid of isomorphism classes is
1650:
1584:
1867:
6363:
2803:
6531:
4463:
5084:
7986:
5282:
375:
2622:
8511:
8275:
6992:
4040:
3606:
3083:
2581:
7767:
5246:
6521:
5690:
3182:
4192:
4871:
2729:
7562:
5775:
8045:
5347:
4402:
3129:
In algebraic geometry, the same construction can be applied to algebraic vector bundles over a smooth scheme. But, there is an alternative construction for any
Noetherian scheme
2836:
2157:
1486:
7240:
7608:
6697:{\displaystyle {\begin{aligned}E_{\infty }^{1,-1}\cong E_{2}^{1,-1}&\cong {\text{CH}}^{1}(C)\\E_{\infty }^{0,0}\cong E_{2}^{0,0}&\cong {\text{CH}}^{0}(C)\end{aligned}}}
5023:
1941:
4688:
2996:
2404:
555:
6031:
1691:
1116:
294:
7803:
5059:
2092:
4545:
8468:
7116:
7111:
5369:
4796:
4239:
4217:
4106:
4084:
4062:
1801:
5654:
2436:
854:
235:
1772:
901:
5495:
1066:
6747:
6742:
6419:
6249:
5897:
5453:
5417:
3728:
3457:
3401:
3365:
3119:
2872:
2678:
2535:
813:
599:
7366:
6197:
3040:
1515:
2224:
1142:
404:
5717:
5309:
2930:
2903:
2291:
2259:
2189:
1899:
944:
255:
6298:
5191:
4574:
1421:
1354:
1197:
3621:
1322:
1168:
769:
3254:
1734:
8322:
8299:
7513:
7446:
7426:
7406:
7386:
7089:
7069:
6459:
6439:
6383:
6269:
6217:
6014:
5929:
5618:
5079:
4422:
4141:
3421:
3147:
3020:
2965:
2642:
2373:
2353:
1711:
984:
964:
5456:
2462:
1392:
1022:
8775:
8513:. The theory was developed by R. W. Thomason in 1980s. Specifically, he proved equivariant analogs of fundamental theorems such as the localization theorem.
6705:
4579:
8218:
The Chern character is useful in part because it facilitates the computation of the Chern class of a tensor product. The Chern character is used in the
3769:
of sheaves as generators of the group, subject to a relation that identifies any extension of two sheaves with their sum. The resulting group is called
5504:
9546:
3265:
5455:, which comes from the fact every vector bundle can be equivalently described as a coherent sheaf. This is done using the Grothendieck group of the
5934:
8334:
4348:{\displaystyle K_{0}\left({\text{Spec}}\left({\frac {\mathbb {F} }{(x^{9})}}\times \mathbb {F} \right)\right)=\mathbb {Z} \oplus \mathbb {Z} }
3958:
which is related to the study of pseudo-isotopies. Much modern research on higher K-theory is related to algebraic geometry and the study of
7216:
One useful application of the
Grothendieck-group is to define virtual vector bundles. For example, if we have an embedding of smooth spaces
7248:
6303:
5383:
One recent technique for computing the
Grothendieck group of spaces with minor singularities comes from evaluating the difference between
8705:
8681:
8557:
5498:
3747:
3612:
115:
8208:{\displaystyle \operatorname {ch} (V)=e^{x_{1}}+\dots +e^{x_{n}}:=\sum _{m=0}^{\infty }{\frac {1}{m!}}(x_{1}^{m}+\dots +x_{n}^{m}).}
6997:
7518:
Another useful application of virtual bundles is with the definition of a virtual tangent bundle of an intersection of spaces: Let
4801:
1949:
7198:
Moreover, the techniques above using the derived category of singularities for isolated singularities can be extended to isolated
2878:, which makes it very accessible. The only required computations for understanding the spectral sequences are computing the group
8219:
5780:
2875:
2624:. We can then apply the Grothendieck completion to get an abelian group from this abelian monoid. This is called the K-theory of
7564:
be projective subvarieties of a smooth projective variety. Then, we can define the virtual tangent bundle of their intersection
9127:
9019:
8992:
8953:
8861:
8612:
6895:
1592:
1526:
5157:{\displaystyle \mathbb {P} ({\mathcal {E}})=\operatorname {Proj} (\operatorname {Sym} ^{\bullet }({\mathcal {E}}^{\vee }))}
1806:
774:
Equivalence classes in this group should be thought of as formal differences of elements in the abelian monoid. This group
5722:
2742:
2687:
and some algebra to get an alternative description of vector bundles over the ring of continuous complex-valued functions
4427:
9180:
4086:
corresponding to the dimension of the vector space. It is an easy exercise to show that the
Grothendieck group is then
8776:"ag.algebraic geometry - Is the algebraic Grothendieck group of a weighted projective space finitely generated ?"
7939:
409:
9534:
9529:
9075:
9045:
5251:
302:
5037:
Another important formula for the
Grothendieck group is the projective bundle formula: given a rank r vector bundle
9037:
2586:
4769:{\displaystyle \mathbb {P} ^{n}=\mathbb {A} ^{n}\coprod \mathbb {A} ^{n-1}\coprod \cdots \coprod \mathbb {A} ^{0}}
9524:
8477:
8241:
6950:
4012:
3875:
3871:
123:
3465:
3045:
2543:
8231:
7616:
2324:
There are a number of basic definitions of K-theory: two coming from topology and two from algebraic geometry.
5200:
9426:
8302:
6464:
5659:
3155:
158:
4146:
8802:
Pavic, Nebojsa; Shinder, Evgeny (2021). "K-theory and the singularity category of quotient singularities".
2690:
7521:
904:
8537:
7991:
5314:
4369:
2808:
2100:
1429:
7219:
2874:. One of the main techniques for computing the Grothendieck group for topological spaces comes from the
9434:
7926:{\displaystyle \operatorname {ch} (L)=\exp(c_{1}(L)):=\sum _{m=0}^{\infty }{\frac {c_{1}(L)^{m}}{m!}}.}
7567:
4982:
3938:
3902:
4664:
2970:
2378:
510:
6017:
1655:
1071:
260:
5040:
2056:
9233:
4468:
3867:
162:
17:
8432:
7094:
5352:
4779:
4222:
4200:
4089:
4067:
4045:
1784:
9519:
9505:
7199:
5623:
3985:
3879:
3832:, then all extensions of locally free sheaves split, so the group has an alternative definition.
2999:
2409:
818:
150:
90:
4366:
One of the most commonly used computations of the
Grothendieck group is with the computation of
1943:
by using the invariance under scaling. For example, we can see from the scaling invariance that
1739:
859:
9454:
9375:
9252:
9240:
9213:
9173:
8664:
5461:
3814:
3743:
2684:
1027:
9449:
6711:
6388:
6222:
5870:
5422:
5386:
4972:{\displaystyle \mathbb {A} ^{n-k_{1}}\cap \mathbb {A} ^{n-k_{2}}=\mathbb {A} ^{n-k_{1}-k_{2}}}
3697:
3426:
3370:
3334:
3088:
2841:
2647:
2474:
777:
563:
9296:
9223:
7336:
6169:
3025:
1904:
1491:
166:
142:
9143:
7051:, we have the sequence of abelian groups above splits, giving the isomorphism. Note that if
4404:
for projective space over a field. This is because the intersection numbers of a projective
3684:{\displaystyle \operatorname {ch} :K_{0}(X)\otimes \mathbb {Q} \to A(X)\otimes \mathbb {Q} }
2194:
1121:
383:
203:
9444:
9396:
9370:
9218:
9002:
8963:
8907:
8892:, Progress in Mathematics, vol. 129, Boston, MA: Birkhäuser Boston, pp. 335–368,
8741:
8547:
7790:
5695:
5287:
3951:
3848:
2908:
2881:
2318:
2264:
2232:
2162:
1872:
917:
240:
94:
70:
58:
8706:"kt.k theory and homology - Grothendieck group for projective space over the dual numbers"
6274:
5167:
4550:
1397:
1330:
1173:
8:
9291:
3789:) when all are coherent sheaves. Either of these two constructions is referred to as the
3731:
1205:
1147:
607:
558:
134:
110:
9495:
9107:
9014:. Cambridge Studies in Advanced Mathematics. Vol. 111. Cambridge University Press.
8745:
6159:{\displaystyle E_{1}^{p,q}=\coprod _{x\in X^{(p)}}K^{-p-q}(k(x))\Rightarrow K_{-p-q}(X)}
5374:
5311:
or
Hirzebruch surfaces. In addition, this can be used to compute the Grothendieck group
3187:
1716:
9489:
9459:
9439:
9360:
9350:
9228:
9208:
9089:
8920:
8893:
8829:
8811:
8581:
8542:
8471:
8307:
8284:
8235:
7454:
7431:
7411:
7391:
7371:
7074:
7054:
6444:
6424:
6368:
6300:
the algebraic function field of the subscheme. This spectral sequence has the property
6254:
6202:
6021:
5999:
5914:
5603:
5597:
5064:
4407:
4126:
3959:
3926:
3790:
3778:
3406:
3132:
3121:
is defined by the application of the
Grothendieck construction on this abelian monoid.
3005:
2950:
2941:
2627:
2358:
2338:
2314:
2306:
1696:
969:
949:
190:
178:
82:
78:
62:
46:
8637:
2805:. We can define equivalence classes of idempotent matrices and form an abelian monoid
2441:
1359:
989:
9567:
9484:
9477:
9343:
9301:
9166:
9123:
9071:
9041:
9015:
8988:
8949:
8867:
8857:
8757:
8618:
8608:
8552:
8278:
7191:{\displaystyle K_{0}(C)\cong \mathbb {Z} \oplus (\mathbb {C} ^{g}/\mathbb {Z} ^{2g})}
4121:
3930:
3914:
3898:
3890:
3860:
3766:
3759:
2945:
2732:
1587:
1356:. This should give us the hint that we should be thinking of the equivalence classes
911:
66:
54:
9509:
8833:
8753:
4776:
since the
Grothendieck group of coherent sheaves on affine spaces are isomorphic to
3042:
of isomorphisms classes of vector bundles is well-defined, giving an abelian monoid
9257:
9203:
9063:
8980:
8885:
8821:
8749:
8522:
4009:
The easiest example of the Grothendieck group is the Grothendieck group of a point
1423:
Another useful observation is the invariance of equivalence classes under scaling:
138:
119:
86:
8660:
2355:
consider the set of isomorphism classes of finite-dimensional vector bundles over
9316:
9311:
8998:
8976:
8959:
8903:
7781:
6885:{\displaystyle 0\to F^{1}(K_{0}(X))\to K_{0}(X)\to K_{0}(X)/F^{1}(K_{0}(X))\to 0}
3947:
3906:
2333:
127:
9499:
3321:{\displaystyle 0\to {\mathcal {E}}'\to {\mathcal {E}}\to {\mathcal {E}}''\to 0.}
9406:
9338:
9115:
8945:
8937:
8577:
8325:
6025:
3974:
3966:
3943:
3844:
3829:
3825:
3755:
3257:
3150:
3124:
170:
50:
9067:
8984:
9561:
9416:
9326:
9306:
9103:
8871:
8761:
8622:
7793:
of a space to (the completion of) its rational cohomology. For a line bundle
7786:
4195:
3981:
3894:
3840:
2935:
174:
9148:
9401:
9321:
9267:
9029:
8825:
5993:
3936:
There followed a period in which there were various partial definitions of
8851:
8602:
3765:. Rather than working directly with the sheaves, he defined a group using
2583:
is an abelian monoid where the unit is given by the trivial vector bundle
2464:. Since isomorphism classes of vector bundles behave well with respect to
9411:
9085:
9055:
8638:"SGA 6 - Formalisme des intersections sur les schema algebriques propres"
3910:
2736:
38:
8419:{\displaystyle K_{i}^{G}(X)=\pi _{i}(B^{+}\operatorname {Coh} ^{G}(X)).}
3913:; this assertion is correct, but was not settled until 20 years later. (
9355:
9286:
9245:
8898:
8729:
8668:
3883:
3802:
2465:
4547:. This makes it possible to do concrete calculations with elements in
3905:, which states that every finitely generated projective module over a
9380:
9153:
9094:
8586:
8532:
8527:
6028:
of finite type over a field, there is a convergent spectral sequence
4654:{\displaystyle K(\mathbb {P} ^{n})={\frac {\mathbb {Z} }{(T^{n+1})}}}
1781:
An illustrative example to look at is the Grothendieck completion of
1586:
and it has the property that it is left adjoint to the corresponding
7323:{\displaystyle 0\to \Omega _{Y}\to \Omega _{X}|_{Y}\to C_{Y/X}\to 0}
6744:
as the desired explicit direct sum since it gives an exact sequence
2940:
There is an analogous construction by considering vector bundles in
9365:
9333:
9282:
9189:
8816:
3989:
3970:
3836:
3459:
is special because there is also a ring structure: we define it as
146:
31:
5379:
of singular spaces and spaces with isolated quotient singularities
3925:
The other historical origin of algebraic K-theory was the work of
2327:
8732:(1969-01-01). "Lectures on the K-functor in algebraic geometry".
5589:{\displaystyle \cdots \to K^{0}(X)\to K_{0}(X)\to K_{sg}(X)\to 0}
4241:, one for each connected component of its spectrum. For example,
1521:
154:
105:
74:
9122:. Grad. Studies in Math. Vol. 145. American Math Society.
8682:"Grothendieck group for projective space over the dual numbers"
6461:
points, the only nontrivial parts of the spectral sequence are
5248:. This formula allows one to compute the Grothendieck group of
3942:. Finally, two useful and equivalent definitions were given by
196:
7044:{\displaystyle {\text{Ext}}_{\text{Ab}}^{1}(\mathbb {Z} ,G)=0}
4861:{\displaystyle \mathbb {A} ^{n-k_{1}},\mathbb {A} ^{n-k_{2}}}
2043:{\displaystyle (4,6)\sim (3,5)\sim (2,4)\sim (1,3)\sim (0,2)}
8888:(1995), "Enumeration of rational curves via torus actions",
3125:
Grothendieck group of coherent sheaves in algebraic geometry
1170:
and apply the equation from the equivalence relation to get
9158:
5985:{\displaystyle K_{0}(C)=\mathbb {Z} \oplus {\text{Pic}}(C)}
5860:{\displaystyle {\text{lcm}}(|G_{1}|,\ldots ,|G_{k}|)^{n-1}}
910:
To get a better understanding of this group, consider some
7988:
is a direct sum of line bundles, with first Chern classes
7789:
can be used to construct a homomorphism of rings from the
2936:
Grothendieck group of vector bundles in algebraic geometry
2468:, we can write these operations on isomorphism classes by
4111:
7772:
Kontsevich uses this construction in one of his papers.
4661:
One technique for determining the Grothendieck group of
6940:{\displaystyle {\text{CH}}^{1}(C)\cong {\text{Pic}}(C)}
3965:
The corresponding constructions involving an auxiliary
1645:{\displaystyle U:\mathbf {AbGrp} \to \mathbf {AbMon} .}
1579:{\displaystyle G:\mathbf {AbMon} \to \mathbf {AbGrp} ,}
5349:
by observing it is a projective bundle over the field
4576:
without having to explicitly know its structure since
4120:
One important property of the Grothendieck group of a
1327:
hence we have an additive inverse for each element in
8480:
8435:
8337:
8310:
8287:
8244:
8056:
7994:
7942:
7806:
7619:
7570:
7524:
7457:
7434:
7414:
7394:
7374:
7339:
7251:
7222:
7119:
7097:
7077:
7057:
7000:
6953:
6898:
6750:
6714:
6529:
6467:
6447:
6427:
6391:
6371:
6306:
6277:
6257:
6225:
6205:
6172:
6034:
6002:
5937:
5917:
5873:
5783:
5725:
5698:
5662:
5626:
5606:
5507:
5464:
5425:
5389:
5355:
5317:
5290:
5254:
5203:
5170:
5087:
5067:
5043:
4985:
4874:
4804:
4782:
4696:
4667:
4582:
4553:
4471:
4430:
4410:
4372:
4247:
4225:
4203:
4149:
4129:
4092:
4070:
4048:
4015:
3754:, meaning "class". Grothendieck needed to work with
3700:
3624:
3468:
3429:
3409:
3373:
3337:
3268:
3190:
3158:
3135:
3091:
3048:
3028:
3008:
2973:
2953:
2911:
2884:
2844:
2811:
2745:
2693:
2650:
2630:
2589:
2546:
2477:
2444:
2412:
2381:
2361:
2341:
2267:
2235:
2197:
2165:
2103:
2059:
1952:
1907:
1875:
1862:{\displaystyle G((\mathbb {N} ,+))=(\mathbb {Z} ,+).}
1809:
1787:
1742:
1719:
1713:
to the underlying abelian monoid of an abelian group
1699:
1658:
1595:
1529:
1494:
1432:
1400:
1362:
1333:
1208:
1176:
1150:
1124:
1074:
1030:
992:
972:
952:
920:
862:
821:
780:
610:
566:
513:
412:
386:
305:
263:
243:
206:
6358:{\displaystyle E_{2}^{p,-p}\cong {\text{CH}}^{p}(X)}
3874:. It played a major role in the second proof of the
2798:{\displaystyle M_{n\times n}(C^{0}(X;\mathbb {C} ))}
4458:{\displaystyle i:X\hookrightarrow \mathbb {P} ^{n}}
8970:
8505:
8462:
8418:
8316:
8293:
8269:
8207:
8039:
7980:
7925:
7761:
7602:
7556:
7507:
7440:
7420:
7400:
7380:
7360:
7322:
7234:
7190:
7105:
7083:
7063:
7043:
6986:
6939:
6884:
6736:
6696:
6515:
6453:
6433:
6413:
6377:
6357:
6292:
6263:
6243:
6211:
6191:
6158:
6008:
5984:
5923:
5902:
5891:
5859:
5769:
5711:
5684:
5648:
5612:
5588:
5489:
5447:
5411:
5363:
5341:
5303:
5276:
5240:
5185:
5156:
5081:, the Grothendieck group of the projective bundle
5073:
5053:
5017:
4971:
4860:
4790:
4768:
4682:
4653:
4568:
4539:
4457:
4416:
4396:
4347:
4233:
4211:
4186:
4135:
4100:
4078:
4056:
4034:
3878:(circa 1962). Furthermore, this approach led to a
3722:
3683:
3600:
3451:
3415:
3395:
3359:
3320:
3248:
3176:
3141:
3113:
3077:
3034:
3014:
2990:
2959:
2924:
2897:
2866:
2830:
2797:
2723:
2672:
2636:
2616:
2575:
2529:
2456:
2430:
2398:
2367:
2347:
2285:
2253:
2218:
2183:
2151:
2086:
2042:
1935:
1893:
1861:
1795:
1766:
1728:
1705:
1685:
1644:
1578:
1509:
1480:
1415:
1386:
1348:
1316:
1191:
1162:
1136:
1110:
1060:
1016:
978:
958:
938:
895:
848:
807:
763:
593:
549:
499:
398:
369:
288:
249:
229:
100:K-theory involves the construction of families of
89:. It can be seen as the study of certain kinds of
9120:The K-book: an introduction to algebraic K-theory
8971:Friedlander, Eric; Grayson, Daniel, eds. (2005).
3828:, the two groups are the same. If it is a smooth
2406:and let the isomorphism class of a vector bundle
145:where it has been conjectured that they classify
9559:
9088:(2006). "K-theory. An elementary introduction".
7981:{\displaystyle V=L_{1}\oplus \dots \oplus L_{n}}
5777:is injective and the cokernel is annihilated by
2066:
500:{\displaystyle a_{1}+'b_{2}+'c=a_{2}+'b_{1}+'c.}
85:. It is also a fundamental tool in the field of
8890:The moduli space of curves (Texel Island, 1994)
8661:http://string.lpthe.jussieu.fr/members.pl?key=7
5501:. It gives a long exact sequence starting with
5277:{\displaystyle \mathbb {P} _{\mathbb {F} }^{n}}
4143:is that it is invariant under reduction, hence
2328:Grothendieck group for compact Hausdorff spaces
1520:The Grothendieck completion can be viewed as a
370:{\displaystyle (a_{1},a_{2})\sim (b_{1},b_{2})}
9036:. Lecture Notes in Mathematics. Vol. 76.
3950:in 1969 and 1972. A variant was also given by
1776:
946:. Here we will denote the identity element of
815:is also associated with a monoid homomorphism
9174:
3694:is an isomorphism of rings. Hence we can use
2838:. Its Grothendieck completion is also called
2617:{\displaystyle \mathbb {R} ^{0}\times X\to X}
1736:there exists a unique abelian group morphism
9062:. Classics in Mathematics. Springer-Verlag.
8849:
8801:
8600:
5028:
2081:
2069:
8506:{\displaystyle \operatorname {Coh} ^{G}(X)}
8270:{\displaystyle \operatorname {Coh} ^{G}(X)}
6987:{\displaystyle CH^{0}(C)\cong \mathbb {Z} }
4035:{\displaystyle {\text{Spec}}(\mathbb {F} )}
3149:. If we look at the isomorphism classes of
9542:
9515:
9181:
9167:
8884:
8047:the Chern character is defined additively
6892:where the left hand term is isomorphic to
3995:
3601:{\displaystyle \cdot =\sum (-1)^{k}\left.}
3078:{\displaystyle ({\text{Vect}}(X),\oplus )}
2576:{\displaystyle ({\text{Vect}}(X),\oplus )}
2305:, the most basic K-theory group (see also
184:
9093:
8944:. Advanced Book Classics (2nd ed.).
8897:
8815:
8607:. Cambridge: Cambridge University Press.
8585:
7762:{\displaystyle ^{vir}=|_{Z}+|_{Z}-|_{Z}.}
7448:we define the virtual conormal bundle as
7172:
7155:
7143:
7099:
7022:
6980:
6947:and the right hand term is isomorphic to
5961:
5600:. Note that vector bundles on a singular
5499:derived noncommutative algebraic geometry
5357:
5326:
5263:
5257:
5089:
4933:
4905:
4877:
4835:
4807:
4784:
4756:
4729:
4714:
4699:
4670:
4611:
4591:
4445:
4381:
4357:
4341:
4333:
4315:
4277:
4227:
4205:
4094:
4072:
4050:
4025:
3929:and others on what later became known as
3677:
3654:
2785:
2714:
2592:
1843:
1820:
1789:
8635:
6385:, essentially giving the computation of
5241:{\displaystyle 1,\xi ,\dots ,\xi ^{n-1}}
9102:
9084:
9054:
8225:
7797:, the Chern character ch is defined by
6516:{\displaystyle E_{1}^{0,q},E_{1}^{1,q}}
6018:Brown-Gersten-Quillen spectral sequence
5685:{\displaystyle X_{sm}\hookrightarrow X}
5284:. This make it possible to compute the
3839:, by applying the same construction to
3177:{\displaystyle \operatorname {Coh} (X)}
2309:). For definitions of higher K-groups K
2229:This shows that we should think of the
14:
9560:
9114:
8936:
8576:
7071:is a smooth projective curve of genus
6219:points, meaning the set of subschemes
4219:-algebra is a direct sum of copies of
4194:. Hence the Grothendieck group of any
4187:{\displaystyle K(X)=K(X_{\text{red}})}
3742:The subject can be said to begin with
9162:
8845:
8843:
8797:
8795:
8728:
8580:(2000). "K-Theory Past and Present".
7242:then there is a short exact sequence
3750:. It takes its name from the German
3746:(1957), who used it to formulate his
2735:. Then, these can be identified with
2724:{\displaystyle C^{0}(X;\mathbb {C} )}
1901:we can find a minimal representative
45:is, roughly speaking, the study of a
9028:
9009:
7557:{\displaystyle Y_{1},Y_{2}\subset X}
5770:{\displaystyle K^{0}(X)\to K_{0}(X)}
3988:strengths and the charges of stable
3917:is another aspect of this analogy.)
8040:{\displaystyle x_{i}=c_{1}(L_{i}),}
7775:
5342:{\displaystyle K(\mathbb {P} ^{n})}
4397:{\displaystyle K(\mathbb {P} ^{n})}
4116:of an Artinian algebra over a field
2876:Atiyah–Hirzebruch spectral sequence
2831:{\displaystyle {\textbf {Idem}}(X)}
2814:
2152:{\displaystyle (a,b)\sim (a-k,b-k)}
1481:{\displaystyle (a,b)\sim (a+k,b+k)}
165:K-theory has been used to classify
24:
8840:
8792:
8303:action of a linear algebraic group
8137:
7875:
7493:
7462:
7272:
7259:
7235:{\displaystyle Y\hookrightarrow X}
7211:
6622:
6539:
5137:
5098:
5046:
4526:
4500:
3578:
3567:
3546:
3491:
3474:
3331:This gives the Grothendieck-group
3303:
3292:
3278:
3234:
3213:
3196:
3022:. Then, as before, the direct sum
2739:matrices in some ring of matrices
1652:That means that, given a morphism
195:The Grothendieck completion of an
25:
9579:
9137:
8669:K-theory and Ramond–Ramond Charge
8558:Grothendieck–Riemann–Roch theorem
7603:{\displaystyle Z=Y_{1}\cap Y_{2}}
5596:where the higher terms come from
5018:{\displaystyle k_{1}+k_{2}\leq n}
4690:comes from its stratification as
3984:, the K-theory classification of
3748:Grothendieck–Riemann–Roch theorem
3613:Grothendieck–Riemann–Roch theorem
116:Grothendieck–Riemann–Roch theorem
9541:
9514:
9504:
9494:
9483:
9473:
9472:
9266:
4683:{\displaystyle \mathbb {P} ^{n}}
4465:and using the push pull formula
4000:
3737:
2991:{\displaystyle {\text{Vect}}(X)}
2399:{\displaystyle {\text{Vect}}(X)}
1635:
1632:
1629:
1626:
1623:
1615:
1612:
1609:
1606:
1603:
1569:
1566:
1563:
1560:
1557:
1549:
1546:
1543:
1540:
1537:
1024:will be the identity element of
550:{\displaystyle G(A)=A^{2}/\sim }
9108:"Vector Bundles & K-Theory"
8913:
8878:
8754:10.1070/rm1969v024n05abeh001357
8220:Hirzebruch–Riemann–Roch theorem
7206:
3920:
3872:extraordinary cohomology theory
3184:we can mod out by the relation
3085:. Then, the Grothendieck group
1686:{\displaystyle \phi :A\to U(B)}
1111:{\displaystyle (0,0)\sim (n,n)}
289:{\displaystyle A^{2}=A\times A}
8921:Robert W. Thomason (1952–1995)
8768:
8722:
8698:
8674:
8653:
8644:
8629:
8594:
8570:
8500:
8494:
8457:
8451:
8410:
8407:
8401:
8375:
8359:
8353:
8264:
8258:
8232:equivariant algebraic K-theory
8199:
8157:
8069:
8063:
8031:
8018:
7900:
7893:
7853:
7850:
7844:
7831:
7819:
7813:
7746:
7741:
7728:
7715:
7710:
7690:
7677:
7672:
7652:
7634:
7620:
7502:
7489:
7483:
7473:
7458:
7408:. If we have a singular space
7314:
7293:
7283:
7268:
7255:
7226:
7185:
7150:
7136:
7130:
7032:
7018:
6973:
6967:
6934:
6928:
6917:
6911:
6876:
6873:
6870:
6864:
6851:
6833:
6827:
6814:
6811:
6805:
6792:
6789:
6786:
6780:
6767:
6754:
6731:
6725:
6708:can then be used to determine
6687:
6681:
6610:
6604:
6408:
6402:
6352:
6346:
6287:
6281:
6235:
6184:
6178:
6153:
6147:
6125:
6122:
6119:
6113:
6107:
6081:
6075:
5979:
5973:
5954:
5948:
5911:For a smooth projective curve
5842:
5837:
5822:
5808:
5793:
5789:
5764:
5758:
5745:
5742:
5736:
5676:
5630:
5580:
5577:
5571:
5555:
5552:
5546:
5533:
5530:
5524:
5511:
5484:
5478:
5442:
5436:
5406:
5400:
5336:
5321:
5180:
5174:
5151:
5148:
5131:
5115:
5103:
5093:
5054:{\displaystyle {\mathcal {E}}}
4645:
4626:
4621:
4615:
4601:
4586:
4563:
4557:
4534:
4531:
4511:
4505:
4485:
4482:
4440:
4391:
4376:
4305:
4292:
4287:
4281:
4181:
4168:
4159:
4153:
4029:
4021:
3717:
3711:
3670:
3664:
3658:
3647:
3641:
3587:
3562:
3519:
3509:
3500:
3485:
3479:
3469:
3446:
3440:
3390:
3384:
3354:
3348:
3312:
3297:
3287:
3272:
3243:
3228:
3222:
3207:
3201:
3191:
3171:
3165:
3108:
3102:
3072:
3063:
3057:
3049:
2998:of all isomorphism classes of
2985:
2979:
2861:
2855:
2825:
2819:
2792:
2789:
2775:
2762:
2718:
2704:
2667:
2661:
2608:
2570:
2561:
2555:
2547:
2524:
2507:
2501:
2490:
2484:
2478:
2451:
2445:
2422:
2393:
2387:
2296:
2280:
2268:
2248:
2236:
2210:
2198:
2178:
2166:
2146:
2122:
2116:
2104:
2087:{\displaystyle k:=\min\{a,b\}}
2037:
2025:
2019:
2007:
2001:
1989:
1983:
1971:
1965:
1953:
1930:
1908:
1888:
1876:
1853:
1839:
1833:
1830:
1816:
1813:
1755:
1752:
1746:
1680:
1674:
1668:
1619:
1553:
1475:
1451:
1445:
1433:
1381:
1378:
1366:
1363:
1343:
1337:
1311:
1308:
1296:
1293:
1287:
1284:
1260:
1257:
1251:
1248:
1236:
1233:
1227:
1224:
1212:
1209:
1105:
1093:
1087:
1075:
1052:
1043:
1037:
1031:
1011:
1008:
996:
993:
933:
921:
887:
884:
872:
869:
866:
843:
837:
831:
802:
793:
787:
781:
755:
752:
690:
687:
681:
678:
652:
649:
643:
640:
614:
611:
588:
579:
573:
567:
523:
517:
364:
338:
332:
306:
224:
207:
13:
1:
8930:
7428:embedded into a smooth space
4540:{\displaystyle i^{*}(\cdot )}
4424:can be computed by embedding
3956:algebraic K-theory of spaces,
3870:they made it the basis of an
2261:as positive integers and the
159:generalized complex manifolds
151:Ramond–Ramond field strengths
137:, K-theory and in particular
9188:
9012:Complex Topological K-Theory
8734:Russian Mathematical Surveys
8604:Complex topological K-theory
8463:{\displaystyle K_{0}^{G}(C)}
8279:equivariant coherent sheaves
7106:{\displaystyle \mathbb {C} }
5907:of a smooth projective curve
5620:are given by vector bundles
5364:{\displaystyle \mathbb {F} }
4791:{\displaystyle \mathbb {Z} }
4234:{\displaystyle \mathbb {Z} }
4212:{\displaystyle \mathbb {F} }
4101:{\displaystyle \mathbb {Z} }
4079:{\displaystyle \mathbb {N} }
4057:{\displaystyle \mathbb {F} }
3992:was first proposed in 1997.
1796:{\displaystyle \mathbb {N} }
7:
8538:List of cohomology theories
8516:
8238:associated to the category
5649:{\displaystyle E\to X_{sm}}
3876:Atiyah–Singer index theorem
2431:{\displaystyle \pi :E\to X}
1777:Example for natural numbers
849:{\displaystyle i:A\to G(A)}
124:Atiyah–Singer index theorem
30:For the hip hop group, see
10:
9584:
9435:Banach fixed-point theorem
7779:
7368:is the conormal bundle of
5931:the Grothendieck group is
4798:, and the intersection of
3969:received the general name
2300:
1767:{\displaystyle G(A)\to B.}
905:certain universal property
896:{\displaystyle a\mapsto ,}
188:
29:
9468:
9425:
9389:
9275:
9264:
9196:
9154:K-theory preprint archive
9144:Grothendieck-Riemann-Roch
9068:10.1007/978-3-540-79890-3
9060:K-theory: an introduction
8985:10.1007/978-3-540-27855-9
5490:{\displaystyle D_{sg}(X)}
5061:over a Noetherian scheme
1061:{\displaystyle (G(A),+).}
8563:
6737:{\displaystyle K_{0}(C)}
6414:{\displaystyle K_{0}(C)}
6244:{\displaystyle x:Y\to X}
6016:. This follows from the
5892:{\displaystyle n=\dim X}
5448:{\displaystyle K_{0}(X)}
5412:{\displaystyle K^{0}(X)}
3973:. It is a major tool of
3939:higher K-theory functors
3893:had used the analogy of
3868:Bott periodicity theorem
3723:{\displaystyle K_{0}(X)}
3452:{\displaystyle K_{0}(X)}
3396:{\displaystyle K^{0}(X)}
3360:{\displaystyle K_{0}(X)}
3114:{\displaystyle K^{0}(X)}
3000:algebraic vector bundles
2867:{\displaystyle K^{0}(X)}
2673:{\displaystyle K^{0}(X)}
2540:It should be clear that
2530:{\displaystyle \oplus =}
808:{\displaystyle (G(A),+)}
594:{\displaystyle (G(A),+)}
177:. For more details, see
163:condensed matter physics
8938:Atiyah, Michael Francis
8328:; thus, by definition,
8281:on an algebraic scheme
7361:{\displaystyle C_{Y/X}}
6199:the set of codimension
6192:{\displaystyle X^{(p)}}
3996:Examples and properties
3866:in 1959, and using the
3367:which is isomorphic to
3035:{\displaystyle \oplus }
2301:This section is about K
1936:{\displaystyle (a',b')}
1510:{\displaystyle k\in A.}
557:has the structure of a
185:Grothendieck completion
81:, it is referred to as
9490:Mathematics portal
9390:Metrics and properties
9376:Second-countable space
8856:. Boston: Birkhäuser.
8826:10.2140/akt.2021.6.381
8507:
8464:
8420:
8318:
8295:
8271:
8209:
8141:
8041:
7982:
7927:
7879:
7763:
7604:
7558:
7509:
7442:
7422:
7402:
7382:
7362:
7324:
7236:
7192:
7107:
7085:
7065:
7045:
6988:
6941:
6886:
6738:
6698:
6517:
6455:
6435:
6415:
6379:
6359:
6294:
6265:
6245:
6213:
6193:
6160:
6010:
5986:
5925:
5893:
5861:
5771:
5713:
5686:
5650:
5614:
5590:
5491:
5449:
5413:
5365:
5343:
5305:
5278:
5242:
5187:
5158:
5075:
5055:
5033:of a projective bundle
5019:
4973:
4862:
4792:
4770:
4684:
4655:
4570:
4541:
4459:
4418:
4398:
4349:
4235:
4213:
4188:
4137:
4102:
4080:
4058:
4036:
3954:in order to study the
3744:Alexander Grothendieck
3724:
3685:
3602:
3453:
3417:
3397:
3361:
3322:
3250:
3178:
3143:
3115:
3079:
3036:
3016:
2992:
2961:
2926:
2899:
2868:
2832:
2799:
2725:
2674:
2638:
2618:
2577:
2531:
2458:
2432:
2400:
2369:
2349:
2293:as negative integers.
2287:
2255:
2220:
2219:{\displaystyle (0,d).}
2185:
2153:
2088:
2044:
1937:
1895:
1863:
1797:
1768:
1730:
1707:
1687:
1646:
1580:
1511:
1482:
1417:
1394:as formal differences
1388:
1350:
1318:
1193:
1164:
1138:
1137:{\displaystyle n\in A}
1112:
1062:
1018:
980:
960:
940:
914:of the abelian monoid
897:
850:
809:
765:
595:
551:
501:
400:
399:{\displaystyle c\in A}
371:
290:
251:
231:
230:{\displaystyle (A,+')}
167:topological insulators
8850:Srinivas, V. (1991).
8601:Park, Efton. (2008).
8508:
8465:
8421:
8319:
8296:
8272:
8210:
8121:
8042:
7983:
7928:
7859:
7764:
7605:
7559:
7510:
7443:
7423:
7403:
7383:
7363:
7325:
7237:
7193:
7108:
7086:
7066:
7046:
6989:
6942:
6887:
6739:
6699:
6518:
6456:
6436:
6416:
6380:
6365:for the Chow ring of
6360:
6295:
6266:
6246:
6214:
6194:
6161:
6011:
5987:
5926:
5894:
5862:
5772:
5714:
5712:{\displaystyle G_{i}}
5687:
5651:
5615:
5591:
5492:
5450:
5414:
5366:
5344:
5306:
5304:{\displaystyle K_{0}}
5279:
5243:
5188:
5159:
5076:
5056:
5020:
4974:
4863:
4793:
4771:
4685:
4656:
4571:
4542:
4460:
4419:
4399:
4350:
4236:
4214:
4189:
4138:
4103:
4081:
4059:
4037:
3725:
3686:
3603:
3454:
3423:is smooth. The group
3418:
3398:
3362:
3323:
3251:
3179:
3144:
3116:
3080:
3037:
3017:
2993:
2962:
2927:
2925:{\displaystyle S^{n}}
2900:
2898:{\displaystyle K^{0}}
2869:
2833:
2800:
2726:
2675:
2639:
2619:
2578:
2532:
2459:
2433:
2401:
2370:
2350:
2288:
2286:{\displaystyle (0,b)}
2256:
2254:{\displaystyle (a,0)}
2221:
2186:
2184:{\displaystyle (c,0)}
2159:which is of the form
2154:
2089:
2045:
1938:
1896:
1894:{\displaystyle (a,b)}
1864:
1798:
1769:
1731:
1708:
1693:of an abelian monoid
1688:
1647:
1581:
1512:
1483:
1418:
1389:
1351:
1319:
1194:
1165:
1139:
1113:
1063:
1019:
981:
961:
941:
939:{\displaystyle (A,+)}
898:
851:
810:
766:
596:
552:
502:
401:
372:
291:
252:
250:{\displaystyle \sim }
232:
143:Type II string theory
27:Branch of mathematics
9445:Invariance of domain
9397:Euler characteristic
9371:Bundle (mathematics)
9010:Park, Efton (2008).
8975:. Berlin, New York:
8973:Handbook of K-Theory
8548:Topological K-theory
8478:
8433:
8335:
8308:
8285:
8242:
8226:Equivariant K-theory
8054:
7992:
7940:
7804:
7791:topological K-theory
7617:
7568:
7522:
7455:
7432:
7412:
7392:
7372:
7337:
7249:
7220:
7117:
7095:
7075:
7055:
6998:
6951:
6896:
6748:
6712:
6527:
6465:
6445:
6425:
6421:. Note that because
6389:
6369:
6304:
6293:{\displaystyle k(x)}
6275:
6255:
6223:
6203:
6170:
6032:
6000:
5935:
5915:
5871:
5781:
5723:
5696:
5660:
5656:on the smooth locus
5624:
5604:
5505:
5462:
5457:Singularity category
5423:
5387:
5353:
5315:
5288:
5252:
5201:
5186:{\displaystyle K(X)}
5168:
5085:
5065:
5041:
4983:
4872:
4802:
4780:
4694:
4665:
4580:
4569:{\displaystyle K(X)}
4551:
4469:
4428:
4408:
4370:
4245:
4223:
4201:
4147:
4127:
4090:
4068:
4046:
4013:
3952:Friedhelm Waldhausen
3849:Friedrich Hirzebruch
3779:locally free sheaves
3698:
3622:
3466:
3427:
3407:
3371:
3335:
3266:
3258:short exact sequence
3188:
3156:
3133:
3089:
3046:
3026:
3006:
2971:
2951:
2909:
2882:
2842:
2809:
2743:
2691:
2648:
2628:
2587:
2544:
2475:
2442:
2410:
2379:
2359:
2339:
2319:Topological K-theory
2265:
2233:
2195:
2163:
2101:
2057:
1950:
1905:
1873:
1807:
1785:
1740:
1717:
1697:
1656:
1593:
1527:
1492:
1430:
1416:{\displaystyle a-b.}
1398:
1360:
1349:{\displaystyle G(A)}
1331:
1206:
1192:{\displaystyle n=n.}
1174:
1148:
1122:
1072:
1028:
990:
970:
950:
918:
860:
819:
778:
608:
564:
511:
410:
384:
303:
261:
241:
204:
71:topological K-theory
9455:Tychonoff's theorem
9450:Poincaré conjecture
9204:General (point-set)
8919:Charles A. Weibel,
8746:1969RuMaS..24....1M
8659:by Ruben Minasian (
8450:
8352:
8198:
8174:
7936:More generally, if
7017:
6706:coniveau filtration
6661:
6637:
6584:
6557:
6512:
6488:
6441:has no codimension
6330:
6055:
5273:
4362:of projective space
3986:Ramond–Ramond field
3767:isomorphism classes
3732:intersection theory
3558:
1317:{\displaystyle +==}
1163:{\displaystyle c=0}
912:equivalence classes
764:{\displaystyle +=.}
257:be the relation on
135:high energy physics
9440:De Rham cohomology
9361:Polyhedral complex
9351:Simplicial complex
9149:Max Karoubi's Page
9034:Algebraic K-Theory
8853:Algebraic K-theory
8804:Annals of K-Theory
8543:Algebraic K-theory
8503:
8472:Grothendieck group
8460:
8436:
8416:
8338:
8314:
8291:
8267:
8236:algebraic K-theory
8205:
8184:
8160:
8037:
7978:
7923:
7759:
7600:
7554:
7505:
7438:
7418:
7398:
7378:
7358:
7320:
7232:
7188:
7103:
7081:
7061:
7041:
7001:
6984:
6937:
6882:
6734:
6694:
6692:
6641:
6617:
6561:
6534:
6513:
6492:
6468:
6451:
6431:
6411:
6375:
6355:
6307:
6290:
6261:
6241:
6209:
6189:
6156:
6087:
6035:
6022:algebraic K-theory
6006:
5982:
5921:
5889:
5857:
5767:
5709:
5682:
5646:
5610:
5586:
5487:
5445:
5409:
5361:
5339:
5301:
5274:
5255:
5238:
5183:
5154:
5071:
5051:
5015:
4969:
4858:
4788:
4766:
4680:
4651:
4566:
4537:
4455:
4414:
4394:
4345:
4231:
4209:
4184:
4133:
4098:
4076:
4054:
4032:
3960:motivic cohomology
3927:J. H. C. Whitehead
3903:Serre's conjecture
3899:projective modules
3791:Grothendieck group
3720:
3681:
3598:
3533:
3449:
3413:
3393:
3357:
3318:
3249:{\displaystyle =+}
3246:
3174:
3139:
3111:
3075:
3032:
3012:
2988:
2957:
2942:algebraic geometry
2922:
2895:
2864:
2828:
2795:
2733:projective modules
2721:
2685:Serre–Swan theorem
2670:
2634:
2614:
2573:
2527:
2454:
2428:
2396:
2365:
2345:
2315:Algebraic K-theory
2307:Grothendieck group
2283:
2251:
2216:
2181:
2149:
2084:
2040:
1933:
1891:
1859:
1803:. We can see that
1793:
1764:
1729:{\displaystyle B,}
1726:
1703:
1683:
1642:
1576:
1507:
1478:
1413:
1384:
1346:
1314:
1189:
1160:
1134:
1108:
1058:
1014:
976:
956:
936:
893:
846:
805:
761:
591:
547:
497:
396:
380:if there exists a
367:
286:
247:
227:
191:Grothendieck group
179:K-theory (physics)
83:algebraic K-theory
79:algebraic geometry
63:algebraic topology
9555:
9554:
9344:fundamental group
9129:978-0-8218-9132-2
9021:978-0-521-85634-8
8994:978-3-540-30436-4
8955:978-0-201-09394-0
8886:Kontsevich, Maxim
8863:978-1-4899-6735-0
8614:978-0-511-38869-9
8553:Operator K-theory
8324:, via Quillen's
8317:{\displaystyle G}
8294:{\displaystyle X}
8155:
7918:
7508:{\displaystyle -}
7441:{\displaystyle X}
7421:{\displaystyle Y}
7401:{\displaystyle X}
7381:{\displaystyle Y}
7084:{\displaystyle g}
7064:{\displaystyle C}
7010:
7005:
6926:
6903:
6673:
6596:
6454:{\displaystyle 2}
6434:{\displaystyle C}
6378:{\displaystyle X}
6338:
6264:{\displaystyle p}
6212:{\displaystyle p}
6059:
6009:{\displaystyle C}
5971:
5924:{\displaystyle C}
5787:
5613:{\displaystyle X}
5074:{\displaystyle X}
4649:
4417:{\displaystyle X}
4309:
4266:
4178:
4136:{\displaystyle X}
4122:Noetherian scheme
4019:
3931:Whitehead torsion
3891:Jean-Pierre Serre
3889:Already in 1955,
3861:topological space
3760:algebraic variety
3416:{\displaystyle X}
3142:{\displaystyle X}
3055:
3015:{\displaystyle X}
2977:
2960:{\displaystyle X}
2946:Noetherian scheme
2816:
2637:{\displaystyle X}
2553:
2385:
2368:{\displaystyle X}
2348:{\displaystyle X}
1706:{\displaystyle A}
1588:forgetful functor
1144:since we can set
979:{\displaystyle 0}
959:{\displaystyle A}
153:and also certain
141:have appeared in
87:operator algebras
67:cohomology theory
55:topological space
16:(Redirected from
9575:
9545:
9544:
9518:
9517:
9508:
9498:
9488:
9487:
9476:
9475:
9270:
9183:
9176:
9169:
9160:
9159:
9133:
9111:
9099:
9097:
9081:
9051:
9025:
9006:
8967:
8924:
8917:
8911:
8910:
8901:
8882:
8876:
8875:
8847:
8838:
8837:
8819:
8799:
8790:
8789:
8787:
8786:
8772:
8766:
8765:
8726:
8720:
8719:
8717:
8716:
8702:
8696:
8695:
8693:
8692:
8686:mathoverflow.net
8678:
8672:
8657:
8651:
8648:
8642:
8641:
8633:
8627:
8626:
8598:
8592:
8591:
8589:
8574:
8523:Bott periodicity
8512:
8510:
8509:
8504:
8490:
8489:
8469:
8467:
8466:
8461:
8449:
8444:
8425:
8423:
8422:
8417:
8397:
8396:
8387:
8386:
8374:
8373:
8351:
8346:
8323:
8321:
8320:
8315:
8300:
8298:
8297:
8292:
8276:
8274:
8273:
8268:
8254:
8253:
8214:
8212:
8211:
8206:
8197:
8192:
8173:
8168:
8156:
8154:
8143:
8140:
8135:
8117:
8116:
8115:
8114:
8091:
8090:
8089:
8088:
8046:
8044:
8043:
8038:
8030:
8029:
8017:
8016:
8004:
8003:
7987:
7985:
7984:
7979:
7977:
7976:
7958:
7957:
7932:
7930:
7929:
7924:
7919:
7917:
7909:
7908:
7907:
7892:
7891:
7881:
7878:
7873:
7843:
7842:
7776:Chern characters
7768:
7766:
7765:
7760:
7755:
7754:
7749:
7740:
7739:
7724:
7723:
7718:
7709:
7708:
7707:
7706:
7686:
7685:
7680:
7671:
7670:
7669:
7668:
7648:
7647:
7632:
7631:
7609:
7607:
7606:
7601:
7599:
7598:
7586:
7585:
7563:
7561:
7560:
7555:
7547:
7546:
7534:
7533:
7514:
7512:
7511:
7506:
7501:
7500:
7482:
7481:
7476:
7470:
7469:
7447:
7445:
7444:
7439:
7427:
7425:
7424:
7419:
7407:
7405:
7404:
7399:
7387:
7385:
7384:
7379:
7367:
7365:
7364:
7359:
7357:
7356:
7352:
7329:
7327:
7326:
7321:
7313:
7312:
7308:
7292:
7291:
7286:
7280:
7279:
7267:
7266:
7241:
7239:
7238:
7233:
7197:
7195:
7194:
7189:
7184:
7183:
7175:
7169:
7164:
7163:
7158:
7146:
7129:
7128:
7112:
7110:
7109:
7104:
7102:
7090:
7088:
7087:
7082:
7070:
7068:
7067:
7062:
7050:
7048:
7047:
7042:
7025:
7016:
7011:
7008:
7006:
7003:
6993:
6991:
6990:
6985:
6983:
6966:
6965:
6946:
6944:
6943:
6938:
6927:
6924:
6910:
6909:
6904:
6901:
6891:
6889:
6888:
6883:
6863:
6862:
6850:
6849:
6840:
6826:
6825:
6804:
6803:
6779:
6778:
6766:
6765:
6743:
6741:
6740:
6735:
6724:
6723:
6703:
6701:
6700:
6695:
6693:
6680:
6679:
6674:
6671:
6660:
6649:
6636:
6625:
6603:
6602:
6597:
6594:
6583:
6569:
6556:
6542:
6522:
6520:
6519:
6514:
6511:
6500:
6487:
6476:
6460:
6458:
6457:
6452:
6440:
6438:
6437:
6432:
6420:
6418:
6417:
6412:
6401:
6400:
6384:
6382:
6381:
6376:
6364:
6362:
6361:
6356:
6345:
6344:
6339:
6336:
6329:
6315:
6299:
6297:
6296:
6291:
6270:
6268:
6267:
6262:
6250:
6248:
6247:
6242:
6218:
6216:
6215:
6210:
6198:
6196:
6195:
6190:
6188:
6187:
6165:
6163:
6162:
6157:
6146:
6145:
6106:
6105:
6086:
6085:
6084:
6054:
6043:
6015:
6013:
6012:
6007:
5991:
5989:
5988:
5983:
5972:
5969:
5964:
5947:
5946:
5930:
5928:
5927:
5922:
5898:
5896:
5895:
5890:
5866:
5864:
5863:
5858:
5856:
5855:
5840:
5835:
5834:
5825:
5811:
5806:
5805:
5796:
5788:
5785:
5776:
5774:
5773:
5768:
5757:
5756:
5735:
5734:
5718:
5716:
5715:
5710:
5708:
5707:
5691:
5689:
5688:
5683:
5675:
5674:
5655:
5653:
5652:
5647:
5645:
5644:
5619:
5617:
5616:
5611:
5595:
5593:
5592:
5587:
5570:
5569:
5545:
5544:
5523:
5522:
5496:
5494:
5493:
5488:
5477:
5476:
5454:
5452:
5451:
5446:
5435:
5434:
5418:
5416:
5415:
5410:
5399:
5398:
5370:
5368:
5367:
5362:
5360:
5348:
5346:
5345:
5340:
5335:
5334:
5329:
5310:
5308:
5307:
5302:
5300:
5299:
5283:
5281:
5280:
5275:
5272:
5267:
5266:
5260:
5247:
5245:
5244:
5239:
5237:
5236:
5193:-module of rank
5192:
5190:
5189:
5184:
5163:
5161:
5160:
5155:
5147:
5146:
5141:
5140:
5127:
5126:
5102:
5101:
5092:
5080:
5078:
5077:
5072:
5060:
5058:
5057:
5052:
5050:
5049:
5024:
5022:
5021:
5016:
5008:
5007:
4995:
4994:
4978:
4976:
4975:
4970:
4968:
4967:
4966:
4965:
4953:
4952:
4936:
4927:
4926:
4925:
4924:
4908:
4899:
4898:
4897:
4896:
4880:
4867:
4865:
4864:
4859:
4857:
4856:
4855:
4854:
4838:
4829:
4828:
4827:
4826:
4810:
4797:
4795:
4794:
4789:
4787:
4775:
4773:
4772:
4767:
4765:
4764:
4759:
4744:
4743:
4732:
4723:
4722:
4717:
4708:
4707:
4702:
4689:
4687:
4686:
4681:
4679:
4678:
4673:
4660:
4658:
4657:
4652:
4650:
4648:
4644:
4643:
4624:
4614:
4608:
4600:
4599:
4594:
4575:
4573:
4572:
4567:
4546:
4544:
4543:
4538:
4530:
4529:
4523:
4522:
4504:
4503:
4497:
4496:
4481:
4480:
4464:
4462:
4461:
4456:
4454:
4453:
4448:
4423:
4421:
4420:
4415:
4403:
4401:
4400:
4395:
4390:
4389:
4384:
4354:
4352:
4351:
4346:
4344:
4336:
4328:
4324:
4323:
4319:
4318:
4310:
4308:
4304:
4303:
4290:
4280:
4274:
4267:
4264:
4257:
4256:
4240:
4238:
4237:
4232:
4230:
4218:
4216:
4215:
4210:
4208:
4193:
4191:
4190:
4185:
4180:
4179:
4176:
4142:
4140:
4139:
4134:
4107:
4105:
4104:
4099:
4097:
4085:
4083:
4082:
4077:
4075:
4063:
4061:
4060:
4055:
4053:
4041:
4039:
4038:
4033:
4028:
4020:
4017:
3756:coherent sheaves
3729:
3727:
3726:
3721:
3710:
3709:
3690:
3688:
3687:
3682:
3680:
3657:
3640:
3639:
3607:
3605:
3604:
3599:
3594:
3590:
3586:
3582:
3581:
3571:
3570:
3557:
3556:
3555:
3550:
3549:
3541:
3527:
3526:
3499:
3495:
3494:
3478:
3477:
3458:
3456:
3455:
3450:
3439:
3438:
3422:
3420:
3419:
3414:
3402:
3400:
3399:
3394:
3383:
3382:
3366:
3364:
3363:
3358:
3347:
3346:
3327:
3325:
3324:
3319:
3311:
3307:
3306:
3296:
3295:
3286:
3282:
3281:
3255:
3253:
3252:
3247:
3242:
3238:
3237:
3221:
3217:
3216:
3200:
3199:
3183:
3181:
3180:
3175:
3151:coherent sheaves
3148:
3146:
3145:
3140:
3120:
3118:
3117:
3112:
3101:
3100:
3084:
3082:
3081:
3076:
3056:
3053:
3041:
3039:
3038:
3033:
3021:
3019:
3018:
3013:
2997:
2995:
2994:
2989:
2978:
2975:
2966:
2964:
2963:
2958:
2931:
2929:
2928:
2923:
2921:
2920:
2905:for the spheres
2904:
2902:
2901:
2896:
2894:
2893:
2873:
2871:
2870:
2865:
2854:
2853:
2837:
2835:
2834:
2829:
2818:
2817:
2804:
2802:
2801:
2796:
2788:
2774:
2773:
2761:
2760:
2730:
2728:
2727:
2722:
2717:
2703:
2702:
2679:
2677:
2676:
2671:
2660:
2659:
2643:
2641:
2640:
2635:
2623:
2621:
2620:
2615:
2601:
2600:
2595:
2582:
2580:
2579:
2574:
2554:
2551:
2536:
2534:
2533:
2528:
2523:
2500:
2463:
2461:
2460:
2457:{\displaystyle }
2455:
2437:
2435:
2434:
2429:
2405:
2403:
2402:
2397:
2386:
2383:
2374:
2372:
2371:
2366:
2354:
2352:
2351:
2346:
2332:Given a compact
2292:
2290:
2289:
2284:
2260:
2258:
2257:
2252:
2225:
2223:
2222:
2217:
2190:
2188:
2187:
2182:
2158:
2156:
2155:
2150:
2093:
2091:
2090:
2085:
2049:
2047:
2046:
2041:
1942:
1940:
1939:
1934:
1929:
1918:
1900:
1898:
1897:
1892:
1868:
1866:
1865:
1860:
1846:
1823:
1802:
1800:
1799:
1794:
1792:
1773:
1771:
1770:
1765:
1735:
1733:
1732:
1727:
1712:
1710:
1709:
1704:
1692:
1690:
1689:
1684:
1651:
1649:
1648:
1643:
1638:
1618:
1585:
1583:
1582:
1577:
1572:
1552:
1516:
1514:
1513:
1508:
1487:
1485:
1484:
1479:
1422:
1420:
1419:
1414:
1393:
1391:
1390:
1387:{\displaystyle }
1385:
1355:
1353:
1352:
1347:
1323:
1321:
1320:
1315:
1198:
1196:
1195:
1190:
1169:
1167:
1166:
1161:
1143:
1141:
1140:
1135:
1117:
1115:
1114:
1109:
1067:
1065:
1064:
1059:
1023:
1021:
1020:
1017:{\displaystyle }
1015:
985:
983:
982:
977:
965:
963:
962:
957:
945:
943:
942:
937:
902:
900:
899:
894:
855:
853:
852:
847:
814:
812:
811:
806:
770:
768:
767:
762:
751:
750:
741:
733:
732:
720:
719:
710:
702:
701:
677:
676:
664:
663:
639:
638:
626:
625:
600:
598:
597:
592:
556:
554:
553:
548:
543:
538:
537:
506:
504:
503:
498:
490:
482:
481:
472:
464:
463:
448:
440:
439:
430:
422:
421:
405:
403:
402:
397:
376:
374:
373:
368:
363:
362:
350:
349:
331:
330:
318:
317:
295:
293:
292:
287:
273:
272:
256:
254:
253:
248:
236:
234:
233:
228:
223:
139:twisted K-theory
128:Adams operations
120:Bott periodicity
21:
9583:
9582:
9578:
9577:
9576:
9574:
9573:
9572:
9558:
9557:
9556:
9551:
9482:
9464:
9460:Urysohn's lemma
9421:
9385:
9271:
9262:
9234:low-dimensional
9192:
9187:
9140:
9130:
9116:Weibel, Charles
9078:
9048:
9022:
8995:
8977:Springer-Verlag
8956:
8933:
8928:
8927:
8918:
8914:
8883:
8879:
8864:
8848:
8841:
8800:
8793:
8784:
8782:
8774:
8773:
8769:
8727:
8723:
8714:
8712:
8704:
8703:
8699:
8690:
8688:
8680:
8679:
8675:
8658:
8654:
8649:
8645:
8634:
8630:
8615:
8599:
8595:
8578:Atiyah, Michael
8575:
8571:
8566:
8519:
8485:
8481:
8479:
8476:
8475:
8445:
8440:
8434:
8431:
8430:
8429:In particular,
8392:
8388:
8382:
8378:
8369:
8365:
8347:
8342:
8336:
8333:
8332:
8309:
8306:
8305:
8286:
8283:
8282:
8249:
8245:
8243:
8240:
8239:
8228:
8193:
8188:
8169:
8164:
8147:
8142:
8136:
8125:
8110:
8106:
8105:
8101:
8084:
8080:
8079:
8075:
8055:
8052:
8051:
8025:
8021:
8012:
8008:
7999:
7995:
7993:
7990:
7989:
7972:
7968:
7953:
7949:
7941:
7938:
7937:
7910:
7903:
7899:
7887:
7883:
7882:
7880:
7874:
7863:
7838:
7834:
7805:
7802:
7801:
7784:
7782:Chern character
7778:
7750:
7745:
7744:
7735:
7731:
7719:
7714:
7713:
7702:
7698:
7697:
7693:
7681:
7676:
7675:
7664:
7660:
7659:
7655:
7637:
7633:
7627:
7623:
7618:
7615:
7614:
7594:
7590:
7581:
7577:
7569:
7566:
7565:
7542:
7538:
7529:
7525:
7523:
7520:
7519:
7496:
7492:
7477:
7472:
7471:
7465:
7461:
7456:
7453:
7452:
7433:
7430:
7429:
7413:
7410:
7409:
7393:
7390:
7389:
7373:
7370:
7369:
7348:
7344:
7340:
7338:
7335:
7334:
7304:
7300:
7296:
7287:
7282:
7281:
7275:
7271:
7262:
7258:
7250:
7247:
7246:
7221:
7218:
7217:
7214:
7212:Virtual bundles
7209:
7176:
7171:
7170:
7165:
7159:
7154:
7153:
7142:
7124:
7120:
7118:
7115:
7114:
7098:
7096:
7093:
7092:
7076:
7073:
7072:
7056:
7053:
7052:
7021:
7012:
7007:
7002:
6999:
6996:
6995:
6979:
6961:
6957:
6952:
6949:
6948:
6923:
6905:
6900:
6899:
6897:
6894:
6893:
6858:
6854:
6845:
6841:
6836:
6821:
6817:
6799:
6795:
6774:
6770:
6761:
6757:
6749:
6746:
6745:
6719:
6715:
6713:
6710:
6709:
6691:
6690:
6675:
6670:
6669:
6662:
6650:
6645:
6626:
6621:
6614:
6613:
6598:
6593:
6592:
6585:
6570:
6565:
6543:
6538:
6530:
6528:
6525:
6524:
6501:
6496:
6477:
6472:
6466:
6463:
6462:
6446:
6443:
6442:
6426:
6423:
6422:
6396:
6392:
6390:
6387:
6386:
6370:
6367:
6366:
6340:
6335:
6334:
6316:
6311:
6305:
6302:
6301:
6276:
6273:
6272:
6256:
6253:
6252:
6251:of codimension
6224:
6221:
6220:
6204:
6201:
6200:
6177:
6173:
6171:
6168:
6167:
6132:
6128:
6092:
6088:
6074:
6070:
6063:
6044:
6039:
6033:
6030:
6029:
6001:
5998:
5997:
5968:
5960:
5942:
5938:
5936:
5933:
5932:
5916:
5913:
5912:
5909:
5906:
5872:
5869:
5868:
5845:
5841:
5836:
5830:
5826:
5821:
5807:
5801:
5797:
5792:
5784:
5782:
5779:
5778:
5752:
5748:
5730:
5726:
5724:
5721:
5720:
5703:
5699:
5697:
5694:
5693:
5667:
5663:
5661:
5658:
5657:
5637:
5633:
5625:
5622:
5621:
5605:
5602:
5601:
5598:higher K-theory
5562:
5558:
5540:
5536:
5518:
5514:
5506:
5503:
5502:
5469:
5465:
5463:
5460:
5459:
5430:
5426:
5424:
5421:
5420:
5394:
5390:
5388:
5385:
5384:
5381:
5378:
5356:
5354:
5351:
5350:
5330:
5325:
5324:
5316:
5313:
5312:
5295:
5291:
5289:
5286:
5285:
5268:
5262:
5261:
5256:
5253:
5250:
5249:
5226:
5222:
5202:
5199:
5198:
5169:
5166:
5165:
5142:
5136:
5135:
5134:
5122:
5118:
5097:
5096:
5088:
5086:
5083:
5082:
5066:
5063:
5062:
5045:
5044:
5042:
5039:
5038:
5035:
5032:
5003:
4999:
4990:
4986:
4984:
4981:
4980:
4961:
4957:
4948:
4944:
4937:
4932:
4931:
4920:
4916:
4909:
4904:
4903:
4892:
4888:
4881:
4876:
4875:
4873:
4870:
4869:
4868:is generically
4850:
4846:
4839:
4834:
4833:
4822:
4818:
4811:
4806:
4805:
4803:
4800:
4799:
4783:
4781:
4778:
4777:
4760:
4755:
4754:
4733:
4728:
4727:
4718:
4713:
4712:
4703:
4698:
4697:
4695:
4692:
4691:
4674:
4669:
4668:
4666:
4663:
4662:
4633:
4629:
4625:
4610:
4609:
4607:
4595:
4590:
4589:
4581:
4578:
4577:
4552:
4549:
4548:
4525:
4524:
4518:
4514:
4499:
4498:
4492:
4488:
4476:
4472:
4470:
4467:
4466:
4449:
4444:
4443:
4429:
4426:
4425:
4409:
4406:
4405:
4385:
4380:
4379:
4371:
4368:
4367:
4364:
4361:
4340:
4332:
4314:
4299:
4295:
4291:
4276:
4275:
4273:
4272:
4268:
4263:
4262:
4258:
4252:
4248:
4246:
4243:
4242:
4226:
4224:
4221:
4220:
4204:
4202:
4199:
4198:
4175:
4171:
4148:
4145:
4144:
4128:
4125:
4124:
4118:
4115:
4093:
4091:
4088:
4087:
4071:
4069:
4066:
4065:
4049:
4047:
4044:
4043:
4024:
4016:
4014:
4011:
4010:
4007:
4004:
3998:
3948:homotopy theory
3923:
3907:polynomial ring
3740:
3705:
3701:
3699:
3696:
3695:
3676:
3653:
3635:
3631:
3623:
3620:
3619:
3615:, we have that
3577:
3576:
3575:
3566:
3565:
3551:
3545:
3544:
3543:
3542:
3537:
3532:
3528:
3522:
3518:
3490:
3489:
3488:
3473:
3472:
3467:
3464:
3463:
3434:
3430:
3428:
3425:
3424:
3408:
3405:
3404:
3378:
3374:
3372:
3369:
3368:
3342:
3338:
3336:
3333:
3332:
3302:
3301:
3300:
3291:
3290:
3277:
3276:
3275:
3267:
3264:
3263:
3233:
3232:
3231:
3212:
3211:
3210:
3195:
3194:
3189:
3186:
3185:
3157:
3154:
3153:
3134:
3131:
3130:
3127:
3096:
3092:
3090:
3087:
3086:
3052:
3047:
3044:
3043:
3027:
3024:
3023:
3007:
3004:
3003:
2974:
2972:
2969:
2968:
2967:there is a set
2952:
2949:
2948:
2938:
2916:
2912:
2910:
2907:
2906:
2889:
2885:
2883:
2880:
2879:
2849:
2845:
2843:
2840:
2839:
2813:
2812:
2810:
2807:
2806:
2784:
2769:
2765:
2750:
2746:
2744:
2741:
2740:
2713:
2698:
2694:
2692:
2689:
2688:
2683:We can use the
2655:
2651:
2649:
2646:
2645:
2644:and is denoted
2629:
2626:
2625:
2596:
2591:
2590:
2588:
2585:
2584:
2550:
2545:
2542:
2541:
2516:
2493:
2476:
2473:
2472:
2443:
2440:
2439:
2411:
2408:
2407:
2382:
2380:
2377:
2376:
2360:
2357:
2356:
2340:
2337:
2336:
2334:Hausdorff space
2330:
2322:
2312:
2304:
2299:
2266:
2263:
2262:
2234:
2231:
2230:
2196:
2193:
2192:
2164:
2161:
2160:
2102:
2099:
2098:
2058:
2055:
2054:
2053:In general, if
1951:
1948:
1947:
1922:
1911:
1906:
1903:
1902:
1874:
1871:
1870:
1842:
1819:
1808:
1805:
1804:
1788:
1786:
1783:
1782:
1779:
1741:
1738:
1737:
1718:
1715:
1714:
1698:
1695:
1694:
1657:
1654:
1653:
1622:
1602:
1594:
1591:
1590:
1556:
1536:
1528:
1525:
1524:
1493:
1490:
1489:
1431:
1428:
1427:
1399:
1396:
1395:
1361:
1358:
1357:
1332:
1329:
1328:
1207:
1204:
1203:
1175:
1172:
1171:
1149:
1146:
1145:
1123:
1120:
1119:
1073:
1070:
1069:
1029:
1026:
1025:
991:
988:
987:
971:
968:
967:
951:
948:
947:
919:
916:
915:
861:
858:
857:
820:
817:
816:
779:
776:
775:
746:
742:
734:
728:
724:
715:
711:
703:
697:
693:
672:
668:
659:
655:
634:
630:
621:
617:
609:
606:
605:
565:
562:
561:
539:
533:
529:
512:
509:
508:
483:
477:
473:
465:
459:
455:
441:
435:
431:
423:
417:
413:
411:
408:
407:
385:
382:
381:
358:
354:
345:
341:
326:
322:
313:
309:
304:
301:
300:
268:
264:
262:
259:
258:
242:
239:
238:
216:
205:
202:
201:
193:
187:
171:superconductors
35:
28:
23:
22:
15:
12:
11:
5:
9581:
9571:
9570:
9553:
9552:
9550:
9549:
9539:
9538:
9537:
9532:
9527:
9512:
9502:
9492:
9480:
9469:
9466:
9465:
9463:
9462:
9457:
9452:
9447:
9442:
9437:
9431:
9429:
9423:
9422:
9420:
9419:
9414:
9409:
9407:Winding number
9404:
9399:
9393:
9391:
9387:
9386:
9384:
9383:
9378:
9373:
9368:
9363:
9358:
9353:
9348:
9347:
9346:
9341:
9339:homotopy group
9331:
9330:
9329:
9324:
9319:
9314:
9309:
9299:
9294:
9289:
9279:
9277:
9273:
9272:
9265:
9263:
9261:
9260:
9255:
9250:
9249:
9248:
9238:
9237:
9236:
9226:
9221:
9216:
9211:
9206:
9200:
9198:
9194:
9193:
9186:
9185:
9178:
9171:
9163:
9157:
9156:
9151:
9146:
9139:
9138:External links
9136:
9135:
9134:
9128:
9112:
9104:Hatcher, Allen
9100:
9082:
9076:
9052:
9046:
9026:
9020:
9007:
8993:
8968:
8954:
8946:Addison-Wesley
8932:
8929:
8926:
8925:
8912:
8899:hep-th/9405035
8877:
8862:
8839:
8810:(3): 381–424.
8791:
8767:
8721:
8697:
8673:
8652:
8643:
8636:Grothendieck.
8628:
8613:
8593:
8568:
8567:
8565:
8562:
8561:
8560:
8555:
8550:
8545:
8540:
8535:
8530:
8525:
8518:
8515:
8502:
8499:
8496:
8493:
8488:
8484:
8459:
8456:
8453:
8448:
8443:
8439:
8427:
8426:
8415:
8412:
8409:
8406:
8403:
8400:
8395:
8391:
8385:
8381:
8377:
8372:
8368:
8364:
8361:
8358:
8355:
8350:
8345:
8341:
8326:Q-construction
8313:
8290:
8266:
8263:
8260:
8257:
8252:
8248:
8227:
8224:
8216:
8215:
8204:
8201:
8196:
8191:
8187:
8183:
8180:
8177:
8172:
8167:
8163:
8159:
8153:
8150:
8146:
8139:
8134:
8131:
8128:
8124:
8120:
8113:
8109:
8104:
8100:
8097:
8094:
8087:
8083:
8078:
8074:
8071:
8068:
8065:
8062:
8059:
8036:
8033:
8028:
8024:
8020:
8015:
8011:
8007:
8002:
7998:
7975:
7971:
7967:
7964:
7961:
7956:
7952:
7948:
7945:
7934:
7933:
7922:
7916:
7913:
7906:
7902:
7898:
7895:
7890:
7886:
7877:
7872:
7869:
7866:
7862:
7858:
7855:
7852:
7849:
7846:
7841:
7837:
7833:
7830:
7827:
7824:
7821:
7818:
7815:
7812:
7809:
7780:Main article:
7777:
7774:
7770:
7769:
7758:
7753:
7748:
7743:
7738:
7734:
7730:
7727:
7722:
7717:
7712:
7705:
7701:
7696:
7692:
7689:
7684:
7679:
7674:
7667:
7663:
7658:
7654:
7651:
7646:
7643:
7640:
7636:
7630:
7626:
7622:
7597:
7593:
7589:
7584:
7580:
7576:
7573:
7553:
7550:
7545:
7541:
7537:
7532:
7528:
7516:
7515:
7504:
7499:
7495:
7491:
7488:
7485:
7480:
7475:
7468:
7464:
7460:
7437:
7417:
7397:
7377:
7355:
7351:
7347:
7343:
7331:
7330:
7319:
7316:
7311:
7307:
7303:
7299:
7295:
7290:
7285:
7278:
7274:
7270:
7265:
7261:
7257:
7254:
7231:
7228:
7225:
7213:
7210:
7208:
7205:
7200:Cohen-Macaulay
7187:
7182:
7179:
7174:
7168:
7162:
7157:
7152:
7149:
7145:
7141:
7138:
7135:
7132:
7127:
7123:
7101:
7080:
7060:
7040:
7037:
7034:
7031:
7028:
7024:
7020:
7015:
6982:
6978:
6975:
6972:
6969:
6964:
6960:
6956:
6936:
6933:
6930:
6922:
6919:
6916:
6913:
6908:
6881:
6878:
6875:
6872:
6869:
6866:
6861:
6857:
6853:
6848:
6844:
6839:
6835:
6832:
6829:
6824:
6820:
6816:
6813:
6810:
6807:
6802:
6798:
6794:
6791:
6788:
6785:
6782:
6777:
6773:
6769:
6764:
6760:
6756:
6753:
6733:
6730:
6727:
6722:
6718:
6689:
6686:
6683:
6678:
6668:
6665:
6663:
6659:
6656:
6653:
6648:
6644:
6640:
6635:
6632:
6629:
6624:
6620:
6616:
6615:
6612:
6609:
6606:
6601:
6591:
6588:
6586:
6582:
6579:
6576:
6573:
6568:
6564:
6560:
6555:
6552:
6549:
6546:
6541:
6537:
6533:
6532:
6510:
6507:
6504:
6499:
6495:
6491:
6486:
6483:
6480:
6475:
6471:
6450:
6430:
6410:
6407:
6404:
6399:
6395:
6374:
6354:
6351:
6348:
6343:
6333:
6328:
6325:
6322:
6319:
6314:
6310:
6289:
6286:
6283:
6280:
6260:
6240:
6237:
6234:
6231:
6228:
6208:
6186:
6183:
6180:
6176:
6155:
6152:
6149:
6144:
6141:
6138:
6135:
6131:
6127:
6124:
6121:
6118:
6115:
6112:
6109:
6104:
6101:
6098:
6095:
6091:
6083:
6080:
6077:
6073:
6069:
6066:
6062:
6058:
6053:
6050:
6047:
6042:
6038:
6026:regular scheme
6005:
5981:
5978:
5975:
5967:
5963:
5959:
5956:
5953:
5950:
5945:
5941:
5920:
5908:
5904:
5901:
5888:
5885:
5882:
5879:
5876:
5854:
5851:
5848:
5844:
5839:
5833:
5829:
5824:
5820:
5817:
5814:
5810:
5804:
5800:
5795:
5791:
5766:
5763:
5760:
5755:
5751:
5747:
5744:
5741:
5738:
5733:
5729:
5706:
5702:
5681:
5678:
5673:
5670:
5666:
5643:
5640:
5636:
5632:
5629:
5609:
5585:
5582:
5579:
5576:
5573:
5568:
5565:
5561:
5557:
5554:
5551:
5548:
5543:
5539:
5535:
5532:
5529:
5526:
5521:
5517:
5513:
5510:
5486:
5483:
5480:
5475:
5472:
5468:
5444:
5441:
5438:
5433:
5429:
5408:
5405:
5402:
5397:
5393:
5380:
5376:
5373:
5359:
5338:
5333:
5328:
5323:
5320:
5298:
5294:
5271:
5265:
5259:
5235:
5232:
5229:
5225:
5221:
5218:
5215:
5212:
5209:
5206:
5182:
5179:
5176:
5173:
5153:
5150:
5145:
5139:
5133:
5130:
5125:
5121:
5117:
5114:
5111:
5108:
5105:
5100:
5095:
5091:
5070:
5048:
5034:
5030:
5027:
5014:
5011:
5006:
5002:
4998:
4993:
4989:
4964:
4960:
4956:
4951:
4947:
4943:
4940:
4935:
4930:
4923:
4919:
4915:
4912:
4907:
4902:
4895:
4891:
4887:
4884:
4879:
4853:
4849:
4845:
4842:
4837:
4832:
4825:
4821:
4817:
4814:
4809:
4786:
4763:
4758:
4753:
4750:
4747:
4742:
4739:
4736:
4731:
4726:
4721:
4716:
4711:
4706:
4701:
4677:
4672:
4647:
4642:
4639:
4636:
4632:
4628:
4623:
4620:
4617:
4613:
4606:
4603:
4598:
4593:
4588:
4585:
4565:
4562:
4559:
4556:
4536:
4533:
4528:
4521:
4517:
4513:
4510:
4507:
4502:
4495:
4491:
4487:
4484:
4479:
4475:
4452:
4447:
4442:
4439:
4436:
4433:
4413:
4393:
4388:
4383:
4378:
4375:
4363:
4359:
4356:
4343:
4339:
4335:
4331:
4327:
4322:
4317:
4313:
4307:
4302:
4298:
4294:
4289:
4286:
4283:
4279:
4271:
4261:
4255:
4251:
4229:
4207:
4183:
4174:
4170:
4167:
4164:
4161:
4158:
4155:
4152:
4132:
4117:
4113:
4110:
4096:
4074:
4052:
4031:
4027:
4023:
4006:
4002:
3999:
3997:
3994:
3975:surgery theory
3967:quadratic form
3944:Daniel Quillen
3922:
3919:
3915:Swan's theorem
3895:vector bundles
3880:noncommutative
3845:Michael Atiyah
3841:vector bundles
3830:affine variety
3826:smooth variety
3739:
3736:
3719:
3716:
3713:
3708:
3704:
3692:
3691:
3679:
3675:
3672:
3669:
3666:
3663:
3660:
3656:
3652:
3649:
3646:
3643:
3638:
3634:
3630:
3627:
3609:
3608:
3597:
3593:
3589:
3585:
3580:
3574:
3569:
3564:
3561:
3554:
3548:
3540:
3536:
3531:
3525:
3521:
3517:
3514:
3511:
3508:
3505:
3502:
3498:
3493:
3487:
3484:
3481:
3476:
3471:
3448:
3445:
3442:
3437:
3433:
3412:
3392:
3389:
3386:
3381:
3377:
3356:
3353:
3350:
3345:
3341:
3329:
3328:
3317:
3314:
3310:
3305:
3299:
3294:
3289:
3285:
3280:
3274:
3271:
3256:if there is a
3245:
3241:
3236:
3230:
3227:
3224:
3220:
3215:
3209:
3206:
3203:
3198:
3193:
3173:
3170:
3167:
3164:
3161:
3138:
3126:
3123:
3110:
3107:
3104:
3099:
3095:
3074:
3071:
3068:
3065:
3062:
3059:
3051:
3031:
3011:
2987:
2984:
2981:
2956:
2937:
2934:
2919:
2915:
2892:
2888:
2863:
2860:
2857:
2852:
2848:
2827:
2824:
2821:
2794:
2791:
2787:
2783:
2780:
2777:
2772:
2768:
2764:
2759:
2756:
2753:
2749:
2720:
2716:
2712:
2709:
2706:
2701:
2697:
2669:
2666:
2663:
2658:
2654:
2633:
2613:
2610:
2607:
2604:
2599:
2594:
2572:
2569:
2566:
2563:
2560:
2557:
2549:
2538:
2537:
2526:
2522:
2519:
2515:
2512:
2509:
2506:
2503:
2499:
2496:
2492:
2489:
2486:
2483:
2480:
2453:
2450:
2447:
2427:
2424:
2421:
2418:
2415:
2395:
2392:
2389:
2364:
2344:
2329:
2326:
2310:
2302:
2298:
2295:
2282:
2279:
2276:
2273:
2270:
2250:
2247:
2244:
2241:
2238:
2227:
2226:
2215:
2212:
2209:
2206:
2203:
2200:
2180:
2177:
2174:
2171:
2168:
2148:
2145:
2142:
2139:
2136:
2133:
2130:
2127:
2124:
2121:
2118:
2115:
2112:
2109:
2106:
2083:
2080:
2077:
2074:
2071:
2068:
2065:
2062:
2051:
2050:
2039:
2036:
2033:
2030:
2027:
2024:
2021:
2018:
2015:
2012:
2009:
2006:
2003:
2000:
1997:
1994:
1991:
1988:
1985:
1982:
1979:
1976:
1973:
1970:
1967:
1964:
1961:
1958:
1955:
1932:
1928:
1925:
1921:
1917:
1914:
1910:
1890:
1887:
1884:
1881:
1878:
1858:
1855:
1852:
1849:
1845:
1841:
1838:
1835:
1832:
1829:
1826:
1822:
1818:
1815:
1812:
1791:
1778:
1775:
1763:
1760:
1757:
1754:
1751:
1748:
1745:
1725:
1722:
1702:
1682:
1679:
1676:
1673:
1670:
1667:
1664:
1661:
1641:
1637:
1634:
1631:
1628:
1625:
1621:
1617:
1614:
1611:
1608:
1605:
1601:
1598:
1575:
1571:
1568:
1565:
1562:
1559:
1555:
1551:
1548:
1545:
1542:
1539:
1535:
1532:
1518:
1517:
1506:
1503:
1500:
1497:
1477:
1474:
1471:
1468:
1465:
1462:
1459:
1456:
1453:
1450:
1447:
1444:
1441:
1438:
1435:
1412:
1409:
1406:
1403:
1383:
1380:
1377:
1374:
1371:
1368:
1365:
1345:
1342:
1339:
1336:
1325:
1324:
1313:
1310:
1307:
1304:
1301:
1298:
1295:
1292:
1289:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1262:
1259:
1256:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1188:
1185:
1182:
1179:
1159:
1156:
1153:
1133:
1130:
1127:
1107:
1104:
1101:
1098:
1095:
1092:
1089:
1086:
1083:
1080:
1077:
1057:
1054:
1051:
1048:
1045:
1042:
1039:
1036:
1033:
1013:
1010:
1007:
1004:
1001:
998:
995:
975:
955:
935:
932:
929:
926:
923:
892:
889:
886:
883:
880:
877:
874:
871:
868:
865:
845:
842:
839:
836:
833:
830:
827:
824:
804:
801:
798:
795:
792:
789:
786:
783:
772:
771:
760:
757:
754:
749:
745:
740:
737:
731:
727:
723:
718:
714:
709:
706:
700:
696:
692:
689:
686:
683:
680:
675:
671:
667:
662:
658:
654:
651:
648:
645:
642:
637:
633:
629:
624:
620:
616:
613:
590:
587:
584:
581:
578:
575:
572:
569:
546:
542:
536:
532:
528:
525:
522:
519:
516:
507:Then, the set
496:
493:
489:
486:
480:
476:
471:
468:
462:
458:
454:
451:
447:
444:
438:
434:
429:
426:
420:
416:
395:
392:
389:
378:
377:
366:
361:
357:
353:
348:
344:
340:
337:
334:
329:
325:
321:
316:
312:
308:
285:
282:
279:
276:
271:
267:
246:
226:
222:
219:
215:
212:
209:
197:abelian monoid
189:Main article:
186:
183:
175:Fermi surfaces
51:vector bundles
26:
9:
6:
4:
3:
2:
9580:
9569:
9566:
9565:
9563:
9548:
9540:
9536:
9533:
9531:
9528:
9526:
9523:
9522:
9521:
9513:
9511:
9507:
9503:
9501:
9497:
9493:
9491:
9486:
9481:
9479:
9471:
9470:
9467:
9461:
9458:
9456:
9453:
9451:
9448:
9446:
9443:
9441:
9438:
9436:
9433:
9432:
9430:
9428:
9424:
9418:
9417:Orientability
9415:
9413:
9410:
9408:
9405:
9403:
9400:
9398:
9395:
9394:
9392:
9388:
9382:
9379:
9377:
9374:
9372:
9369:
9367:
9364:
9362:
9359:
9357:
9354:
9352:
9349:
9345:
9342:
9340:
9337:
9336:
9335:
9332:
9328:
9325:
9323:
9320:
9318:
9315:
9313:
9310:
9308:
9305:
9304:
9303:
9300:
9298:
9295:
9293:
9290:
9288:
9284:
9281:
9280:
9278:
9274:
9269:
9259:
9256:
9254:
9253:Set-theoretic
9251:
9247:
9244:
9243:
9242:
9239:
9235:
9232:
9231:
9230:
9227:
9225:
9222:
9220:
9217:
9215:
9214:Combinatorial
9212:
9210:
9207:
9205:
9202:
9201:
9199:
9195:
9191:
9184:
9179:
9177:
9172:
9170:
9165:
9164:
9161:
9155:
9152:
9150:
9147:
9145:
9142:
9141:
9131:
9125:
9121:
9117:
9113:
9109:
9105:
9101:
9096:
9091:
9087:
9083:
9079:
9077:0-387-08090-2
9073:
9069:
9065:
9061:
9057:
9053:
9049:
9047:3-540-04245-8
9043:
9039:
9035:
9031:
9027:
9023:
9017:
9013:
9008:
9004:
9000:
8996:
8990:
8986:
8982:
8978:
8974:
8969:
8965:
8961:
8957:
8951:
8947:
8943:
8939:
8935:
8934:
8922:
8916:
8909:
8905:
8900:
8895:
8891:
8887:
8881:
8873:
8869:
8865:
8859:
8855:
8854:
8846:
8844:
8835:
8831:
8827:
8823:
8818:
8813:
8809:
8805:
8798:
8796:
8781:
8777:
8771:
8763:
8759:
8755:
8751:
8747:
8743:
8739:
8735:
8731:
8730:Manin, Yuri I
8725:
8711:
8707:
8701:
8687:
8683:
8677:
8670:
8666:
8665:Gregory Moore
8662:
8656:
8650:Karoubi, 2006
8647:
8639:
8632:
8624:
8620:
8616:
8610:
8606:
8605:
8597:
8588:
8583:
8579:
8573:
8569:
8559:
8556:
8554:
8551:
8549:
8546:
8544:
8541:
8539:
8536:
8534:
8531:
8529:
8526:
8524:
8521:
8520:
8514:
8497:
8491:
8486:
8482:
8473:
8454:
8446:
8441:
8437:
8413:
8404:
8398:
8393:
8389:
8383:
8379:
8370:
8366:
8362:
8356:
8348:
8343:
8339:
8331:
8330:
8329:
8327:
8311:
8304:
8288:
8280:
8261:
8255:
8250:
8246:
8237:
8233:
8223:
8221:
8202:
8194:
8189:
8185:
8181:
8178:
8175:
8170:
8165:
8161:
8151:
8148:
8144:
8132:
8129:
8126:
8122:
8118:
8111:
8107:
8102:
8098:
8095:
8092:
8085:
8081:
8076:
8072:
8066:
8060:
8057:
8050:
8049:
8048:
8034:
8026:
8022:
8013:
8009:
8005:
8000:
7996:
7973:
7969:
7965:
7962:
7959:
7954:
7950:
7946:
7943:
7920:
7914:
7911:
7904:
7896:
7888:
7884:
7870:
7867:
7864:
7860:
7856:
7847:
7839:
7835:
7828:
7825:
7822:
7816:
7810:
7807:
7800:
7799:
7798:
7796:
7792:
7788:
7787:Chern classes
7783:
7773:
7756:
7751:
7736:
7732:
7725:
7720:
7703:
7699:
7694:
7687:
7682:
7665:
7661:
7656:
7649:
7644:
7641:
7638:
7628:
7624:
7613:
7612:
7611:
7595:
7591:
7587:
7582:
7578:
7574:
7571:
7551:
7548:
7543:
7539:
7535:
7530:
7526:
7497:
7486:
7478:
7466:
7451:
7450:
7449:
7435:
7415:
7395:
7375:
7353:
7349:
7345:
7341:
7317:
7309:
7305:
7301:
7297:
7288:
7276:
7263:
7252:
7245:
7244:
7243:
7229:
7223:
7204:
7201:
7180:
7177:
7166:
7160:
7147:
7139:
7133:
7125:
7121:
7078:
7058:
7038:
7035:
7029:
7026:
7013:
6976:
6970:
6962:
6958:
6954:
6931:
6920:
6914:
6906:
6879:
6867:
6859:
6855:
6846:
6842:
6837:
6830:
6822:
6818:
6808:
6800:
6796:
6783:
6775:
6771:
6762:
6758:
6751:
6728:
6720:
6716:
6707:
6684:
6676:
6666:
6664:
6657:
6654:
6651:
6646:
6642:
6638:
6633:
6630:
6627:
6618:
6607:
6599:
6589:
6587:
6580:
6577:
6574:
6571:
6566:
6562:
6558:
6553:
6550:
6547:
6544:
6535:
6508:
6505:
6502:
6497:
6493:
6489:
6484:
6481:
6478:
6473:
6469:
6448:
6428:
6405:
6397:
6393:
6372:
6349:
6341:
6331:
6326:
6323:
6320:
6317:
6312:
6308:
6284:
6278:
6258:
6238:
6232:
6229:
6226:
6206:
6181:
6174:
6150:
6142:
6139:
6136:
6133:
6129:
6116:
6110:
6102:
6099:
6096:
6093:
6089:
6078:
6071:
6067:
6064:
6060:
6056:
6051:
6048:
6045:
6040:
6036:
6027:
6023:
6019:
6003:
5995:
5976:
5965:
5957:
5951:
5943:
5939:
5918:
5900:
5886:
5883:
5880:
5877:
5874:
5852:
5849:
5846:
5831:
5827:
5818:
5815:
5812:
5802:
5798:
5761:
5753:
5749:
5739:
5731:
5727:
5719:then the map
5704:
5700:
5679:
5671:
5668:
5664:
5641:
5638:
5634:
5627:
5607:
5599:
5583:
5574:
5566:
5563:
5559:
5549:
5541:
5537:
5527:
5519:
5515:
5508:
5500:
5481:
5473:
5470:
5466:
5458:
5439:
5431:
5427:
5403:
5395:
5391:
5372:
5331:
5318:
5296:
5292:
5269:
5233:
5230:
5227:
5223:
5219:
5216:
5213:
5210:
5207:
5204:
5196:
5177:
5171:
5143:
5128:
5123:
5119:
5112:
5109:
5106:
5068:
5026:
5012:
5009:
5004:
5000:
4996:
4991:
4987:
4962:
4958:
4954:
4949:
4945:
4941:
4938:
4928:
4921:
4917:
4913:
4910:
4900:
4893:
4889:
4885:
4882:
4851:
4847:
4843:
4840:
4830:
4823:
4819:
4815:
4812:
4761:
4751:
4748:
4745:
4740:
4737:
4734:
4724:
4719:
4709:
4704:
4675:
4640:
4637:
4634:
4630:
4618:
4604:
4596:
4583:
4560:
4554:
4519:
4515:
4508:
4493:
4489:
4477:
4473:
4450:
4437:
4434:
4431:
4411:
4386:
4373:
4355:
4337:
4329:
4325:
4320:
4311:
4300:
4296:
4284:
4269:
4259:
4253:
4249:
4197:
4172:
4165:
4162:
4156:
4150:
4130:
4123:
4109:
3993:
3991:
3987:
3983:
3982:string theory
3978:
3976:
3972:
3968:
3963:
3961:
3957:
3953:
3949:
3945:
3941:
3940:
3934:
3932:
3928:
3918:
3916:
3912:
3908:
3904:
3901:to formulate
3900:
3896:
3892:
3887:
3885:
3882:K-theory for
3881:
3877:
3873:
3869:
3865:
3862:
3858:
3854:
3850:
3846:
3842:
3838:
3833:
3831:
3827:
3823:
3818:
3816:
3812:
3808:
3805:behavior and
3804:
3803:cohomological
3800:
3796:
3792:
3788:
3784:
3781:are used, or
3780:
3776:
3772:
3768:
3764:
3761:
3757:
3753:
3749:
3745:
3738:Early history
3735:
3733:
3714:
3706:
3702:
3673:
3667:
3661:
3650:
3644:
3636:
3632:
3628:
3625:
3618:
3617:
3616:
3614:
3595:
3591:
3583:
3572:
3559:
3552:
3538:
3534:
3529:
3523:
3515:
3512:
3506:
3503:
3496:
3482:
3462:
3461:
3460:
3443:
3435:
3431:
3410:
3387:
3379:
3375:
3351:
3343:
3339:
3315:
3308:
3283:
3269:
3262:
3261:
3260:
3259:
3239:
3225:
3218:
3204:
3168:
3162:
3159:
3152:
3136:
3122:
3105:
3097:
3093:
3069:
3066:
3060:
3029:
3009:
3001:
2982:
2954:
2947:
2943:
2933:
2917:
2913:
2890:
2886:
2877:
2858:
2850:
2846:
2822:
2781:
2778:
2770:
2766:
2757:
2754:
2751:
2747:
2738:
2734:
2710:
2707:
2699:
2695:
2686:
2681:
2664:
2656:
2652:
2631:
2611:
2605:
2602:
2597:
2567:
2564:
2558:
2520:
2517:
2513:
2510:
2504:
2497:
2494:
2487:
2481:
2471:
2470:
2469:
2467:
2448:
2425:
2419:
2416:
2413:
2390:
2362:
2342:
2335:
2325:
2320:
2316:
2308:
2294:
2277:
2274:
2271:
2245:
2242:
2239:
2213:
2207:
2204:
2201:
2175:
2172:
2169:
2143:
2140:
2137:
2134:
2131:
2128:
2125:
2119:
2113:
2110:
2107:
2097:
2096:
2095:
2078:
2075:
2072:
2063:
2060:
2034:
2031:
2028:
2022:
2016:
2013:
2010:
2004:
1998:
1995:
1992:
1986:
1980:
1977:
1974:
1968:
1962:
1959:
1956:
1946:
1945:
1944:
1926:
1923:
1919:
1915:
1912:
1885:
1882:
1879:
1869:For any pair
1856:
1850:
1847:
1836:
1827:
1824:
1810:
1774:
1761:
1758:
1749:
1743:
1723:
1720:
1700:
1677:
1671:
1665:
1662:
1659:
1639:
1599:
1596:
1589:
1573:
1533:
1530:
1523:
1504:
1501:
1498:
1495:
1472:
1469:
1466:
1463:
1460:
1457:
1454:
1448:
1442:
1439:
1436:
1426:
1425:
1424:
1410:
1407:
1404:
1401:
1375:
1372:
1369:
1340:
1334:
1305:
1302:
1299:
1290:
1281:
1278:
1275:
1272:
1269:
1266:
1263:
1254:
1245:
1242:
1239:
1230:
1221:
1218:
1215:
1202:
1201:
1200:
1199:This implies
1186:
1183:
1180:
1177:
1157:
1154:
1151:
1131:
1128:
1125:
1102:
1099:
1096:
1090:
1084:
1081:
1078:
1055:
1049:
1046:
1040:
1034:
1005:
1002:
999:
973:
953:
930:
927:
924:
913:
908:
906:
890:
881:
878:
875:
863:
840:
834:
828:
825:
822:
799:
796:
790:
784:
758:
747:
743:
738:
735:
729:
725:
721:
716:
712:
707:
704:
698:
694:
684:
673:
669:
665:
660:
656:
646:
635:
631:
627:
622:
618:
604:
603:
602:
585:
582:
576:
570:
560:
544:
540:
534:
530:
526:
520:
514:
494:
491:
487:
484:
478:
474:
469:
466:
460:
456:
452:
449:
445:
442:
436:
432:
427:
424:
418:
414:
393:
390:
387:
359:
355:
351:
346:
342:
335:
327:
323:
319:
314:
310:
299:
298:
297:
283:
280:
277:
274:
269:
265:
244:
220:
217:
213:
210:
198:
192:
182:
180:
176:
172:
168:
164:
160:
156:
152:
148:
144:
140:
136:
131:
129:
125:
121:
117:
112:
107:
103:
98:
96:
92:
88:
84:
80:
76:
72:
68:
64:
60:
56:
52:
49:generated by
48:
44:
40:
33:
19:
9547:Publications
9412:Chern number
9402:Betti number
9285: /
9276:Key concepts
9224:Differential
9119:
9095:math/0602082
9086:Karoubi, Max
9059:
9056:Karoubi, Max
9033:
9011:
8972:
8941:
8915:
8889:
8880:
8852:
8807:
8803:
8783:. Retrieved
8780:MathOverflow
8779:
8770:
8737:
8733:
8724:
8713:. Retrieved
8710:MathOverflow
8709:
8700:
8689:. Retrieved
8685:
8676:
8655:
8646:
8631:
8603:
8596:
8587:math/0012213
8572:
8428:
8229:
8217:
7935:
7794:
7785:
7771:
7517:
7332:
7215:
7207:Applications
5994:Picard group
5910:
5382:
5194:
5036:
4365:
4119:
4042:for a field
4008:
3979:
3964:
3955:
3937:
3935:
3924:
3921:Developments
3888:
3863:
3856:
3852:
3834:
3821:
3819:
3810:
3806:
3798:
3794:
3786:
3782:
3777:) when only
3774:
3770:
3762:
3751:
3741:
3693:
3610:
3330:
3128:
2939:
2682:
2539:
2331:
2323:
2228:
2052:
1780:
1519:
1326:
909:
903:which has a
773:
379:
194:
132:
101:
99:
42:
36:
9510:Wikiversity
9427:Key results
9030:Swan, R. G.
8740:(5): 1–89.
5197:with basis
3884:C*-algebras
3815:homological
2466:direct sums
2438:be denoted
2297:Definitions
296:defined by
173:and stable
39:mathematics
9356:CW complex
9297:Continuity
9287:Closed set
9246:cohomology
8931:References
8817:1809.10919
8785:2020-10-20
8715:2020-10-20
8691:2017-04-16
5164:is a free
4005:of a field
3817:behavior.
3611:Using the
2737:idempotent
2375:, denoted
406:such that
126:, and the
91:invariants
65:, it is a
9535:geometric
9530:algebraic
9381:Cobordism
9317:Hausdorff
9312:connected
9229:Geometric
9219:Continuum
9209:Algebraic
8872:624583210
8762:0036-0279
8623:227161674
8533:KR-theory
8528:KK-theory
8492:
8399:
8367:π
8256:
8179:⋯
8138:∞
8123:∑
8096:⋯
8061:
7966:⊕
7963:⋯
7960:⊕
7876:∞
7861:∑
7829:
7811:
7726:−
7588:∩
7549:⊂
7494:Ω
7487:−
7463:Ω
7315:→
7294:→
7273:Ω
7269:→
7260:Ω
7256:→
7227:↪
7148:⊕
7140:≅
6977:≅
6921:≅
6877:→
6815:→
6793:→
6755:→
6667:≅
6639:≅
6623:∞
6590:≅
6578:−
6559:≅
6551:−
6540:∞
6332:≅
6324:−
6236:→
6140:−
6134:−
6126:⇒
6100:−
6094:−
6068:∈
6061:∐
5966:⊕
5884:
5850:−
5816:…
5746:→
5677:↪
5631:→
5581:→
5556:→
5534:→
5512:→
5509:⋯
5231:−
5224:ξ
5217:…
5211:ξ
5144:∨
5129:
5124:∙
5113:
5010:≤
4955:−
4942:−
4914:−
4901:∩
4886:−
4844:−
4816:−
4752:∐
4749:⋯
4746:∐
4738:−
4725:∐
4520:∗
4509:⋅
4494:∗
4478:∗
4441:↪
4338:⊕
4312:×
3674:⊗
3659:→
3651:⊗
3560:
3513:−
3507:∑
3483:⋅
3313:→
3298:→
3288:→
3273:→
3163:
3070:⊕
3030:⊕
2755:×
2609:→
2603:×
2568:⊕
2514:⊕
2488:⊕
2423:→
2414:π
2141:−
2129:−
2120:∼
2023:∼
2005:∼
1987:∼
1969:∼
1756:→
1669:→
1660:ϕ
1620:→
1554:→
1499:∈
1449:∼
1405:−
1129:∈
1091:∼
867:↦
856:given by
832:→
545:∼
391:∈
336:∼
281:×
245:∼
93:of large
69:known as
9568:K-theory
9562:Category
9500:Wikibook
9478:Category
9366:Manifold
9334:Homotopy
9292:Interior
9283:Open set
9241:Homology
9190:Topology
9118:(2013).
9106:(2003).
9058:(1978).
9038:Springer
9032:(1968).
8942:K-theory
8940:(1989).
8834:85502709
8517:See also
6994:. Since
6523:, hence
6024:. For a
4196:Artinian
3990:D-branes
3971:L-theory
3859:) for a
3851:defined
3837:topology
3584:′
3497:′
3309:″
3284:′
3240:″
3219:′
2944:. For a
2521:′
2498:′
1927:′
1916:′
1488:for any
1118:for any
986:so that
739:′
708:′
488:′
470:′
446:′
428:′
221:′
147:D-branes
106:functors
95:matrices
43:K-theory
32:K Theory
18:K theory
9525:general
9327:uniform
9307:compact
9258:Digital
9003:2182598
8964:1043170
8908:1363062
8742:Bibcode
8663:), and
8470:is the
7113:, then
1522:functor
1068:First,
601:where:
155:spinors
75:algebra
53:over a
9520:Topics
9322:metric
9197:Fields
9126:
9074:
9044:
9018:
9001:
8991:
8962:
8952:
8906:
8870:
8860:
8832:
8760:
8621:
8611:
8234:is an
7333:where
6271:, and
3946:using
3813:) has
3801:) has
3758:on an
3752:Klasse
2313:, see
122:, the
111:groups
59:scheme
9302:Space
9090:arXiv
8894:arXiv
8830:S2CID
8812:arXiv
8582:arXiv
8564:Notes
8301:with
7091:over
5497:from
3897:with
3824:is a
2094:then
559:group
161:. In
73:. In
61:. In
9124:ISBN
9072:ISBN
9042:ISBN
9016:ISBN
8989:ISBN
8950:ISBN
8868:OCLC
8858:ISBN
8758:ISSN
8619:OCLC
8609:ISBN
8230:The
6704:The
6166:for
5992:for
5867:for
5419:and
5110:Proj
4979:for
4265:Spec
4018:Spec
3911:free
3847:and
3730:for
3054:Vect
2976:Vect
2815:Idem
2552:Vect
2384:Vect
2317:and
237:let
77:and
47:ring
9064:doi
8981:doi
8822:doi
8750:doi
8667:in
8483:Coh
8474:of
8390:Coh
8277:of
8247:Coh
7826:exp
7610:as
7388:in
7004:Ext
6925:Pic
6020:of
5996:of
5970:Pic
5881:dim
5786:lcm
5120:Sym
4177:red
3980:In
3909:is
3835:In
3820:If
3535:Tor
3403:if
3160:Coh
3002:on
2731:as
2191:or
2067:min
966:by
157:on
133:In
57:or
37:In
9564::
9070:.
9040:.
8999:MR
8997:.
8987:.
8979:.
8960:MR
8958:.
8948:.
8904:MR
8902:,
8866:.
8842:^
8828:.
8820:.
8806:.
8794:^
8778:.
8756:.
8748:.
8738:24
8736:.
8708:.
8684:.
8617:.
8222:.
8119::=
8058:ch
7857::=
7808:ch
7009:Ab
6902:CH
6672:CH
6595:CH
6337:CH
5899:.
5371:.
5025:.
4108:.
3977:.
3962:.
3933:.
3886:.
3843:,
3793:;
3734:.
3626:ch
3316:0.
2932:.
2680:.
2064::=
907:.
181:.
169:,
149:,
130:.
118:,
97:.
41:,
9182:e
9175:t
9168:v
9132:.
9110:.
9098:.
9092::
9080:.
9066::
9050:.
9024:.
9005:.
8983::
8966:.
8923:.
8896::
8874:.
8836:.
8824::
8814::
8808:6
8788:.
8764:.
8752::
8744::
8718:.
8694:.
8671:.
8640:.
8625:.
8590:.
8584::
8501:)
8498:X
8495:(
8487:G
8458:)
8455:C
8452:(
8447:G
8442:0
8438:K
8414:.
8411:)
8408:)
8405:X
8402:(
8394:G
8384:+
8380:B
8376:(
8371:i
8363:=
8360:)
8357:X
8354:(
8349:G
8344:i
8340:K
8312:G
8289:X
8265:)
8262:X
8259:(
8251:G
8203:.
8200:)
8195:m
8190:n
8186:x
8182:+
8176:+
8171:m
8166:1
8162:x
8158:(
8152:!
8149:m
8145:1
8133:0
8130:=
8127:m
8112:n
8108:x
8103:e
8099:+
8093:+
8086:1
8082:x
8077:e
8073:=
8070:)
8067:V
8064:(
8035:,
8032:)
8027:i
8023:L
8019:(
8014:1
8010:c
8006:=
8001:i
7997:x
7974:n
7970:L
7955:1
7951:L
7947:=
7944:V
7921:.
7915:!
7912:m
7905:m
7901:)
7897:L
7894:(
7889:1
7885:c
7871:0
7868:=
7865:m
7854:)
7851:)
7848:L
7845:(
7840:1
7836:c
7832:(
7823:=
7820:)
7817:L
7814:(
7795:L
7757:.
7752:Z
7747:|
7742:]
7737:X
7733:T
7729:[
7721:Z
7716:|
7711:]
7704:2
7700:Y
7695:T
7691:[
7688:+
7683:Z
7678:|
7673:]
7666:1
7662:Y
7657:T
7653:[
7650:=
7645:r
7642:i
7639:v
7635:]
7629:Z
7625:T
7621:[
7596:2
7592:Y
7583:1
7579:Y
7575:=
7572:Z
7552:X
7544:2
7540:Y
7536:,
7531:1
7527:Y
7503:]
7498:Y
7490:[
7484:]
7479:Y
7474:|
7467:X
7459:[
7436:X
7416:Y
7396:X
7376:Y
7354:X
7350:/
7346:Y
7342:C
7318:0
7310:X
7306:/
7302:Y
7298:C
7289:Y
7284:|
7277:X
7264:Y
7253:0
7230:X
7224:Y
7186:)
7181:g
7178:2
7173:Z
7167:/
7161:g
7156:C
7151:(
7144:Z
7137:)
7134:C
7131:(
7126:0
7122:K
7100:C
7079:g
7059:C
7039:0
7036:=
7033:)
7030:G
7027:,
7023:Z
7019:(
7014:1
6981:Z
6974:)
6971:C
6968:(
6963:0
6959:H
6955:C
6935:)
6932:C
6929:(
6918:)
6915:C
6912:(
6907:1
6880:0
6874:)
6871:)
6868:X
6865:(
6860:0
6856:K
6852:(
6847:1
6843:F
6838:/
6834:)
6831:X
6828:(
6823:0
6819:K
6812:)
6809:X
6806:(
6801:0
6797:K
6790:)
6787:)
6784:X
6781:(
6776:0
6772:K
6768:(
6763:1
6759:F
6752:0
6732:)
6729:C
6726:(
6721:0
6717:K
6688:)
6685:C
6682:(
6677:0
6658:0
6655:,
6652:0
6647:2
6643:E
6634:0
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225:)
218:+
214:,
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208:(
104:-
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34:.
20:)
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