9474:
9257:
9495:
9463:
9532:
9505:
9485:
6691:
4342:
8202:
5151:
6515:
4763:
188:
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
102:
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
7920:
3315:
4966:
3678:
97:
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
6153:
7185:
4233:
6879:
8413:
4648:
7317:
494:
5583:
5681:. 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
7191:
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.
7038:
4855:
2037:
8042:
5979:
5854:
4682:
6934:
4053:. 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
1639:
1573:
1856:
6352:
2792:
6520:
4452:
5073:
7975:
5271:
364:
2611:
8500:
8264:
6981:
4029:
3595:
3072:
2570:
7756:
5235:
6510:
5679:
3171:
4181:
4860:
2718:
7551:
5764:
8034:
5336:
4391:
3118:
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
2825:
2146:
1475:
7229:
7597:
6686:{\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}}}
5012:
1930:
4677:
2985:
2393:
544:
6020:
1680:
1105:
283:
7792:
5048:
2081:
4534:
8457:
7105:
7100:
5358:
4785:
4228:
4206:
4095:
4073:
4051:
1790:
5643:
2425:
843:
224:
1761:
890:
5484:
1055:
6736:
6731:
6408:
6238:
5886:
5442:
5406:
3717:
3446:
3390:
3354:
3108:
2861:
2667:
2524:
802:
588:
7355:
6186:
3029:
1504:
2213:
1131:
393:
5706:
5298:
2919:
2892:
2280:
2248:
2178:
1888:
933:
244:
6287:
5180:
4563:
1410:
1343:
1186:
3610:
1311:
1157:
758:
3243:
1723:
8311:
8288:
7502:
7435:
7415:
7395:
7375:
7078:
7058:
6448:
6428:
6372:
6258:
6206:
6003:
5918:
5607:
5068:
4411:
4130:
3410:
3136:
3009:
2954:
2631:
2362:
2342:
1700:
973:
953:
5445:
2451:
1381:
1011:
8764:
8502:. The theory was developed by R. W. Thomason in 1980s. Specifically, he proved equivariant analogs of fundamental theorems such as the localization theorem.
6694:
4568:
8207:
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
3758:
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
5493:
9535:
3254:
5444:, 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
5923:
8323:
4337:{\displaystyle K_{0}\left({\text{Spec}}\left({\frac {\mathbb {F} }{(x^{9})}}\times \mathbb {F} \right)\right)=\mathbb {Z} \oplus \mathbb {Z} }
3947:
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
7205:
One useful application of the
Grothendieck-group is to define virtual vector bundles. For example, if we have an embedding of smooth spaces
7237:
6292:
5372:
One recent technique for computing the
Grothendieck group of spaces with minor singularities comes from evaluating the difference between
8694:
8670:
8546:
5487:
3736:
3601:
104:
8197:{\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}).}
6986:
7507:
Another useful application of virtual bundles is with the definition of a virtual tangent bundle of an intersection of spaces: Let
4790:
1938:
7187:
Moreover, the techniques above using the derived category of singularities for isolated singularities can be extended to isolated
2867:, which makes it very accessible. The only required computations for understanding the spectral sequences are computing the group
8208:
5769:
2864:
2613:. We can then apply the Grothendieck completion to get an abelian group from this abelian monoid. This is called the K-theory of
7553:
be projective subvarieties of a smooth projective variety. Then, we can define the virtual tangent bundle of their intersection
9116:
9008:
8981:
8942:
8850:
8601:
6884:
1581:
1515:
5146:{\displaystyle \mathbb {P} ({\mathcal {E}})=\operatorname {Proj} (\operatorname {Sym} ^{\bullet }({\mathcal {E}}^{\vee }))}
1795:
763:
Equivalence classes in this group should be thought of as formal differences of elements in the abelian monoid. This group
5711:
2731:
2676:
and some algebra to get an alternative description of vector bundles over the ring of continuous complex-valued functions
4416:
9169:
4075:
corresponding to the dimension of the vector space. It is an easy exercise to show that the
Grothendieck group is then
8765:"ag.algebraic geometry - Is the algebraic Grothendieck group of a weighted projective space finitely generated ?"
7928:
398:
9523:
9518:
9064:
9034:
5240:
291:
5026:
Another important formula for the
Grothendieck group is the projective bundle formula: given a rank r vector bundle
9026:
2575:
4758:{\displaystyle \mathbb {P} ^{n}=\mathbb {A} ^{n}\coprod \mathbb {A} ^{n-1}\coprod \cdots \coprod \mathbb {A} ^{0}}
9513:
8466:
8230:
6939:
4001:
3864:
3860:
112:
3454:
3034:
2532:
8220:
7605:
2313:
There are a number of basic definitions of K-theory: two coming from topology and two from algebraic geometry.
5189:
9415:
8291:
6453:
5648:
3144:
147:
4135:
8791:
Pavic, Nebojsa; Shinder, Evgeny (2021). "K-theory and the singularity category of quotient singularities".
2679:
7510:
893:
8526:
7980:
5303:
4358:
2797:
2089:
1418:
7208:
2863:. One of the main techniques for computing the Grothendieck group for topological spaces comes from the
9423:
7915:{\displaystyle \operatorname {ch} (L)=\exp(c_{1}(L)):=\sum _{m=0}^{\infty }{\frac {c_{1}(L)^{m}}{m!}}.}
7556:
4971:
3927:
3891:
4653:
2959:
2367:
499:
6006:
1644:
1060:
249:
5029:
2045:
9222:
4457:
3856:
151:
8421:
7083:
5341:
4768:
4211:
4189:
4078:
4056:
4034:
1773:
9508:
9494:
7188:
5612:
3974:
3868:
3821:, then all extensions of locally free sheaves split, so the group has an alternative definition.
2988:
2398:
807:
139:
79:
4355:
One of the most commonly used computations of the
Grothendieck group is with the computation of
1932:
by using the invariance under scaling. For example, we can see from the scaling invariance that
1728:
848:
9443:
9364:
9241:
9229:
9202:
9162:
8653:
5450:
3803:
3732:
2673:
1016:
9438:
6700:
6377:
6211:
5859:
5411:
5375:
4961:{\displaystyle \mathbb {A} ^{n-k_{1}}\cap \mathbb {A} ^{n-k_{2}}=\mathbb {A} ^{n-k_{1}-k_{2}}}
3686:
3415:
3359:
3323:
3077:
2830:
2636:
2463:
766:
552:
9285:
9212:
7325:
6158:
3014:
1893:
1480:
155:
131:
9132:
7040:, we have the sequence of abelian groups above splits, giving the isomorphism. Note that if
4393:
for projective space over a field. This is because the intersection numbers of a projective
3673:{\displaystyle \operatorname {ch} :K_{0}(X)\otimes \mathbb {Q} \to A(X)\otimes \mathbb {Q} }
2183:
1110:
372:
192:
9433:
9385:
9359:
9207:
8991:
8952:
8896:
8881:, Progress in Mathematics, vol. 129, Boston, MA: Birkhäuser Boston, pp. 335–368,
8730:
8536:
7779:
5684:
5276:
3940:
3837:
2897:
2870:
2307:
2253:
2221:
2151:
1861:
906:
229:
83:
59:
47:
8695:"kt.k theory and homology - Grothendieck group for projective space over the dual numbers"
6263:
5156:
4539:
1386:
1319:
1162:
8:
9280:
3778:) when all are coherent sheaves. Either of these two constructions is referred to as the
3720:
1194:
1136:
596:
547:
123:
99:
9484:
9096:
9003:. Cambridge Studies in Advanced Mathematics. Vol. 111. Cambridge University Press.
8734:
6148:{\displaystyle E_{1}^{p,q}=\coprod _{x\in X^{(p)}}K^{-p-q}(k(x))\Rightarrow K_{-p-q}(X)}
5363:
5300:
or
Hirzebruch surfaces. In addition, this can be used to compute the Grothendieck group
3176:
1705:
9478:
9448:
9428:
9349:
9339:
9217:
9197:
9078:
8909:
8882:
8818:
8800:
8570:
8531:
8460:
8296:
8273:
8224:
7443:
7420:
7400:
7380:
7360:
7063:
7043:
6433:
6413:
6357:
6289:
the algebraic function field of the subscheme. This spectral sequence has the property
6243:
6191:
6010:
5988:
5903:
5592:
5586:
5053:
4396:
4115:
3948:
3915:
3779:
3767:
3395:
3121:
3110:
is defined by the application of the
Grothendieck construction on this abelian monoid.
2994:
2939:
2930:
2616:
2347:
2327:
2303:
2295:
1685:
958:
938:
179:
167:
71:
67:
51:
35:
8626:
2794:. We can define equivalence classes of idempotent matrices and form an abelian monoid
2430:
1348:
978:
9556:
9473:
9466:
9332:
9290:
9155:
9112:
9060:
9030:
9004:
8977:
8938:
8856:
8846:
8746:
8607:
8597:
8541:
8267:
7180:{\displaystyle K_{0}(C)\cong \mathbb {Z} \oplus (\mathbb {C} ^{g}/\mathbb {Z} ^{2g})}
4110:
3919:
3903:
3887:
3879:
3849:
3755:
3748:
2934:
2721:
1576:
1345:. This should give us the hint that we should be thinking of the equivalence classes
900:
55:
43:
9498:
8822:
8742:
4765:
since the
Grothendieck group of coherent sheaves on affine spaces are isomorphic to
3031:
of isomorphisms classes of vector bundles is well-defined, giving an abelian monoid
9246:
9192:
9052:
8969:
8874:
8810:
8738:
8511:
3998:
The easiest example of the Grothendieck group is the Grothendieck group of a point
1412:
Another useful observation is the invariance of equivalence classes under scaling:
127:
108:
75:
8649:
2344:
consider the set of isomorphism classes of finite-dimensional vector bundles over
9305:
9300:
8987:
8965:
8948:
8892:
7770:
6874:{\displaystyle 0\to F^{1}(K_{0}(X))\to K_{0}(X)\to K_{0}(X)/F^{1}(K_{0}(X))\to 0}
3936:
3895:
2322:
116:
9488:
3310:{\displaystyle 0\to {\mathcal {E}}'\to {\mathcal {E}}\to {\mathcal {E}}''\to 0.}
9395:
9327:
9104:
8934:
8926:
8566:
8314:
6014:
3963:
3955:
3932:
3833:
3818:
3814:
3744:
3246:
3139:
3113:
159:
39:
9056:
8973:
9550:
9405:
9315:
9295:
9092:
8860:
8750:
8611:
7782:
of a space to (the completion of) its rational cohomology. For a line bundle
7775:
4184:
3970:
3883:
3829:
2924:
163:
9137:
9390:
9310:
9256:
9018:
8814:
5982:
3925:
There followed a period in which there were various partial definitions of
8840:
8591:
3754:. Rather than working directly with the sheaves, he defined a group using
2572:
is an abelian monoid where the unit is given by the trivial vector bundle
2453:. Since isomorphism classes of vector bundles behave well with respect to
9400:
9074:
9044:
8627:"SGA 6 - Formalisme des intersections sur les schema algebriques propres"
3899:
2725:
27:
8408:{\displaystyle K_{i}^{G}(X)=\pi _{i}(B^{+}\operatorname {Coh} ^{G}(X)).}
3902:; this assertion is correct, but was not settled until 20 years later. (
9344:
9275:
9234:
8887:
8718:
8657:
3872:
3791:
2454:
4536:. This makes it possible to do concrete calculations with elements in
3894:, which states that every finitely generated projective module over a
9369:
9142:
9083:
8575:
8521:
8516:
6017:
of finite type over a field, there is a convergent spectral sequence
4643:{\displaystyle K(\mathbb {P} ^{n})={\frac {\mathbb {Z} }{(T^{n+1})}}}
1770:
An illustrative example to look at is the Grothendieck completion of
1575:
and it has the property that it is left adjoint to the corresponding
7312:{\displaystyle 0\to \Omega _{Y}\to \Omega _{X}|_{Y}\to C_{Y/X}\to 0}
6733:
as the desired explicit direct sum since it gives an exact sequence
2929:
There is an analogous construction by considering vector bundles in
9354:
9322:
9271:
9178:
8805:
3978:
3959:
3825:
3448:
is special because there is also a ring structure: we define it as
135:
20:
5368:
of singular spaces and spaces with isolated quotient singularities
3914:
The other historical origin of algebraic K-theory was the work of
2316:
8721:(1969-01-01). "Lectures on the K-functor in algebraic geometry".
5578:{\displaystyle \cdots \to K^{0}(X)\to K_{0}(X)\to K_{sg}(X)\to 0}
4230:, one for each connected component of its spectrum. For example,
1510:
143:
94:
63:
9111:. Grad. Studies in Math. Vol. 145. American Math Society.
8671:"Grothendieck group for projective space over the dual numbers"
6450:
points, the only nontrivial parts of the spectral sequence are
5237:. This formula allows one to compute the Grothendieck group of
3931:. Finally, two useful and equivalent definitions were given by
185:
7033:{\displaystyle {\text{Ext}}_{\text{Ab}}^{1}(\mathbb {Z} ,G)=0}
4850:{\displaystyle \mathbb {A} ^{n-k_{1}},\mathbb {A} ^{n-k_{2}}}
2032:{\displaystyle (4,6)\sim (3,5)\sim (2,4)\sim (1,3)\sim (0,2)}
8877:(1995), "Enumeration of rational curves via torus actions",
3114:
Grothendieck group of coherent sheaves in algebraic geometry
1159:
and apply the equation from the equivalence relation to get
9147:
5974:{\displaystyle K_{0}(C)=\mathbb {Z} \oplus {\text{Pic}}(C)}
5849:{\displaystyle {\text{lcm}}(|G_{1}|,\ldots ,|G_{k}|)^{n-1}}
899:
To get a better understanding of this group, consider some
7977:
is a direct sum of line bundles, with first Chern classes
7778:
can be used to construct a homomorphism of rings from the
2925:
Grothendieck group of vector bundles in algebraic geometry
2457:, we can write these operations on isomorphism classes by
4100:
7761:
Kontsevich uses this construction in one of his papers.
4650:
One technique for determining the Grothendieck group of
6929:{\displaystyle {\text{CH}}^{1}(C)\cong {\text{Pic}}(C)}
3954:
The corresponding constructions involving an auxiliary
1634:{\displaystyle U:\mathbf {AbGrp} \to \mathbf {AbMon} .}
1568:{\displaystyle G:\mathbf {AbMon} \to \mathbf {AbGrp} ,}
5338:
by observing it is a projective bundle over the field
4565:
without having to explicitly know its structure since
4109:
One important property of the Grothendieck group of a
1316:
hence we have an additive inverse for each element in
8469:
8424:
8326:
8299:
8276:
8233:
8045:
7983:
7931:
7795:
7608:
7559:
7513:
7446:
7423:
7403:
7383:
7363:
7328:
7240:
7211:
7108:
7086:
7066:
7046:
6989:
6942:
6887:
6739:
6703:
6518:
6456:
6436:
6416:
6380:
6360:
6295:
6266:
6246:
6214:
6194:
6161:
6023:
5991:
5926:
5906:
5862:
5772:
5714:
5687:
5651:
5615:
5595:
5496:
5453:
5414:
5378:
5344:
5306:
5279:
5243:
5192:
5159:
5076:
5056:
5032:
4974:
4863:
4793:
4771:
4685:
4656:
4571:
4542:
4460:
4419:
4399:
4361:
4236:
4214:
4192:
4138:
4118:
4081:
4059:
4037:
4004:
3743:, meaning "class". Grothendieck needed to work with
3689:
3613:
3457:
3418:
3398:
3362:
3326:
3257:
3179:
3147:
3124:
3080:
3037:
3017:
2997:
2962:
2942:
2900:
2873:
2833:
2800:
2734:
2682:
2639:
2619:
2578:
2535:
2466:
2433:
2401:
2370:
2350:
2330:
2256:
2224:
2186:
2154:
2092:
2048:
1941:
1896:
1864:
1851:{\displaystyle G((\mathbb {N} ,+))=(\mathbb {Z} ,+).}
1798:
1776:
1731:
1708:
1702:
to the underlying abelian monoid of an abelian group
1688:
1647:
1584:
1518:
1483:
1421:
1389:
1351:
1322:
1197:
1165:
1139:
1113:
1063:
1019:
981:
961:
941:
909:
851:
810:
769:
599:
555:
502:
401:
375:
294:
252:
232:
195:
6347:{\displaystyle E_{2}^{p,-p}\cong {\text{CH}}^{p}(X)}
3863:. It played a major role in the second proof of the
2787:{\displaystyle M_{n\times n}(C^{0}(X;\mathbb {C} ))}
4447:{\displaystyle i:X\hookrightarrow \mathbb {P} ^{n}}
8959:
8494:
8451:
8407:
8305:
8282:
8258:
8196:
8028:
7969:
7914:
7750:
7591:
7545:
7496:
7429:
7409:
7389:
7369:
7349:
7311:
7223:
7179:
7094:
7072:
7052:
7032:
6975:
6928:
6873:
6725:
6685:
6504:
6442:
6422:
6402:
6366:
6346:
6281:
6252:
6232:
6200:
6180:
6147:
5997:
5973:
5912:
5891:
5880:
5848:
5758:
5700:
5673:
5637:
5601:
5577:
5478:
5436:
5400:
5352:
5330:
5292:
5265:
5229:
5174:
5145:
5070:, the Grothendieck group of the projective bundle
5062:
5042:
5006:
4960:
4849:
4779:
4757:
4671:
4642:
4557:
4528:
4446:
4405:
4385:
4336:
4222:
4200:
4175:
4124:
4089:
4067:
4045:
4023:
3867:(circa 1962). Furthermore, this approach led to a
3711:
3672:
3589:
3440:
3404:
3384:
3348:
3309:
3237:
3165:
3130:
3102:
3066:
3023:
3003:
2979:
2948:
2913:
2886:
2855:
2819:
2786:
2712:
2661:
2625:
2605:
2564:
2518:
2445:
2419:
2387:
2356:
2336:
2274:
2242:
2207:
2172:
2140:
2075:
2031:
1924:
1882:
1850:
1784:
1755:
1717:
1694:
1674:
1633:
1567:
1498:
1469:
1404:
1375:
1337:
1305:
1180:
1151:
1125:
1099:
1049:
1005:
967:
947:
927:
884:
837:
796:
752:
582:
538:
488:
387:
358:
277:
238:
218:
89:K-theory involves the construction of families of
78:. It can be seen as the study of certain kinds of
9109:The K-book: an introduction to algebraic K-theory
8960:Friedlander, Eric; Grayson, Daniel, eds. (2005).
3817:, the two groups are the same. If it is a smooth
2395:and let the isomorphism class of a vector bundle
134:where it has been conjectured that they classify
9548:
9077:(2006). "K-theory. An elementary introduction".
7970:{\displaystyle V=L_{1}\oplus \dots \oplus L_{n}}
5766:is injective and the cokernel is annihilated by
2055:
489:{\displaystyle a_{1}+'b_{2}+'c=a_{2}+'b_{1}+'c.}
74:. It is also a fundamental tool in the field of
8879:The moduli space of curves (Texel Island, 1994)
8650:http://string.lpthe.jussieu.fr/members.pl?key=7
5490:. It gives a long exact sequence starting with
5266:{\displaystyle \mathbb {P} _{\mathbb {F} }^{n}}
4132:is that it is invariant under reduction, hence
2317:Grothendieck group for compact Hausdorff spaces
1509:The Grothendieck completion can be viewed as a
359:{\displaystyle (a_{1},a_{2})\sim (b_{1},b_{2})}
9025:. Lecture Notes in Mathematics. Vol. 76.
3939:in 1969 and 1972. A variant was also given by
1765:
935:. Here we will denote the identity element of
804:is also associated with a monoid homomorphism
9163:
3683:is an isomorphism of rings. Hence we can use
2827:. Its Grothendieck completion is also called
2606:{\displaystyle \mathbb {R} ^{0}\times X\to X}
1725:there exists a unique abelian group morphism
9051:. Classics in Mathematics. Springer-Verlag.
8838:
8790:
8589:
5017:
2070:
2058:
8495:{\displaystyle \operatorname {Coh} ^{G}(X)}
8259:{\displaystyle \operatorname {Coh} ^{G}(X)}
6976:{\displaystyle CH^{0}(C)\cong \mathbb {Z} }
4024:{\displaystyle {\text{Spec}}(\mathbb {F} )}
3138:. If we look at the isomorphism classes of
9531:
9504:
9170:
9156:
8873:
8036:the Chern character is defined additively
6881:where the left hand term is isomorphic to
3984:
3590:{\displaystyle \cdot =\sum (-1)^{k}\left.}
3067:{\displaystyle ({\text{Vect}}(X),\oplus )}
2565:{\displaystyle ({\text{Vect}}(X),\oplus )}
2294:, the most basic K-theory group (see also
173:
9082:
8933:. Advanced Book Classics (2nd ed.).
8886:
8804:
8596:. Cambridge: Cambridge University Press.
8574:
7751:{\displaystyle ^{vir}=|_{Z}+|_{Z}-|_{Z}.}
7437:we define the virtual conormal bundle as
7161:
7144:
7132:
7088:
7011:
6969:
6936:and the right hand term is isomorphic to
5950:
5589:. Note that vector bundles on a singular
5488:derived noncommutative algebraic geometry
5346:
5315:
5252:
5246:
5078:
4922:
4894:
4866:
4824:
4796:
4773:
4745:
4718:
4703:
4688:
4659:
4600:
4580:
4434:
4370:
4346:
4330:
4322:
4304:
4266:
4216:
4194:
4083:
4061:
4039:
4014:
3918:and others on what later became known as
3666:
3643:
2774:
2703:
2581:
1832:
1809:
1778:
8624:
6374:, essentially giving the computation of
5230:{\displaystyle 1,\xi ,\dots ,\xi ^{n-1}}
9091:
9073:
9043:
8214:
7786:, the Chern character ch is defined by
6505:{\displaystyle E_{1}^{0,q},E_{1}^{1,q}}
6007:Brown-Gersten-Quillen spectral sequence
5674:{\displaystyle X_{sm}\hookrightarrow X}
5273:. This make it possible to compute the
3828:, by applying the same construction to
3166:{\displaystyle \operatorname {Coh} (X)}
2298:). For definitions of higher K-groups K
2218:This shows that we should think of the
9549:
9103:
8925:
8565:
7060:is a smooth projective curve of genus
6208:points, meaning the set of subschemes
4208:-algebra is a direct sum of copies of
4183:. Hence the Grothendieck group of any
4176:{\displaystyle K(X)=K(X_{\text{red}})}
3731:The subject can be said to begin with
9151:
8834:
8832:
8786:
8784:
8717:
8569:(2000). "K-Theory Past and Present".
7231:then there is a short exact sequence
3739:. It takes its name from the German
3735:(1957), who used it to formulate his
2724:. Then, these can be identified with
2713:{\displaystyle C^{0}(X;\mathbb {C} )}
1890:we can find a minimal representative
34:is, roughly speaking, the study of a
9017:
8998:
7546:{\displaystyle Y_{1},Y_{2}\subset X}
5759:{\displaystyle K^{0}(X)\to K_{0}(X)}
3977:strengths and the charges of stable
3906:is another aspect of this analogy.)
8029:{\displaystyle x_{i}=c_{1}(L_{i}),}
7764:
5331:{\displaystyle K(\mathbb {P} ^{n})}
4386:{\displaystyle K(\mathbb {P} ^{n})}
4105:of an Artinian algebra over a field
2865:Atiyah–Hirzebruch spectral sequence
2820:{\displaystyle {\textbf {Idem}}(X)}
2803:
2141:{\displaystyle (a,b)\sim (a-k,b-k)}
1470:{\displaystyle (a,b)\sim (a+k,b+k)}
154:K-theory has been used to classify
13:
8829:
8781:
8292:action of a linear algebraic group
8126:
7864:
7482:
7451:
7261:
7248:
7224:{\displaystyle Y\hookrightarrow X}
7200:
6611:
6528:
5126:
5087:
5035:
4515:
4489:
3567:
3556:
3535:
3480:
3463:
3320:This gives the Grothendieck-group
3292:
3281:
3267:
3223:
3202:
3185:
3011:. Then, as before, the direct sum
2728:matrices in some ring of matrices
1641:That means that, given a morphism
184:The Grothendieck completion of an
14:
9568:
9126:
8658:K-theory and Ramond–Ramond Charge
8547:Grothendieck–Riemann–Roch theorem
7592:{\displaystyle Z=Y_{1}\cap Y_{2}}
5585:where the higher terms come from
5007:{\displaystyle k_{1}+k_{2}\leq n}
4679:comes from its stratification as
3973:, the K-theory classification of
3737:Grothendieck–Riemann–Roch theorem
3602:Grothendieck–Riemann–Roch theorem
105:Grothendieck–Riemann–Roch theorem
9530:
9503:
9493:
9483:
9472:
9462:
9461:
9255:
4672:{\displaystyle \mathbb {P} ^{n}}
4454:and using the push pull formula
3989:
3726:
2980:{\displaystyle {\text{Vect}}(X)}
2388:{\displaystyle {\text{Vect}}(X)}
1624:
1621:
1618:
1615:
1612:
1604:
1601:
1598:
1595:
1592:
1558:
1555:
1552:
1549:
1546:
1538:
1535:
1532:
1529:
1526:
1013:will be the identity element of
539:{\displaystyle G(A)=A^{2}/\sim }
9097:"Vector Bundles & K-Theory"
8902:
8867:
8743:10.1070/rm1969v024n05abeh001357
8209:Hirzebruch–Riemann–Roch theorem
7195:
3909:
3861:extraordinary cohomology theory
3173:we can mod out by the relation
3074:. Then, the Grothendieck group
1675:{\displaystyle \phi :A\to U(B)}
1100:{\displaystyle (0,0)\sim (n,n)}
278:{\displaystyle A^{2}=A\times A}
8910:Robert W. Thomason (1952–1995)
8757:
8711:
8687:
8663:
8642:
8633:
8618:
8583:
8559:
8489:
8483:
8446:
8440:
8399:
8396:
8390:
8364:
8348:
8342:
8253:
8247:
8221:equivariant algebraic K-theory
8188:
8146:
8058:
8052:
8020:
8007:
7889:
7882:
7842:
7839:
7833:
7820:
7808:
7802:
7735:
7730:
7717:
7704:
7699:
7679:
7666:
7661:
7641:
7623:
7609:
7491:
7478:
7472:
7462:
7447:
7397:. If we have a singular space
7303:
7282:
7272:
7257:
7244:
7215:
7174:
7139:
7125:
7119:
7021:
7007:
6962:
6956:
6923:
6917:
6906:
6900:
6865:
6862:
6859:
6853:
6840:
6822:
6816:
6803:
6800:
6794:
6781:
6778:
6775:
6769:
6756:
6743:
6720:
6714:
6697:can then be used to determine
6676:
6670:
6599:
6593:
6397:
6391:
6341:
6335:
6276:
6270:
6224:
6173:
6167:
6142:
6136:
6114:
6111:
6108:
6102:
6096:
6070:
6064:
5968:
5962:
5943:
5937:
5900:For a smooth projective curve
5831:
5826:
5811:
5797:
5782:
5778:
5753:
5747:
5734:
5731:
5725:
5665:
5619:
5569:
5566:
5560:
5544:
5541:
5535:
5522:
5519:
5513:
5500:
5473:
5467:
5431:
5425:
5395:
5389:
5325:
5310:
5169:
5163:
5140:
5137:
5120:
5104:
5092:
5082:
5043:{\displaystyle {\mathcal {E}}}
4634:
4615:
4610:
4604:
4590:
4575:
4552:
4546:
4523:
4520:
4500:
4494:
4474:
4471:
4429:
4380:
4365:
4294:
4281:
4276:
4270:
4170:
4157:
4148:
4142:
4018:
4010:
3706:
3700:
3659:
3653:
3647:
3636:
3630:
3576:
3551:
3508:
3498:
3489:
3474:
3468:
3458:
3435:
3429:
3379:
3373:
3343:
3337:
3301:
3286:
3276:
3261:
3232:
3217:
3211:
3196:
3190:
3180:
3160:
3154:
3097:
3091:
3061:
3052:
3046:
3038:
2987:of all isomorphism classes of
2974:
2968:
2850:
2844:
2814:
2808:
2781:
2778:
2764:
2751:
2707:
2693:
2656:
2650:
2597:
2559:
2550:
2544:
2536:
2513:
2496:
2490:
2479:
2473:
2467:
2440:
2434:
2411:
2382:
2376:
2285:
2269:
2257:
2237:
2225:
2199:
2187:
2167:
2155:
2135:
2111:
2105:
2093:
2076:{\displaystyle k:=\min\{a,b\}}
2026:
2014:
2008:
1996:
1990:
1978:
1972:
1960:
1954:
1942:
1919:
1897:
1877:
1865:
1842:
1828:
1822:
1819:
1805:
1802:
1744:
1741:
1735:
1669:
1663:
1657:
1608:
1542:
1464:
1440:
1434:
1422:
1370:
1367:
1355:
1352:
1332:
1326:
1300:
1297:
1285:
1282:
1276:
1273:
1249:
1246:
1240:
1237:
1225:
1222:
1216:
1213:
1201:
1198:
1094:
1082:
1076:
1064:
1041:
1032:
1026:
1020:
1000:
997:
985:
982:
922:
910:
876:
873:
861:
858:
855:
832:
826:
820:
791:
782:
776:
770:
744:
741:
679:
676:
670:
667:
641:
638:
632:
629:
603:
600:
577:
568:
562:
556:
512:
506:
353:
327:
321:
295:
213:
196:
1:
8919:
7417:embedded into a smooth space
4529:{\displaystyle i^{*}(\cdot )}
4413:can be computed by embedding
3945:algebraic K-theory of spaces,
3859:they made it the basis of an
2250:as positive integers and the
148:generalized complex manifolds
140:Ramond–Ramond field strengths
126:, K-theory and in particular
9177:
9001:Complex Topological K-Theory
8723:Russian Mathematical Surveys
8593:Complex topological K-theory
8452:{\displaystyle K_{0}^{G}(C)}
8268:equivariant coherent sheaves
7095:{\displaystyle \mathbb {C} }
5896:of a smooth projective curve
5609:are given by vector bundles
5353:{\displaystyle \mathbb {F} }
4780:{\displaystyle \mathbb {Z} }
4223:{\displaystyle \mathbb {Z} }
4201:{\displaystyle \mathbb {F} }
4090:{\displaystyle \mathbb {Z} }
4068:{\displaystyle \mathbb {N} }
4046:{\displaystyle \mathbb {F} }
3981:was first proposed in 1997.
1785:{\displaystyle \mathbb {N} }
7:
8527:List of cohomology theories
8505:
8227:associated to the category
5638:{\displaystyle E\to X_{sm}}
3865:Atiyah–Singer index theorem
2420:{\displaystyle \pi :E\to X}
1766:Example for natural numbers
838:{\displaystyle i:A\to G(A)}
113:Atiyah–Singer index theorem
19:For the hip hop group, see
10:
9573:
9424:Banach fixed-point theorem
7768:
7357:is the conormal bundle of
5920:the Grothendieck group is
4787:, and the intersection of
3958:received the general name
2289:
1756:{\displaystyle G(A)\to B.}
894:certain universal property
885:{\displaystyle a\mapsto ,}
177:
18:
9457:
9414:
9378:
9264:
9253:
9185:
9143:K-theory preprint archive
9133:Grothendieck-Riemann-Roch
9057:10.1007/978-3-540-79890-3
9049:K-theory: an introduction
8974:10.1007/978-3-540-27855-9
5479:{\displaystyle D_{sg}(X)}
5050:over a Noetherian scheme
1050:{\displaystyle (G(A),+).}
8552:
6726:{\displaystyle K_{0}(C)}
6403:{\displaystyle K_{0}(C)}
6233:{\displaystyle x:Y\to X}
6005:. This follows from the
5881:{\displaystyle n=\dim X}
5437:{\displaystyle K_{0}(X)}
5401:{\displaystyle K^{0}(X)}
3962:. It is a major tool of
3928:higher K-theory functors
3882:had used the analogy of
3857:Bott periodicity theorem
3712:{\displaystyle K_{0}(X)}
3441:{\displaystyle K_{0}(X)}
3385:{\displaystyle K^{0}(X)}
3349:{\displaystyle K_{0}(X)}
3103:{\displaystyle K^{0}(X)}
2989:algebraic vector bundles
2856:{\displaystyle K^{0}(X)}
2662:{\displaystyle K^{0}(X)}
2529:It should be clear that
2519:{\displaystyle \oplus =}
797:{\displaystyle (G(A),+)}
583:{\displaystyle (G(A),+)}
166:. For more details, see
152:condensed matter physics
8927:Atiyah, Michael Francis
8317:; thus, by definition,
8270:on an algebraic scheme
7350:{\displaystyle C_{Y/X}}
6188:the set of codimension
6181:{\displaystyle X^{(p)}}
3985:Examples and properties
3855:in 1959, and using the
3356:which is isomorphic to
3024:{\displaystyle \oplus }
2290:This section is about K
1925:{\displaystyle (a',b')}
1499:{\displaystyle k\in A.}
546:has the structure of a
174:Grothendieck completion
70:, it is referred to as
9479:Mathematics portal
9379:Metrics and properties
9365:Second-countable space
8845:. Boston: Birkhäuser.
8815:10.2140/akt.2021.6.381
8496:
8453:
8409:
8307:
8284:
8260:
8198:
8130:
8030:
7971:
7916:
7868:
7752:
7593:
7547:
7498:
7431:
7411:
7391:
7371:
7351:
7313:
7225:
7181:
7096:
7074:
7054:
7034:
6977:
6930:
6875:
6727:
6687:
6506:
6444:
6424:
6404:
6368:
6348:
6283:
6254:
6234:
6202:
6182:
6149:
5999:
5975:
5914:
5882:
5850:
5760:
5702:
5675:
5639:
5603:
5579:
5480:
5438:
5402:
5354:
5332:
5294:
5267:
5231:
5176:
5147:
5064:
5044:
5022:of a projective bundle
5008:
4962:
4851:
4781:
4759:
4673:
4644:
4559:
4530:
4448:
4407:
4387:
4338:
4224:
4202:
4177:
4126:
4091:
4069:
4047:
4025:
3943:in order to study the
3733:Alexander Grothendieck
3713:
3674:
3591:
3442:
3406:
3386:
3350:
3311:
3239:
3167:
3132:
3104:
3068:
3025:
3005:
2981:
2950:
2915:
2888:
2857:
2821:
2788:
2714:
2663:
2627:
2607:
2566:
2520:
2447:
2421:
2389:
2358:
2338:
2282:as negative integers.
2276:
2244:
2209:
2208:{\displaystyle (0,d).}
2174:
2142:
2077:
2033:
1926:
1884:
1852:
1786:
1757:
1719:
1696:
1676:
1635:
1569:
1500:
1471:
1406:
1383:as formal differences
1377:
1339:
1307:
1182:
1153:
1127:
1126:{\displaystyle n\in A}
1101:
1051:
1007:
969:
949:
929:
903:of the abelian monoid
886:
839:
798:
754:
584:
540:
490:
389:
388:{\displaystyle c\in A}
360:
279:
240:
220:
219:{\displaystyle (A,+')}
156:topological insulators
8839:Srinivas, V. (1991).
8590:Park, Efton. (2008).
8497:
8454:
8410:
8308:
8285:
8261:
8199:
8110:
8031:
7972:
7917:
7848:
7753:
7594:
7548:
7499:
7432:
7412:
7392:
7372:
7352:
7314:
7226:
7182:
7097:
7075:
7055:
7035:
6978:
6931:
6876:
6728:
6688:
6507:
6445:
6425:
6405:
6369:
6354:for the Chow ring of
6349:
6284:
6255:
6235:
6203:
6183:
6150:
6000:
5976:
5915:
5883:
5851:
5761:
5703:
5701:{\displaystyle G_{i}}
5676:
5640:
5604:
5580:
5481:
5439:
5403:
5355:
5333:
5295:
5293:{\displaystyle K_{0}}
5268:
5232:
5177:
5148:
5065:
5045:
5009:
4963:
4852:
4782:
4760:
4674:
4645:
4560:
4531:
4449:
4408:
4388:
4339:
4225:
4203:
4178:
4127:
4092:
4070:
4048:
4026:
3714:
3675:
3592:
3443:
3412:is smooth. The group
3407:
3387:
3351:
3312:
3240:
3168:
3133:
3105:
3069:
3026:
3006:
2982:
2951:
2916:
2914:{\displaystyle S^{n}}
2889:
2887:{\displaystyle K^{0}}
2858:
2822:
2789:
2715:
2664:
2628:
2608:
2567:
2521:
2448:
2422:
2390:
2359:
2339:
2277:
2275:{\displaystyle (0,b)}
2245:
2243:{\displaystyle (a,0)}
2210:
2175:
2173:{\displaystyle (c,0)}
2148:which is of the form
2143:
2078:
2034:
1927:
1885:
1883:{\displaystyle (a,b)}
1853:
1787:
1758:
1720:
1697:
1682:of an abelian monoid
1677:
1636:
1570:
1501:
1472:
1407:
1378:
1340:
1308:
1183:
1154:
1128:
1102:
1052:
1008:
970:
950:
930:
928:{\displaystyle (A,+)}
887:
840:
799:
755:
585:
541:
491:
390:
361:
280:
241:
239:{\displaystyle \sim }
221:
132:Type II string theory
16:Branch of mathematics
9434:Invariance of domain
9386:Euler characteristic
9360:Bundle (mathematics)
8999:Park, Efton (2008).
8964:. Berlin, New York:
8962:Handbook of K-Theory
8537:Topological K-theory
8467:
8422:
8324:
8297:
8274:
8231:
8215:Equivariant K-theory
8043:
7981:
7929:
7793:
7780:topological K-theory
7606:
7557:
7511:
7444:
7421:
7401:
7381:
7361:
7326:
7238:
7209:
7106:
7084:
7064:
7044:
6987:
6940:
6885:
6737:
6701:
6516:
6454:
6434:
6414:
6410:. Note that because
6378:
6358:
6293:
6282:{\displaystyle k(x)}
6264:
6244:
6212:
6192:
6159:
6021:
5989:
5924:
5904:
5860:
5770:
5712:
5685:
5649:
5645:on the smooth locus
5613:
5593:
5494:
5451:
5446:Singularity category
5412:
5376:
5342:
5304:
5277:
5241:
5190:
5175:{\displaystyle K(X)}
5157:
5074:
5054:
5030:
4972:
4861:
4791:
4769:
4683:
4654:
4569:
4558:{\displaystyle K(X)}
4540:
4458:
4417:
4397:
4359:
4234:
4212:
4190:
4136:
4116:
4079:
4057:
4035:
4002:
3941:Friedhelm Waldhausen
3838:Friedrich Hirzebruch
3768:locally free sheaves
3687:
3611:
3455:
3416:
3396:
3360:
3324:
3255:
3247:short exact sequence
3177:
3145:
3122:
3078:
3035:
3015:
2995:
2960:
2940:
2898:
2871:
2831:
2798:
2732:
2680:
2637:
2617:
2576:
2533:
2464:
2431:
2399:
2368:
2348:
2328:
2308:Topological K-theory
2254:
2222:
2184:
2152:
2090:
2046:
1939:
1894:
1862:
1796:
1774:
1729:
1706:
1686:
1645:
1582:
1516:
1481:
1419:
1405:{\displaystyle a-b.}
1387:
1349:
1338:{\displaystyle G(A)}
1320:
1195:
1181:{\displaystyle n=n.}
1163:
1137:
1111:
1061:
1017:
979:
959:
939:
907:
849:
808:
767:
597:
553:
500:
399:
373:
292:
250:
230:
193:
60:topological K-theory
9444:Tychonoff's theorem
9439:Poincaré conjecture
9193:General (point-set)
8908:Charles A. Weibel,
8735:1969RuMaS..24....1M
8648:by Ruben Minasian (
8439:
8341:
8187:
8163:
7925:More generally, if
7006:
6695:coniveau filtration
6650:
6626:
6573:
6546:
6501:
6477:
6430:has no codimension
6319:
6044:
5262:
4351:of projective space
3975:Ramond–Ramond field
3756:isomorphism classes
3721:intersection theory
3547:
1306:{\displaystyle +==}
1152:{\displaystyle c=0}
901:equivalence classes
753:{\displaystyle +=.}
246:be the relation on
124:high energy physics
9429:De Rham cohomology
9350:Polyhedral complex
9340:Simplicial complex
9138:Max Karoubi's Page
9023:Algebraic K-Theory
8842:Algebraic K-theory
8793:Annals of K-Theory
8532:Algebraic K-theory
8492:
8461:Grothendieck group
8449:
8425:
8405:
8327:
8303:
8280:
8256:
8225:algebraic K-theory
8194:
8173:
8149:
8026:
7967:
7912:
7748:
7589:
7543:
7494:
7427:
7407:
7387:
7367:
7347:
7309:
7221:
7177:
7092:
7070:
7050:
7030:
6990:
6973:
6926:
6871:
6723:
6683:
6681:
6630:
6606:
6550:
6523:
6502:
6481:
6457:
6440:
6420:
6400:
6364:
6344:
6296:
6279:
6250:
6230:
6198:
6178:
6145:
6076:
6024:
6011:algebraic K-theory
5995:
5971:
5910:
5878:
5846:
5756:
5698:
5671:
5635:
5599:
5575:
5476:
5434:
5398:
5350:
5328:
5290:
5263:
5244:
5227:
5172:
5143:
5060:
5040:
5004:
4958:
4847:
4777:
4755:
4669:
4640:
4555:
4526:
4444:
4403:
4383:
4334:
4220:
4198:
4173:
4122:
4087:
4065:
4043:
4021:
3949:motivic cohomology
3916:J. H. C. Whitehead
3892:Serre's conjecture
3888:projective modules
3780:Grothendieck group
3709:
3670:
3587:
3522:
3438:
3402:
3382:
3346:
3307:
3238:{\displaystyle =+}
3235:
3163:
3128:
3100:
3064:
3021:
3001:
2977:
2946:
2931:algebraic geometry
2911:
2884:
2853:
2817:
2784:
2722:projective modules
2710:
2674:Serre–Swan theorem
2659:
2623:
2603:
2562:
2516:
2443:
2417:
2385:
2354:
2334:
2304:Algebraic K-theory
2296:Grothendieck group
2272:
2240:
2205:
2170:
2138:
2073:
2029:
1922:
1880:
1848:
1792:. We can see that
1782:
1753:
1718:{\displaystyle B,}
1715:
1692:
1672:
1631:
1565:
1496:
1467:
1402:
1373:
1335:
1303:
1178:
1149:
1123:
1097:
1047:
1003:
965:
945:
925:
882:
835:
794:
750:
580:
536:
486:
385:
369:if there exists a
356:
275:
236:
216:
180:Grothendieck group
168:K-theory (physics)
72:algebraic K-theory
68:algebraic geometry
52:algebraic topology
9544:
9543:
9333:fundamental group
9118:978-0-8218-9132-2
9010:978-0-521-85634-8
8983:978-3-540-30436-4
8944:978-0-201-09394-0
8875:Kontsevich, Maxim
8852:978-1-4899-6735-0
8603:978-0-511-38869-9
8542:Operator K-theory
8313:, via Quillen's
8306:{\displaystyle G}
8283:{\displaystyle X}
8144:
7907:
7497:{\displaystyle -}
7430:{\displaystyle X}
7410:{\displaystyle Y}
7390:{\displaystyle X}
7370:{\displaystyle Y}
7073:{\displaystyle g}
7053:{\displaystyle C}
6999:
6994:
6915:
6892:
6662:
6585:
6443:{\displaystyle 2}
6423:{\displaystyle C}
6367:{\displaystyle X}
6327:
6253:{\displaystyle p}
6201:{\displaystyle p}
6048:
5998:{\displaystyle C}
5960:
5913:{\displaystyle C}
5776:
5602:{\displaystyle X}
5063:{\displaystyle X}
4638:
4406:{\displaystyle X}
4298:
4255:
4167:
4125:{\displaystyle X}
4111:Noetherian scheme
4008:
3920:Whitehead torsion
3880:Jean-Pierre Serre
3878:Already in 1955,
3850:topological space
3749:algebraic variety
3405:{\displaystyle X}
3131:{\displaystyle X}
3044:
3004:{\displaystyle X}
2966:
2949:{\displaystyle X}
2935:Noetherian scheme
2805:
2626:{\displaystyle X}
2542:
2374:
2357:{\displaystyle X}
2337:{\displaystyle X}
1695:{\displaystyle A}
1577:forgetful functor
1133:since we can set
968:{\displaystyle 0}
948:{\displaystyle A}
142:and also certain
130:have appeared in
76:operator algebras
56:cohomology theory
44:topological space
9564:
9534:
9533:
9507:
9506:
9497:
9487:
9477:
9476:
9465:
9464:
9259:
9172:
9165:
9158:
9149:
9148:
9122:
9100:
9088:
9086:
9070:
9040:
9014:
8995:
8956:
8913:
8906:
8900:
8899:
8890:
8871:
8865:
8864:
8836:
8827:
8826:
8808:
8788:
8779:
8778:
8776:
8775:
8761:
8755:
8754:
8715:
8709:
8708:
8706:
8705:
8691:
8685:
8684:
8682:
8681:
8675:mathoverflow.net
8667:
8661:
8646:
8640:
8637:
8631:
8630:
8622:
8616:
8615:
8587:
8581:
8580:
8578:
8563:
8512:Bott periodicity
8501:
8499:
8498:
8493:
8479:
8478:
8458:
8456:
8455:
8450:
8438:
8433:
8414:
8412:
8411:
8406:
8386:
8385:
8376:
8375:
8363:
8362:
8340:
8335:
8312:
8310:
8309:
8304:
8289:
8287:
8286:
8281:
8265:
8263:
8262:
8257:
8243:
8242:
8203:
8201:
8200:
8195:
8186:
8181:
8162:
8157:
8145:
8143:
8132:
8129:
8124:
8106:
8105:
8104:
8103:
8080:
8079:
8078:
8077:
8035:
8033:
8032:
8027:
8019:
8018:
8006:
8005:
7993:
7992:
7976:
7974:
7973:
7968:
7966:
7965:
7947:
7946:
7921:
7919:
7918:
7913:
7908:
7906:
7898:
7897:
7896:
7881:
7880:
7870:
7867:
7862:
7832:
7831:
7765:Chern characters
7757:
7755:
7754:
7749:
7744:
7743:
7738:
7729:
7728:
7713:
7712:
7707:
7698:
7697:
7696:
7695:
7675:
7674:
7669:
7660:
7659:
7658:
7657:
7637:
7636:
7621:
7620:
7598:
7596:
7595:
7590:
7588:
7587:
7575:
7574:
7552:
7550:
7549:
7544:
7536:
7535:
7523:
7522:
7503:
7501:
7500:
7495:
7490:
7489:
7471:
7470:
7465:
7459:
7458:
7436:
7434:
7433:
7428:
7416:
7414:
7413:
7408:
7396:
7394:
7393:
7388:
7376:
7374:
7373:
7368:
7356:
7354:
7353:
7348:
7346:
7345:
7341:
7318:
7316:
7315:
7310:
7302:
7301:
7297:
7281:
7280:
7275:
7269:
7268:
7256:
7255:
7230:
7228:
7227:
7222:
7186:
7184:
7183:
7178:
7173:
7172:
7164:
7158:
7153:
7152:
7147:
7135:
7118:
7117:
7101:
7099:
7098:
7093:
7091:
7079:
7077:
7076:
7071:
7059:
7057:
7056:
7051:
7039:
7037:
7036:
7031:
7014:
7005:
7000:
6997:
6995:
6992:
6982:
6980:
6979:
6974:
6972:
6955:
6954:
6935:
6933:
6932:
6927:
6916:
6913:
6899:
6898:
6893:
6890:
6880:
6878:
6877:
6872:
6852:
6851:
6839:
6838:
6829:
6815:
6814:
6793:
6792:
6768:
6767:
6755:
6754:
6732:
6730:
6729:
6724:
6713:
6712:
6692:
6690:
6689:
6684:
6682:
6669:
6668:
6663:
6660:
6649:
6638:
6625:
6614:
6592:
6591:
6586:
6583:
6572:
6558:
6545:
6531:
6511:
6509:
6508:
6503:
6500:
6489:
6476:
6465:
6449:
6447:
6446:
6441:
6429:
6427:
6426:
6421:
6409:
6407:
6406:
6401:
6390:
6389:
6373:
6371:
6370:
6365:
6353:
6351:
6350:
6345:
6334:
6333:
6328:
6325:
6318:
6304:
6288:
6286:
6285:
6280:
6259:
6257:
6256:
6251:
6239:
6237:
6236:
6231:
6207:
6205:
6204:
6199:
6187:
6185:
6184:
6179:
6177:
6176:
6154:
6152:
6151:
6146:
6135:
6134:
6095:
6094:
6075:
6074:
6073:
6043:
6032:
6004:
6002:
6001:
5996:
5980:
5978:
5977:
5972:
5961:
5958:
5953:
5936:
5935:
5919:
5917:
5916:
5911:
5887:
5885:
5884:
5879:
5855:
5853:
5852:
5847:
5845:
5844:
5829:
5824:
5823:
5814:
5800:
5795:
5794:
5785:
5777:
5774:
5765:
5763:
5762:
5757:
5746:
5745:
5724:
5723:
5707:
5705:
5704:
5699:
5697:
5696:
5680:
5678:
5677:
5672:
5664:
5663:
5644:
5642:
5641:
5636:
5634:
5633:
5608:
5606:
5605:
5600:
5584:
5582:
5581:
5576:
5559:
5558:
5534:
5533:
5512:
5511:
5485:
5483:
5482:
5477:
5466:
5465:
5443:
5441:
5440:
5435:
5424:
5423:
5407:
5405:
5404:
5399:
5388:
5387:
5359:
5357:
5356:
5351:
5349:
5337:
5335:
5334:
5329:
5324:
5323:
5318:
5299:
5297:
5296:
5291:
5289:
5288:
5272:
5270:
5269:
5264:
5261:
5256:
5255:
5249:
5236:
5234:
5233:
5228:
5226:
5225:
5182:-module of rank
5181:
5179:
5178:
5173:
5152:
5150:
5149:
5144:
5136:
5135:
5130:
5129:
5116:
5115:
5091:
5090:
5081:
5069:
5067:
5066:
5061:
5049:
5047:
5046:
5041:
5039:
5038:
5013:
5011:
5010:
5005:
4997:
4996:
4984:
4983:
4967:
4965:
4964:
4959:
4957:
4956:
4955:
4954:
4942:
4941:
4925:
4916:
4915:
4914:
4913:
4897:
4888:
4887:
4886:
4885:
4869:
4856:
4854:
4853:
4848:
4846:
4845:
4844:
4843:
4827:
4818:
4817:
4816:
4815:
4799:
4786:
4784:
4783:
4778:
4776:
4764:
4762:
4761:
4756:
4754:
4753:
4748:
4733:
4732:
4721:
4712:
4711:
4706:
4697:
4696:
4691:
4678:
4676:
4675:
4670:
4668:
4667:
4662:
4649:
4647:
4646:
4641:
4639:
4637:
4633:
4632:
4613:
4603:
4597:
4589:
4588:
4583:
4564:
4562:
4561:
4556:
4535:
4533:
4532:
4527:
4519:
4518:
4512:
4511:
4493:
4492:
4486:
4485:
4470:
4469:
4453:
4451:
4450:
4445:
4443:
4442:
4437:
4412:
4410:
4409:
4404:
4392:
4390:
4389:
4384:
4379:
4378:
4373:
4343:
4341:
4340:
4335:
4333:
4325:
4317:
4313:
4312:
4308:
4307:
4299:
4297:
4293:
4292:
4279:
4269:
4263:
4256:
4253:
4246:
4245:
4229:
4227:
4226:
4221:
4219:
4207:
4205:
4204:
4199:
4197:
4182:
4180:
4179:
4174:
4169:
4168:
4165:
4131:
4129:
4128:
4123:
4096:
4094:
4093:
4088:
4086:
4074:
4072:
4071:
4066:
4064:
4052:
4050:
4049:
4044:
4042:
4030:
4028:
4027:
4022:
4017:
4009:
4006:
3745:coherent sheaves
3718:
3716:
3715:
3710:
3699:
3698:
3679:
3677:
3676:
3671:
3669:
3646:
3629:
3628:
3596:
3594:
3593:
3588:
3583:
3579:
3575:
3571:
3570:
3560:
3559:
3546:
3545:
3544:
3539:
3538:
3530:
3516:
3515:
3488:
3484:
3483:
3467:
3466:
3447:
3445:
3444:
3439:
3428:
3427:
3411:
3409:
3408:
3403:
3391:
3389:
3388:
3383:
3372:
3371:
3355:
3353:
3352:
3347:
3336:
3335:
3316:
3314:
3313:
3308:
3300:
3296:
3295:
3285:
3284:
3275:
3271:
3270:
3244:
3242:
3241:
3236:
3231:
3227:
3226:
3210:
3206:
3205:
3189:
3188:
3172:
3170:
3169:
3164:
3140:coherent sheaves
3137:
3135:
3134:
3129:
3109:
3107:
3106:
3101:
3090:
3089:
3073:
3071:
3070:
3065:
3045:
3042:
3030:
3028:
3027:
3022:
3010:
3008:
3007:
3002:
2986:
2984:
2983:
2978:
2967:
2964:
2955:
2953:
2952:
2947:
2920:
2918:
2917:
2912:
2910:
2909:
2894:for the spheres
2893:
2891:
2890:
2885:
2883:
2882:
2862:
2860:
2859:
2854:
2843:
2842:
2826:
2824:
2823:
2818:
2807:
2806:
2793:
2791:
2790:
2785:
2777:
2763:
2762:
2750:
2749:
2719:
2717:
2716:
2711:
2706:
2692:
2691:
2668:
2666:
2665:
2660:
2649:
2648:
2632:
2630:
2629:
2624:
2612:
2610:
2609:
2604:
2590:
2589:
2584:
2571:
2569:
2568:
2563:
2543:
2540:
2525:
2523:
2522:
2517:
2512:
2489:
2452:
2450:
2449:
2446:{\displaystyle }
2444:
2426:
2424:
2423:
2418:
2394:
2392:
2391:
2386:
2375:
2372:
2363:
2361:
2360:
2355:
2343:
2341:
2340:
2335:
2321:Given a compact
2281:
2279:
2278:
2273:
2249:
2247:
2246:
2241:
2214:
2212:
2211:
2206:
2179:
2177:
2176:
2171:
2147:
2145:
2144:
2139:
2082:
2080:
2079:
2074:
2038:
2036:
2035:
2030:
1931:
1929:
1928:
1923:
1918:
1907:
1889:
1887:
1886:
1881:
1857:
1855:
1854:
1849:
1835:
1812:
1791:
1789:
1788:
1783:
1781:
1762:
1760:
1759:
1754:
1724:
1722:
1721:
1716:
1701:
1699:
1698:
1693:
1681:
1679:
1678:
1673:
1640:
1638:
1637:
1632:
1627:
1607:
1574:
1572:
1571:
1566:
1561:
1541:
1505:
1503:
1502:
1497:
1476:
1474:
1473:
1468:
1411:
1409:
1408:
1403:
1382:
1380:
1379:
1376:{\displaystyle }
1374:
1344:
1342:
1341:
1336:
1312:
1310:
1309:
1304:
1187:
1185:
1184:
1179:
1158:
1156:
1155:
1150:
1132:
1130:
1129:
1124:
1106:
1104:
1103:
1098:
1056:
1054:
1053:
1048:
1012:
1010:
1009:
1006:{\displaystyle }
1004:
974:
972:
971:
966:
954:
952:
951:
946:
934:
932:
931:
926:
891:
889:
888:
883:
844:
842:
841:
836:
803:
801:
800:
795:
759:
757:
756:
751:
740:
739:
730:
722:
721:
709:
708:
699:
691:
690:
666:
665:
653:
652:
628:
627:
615:
614:
589:
587:
586:
581:
545:
543:
542:
537:
532:
527:
526:
495:
493:
492:
487:
479:
471:
470:
461:
453:
452:
437:
429:
428:
419:
411:
410:
394:
392:
391:
386:
365:
363:
362:
357:
352:
351:
339:
338:
320:
319:
307:
306:
284:
282:
281:
276:
262:
261:
245:
243:
242:
237:
225:
223:
222:
217:
212:
128:twisted K-theory
117:Adams operations
109:Bott periodicity
9572:
9571:
9567:
9566:
9565:
9563:
9562:
9561:
9547:
9546:
9545:
9540:
9471:
9453:
9449:Urysohn's lemma
9410:
9374:
9260:
9251:
9223:low-dimensional
9181:
9176:
9129:
9119:
9105:Weibel, Charles
9067:
9037:
9011:
8984:
8966:Springer-Verlag
8945:
8922:
8917:
8916:
8907:
8903:
8872:
8868:
8853:
8837:
8830:
8789:
8782:
8773:
8771:
8763:
8762:
8758:
8716:
8712:
8703:
8701:
8693:
8692:
8688:
8679:
8677:
8669:
8668:
8664:
8647:
8643:
8638:
8634:
8623:
8619:
8604:
8588:
8584:
8567:Atiyah, Michael
8564:
8560:
8555:
8508:
8474:
8470:
8468:
8465:
8464:
8434:
8429:
8423:
8420:
8419:
8418:In particular,
8381:
8377:
8371:
8367:
8358:
8354:
8336:
8331:
8325:
8322:
8321:
8298:
8295:
8294:
8275:
8272:
8271:
8238:
8234:
8232:
8229:
8228:
8217:
8182:
8177:
8158:
8153:
8136:
8131:
8125:
8114:
8099:
8095:
8094:
8090:
8073:
8069:
8068:
8064:
8044:
8041:
8040:
8014:
8010:
8001:
7997:
7988:
7984:
7982:
7979:
7978:
7961:
7957:
7942:
7938:
7930:
7927:
7926:
7899:
7892:
7888:
7876:
7872:
7871:
7869:
7863:
7852:
7827:
7823:
7794:
7791:
7790:
7773:
7771:Chern character
7767:
7739:
7734:
7733:
7724:
7720:
7708:
7703:
7702:
7691:
7687:
7686:
7682:
7670:
7665:
7664:
7653:
7649:
7648:
7644:
7626:
7622:
7616:
7612:
7607:
7604:
7603:
7583:
7579:
7570:
7566:
7558:
7555:
7554:
7531:
7527:
7518:
7514:
7512:
7509:
7508:
7485:
7481:
7466:
7461:
7460:
7454:
7450:
7445:
7442:
7441:
7422:
7419:
7418:
7402:
7399:
7398:
7382:
7379:
7378:
7362:
7359:
7358:
7337:
7333:
7329:
7327:
7324:
7323:
7293:
7289:
7285:
7276:
7271:
7270:
7264:
7260:
7251:
7247:
7239:
7236:
7235:
7210:
7207:
7206:
7203:
7201:Virtual bundles
7198:
7165:
7160:
7159:
7154:
7148:
7143:
7142:
7131:
7113:
7109:
7107:
7104:
7103:
7087:
7085:
7082:
7081:
7065:
7062:
7061:
7045:
7042:
7041:
7010:
7001:
6996:
6991:
6988:
6985:
6984:
6968:
6950:
6946:
6941:
6938:
6937:
6912:
6894:
6889:
6888:
6886:
6883:
6882:
6847:
6843:
6834:
6830:
6825:
6810:
6806:
6788:
6784:
6763:
6759:
6750:
6746:
6738:
6735:
6734:
6708:
6704:
6702:
6699:
6698:
6680:
6679:
6664:
6659:
6658:
6651:
6639:
6634:
6615:
6610:
6603:
6602:
6587:
6582:
6581:
6574:
6559:
6554:
6532:
6527:
6519:
6517:
6514:
6513:
6490:
6485:
6466:
6461:
6455:
6452:
6451:
6435:
6432:
6431:
6415:
6412:
6411:
6385:
6381:
6379:
6376:
6375:
6359:
6356:
6355:
6329:
6324:
6323:
6305:
6300:
6294:
6291:
6290:
6265:
6262:
6261:
6245:
6242:
6241:
6240:of codimension
6213:
6210:
6209:
6193:
6190:
6189:
6166:
6162:
6160:
6157:
6156:
6121:
6117:
6081:
6077:
6063:
6059:
6052:
6033:
6028:
6022:
6019:
6018:
5990:
5987:
5986:
5957:
5949:
5931:
5927:
5925:
5922:
5921:
5905:
5902:
5901:
5898:
5895:
5861:
5858:
5857:
5834:
5830:
5825:
5819:
5815:
5810:
5796:
5790:
5786:
5781:
5773:
5771:
5768:
5767:
5741:
5737:
5719:
5715:
5713:
5710:
5709:
5692:
5688:
5686:
5683:
5682:
5656:
5652:
5650:
5647:
5646:
5626:
5622:
5614:
5611:
5610:
5594:
5591:
5590:
5587:higher K-theory
5551:
5547:
5529:
5525:
5507:
5503:
5495:
5492:
5491:
5458:
5454:
5452:
5449:
5448:
5419:
5415:
5413:
5410:
5409:
5383:
5379:
5377:
5374:
5373:
5370:
5367:
5345:
5343:
5340:
5339:
5319:
5314:
5313:
5305:
5302:
5301:
5284:
5280:
5278:
5275:
5274:
5257:
5251:
5250:
5245:
5242:
5239:
5238:
5215:
5211:
5191:
5188:
5187:
5158:
5155:
5154:
5131:
5125:
5124:
5123:
5111:
5107:
5086:
5085:
5077:
5075:
5072:
5071:
5055:
5052:
5051:
5034:
5033:
5031:
5028:
5027:
5024:
5021:
4992:
4988:
4979:
4975:
4973:
4970:
4969:
4950:
4946:
4937:
4933:
4926:
4921:
4920:
4909:
4905:
4898:
4893:
4892:
4881:
4877:
4870:
4865:
4864:
4862:
4859:
4858:
4857:is generically
4839:
4835:
4828:
4823:
4822:
4811:
4807:
4800:
4795:
4794:
4792:
4789:
4788:
4772:
4770:
4767:
4766:
4749:
4744:
4743:
4722:
4717:
4716:
4707:
4702:
4701:
4692:
4687:
4686:
4684:
4681:
4680:
4663:
4658:
4657:
4655:
4652:
4651:
4622:
4618:
4614:
4599:
4598:
4596:
4584:
4579:
4578:
4570:
4567:
4566:
4541:
4538:
4537:
4514:
4513:
4507:
4503:
4488:
4487:
4481:
4477:
4465:
4461:
4459:
4456:
4455:
4438:
4433:
4432:
4418:
4415:
4414:
4398:
4395:
4394:
4374:
4369:
4368:
4360:
4357:
4356:
4353:
4350:
4329:
4321:
4303:
4288:
4284:
4280:
4265:
4264:
4262:
4261:
4257:
4252:
4251:
4247:
4241:
4237:
4235:
4232:
4231:
4215:
4213:
4210:
4209:
4193:
4191:
4188:
4187:
4164:
4160:
4137:
4134:
4133:
4117:
4114:
4113:
4107:
4104:
4082:
4080:
4077:
4076:
4060:
4058:
4055:
4054:
4038:
4036:
4033:
4032:
4013:
4005:
4003:
4000:
3999:
3996:
3993:
3987:
3937:homotopy theory
3912:
3896:polynomial ring
3729:
3694:
3690:
3688:
3685:
3684:
3665:
3642:
3624:
3620:
3612:
3609:
3608:
3604:, we have that
3566:
3565:
3564:
3555:
3554:
3540:
3534:
3533:
3532:
3531:
3526:
3521:
3517:
3511:
3507:
3479:
3478:
3477:
3462:
3461:
3456:
3453:
3452:
3423:
3419:
3417:
3414:
3413:
3397:
3394:
3393:
3367:
3363:
3361:
3358:
3357:
3331:
3327:
3325:
3322:
3321:
3291:
3290:
3289:
3280:
3279:
3266:
3265:
3264:
3256:
3253:
3252:
3222:
3221:
3220:
3201:
3200:
3199:
3184:
3183:
3178:
3175:
3174:
3146:
3143:
3142:
3123:
3120:
3119:
3116:
3085:
3081:
3079:
3076:
3075:
3041:
3036:
3033:
3032:
3016:
3013:
3012:
2996:
2993:
2992:
2963:
2961:
2958:
2957:
2956:there is a set
2941:
2938:
2937:
2927:
2905:
2901:
2899:
2896:
2895:
2878:
2874:
2872:
2869:
2868:
2838:
2834:
2832:
2829:
2828:
2802:
2801:
2799:
2796:
2795:
2773:
2758:
2754:
2739:
2735:
2733:
2730:
2729:
2702:
2687:
2683:
2681:
2678:
2677:
2672:We can use the
2644:
2640:
2638:
2635:
2634:
2633:and is denoted
2618:
2615:
2614:
2585:
2580:
2579:
2577:
2574:
2573:
2539:
2534:
2531:
2530:
2505:
2482:
2465:
2462:
2461:
2432:
2429:
2428:
2400:
2397:
2396:
2371:
2369:
2366:
2365:
2349:
2346:
2345:
2329:
2326:
2325:
2323:Hausdorff space
2319:
2311:
2301:
2293:
2288:
2255:
2252:
2251:
2223:
2220:
2219:
2185:
2182:
2181:
2153:
2150:
2149:
2091:
2088:
2087:
2047:
2044:
2043:
2042:In general, if
1940:
1937:
1936:
1911:
1900:
1895:
1892:
1891:
1863:
1860:
1859:
1831:
1808:
1797:
1794:
1793:
1777:
1775:
1772:
1771:
1768:
1730:
1727:
1726:
1707:
1704:
1703:
1687:
1684:
1683:
1646:
1643:
1642:
1611:
1591:
1583:
1580:
1579:
1545:
1525:
1517:
1514:
1513:
1482:
1479:
1478:
1420:
1417:
1416:
1388:
1385:
1384:
1350:
1347:
1346:
1321:
1318:
1317:
1196:
1193:
1192:
1164:
1161:
1160:
1138:
1135:
1134:
1112:
1109:
1108:
1062:
1059:
1058:
1018:
1015:
1014:
980:
977:
976:
960:
957:
956:
940:
937:
936:
908:
905:
904:
850:
847:
846:
809:
806:
805:
768:
765:
764:
735:
731:
723:
717:
713:
704:
700:
692:
686:
682:
661:
657:
648:
644:
623:
619:
610:
606:
598:
595:
594:
554:
551:
550:
528:
522:
518:
501:
498:
497:
472:
466:
462:
454:
448:
444:
430:
424:
420:
412:
406:
402:
400:
397:
396:
374:
371:
370:
347:
343:
334:
330:
315:
311:
302:
298:
293:
290:
289:
257:
253:
251:
248:
247:
231:
228:
227:
205:
194:
191:
190:
182:
176:
160:superconductors
24:
17:
12:
11:
5:
9570:
9560:
9559:
9542:
9541:
9539:
9538:
9528:
9527:
9526:
9521:
9516:
9501:
9491:
9481:
9469:
9458:
9455:
9454:
9452:
9451:
9446:
9441:
9436:
9431:
9426:
9420:
9418:
9412:
9411:
9409:
9408:
9403:
9398:
9396:Winding number
9393:
9388:
9382:
9380:
9376:
9375:
9373:
9372:
9367:
9362:
9357:
9352:
9347:
9342:
9337:
9336:
9335:
9330:
9328:homotopy group
9320:
9319:
9318:
9313:
9308:
9303:
9298:
9288:
9283:
9278:
9268:
9266:
9262:
9261:
9254:
9252:
9250:
9249:
9244:
9239:
9238:
9237:
9227:
9226:
9225:
9215:
9210:
9205:
9200:
9195:
9189:
9187:
9183:
9182:
9175:
9174:
9167:
9160:
9152:
9146:
9145:
9140:
9135:
9128:
9127:External links
9125:
9124:
9123:
9117:
9101:
9093:Hatcher, Allen
9089:
9071:
9065:
9041:
9035:
9015:
9009:
8996:
8982:
8957:
8943:
8935:Addison-Wesley
8921:
8918:
8915:
8914:
8901:
8888:hep-th/9405035
8866:
8851:
8828:
8799:(3): 381–424.
8780:
8756:
8710:
8686:
8662:
8641:
8632:
8625:Grothendieck.
8617:
8602:
8582:
8557:
8556:
8554:
8551:
8550:
8549:
8544:
8539:
8534:
8529:
8524:
8519:
8514:
8507:
8504:
8491:
8488:
8485:
8482:
8477:
8473:
8448:
8445:
8442:
8437:
8432:
8428:
8416:
8415:
8404:
8401:
8398:
8395:
8392:
8389:
8384:
8380:
8374:
8370:
8366:
8361:
8357:
8353:
8350:
8347:
8344:
8339:
8334:
8330:
8315:Q-construction
8302:
8279:
8255:
8252:
8249:
8246:
8241:
8237:
8216:
8213:
8205:
8204:
8193:
8190:
8185:
8180:
8176:
8172:
8169:
8166:
8161:
8156:
8152:
8148:
8142:
8139:
8135:
8128:
8123:
8120:
8117:
8113:
8109:
8102:
8098:
8093:
8089:
8086:
8083:
8076:
8072:
8067:
8063:
8060:
8057:
8054:
8051:
8048:
8025:
8022:
8017:
8013:
8009:
8004:
8000:
7996:
7991:
7987:
7964:
7960:
7956:
7953:
7950:
7945:
7941:
7937:
7934:
7923:
7922:
7911:
7905:
7902:
7895:
7891:
7887:
7884:
7879:
7875:
7866:
7861:
7858:
7855:
7851:
7847:
7844:
7841:
7838:
7835:
7830:
7826:
7822:
7819:
7816:
7813:
7810:
7807:
7804:
7801:
7798:
7769:Main article:
7766:
7763:
7759:
7758:
7747:
7742:
7737:
7732:
7727:
7723:
7719:
7716:
7711:
7706:
7701:
7694:
7690:
7685:
7681:
7678:
7673:
7668:
7663:
7656:
7652:
7647:
7643:
7640:
7635:
7632:
7629:
7625:
7619:
7615:
7611:
7586:
7582:
7578:
7573:
7569:
7565:
7562:
7542:
7539:
7534:
7530:
7526:
7521:
7517:
7505:
7504:
7493:
7488:
7484:
7480:
7477:
7474:
7469:
7464:
7457:
7453:
7449:
7426:
7406:
7386:
7366:
7344:
7340:
7336:
7332:
7320:
7319:
7308:
7305:
7300:
7296:
7292:
7288:
7284:
7279:
7274:
7267:
7263:
7259:
7254:
7250:
7246:
7243:
7220:
7217:
7214:
7202:
7199:
7197:
7194:
7189:Cohen-Macaulay
7176:
7171:
7168:
7163:
7157:
7151:
7146:
7141:
7138:
7134:
7130:
7127:
7124:
7121:
7116:
7112:
7090:
7069:
7049:
7029:
7026:
7023:
7020:
7017:
7013:
7009:
7004:
6971:
6967:
6964:
6961:
6958:
6953:
6949:
6945:
6925:
6922:
6919:
6911:
6908:
6905:
6902:
6897:
6870:
6867:
6864:
6861:
6858:
6855:
6850:
6846:
6842:
6837:
6833:
6828:
6824:
6821:
6818:
6813:
6809:
6805:
6802:
6799:
6796:
6791:
6787:
6783:
6780:
6777:
6774:
6771:
6766:
6762:
6758:
6753:
6749:
6745:
6742:
6722:
6719:
6716:
6711:
6707:
6678:
6675:
6672:
6667:
6657:
6654:
6652:
6648:
6645:
6642:
6637:
6633:
6629:
6624:
6621:
6618:
6613:
6609:
6605:
6604:
6601:
6598:
6595:
6590:
6580:
6577:
6575:
6571:
6568:
6565:
6562:
6557:
6553:
6549:
6544:
6541:
6538:
6535:
6530:
6526:
6522:
6521:
6499:
6496:
6493:
6488:
6484:
6480:
6475:
6472:
6469:
6464:
6460:
6439:
6419:
6399:
6396:
6393:
6388:
6384:
6363:
6343:
6340:
6337:
6332:
6322:
6317:
6314:
6311:
6308:
6303:
6299:
6278:
6275:
6272:
6269:
6249:
6229:
6226:
6223:
6220:
6217:
6197:
6175:
6172:
6169:
6165:
6144:
6141:
6138:
6133:
6130:
6127:
6124:
6120:
6116:
6113:
6110:
6107:
6104:
6101:
6098:
6093:
6090:
6087:
6084:
6080:
6072:
6069:
6066:
6062:
6058:
6055:
6051:
6047:
6042:
6039:
6036:
6031:
6027:
6015:regular scheme
5994:
5970:
5967:
5964:
5956:
5952:
5948:
5945:
5942:
5939:
5934:
5930:
5909:
5897:
5893:
5890:
5877:
5874:
5871:
5868:
5865:
5843:
5840:
5837:
5833:
5828:
5822:
5818:
5813:
5809:
5806:
5803:
5799:
5793:
5789:
5784:
5780:
5755:
5752:
5749:
5744:
5740:
5736:
5733:
5730:
5727:
5722:
5718:
5695:
5691:
5670:
5667:
5662:
5659:
5655:
5632:
5629:
5625:
5621:
5618:
5598:
5574:
5571:
5568:
5565:
5562:
5557:
5554:
5550:
5546:
5543:
5540:
5537:
5532:
5528:
5524:
5521:
5518:
5515:
5510:
5506:
5502:
5499:
5475:
5472:
5469:
5464:
5461:
5457:
5433:
5430:
5427:
5422:
5418:
5397:
5394:
5391:
5386:
5382:
5369:
5365:
5362:
5348:
5327:
5322:
5317:
5312:
5309:
5287:
5283:
5260:
5254:
5248:
5224:
5221:
5218:
5214:
5210:
5207:
5204:
5201:
5198:
5195:
5171:
5168:
5165:
5162:
5142:
5139:
5134:
5128:
5122:
5119:
5114:
5110:
5106:
5103:
5100:
5097:
5094:
5089:
5084:
5080:
5059:
5037:
5023:
5019:
5016:
5003:
5000:
4995:
4991:
4987:
4982:
4978:
4953:
4949:
4945:
4940:
4936:
4932:
4929:
4924:
4919:
4912:
4908:
4904:
4901:
4896:
4891:
4884:
4880:
4876:
4873:
4868:
4842:
4838:
4834:
4831:
4826:
4821:
4814:
4810:
4806:
4803:
4798:
4775:
4752:
4747:
4742:
4739:
4736:
4731:
4728:
4725:
4720:
4715:
4710:
4705:
4700:
4695:
4690:
4666:
4661:
4636:
4631:
4628:
4625:
4621:
4617:
4612:
4609:
4606:
4602:
4595:
4592:
4587:
4582:
4577:
4574:
4554:
4551:
4548:
4545:
4525:
4522:
4517:
4510:
4506:
4502:
4499:
4496:
4491:
4484:
4480:
4476:
4473:
4468:
4464:
4441:
4436:
4431:
4428:
4425:
4422:
4402:
4382:
4377:
4372:
4367:
4364:
4352:
4348:
4345:
4332:
4328:
4324:
4320:
4316:
4311:
4306:
4302:
4296:
4291:
4287:
4283:
4278:
4275:
4272:
4268:
4260:
4250:
4244:
4240:
4218:
4196:
4172:
4163:
4159:
4156:
4153:
4150:
4147:
4144:
4141:
4121:
4106:
4102:
4099:
4085:
4063:
4041:
4020:
4016:
4012:
3995:
3991:
3988:
3986:
3983:
3964:surgery theory
3956:quadratic form
3933:Daniel Quillen
3911:
3908:
3904:Swan's theorem
3884:vector bundles
3869:noncommutative
3834:Michael Atiyah
3830:vector bundles
3819:affine variety
3815:smooth variety
3728:
3725:
3708:
3705:
3702:
3697:
3693:
3681:
3680:
3668:
3664:
3661:
3658:
3655:
3652:
3649:
3645:
3641:
3638:
3635:
3632:
3627:
3623:
3619:
3616:
3598:
3597:
3586:
3582:
3578:
3574:
3569:
3563:
3558:
3553:
3550:
3543:
3537:
3529:
3525:
3520:
3514:
3510:
3506:
3503:
3500:
3497:
3494:
3491:
3487:
3482:
3476:
3473:
3470:
3465:
3460:
3437:
3434:
3431:
3426:
3422:
3401:
3381:
3378:
3375:
3370:
3366:
3345:
3342:
3339:
3334:
3330:
3318:
3317:
3306:
3303:
3299:
3294:
3288:
3283:
3278:
3274:
3269:
3263:
3260:
3245:if there is a
3234:
3230:
3225:
3219:
3216:
3213:
3209:
3204:
3198:
3195:
3192:
3187:
3182:
3162:
3159:
3156:
3153:
3150:
3127:
3115:
3112:
3099:
3096:
3093:
3088:
3084:
3063:
3060:
3057:
3054:
3051:
3048:
3040:
3020:
3000:
2976:
2973:
2970:
2945:
2926:
2923:
2908:
2904:
2881:
2877:
2852:
2849:
2846:
2841:
2837:
2816:
2813:
2810:
2783:
2780:
2776:
2772:
2769:
2766:
2761:
2757:
2753:
2748:
2745:
2742:
2738:
2709:
2705:
2701:
2698:
2695:
2690:
2686:
2658:
2655:
2652:
2647:
2643:
2622:
2602:
2599:
2596:
2593:
2588:
2583:
2561:
2558:
2555:
2552:
2549:
2546:
2538:
2527:
2526:
2515:
2511:
2508:
2504:
2501:
2498:
2495:
2492:
2488:
2485:
2481:
2478:
2475:
2472:
2469:
2442:
2439:
2436:
2416:
2413:
2410:
2407:
2404:
2384:
2381:
2378:
2353:
2333:
2318:
2315:
2299:
2291:
2287:
2284:
2271:
2268:
2265:
2262:
2259:
2239:
2236:
2233:
2230:
2227:
2216:
2215:
2204:
2201:
2198:
2195:
2192:
2189:
2169:
2166:
2163:
2160:
2157:
2137:
2134:
2131:
2128:
2125:
2122:
2119:
2116:
2113:
2110:
2107:
2104:
2101:
2098:
2095:
2072:
2069:
2066:
2063:
2060:
2057:
2054:
2051:
2040:
2039:
2028:
2025:
2022:
2019:
2016:
2013:
2010:
2007:
2004:
2001:
1998:
1995:
1992:
1989:
1986:
1983:
1980:
1977:
1974:
1971:
1968:
1965:
1962:
1959:
1956:
1953:
1950:
1947:
1944:
1921:
1917:
1914:
1910:
1906:
1903:
1899:
1879:
1876:
1873:
1870:
1867:
1847:
1844:
1841:
1838:
1834:
1830:
1827:
1824:
1821:
1818:
1815:
1811:
1807:
1804:
1801:
1780:
1767:
1764:
1752:
1749:
1746:
1743:
1740:
1737:
1734:
1714:
1711:
1691:
1671:
1668:
1665:
1662:
1659:
1656:
1653:
1650:
1630:
1626:
1623:
1620:
1617:
1614:
1610:
1606:
1603:
1600:
1597:
1594:
1590:
1587:
1564:
1560:
1557:
1554:
1551:
1548:
1544:
1540:
1537:
1534:
1531:
1528:
1524:
1521:
1507:
1506:
1495:
1492:
1489:
1486:
1466:
1463:
1460:
1457:
1454:
1451:
1448:
1445:
1442:
1439:
1436:
1433:
1430:
1427:
1424:
1401:
1398:
1395:
1392:
1372:
1369:
1366:
1363:
1360:
1357:
1354:
1334:
1331:
1328:
1325:
1314:
1313:
1302:
1299:
1296:
1293:
1290:
1287:
1284:
1281:
1278:
1275:
1272:
1269:
1266:
1263:
1260:
1257:
1254:
1251:
1248:
1245:
1242:
1239:
1236:
1233:
1230:
1227:
1224:
1221:
1218:
1215:
1212:
1209:
1206:
1203:
1200:
1177:
1174:
1171:
1168:
1148:
1145:
1142:
1122:
1119:
1116:
1096:
1093:
1090:
1087:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
1046:
1043:
1040:
1037:
1034:
1031:
1028:
1025:
1022:
1002:
999:
996:
993:
990:
987:
984:
964:
944:
924:
921:
918:
915:
912:
881:
878:
875:
872:
869:
866:
863:
860:
857:
854:
834:
831:
828:
825:
822:
819:
816:
813:
793:
790:
787:
784:
781:
778:
775:
772:
761:
760:
749:
746:
743:
738:
734:
729:
726:
720:
716:
712:
707:
703:
698:
695:
689:
685:
681:
678:
675:
672:
669:
664:
660:
656:
651:
647:
643:
640:
637:
634:
631:
626:
622:
618:
613:
609:
605:
602:
579:
576:
573:
570:
567:
564:
561:
558:
535:
531:
525:
521:
517:
514:
511:
508:
505:
496:Then, the set
485:
482:
478:
475:
469:
465:
460:
457:
451:
447:
443:
440:
436:
433:
427:
423:
418:
415:
409:
405:
384:
381:
378:
367:
366:
355:
350:
346:
342:
337:
333:
329:
326:
323:
318:
314:
310:
305:
301:
297:
274:
271:
268:
265:
260:
256:
235:
215:
211:
208:
204:
201:
198:
186:abelian monoid
178:Main article:
175:
172:
164:Fermi surfaces
40:vector bundles
15:
9:
6:
4:
3:
2:
9569:
9558:
9555:
9554:
9552:
9537:
9529:
9525:
9522:
9520:
9517:
9515:
9512:
9511:
9510:
9502:
9500:
9496:
9492:
9490:
9486:
9482:
9480:
9475:
9470:
9468:
9460:
9459:
9456:
9450:
9447:
9445:
9442:
9440:
9437:
9435:
9432:
9430:
9427:
9425:
9422:
9421:
9419:
9417:
9413:
9407:
9406:Orientability
9404:
9402:
9399:
9397:
9394:
9392:
9389:
9387:
9384:
9383:
9381:
9377:
9371:
9368:
9366:
9363:
9361:
9358:
9356:
9353:
9351:
9348:
9346:
9343:
9341:
9338:
9334:
9331:
9329:
9326:
9325:
9324:
9321:
9317:
9314:
9312:
9309:
9307:
9304:
9302:
9299:
9297:
9294:
9293:
9292:
9289:
9287:
9284:
9282:
9279:
9277:
9273:
9270:
9269:
9267:
9263:
9258:
9248:
9245:
9243:
9242:Set-theoretic
9240:
9236:
9233:
9232:
9231:
9228:
9224:
9221:
9220:
9219:
9216:
9214:
9211:
9209:
9206:
9204:
9203:Combinatorial
9201:
9199:
9196:
9194:
9191:
9190:
9188:
9184:
9180:
9173:
9168:
9166:
9161:
9159:
9154:
9153:
9150:
9144:
9141:
9139:
9136:
9134:
9131:
9130:
9120:
9114:
9110:
9106:
9102:
9098:
9094:
9090:
9085:
9080:
9076:
9072:
9068:
9066:0-387-08090-2
9062:
9058:
9054:
9050:
9046:
9042:
9038:
9036:3-540-04245-8
9032:
9028:
9024:
9020:
9016:
9012:
9006:
9002:
8997:
8993:
8989:
8985:
8979:
8975:
8971:
8967:
8963:
8958:
8954:
8950:
8946:
8940:
8936:
8932:
8928:
8924:
8923:
8911:
8905:
8898:
8894:
8889:
8884:
8880:
8876:
8870:
8862:
8858:
8854:
8848:
8844:
8843:
8835:
8833:
8824:
8820:
8816:
8812:
8807:
8802:
8798:
8794:
8787:
8785:
8770:
8766:
8760:
8752:
8748:
8744:
8740:
8736:
8732:
8728:
8724:
8720:
8719:Manin, Yuri I
8714:
8700:
8696:
8690:
8676:
8672:
8666:
8659:
8655:
8654:Gregory Moore
8651:
8645:
8639:Karoubi, 2006
8636:
8628:
8621:
8613:
8609:
8605:
8599:
8595:
8594:
8586:
8577:
8572:
8568:
8562:
8558:
8548:
8545:
8543:
8540:
8538:
8535:
8533:
8530:
8528:
8525:
8523:
8520:
8518:
8515:
8513:
8510:
8509:
8503:
8486:
8480:
8475:
8471:
8462:
8443:
8435:
8430:
8426:
8402:
8393:
8387:
8382:
8378:
8372:
8368:
8359:
8355:
8351:
8345:
8337:
8332:
8328:
8320:
8319:
8318:
8316:
8300:
8293:
8277:
8269:
8250:
8244:
8239:
8235:
8226:
8222:
8212:
8210:
8191:
8183:
8178:
8174:
8170:
8167:
8164:
8159:
8154:
8150:
8140:
8137:
8133:
8121:
8118:
8115:
8111:
8107:
8100:
8096:
8091:
8087:
8084:
8081:
8074:
8070:
8065:
8061:
8055:
8049:
8046:
8039:
8038:
8037:
8023:
8015:
8011:
8002:
7998:
7994:
7989:
7985:
7962:
7958:
7954:
7951:
7948:
7943:
7939:
7935:
7932:
7909:
7903:
7900:
7893:
7885:
7877:
7873:
7859:
7856:
7853:
7849:
7845:
7836:
7828:
7824:
7817:
7814:
7811:
7805:
7799:
7796:
7789:
7788:
7787:
7785:
7781:
7777:
7776:Chern classes
7772:
7762:
7745:
7740:
7725:
7721:
7714:
7709:
7692:
7688:
7683:
7676:
7671:
7654:
7650:
7645:
7638:
7633:
7630:
7627:
7617:
7613:
7602:
7601:
7600:
7584:
7580:
7576:
7571:
7567:
7563:
7560:
7540:
7537:
7532:
7528:
7524:
7519:
7515:
7486:
7475:
7467:
7455:
7440:
7439:
7438:
7424:
7404:
7384:
7364:
7342:
7338:
7334:
7330:
7306:
7298:
7294:
7290:
7286:
7277:
7265:
7252:
7241:
7234:
7233:
7232:
7218:
7212:
7193:
7190:
7169:
7166:
7155:
7149:
7136:
7128:
7122:
7114:
7110:
7067:
7047:
7027:
7024:
7018:
7015:
7002:
6965:
6959:
6951:
6947:
6943:
6920:
6909:
6903:
6895:
6868:
6856:
6848:
6844:
6835:
6831:
6826:
6819:
6811:
6807:
6797:
6789:
6785:
6772:
6764:
6760:
6751:
6747:
6740:
6717:
6709:
6705:
6696:
6673:
6665:
6655:
6653:
6646:
6643:
6640:
6635:
6631:
6627:
6622:
6619:
6616:
6607:
6596:
6588:
6578:
6576:
6569:
6566:
6563:
6560:
6555:
6551:
6547:
6542:
6539:
6536:
6533:
6524:
6497:
6494:
6491:
6486:
6482:
6478:
6473:
6470:
6467:
6462:
6458:
6437:
6417:
6394:
6386:
6382:
6361:
6338:
6330:
6320:
6315:
6312:
6309:
6306:
6301:
6297:
6273:
6267:
6247:
6227:
6221:
6218:
6215:
6195:
6170:
6163:
6139:
6131:
6128:
6125:
6122:
6118:
6105:
6099:
6091:
6088:
6085:
6082:
6078:
6067:
6060:
6056:
6053:
6049:
6045:
6040:
6037:
6034:
6029:
6025:
6016:
6012:
6008:
5992:
5984:
5965:
5954:
5946:
5940:
5932:
5928:
5907:
5889:
5875:
5872:
5869:
5866:
5863:
5841:
5838:
5835:
5820:
5816:
5807:
5804:
5801:
5791:
5787:
5750:
5742:
5738:
5728:
5720:
5716:
5708:then the map
5693:
5689:
5668:
5660:
5657:
5653:
5630:
5627:
5623:
5616:
5596:
5588:
5572:
5563:
5555:
5552:
5548:
5538:
5530:
5526:
5516:
5508:
5504:
5497:
5489:
5470:
5462:
5459:
5455:
5447:
5428:
5420:
5416:
5392:
5384:
5380:
5361:
5320:
5307:
5285:
5281:
5258:
5222:
5219:
5216:
5212:
5208:
5205:
5202:
5199:
5196:
5193:
5185:
5166:
5160:
5132:
5117:
5112:
5108:
5101:
5098:
5095:
5057:
5015:
5001:
4998:
4993:
4989:
4985:
4980:
4976:
4951:
4947:
4943:
4938:
4934:
4930:
4927:
4917:
4910:
4906:
4902:
4899:
4889:
4882:
4878:
4874:
4871:
4840:
4836:
4832:
4829:
4819:
4812:
4808:
4804:
4801:
4750:
4740:
4737:
4734:
4729:
4726:
4723:
4713:
4708:
4698:
4693:
4664:
4629:
4626:
4623:
4619:
4607:
4593:
4585:
4572:
4549:
4543:
4508:
4504:
4497:
4482:
4478:
4466:
4462:
4439:
4426:
4423:
4420:
4400:
4375:
4362:
4344:
4326:
4318:
4314:
4309:
4300:
4289:
4285:
4273:
4258:
4248:
4242:
4238:
4186:
4161:
4154:
4151:
4145:
4139:
4119:
4112:
4098:
3982:
3980:
3976:
3972:
3971:string theory
3967:
3965:
3961:
3957:
3952:
3950:
3946:
3942:
3938:
3934:
3930:
3929:
3923:
3921:
3917:
3907:
3905:
3901:
3897:
3893:
3890:to formulate
3889:
3885:
3881:
3876:
3874:
3871:K-theory for
3870:
3866:
3862:
3858:
3854:
3851:
3847:
3843:
3839:
3835:
3831:
3827:
3822:
3820:
3816:
3812:
3807:
3805:
3801:
3797:
3794:behavior and
3793:
3792:cohomological
3789:
3785:
3781:
3777:
3773:
3770:are used, or
3769:
3765:
3761:
3757:
3753:
3750:
3746:
3742:
3738:
3734:
3727:Early history
3724:
3722:
3703:
3695:
3691:
3662:
3656:
3650:
3639:
3633:
3625:
3621:
3617:
3614:
3607:
3606:
3605:
3603:
3584:
3580:
3572:
3561:
3548:
3541:
3527:
3523:
3518:
3512:
3504:
3501:
3495:
3492:
3485:
3471:
3451:
3450:
3449:
3432:
3424:
3420:
3399:
3376:
3368:
3364:
3340:
3332:
3328:
3304:
3297:
3272:
3258:
3251:
3250:
3249:
3248:
3228:
3214:
3207:
3193:
3157:
3151:
3148:
3141:
3125:
3111:
3094:
3086:
3082:
3058:
3055:
3049:
3018:
2998:
2990:
2971:
2943:
2936:
2932:
2922:
2906:
2902:
2879:
2875:
2866:
2847:
2839:
2835:
2811:
2770:
2767:
2759:
2755:
2746:
2743:
2740:
2736:
2727:
2723:
2699:
2696:
2688:
2684:
2675:
2670:
2653:
2645:
2641:
2620:
2600:
2594:
2591:
2586:
2556:
2553:
2547:
2509:
2506:
2502:
2499:
2493:
2486:
2483:
2476:
2470:
2460:
2459:
2458:
2456:
2437:
2414:
2408:
2405:
2402:
2379:
2351:
2331:
2324:
2314:
2309:
2305:
2297:
2283:
2266:
2263:
2260:
2234:
2231:
2228:
2202:
2196:
2193:
2190:
2164:
2161:
2158:
2132:
2129:
2126:
2123:
2120:
2117:
2114:
2108:
2102:
2099:
2096:
2086:
2085:
2084:
2067:
2064:
2061:
2052:
2049:
2023:
2020:
2017:
2011:
2005:
2002:
1999:
1993:
1987:
1984:
1981:
1975:
1969:
1966:
1963:
1957:
1951:
1948:
1945:
1935:
1934:
1933:
1915:
1912:
1908:
1904:
1901:
1874:
1871:
1868:
1858:For any pair
1845:
1839:
1836:
1825:
1816:
1813:
1799:
1763:
1750:
1747:
1738:
1732:
1712:
1709:
1689:
1666:
1660:
1654:
1651:
1648:
1628:
1588:
1585:
1578:
1562:
1522:
1519:
1512:
1493:
1490:
1487:
1484:
1461:
1458:
1455:
1452:
1449:
1446:
1443:
1437:
1431:
1428:
1425:
1415:
1414:
1413:
1399:
1396:
1393:
1390:
1364:
1361:
1358:
1329:
1323:
1294:
1291:
1288:
1279:
1270:
1267:
1264:
1261:
1258:
1255:
1252:
1243:
1234:
1231:
1228:
1219:
1210:
1207:
1204:
1191:
1190:
1189:
1188:This implies
1175:
1172:
1169:
1166:
1146:
1143:
1140:
1120:
1117:
1114:
1091:
1088:
1085:
1079:
1073:
1070:
1067:
1044:
1038:
1035:
1029:
1023:
994:
991:
988:
962:
942:
919:
916:
913:
902:
897:
895:
879:
870:
867:
864:
852:
829:
823:
817:
814:
811:
788:
785:
779:
773:
747:
736:
732:
727:
724:
718:
714:
710:
705:
701:
696:
693:
687:
683:
673:
662:
658:
654:
649:
645:
635:
624:
620:
616:
611:
607:
593:
592:
591:
574:
571:
565:
559:
549:
533:
529:
523:
519:
515:
509:
503:
483:
480:
476:
473:
467:
463:
458:
455:
449:
445:
441:
438:
434:
431:
425:
421:
416:
413:
407:
403:
382:
379:
376:
348:
344:
340:
335:
331:
324:
316:
312:
308:
303:
299:
288:
287:
286:
272:
269:
266:
263:
258:
254:
233:
209:
206:
202:
199:
187:
181:
171:
169:
165:
161:
157:
153:
149:
145:
141:
137:
133:
129:
125:
120:
118:
114:
110:
106:
101:
96:
92:
87:
85:
81:
77:
73:
69:
65:
61:
57:
53:
49:
45:
41:
38:generated by
37:
33:
29:
22:
9536:Publications
9401:Chern number
9391:Betti number
9274: /
9265:Key concepts
9213:Differential
9108:
9084:math/0602082
9075:Karoubi, Max
9048:
9045:Karoubi, Max
9022:
9000:
8961:
8930:
8904:
8878:
8869:
8841:
8796:
8792:
8772:. Retrieved
8769:MathOverflow
8768:
8759:
8726:
8722:
8713:
8702:. Retrieved
8699:MathOverflow
8698:
8689:
8678:. Retrieved
8674:
8665:
8644:
8635:
8620:
8592:
8585:
8576:math/0012213
8561:
8417:
8218:
8206:
7924:
7783:
7774:
7760:
7506:
7321:
7204:
7196:Applications
5983:Picard group
5899:
5371:
5183:
5025:
4354:
4108:
4031:for a field
3997:
3968:
3953:
3944:
3926:
3924:
3913:
3910:Developments
3877:
3852:
3845:
3841:
3823:
3810:
3808:
3799:
3795:
3787:
3783:
3775:
3771:
3766:) when only
3763:
3759:
3751:
3740:
3730:
3682:
3599:
3319:
3117:
2928:
2671:
2528:
2320:
2312:
2217:
2041:
1769:
1508:
1315:
898:
892:which has a
762:
368:
183:
121:
90:
88:
31:
25:
9499:Wikiversity
9416:Key results
9019:Swan, R. G.
8729:(5): 1–89.
5186:with basis
3873:C*-algebras
3804:homological
2455:direct sums
2427:be denoted
2286:Definitions
285:defined by
162:and stable
28:mathematics
9345:CW complex
9286:Continuity
9276:Closed set
9235:cohomology
8920:References
8806:1809.10919
8774:2020-10-20
8704:2020-10-20
8680:2017-04-16
5153:is a free
3994:of a field
3806:behavior.
3600:Using the
2726:idempotent
2364:, denoted
395:such that
115:, and the
80:invariants
54:, it is a
9524:geometric
9519:algebraic
9370:Cobordism
9306:Hausdorff
9301:connected
9218:Geometric
9208:Continuum
9198:Algebraic
8861:624583210
8751:0036-0279
8612:227161674
8522:KR-theory
8517:KK-theory
8481:
8388:
8356:π
8245:
8168:⋯
8127:∞
8112:∑
8085:⋯
8050:
7955:⊕
7952:⋯
7949:⊕
7865:∞
7850:∑
7818:
7800:
7715:−
7577:∩
7538:⊂
7483:Ω
7476:−
7452:Ω
7304:→
7283:→
7262:Ω
7258:→
7249:Ω
7245:→
7216:↪
7137:⊕
7129:≅
6966:≅
6910:≅
6866:→
6804:→
6782:→
6744:→
6656:≅
6628:≅
6612:∞
6579:≅
6567:−
6548:≅
6540:−
6529:∞
6321:≅
6313:−
6225:→
6129:−
6123:−
6115:⇒
6089:−
6083:−
6057:∈
6050:∐
5955:⊕
5873:
5839:−
5805:…
5735:→
5666:↪
5620:→
5570:→
5545:→
5523:→
5501:→
5498:⋯
5220:−
5213:ξ
5206:…
5200:ξ
5133:∨
5118:
5113:∙
5102:
4999:≤
4944:−
4931:−
4903:−
4890:∩
4875:−
4833:−
4805:−
4741:∐
4738:⋯
4735:∐
4727:−
4714:∐
4509:∗
4498:⋅
4483:∗
4467:∗
4430:↪
4327:⊕
4301:×
3663:⊗
3648:→
3640:⊗
3549:
3502:−
3496:∑
3472:⋅
3302:→
3287:→
3277:→
3262:→
3152:
3059:⊕
3019:⊕
2744:×
2598:→
2592:×
2557:⊕
2503:⊕
2477:⊕
2412:→
2403:π
2130:−
2118:−
2109:∼
2012:∼
1994:∼
1976:∼
1958:∼
1745:→
1658:→
1649:ϕ
1609:→
1543:→
1488:∈
1438:∼
1394:−
1118:∈
1080:∼
856:↦
845:given by
821:→
534:∼
380:∈
325:∼
270:×
234:∼
82:of large
58:known as
9557:K-theory
9551:Category
9489:Wikibook
9467:Category
9355:Manifold
9323:Homotopy
9281:Interior
9272:Open set
9230:Homology
9179:Topology
9107:(2013).
9095:(2003).
9047:(1978).
9027:Springer
9021:(1968).
8931:K-theory
8929:(1989).
8823:85502709
8506:See also
6983:. Since
6512:, hence
6013:. For a
4185:Artinian
3979:D-branes
3960:L-theory
3848:) for a
3840:defined
3826:topology
3573:′
3486:′
3298:″
3273:′
3229:″
3208:′
2933:. For a
2510:′
2487:′
1916:′
1905:′
1477:for any
1107:for any
975:so that
728:′
697:′
477:′
459:′
435:′
417:′
210:′
136:D-branes
95:functors
84:matrices
32:K-theory
21:K Theory
9514:general
9316:uniform
9296:compact
9247:Digital
8992:2182598
8953:1043170
8897:1363062
8731:Bibcode
8652:), and
8459:is the
7102:, then
1511:functor
1057:First,
590:where:
144:spinors
64:algebra
42:over a
9509:Topics
9311:metric
9186:Fields
9115:
9063:
9033:
9007:
8990:
8980:
8951:
8941:
8895:
8859:
8849:
8821:
8749:
8610:
8600:
8223:is an
7322:where
6260:, and
3935:using
3802:) has
3790:) has
3747:on an
3741:Klasse
2302:, see
111:, the
100:groups
48:scheme
9291:Space
9079:arXiv
8883:arXiv
8819:S2CID
8801:arXiv
8571:arXiv
8553:Notes
8290:with
7080:over
5486:from
3886:with
3813:is a
2083:then
548:group
150:. In
62:. In
50:. In
9113:ISBN
9061:ISBN
9031:ISBN
9005:ISBN
8978:ISBN
8939:ISBN
8857:OCLC
8847:ISBN
8747:ISSN
8608:OCLC
8598:ISBN
8219:The
6693:The
6155:for
5981:for
5856:for
5408:and
5099:Proj
4968:for
4254:Spec
4007:Spec
3900:free
3836:and
3719:for
3043:Vect
2965:Vect
2804:Idem
2541:Vect
2373:Vect
2306:and
226:let
66:and
36:ring
9053:doi
8970:doi
8811:doi
8739:doi
8656:in
8472:Coh
8463:of
8379:Coh
8266:of
8236:Coh
7815:exp
7599:as
7377:in
6993:Ext
6914:Pic
6009:of
5985:of
5959:Pic
5870:dim
5775:lcm
5109:Sym
4166:red
3969:In
3898:is
3824:In
3809:If
3524:Tor
3392:if
3149:Coh
2991:on
2720:as
2180:or
2056:min
955:by
146:on
122:In
46:or
26:In
9553::
9059:.
9029:.
8988:MR
8986:.
8976:.
8968:.
8949:MR
8947:.
8937:.
8893:MR
8891:,
8855:.
8831:^
8817:.
8809:.
8795:.
8783:^
8767:.
8745:.
8737:.
8727:24
8725:.
8697:.
8673:.
8606:.
8211:.
8108::=
8047:ch
7846::=
7797:ch
6998:Ab
6891:CH
6661:CH
6584:CH
6326:CH
5888:.
5360:.
5014:.
4097:.
3966:.
3951:.
3922:.
3875:.
3832:,
3782:;
3723:.
3615:ch
3305:0.
2921:.
2669:.
2053::=
896:.
170:.
158:,
138:,
119:.
107:,
86:.
30:,
9171:e
9164:t
9157:v
9121:.
9099:.
9087:.
9081::
9069:.
9055::
9039:.
9013:.
8994:.
8972::
8955:.
8912:.
8885::
8863:.
8825:.
8813::
8803::
8797:6
8777:.
8753:.
8741::
8733::
8707:.
8683:.
8660:.
8629:.
8614:.
8579:.
8573::
8490:)
8487:X
8484:(
8476:G
8447:)
8444:C
8441:(
8436:G
8431:0
8427:K
8403:.
8400:)
8397:)
8394:X
8391:(
8383:G
8373:+
8369:B
8365:(
8360:i
8352:=
8349:)
8346:X
8343:(
8338:G
8333:i
8329:K
8301:G
8278:X
8254:)
8251:X
8248:(
8240:G
8192:.
8189:)
8184:m
8179:n
8175:x
8171:+
8165:+
8160:m
8155:1
8151:x
8147:(
8141:!
8138:m
8134:1
8122:0
8119:=
8116:m
8101:n
8097:x
8092:e
8088:+
8082:+
8075:1
8071:x
8066:e
8062:=
8059:)
8056:V
8053:(
8024:,
8021:)
8016:i
8012:L
8008:(
8003:1
7999:c
7995:=
7990:i
7986:x
7963:n
7959:L
7944:1
7940:L
7936:=
7933:V
7910:.
7904:!
7901:m
7894:m
7890:)
7886:L
7883:(
7878:1
7874:c
7860:0
7857:=
7854:m
7843:)
7840:)
7837:L
7834:(
7829:1
7825:c
7821:(
7812:=
7809:)
7806:L
7803:(
7784:L
7746:.
7741:Z
7736:|
7731:]
7726:X
7722:T
7718:[
7710:Z
7705:|
7700:]
7693:2
7689:Y
7684:T
7680:[
7677:+
7672:Z
7667:|
7662:]
7655:1
7651:Y
7646:T
7642:[
7639:=
7634:r
7631:i
7628:v
7624:]
7618:Z
7614:T
7610:[
7585:2
7581:Y
7572:1
7568:Y
7564:=
7561:Z
7541:X
7533:2
7529:Y
7525:,
7520:1
7516:Y
7492:]
7487:Y
7479:[
7473:]
7468:Y
7463:|
7456:X
7448:[
7425:X
7405:Y
7385:X
7365:Y
7343:X
7339:/
7335:Y
7331:C
7307:0
7299:X
7295:/
7291:Y
7287:C
7278:Y
7273:|
7266:X
7253:Y
7242:0
7219:X
7213:Y
7175:)
7170:g
7167:2
7162:Z
7156:/
7150:g
7145:C
7140:(
7133:Z
7126:)
7123:C
7120:(
7115:0
7111:K
7089:C
7068:g
7048:C
7028:0
7025:=
7022:)
7019:G
7016:,
7012:Z
7008:(
7003:1
6970:Z
6963:)
6960:C
6957:(
6952:0
6948:H
6944:C
6924:)
6921:C
6918:(
6907:)
6904:C
6901:(
6896:1
6869:0
6863:)
6860:)
6857:X
6854:(
6849:0
6845:K
6841:(
6836:1
6832:F
6827:/
6823:)
6820:X
6817:(
6812:0
6808:K
6801:)
6798:X
6795:(
6790:0
6786:K
6779:)
6776:)
6773:X
6770:(
6765:0
6761:K
6757:(
6752:1
6748:F
6741:0
6721:)
6718:C
6715:(
6710:0
6706:K
6677:)
6674:C
6671:(
6666:0
6647:0
6644:,
6641:0
6636:2
6632:E
6623:0
6620:,
6617:0
6608:E
6600:)
6597:C
6594:(
6589:1
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93:-
91:K
23:.
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