2856:, Bokstedt, Hsiang, and Madsen introduced topological cyclic homology, which bore the same relationship to topological Hochschild homology as cyclic homology did to Hochschild homology. The Dennis trace map to topological Hochschild homology factors through topological cyclic homology, providing an even more detailed tool for calculations. In 1996, Dundas, Goodwillie, and McCarthy proved that topological cyclic homology has in a precise sense the same local structure as algebraic
2826:-theory and Hochschild homology. He called this theory topological Hochschild homology because its ground ring should be the sphere spectrum (considered as a ring whose operations are defined only up to homotopy). In the mid-1980s, Bokstedt gave a definition of topological Hochschild homology that satisfied nearly all of Goodwillie's conjectural properties, and this made possible further computations of
5889:
1753:-theory now appeared to be a homology theory for rings and a cohomology theory for varieties. However, many of its basic theorems carried the hypothesis that the ring or variety in question was regular. One of the basic expected relations was a long exact sequence (called the "localization sequence") relating the
795:
in such a way that each additional simplex or cell deformation retracts into a subdivision of the old space. Part of the motivation for this definition is that a subdivision of a triangulation is simple homotopy equivalent to the original triangulation, and therefore two triangulations that share a
772:
had attempted to define the Betti numbers of a manifold in terms of a triangulation. His methods, however, had a serious gap: Poincaré could not prove that two triangulations of a manifold always yielded the same Betti numbers. It was clearly true that Betti numbers were unchanged by subdividing
381:
of the vector bundle. This is a generalization because on a projective
Riemann surface, the Euler characteristic of a line bundle equals the difference in dimensions mentioned previously, the Euler characteristic of the trivial bundle is one minus the genus, and the only nontrivial characteristic
1742:
All abelian categories are exact categories, but not all exact categories are abelian. Because
Quillen was able to work in this more general situation, he was able to use exact categories as tools in his proofs. This technique allowed him to prove many of the basic theorems of algebraic
5711:
3098:
4583:
5654:, it gives a presentation for the unstable K-theory. This presentation is different from the one given here only for symplectic root systems. For non-symplectic root systems, the unstable second K-group with respect to the root system is exactly the stable K-group for GL(
773:
the triangulation, and therefore it was clear that any two triangulations that shared a common subdivision had the same Betti numbers. What was not known was that any two triangulations admitted a common subdivision. This hypothesis became a conjecture known as the
1291:
6918:
5645:
990:. In addition to providing a coherent exposition of the results then known, Bass improved many of the statements of the theorems. Of particular note is that Bass, building on his earlier work with Murthy, provided the first proof of what is now known as the
5415:
1640:). However, Segal's approach was only able to impose relations for split exact sequences, not general exact sequences. In the category of projective modules over a ring, every short exact sequence splits, and so Î-objects could be used to define the
356:
is projective, then these subspaces are finite dimensional. The
RiemannâRoch theorem states that the difference in dimensions between these subspaces is equal to the degree of the line bundle (a measure of twistedness) plus one minus the genus of
6041:
1655:, a category satisfying certain formal properties similar to, but slightly weaker than, the properties satisfied by a category of modules or vector bundles. From this he constructed an auxiliary category using a new device called his "
2772:-theory had to be reformulated. This was done by Thomason in a lengthy monograph which he co-credited to his dead friend Thomas Trobaugh, who he said gave him a key idea in a dream. Thomason combined Waldhausen's construction of
4135:
4865:
816:
is the fundamental group of the target complex. Whitehead found examples of non-trivial torsion and thereby proved that some homotopy equivalences were not simple. The
Whitehead group was later discovered to be a quotient of
4394:
4067:
3811:
1644:-theory of a ring. However, there are non-split short exact sequences in the category of vector bundles on a variety and in the category of all modules over a ring, so Segal's approach did not apply to all cases of interest.
7490:
5884:{\displaystyle K_{2}F\rightarrow \oplus _{\mathbf {p} }K_{1}A/{\mathbf {p} }\rightarrow K_{1}A\rightarrow K_{1}F\rightarrow \oplus _{\mathbf {p} }K_{0}A/{\mathbf {p} }\rightarrow K_{0}A\rightarrow K_{0}F\rightarrow 0\ }
7802:
is a diagram where the arrow on the left is a covering map (hence surjective) and the arrow on the right is injective. This category can then be turned into a topological space using the classifying space construction
2587:
8597:
5261:
3165:
3005:
7692:
6283:
7240:
4408:
11337:, in Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 1, pp. 35â60, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978.
895:, but elements of that group coming from elementary matrices (matrices corresponding to elementary row or column operations) define equivalent gluings. Motivated by this, the BassâSchanuel definition of
6768:
1352:, and Gersten proved that his and Swan's theories were equivalent, but the two theories were not known to satisfy all the expected properties. Nobile and Villamayor also proposed a definition of higher
6376:
4780:
3678:
507:
7968:
6494:
1182:
6592:
2822:-theory with finite coefficients, it was less successful for rational calculations. Goodwillie, motivated by his "calculus of functors", conjectured the existence of a theory intermediate to
796:
common subdivision must be simple homotopy equivalent. Whitehead proved that simple homotopy equivalence is a finer invariant than homotopy equivalence by introducing an invariant called the
2329:-theory of that open subset. Brown developed such a theory for his thesis. Simultaneously, Gersten had the same idea. At a Seattle conference in autumn of 1972, they together discovered a
1828:-theory of regular varieties had a localization exact sequence. Since this sequence was fundamental to many of the facts in the subject, regularity hypotheses pervaded early work on higher
3284:
2382:
377:
on an algebraic variety (which is the alternating sum of the dimensions of its cohomology groups) equals the Euler characteristic of the trivial bundle plus a correction factor coming from
6814:
5526:
2791:
of sheaves, there was a simple description of when a complex of sheaves could be extended from an open subset of a variety to the whole variety. By applying
Waldhausen's construction of
2446:
7797:
1886:
in his thesis found an invariant similar to Wall's that gives an obstruction to an open manifold being the interior of a compact manifold with boundary. If two manifolds with boundary
5306:
1625:. Where Grothendieck worked with isomorphism classes of bundles, Segal worked with the bundles themselves and used isomorphisms of the bundles as part of his data. This results in a
527:
Grothendieck took the perspective that the
RiemannâRoch theorem is a statement about morphisms of varieties, not the varieties themselves. He proved that there is a homomorphism from
7329:
708:
seemed to satisfy the necessary properties to be the beginning of a cohomology theory of algebraic varieties and of non-commutative rings, there was no clear definition of the higher
701:
except the normalization axiom. The setting of algebraic varieties, however, is much more rigid, and the flexible constructions used in topology were not available. While the group
7374:
6689:
1589:
had a known and accepted definition it was possible to sidestep this difficulty, but it remained technically awkward. Conceptually, the problem was that the definition sprung from
7762:
2362:
3237:
3208:
2089:-cobordisms is the same as a weaker notion called pseudo-isotopy. Hatcher and Wagoner studied the components of the space of pseudo-isotopies and related it to a quotient of
8018:
2149:). This space contains strictly more information than the Whitehead group; for example, the connected component of the trivial cobordism describes the possible cylinders on
6138:
6948:
631:
is a point, a vector bundle is a vector space, the class of a vector space is its dimension, and the
GrothendieckâRiemannâRoch theorem specializes to Hirzebruch's theorem.
6989:
6633:
6535:
11131:
11035:
9502:
8078:
65:. They contain detailed information about the original object but are notoriously difficult to compute; for example, an important outstanding problem is to compute the
5921:
1765:. Quillen was unable to prove the existence of the localization sequence in full generality. He was, however, able to prove its existence for a related theory called
6798:
7540:-construction gives the same results as the +-construction, but it applies in more general situations. Moreover, the definition is more direct in the sense that the
2777:
8045:
8157:
The following variant of this is also used: instead of finitely generated projective (= locally free) modules, take finitely generated modules. The resulting
7827:
7860:
7726:
7616:
7991:
7887:
7636:
7593:
7569:
4957:
in general for PIDs, thus providing one of the rare mathematical features of
Euclidean domains that do not generalize to all PIDs. An explicit PID such that SK
4210:
2749:-theory admitted a spectral sequence similar to the one conjectured by Quillen. Thomason proved around 1980 that after inverting the Bott element, algebraic
11204:
10819:
4078:
152:-groups were discovered first, in the sense that adequate descriptions of these groups in terms of other algebraic structures were found. For example, if
4795:
4276:
3947:
3693:
1747:-theory. Additionally, it was possible to prove that the earlier definitions of Swan and Gersten were equivalent to Quillen's under certain conditions.
11248:
11328:
11324:
7393:
1081:
of a
Chevalley group over a field and gave an explicit presentation of this group in terms of generators and relations. In the case of the group E
891:, where two trivial vector bundles on two halves of a space are glued along a common strip of the space. This gluing data is expressed using the
2514:
5205:
5136:) is trivial for any central simple algebra over a number field, but Platonov has given examples of algebras of degree prime squared for which S
3109:
3093:{\displaystyle {\tilde {K}}_{0}\left(A\right)=\bigcap \limits _{{\mathfrak {p}}{\text{ prime ideal of }}A}\mathrm {Ker} \dim _{\mathfrak {p}},}
10706:
2717:-theory, Ă©tale cohomology is highly computable, so Ă©tale Chern classes provided an effective tool for detecting the existence of elements in
10884:
2615:
of the ring of integers. Quillen therefore generalized
Lichtenbaum's conjecture, predicting the existence of a spectral sequence like the
7038:, after a few years during which several incompatible definitions were suggested. The object of the program was to find definitions of
6183:
10299:
Seiler, Wolfgang (1988), "λ-Rings and Adams Operations in Algebraic K-Theory", in Rapoport, M.; Schneider, P.; Schappacher, N. (eds.),
4578:{\displaystyle K_{1}(A,I)\rightarrow K_{1}(A)\rightarrow K_{1}(A/I)\rightarrow K_{0}(A,I)\rightarrow K_{0}(A)\rightarrow K_{0}(A/I)\ .}
2138:-cobordism theorem can be reinterpreted as the statement that the set of connected components of this space is the Whitehead group of
8682:
7162:
992:
394:
103:
6705:
2616:
366:
6309:
2058:-cobordism theorem because the simple connectedness hypotheses imply that the relevant Whitehead group is trivial. In fact the
1796:, modulo relations coming from exact sequences of coherent sheaves. In the categorical framework adopted by later authors, the
524:), and they imply that it is the universal way to assign invariants to vector bundles in a way compatible with exact sequences.
5662:
of universal type for a given root system. This construction yields the kernel of the Steinberg extension for the root systems
4696:
3588:
2768:-theory gave the correct groups, it was not known that these groups had all of the envisaged properties. For this, algebraic
1812:-theory of its category of coherent sheaves. Not only could Quillen prove the existence of a localization exact sequence for
11306:
10572:
10503:
10477:
10447:
10378:
10330:
10308:
10270:
10182:
10050:
10004:
9940:
9910:
9853:
9761:
9706:
9212:
7652:
8691:
7895:
1286:{\displaystyle F^{\times }\otimes _{\mathbf {Z} }F^{\times }/\langle x\otimes (1-x)\colon x\in F\setminus \{0,1\}\rangle .}
6440:
627:
and then compute the pushforward for Chow groups. The GrothendieckâRiemannâRoch theorem says that these are equal. When
266:, and the solvability of quadratic equations over completions. In contrast, finding the correct definition of the higher
5658:). Unstable second K-groups (in this context) are defined by taking the kernel of the universal central extension of the
10542:
10240:
8577:
6551:
2787:
was described in terms of complexes of sheaves on algebraic varieties. Thomason discovered that if one worked with in
17:
1604:
knows only about gluing vector bundles, not about the vector bundles themselves, it was impossible for it to describe
6913:{\displaystyle \partial ^{n}:k^{*}\times \cdots \times k^{*}\rightarrow H^{n}\left({k,\mu _{m}^{\otimes n}}\right)\ }
5640:{\displaystyle K_{2}(k)=k^{\times }\otimes _{\mathbf {Z} }k^{\times }/\langle a\otimes (1-a)\mid a\not =0,1\rangle .}
7144:-construction", the latter subsequently modified in different ways. The two constructions yield the same K-groups.
3242:
1848:-theory to topology was Whitehead's construction of Whitehead torsion. A closely related construction was found by
10880:"La stratification naturelle des espaces de fonctions differentiables reelles et le theoreme de la pseudo-isotopie"
3239:(this module is indeed free, as any finitely generated projective module over a local ring is free). This subgroup
1406:", and it neither appeared to generalize to all rings nor did it appear to be the correct definition of the higher
5410:{\displaystyle K_{2}(\mathbf {Q} )=(\mathbf {Z} /4)^{*}\times \prod _{p{\text{ odd prime}}}(\mathbf {Z} /p)^{*}\ }
4686:) (unit in the upper left corner), and hence is onto, and has the special Whitehead group as kernel, yielding the
2647:. In the case studied by Lichtenbaum, the spectral sequence would degenerate, yielding Lichtenbaum's conjecture.
10841:
10611:
8280:-groups have proved particularly difficult to compute except in a few isolated but interesting cases. (See also:
2398:
677:
11289:
Thomason, Robert W.; Trobaugh, Thomas (1990), "Higher Algebraic K-Theory of Schemes and of Derived Categories",
7770:
721:). Even as such definitions were developed, technical issues surrounding restriction and gluing usually forced
8614:-groups for smooth varieties over finite fields, and states that in this case the groups vanish up to torsion.
455:
7286:
1618:-theory under the name of Î-objects. Segal's approach is a homotopy analog of Grothendieck's construction of
10641:
10615:
10552:
10404:
10250:
9996:
8593:
7334:
6642:
3938:
2314:
2306:
1894:
have isomorphic interiors (in TOP, PL, or DIFF as appropriate), then the isomorphism between them defines an
698:
4220:
to be one which is the sum of an identity matrix and a single off-diagonal element (this is a subset of the
7099:
7007:
5425:
2880:-groups were discovered first, and given various ad hoc descriptions, which remain useful. Throughout, let
2040:-cobordism theorem, due independently to Mazur, Stallings, and Barden, explains the general situation: An
1067:-groups for certain categories and proved that his definitions yielded that same groups as those of Bass.
11486:
11481:
7729:
1614:
Inspired by conversations with Quillen, Segal soon introduced another approach to constructing algebraic
1078:
7735:
3377:
3325:
to a ring with unity obtaining by adjoining an identity element (0,1). There is a short exact sequence
2818:-theory to Hochschild homology. While the Dennis trace map seemed to be successful for calculations of
2336:
1777:-theory had been defined early in the development of the subject by Grothendieck. Grothendieck defined
10469:
10224:
10134:
9902:
6080:
5181:
5161:
4687:
3213:
3184:
2764:-theory on singular varieties still lacked adequate foundations. While it was believed that Quillen's
11243:, ÌColloq. Theorie des Groupes Algebriques, Gauthier-Villars, Paris, 1962, pp. 113â127. (French)
10163:
Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972)
8243:
7996:
3505:
2917:
1410:-theory of fields. Much later, it was discovered by Nesterenko and Suslin and by Totaro that Milnor
1305:
6094:
2611:-groups of the ring of integers of the field. These special values were known to be related to the
1856:
dominated by a finite complex has a generalized Euler characteristic taking values in a quotient of
11171:, Proc. Intern. Congress Math., Vancouver, 1974, vol. I, Canad. Math. Soc., 1975, pp. 171â176.
8573:
6926:
3879:
of projective modules is again projective, and so tensor product induces a multiplication turning K
3410:
2600:. While progress has been made on Gersten's conjecture since then, the general case remains open.
2062:-cobordism theorem implies that there is a bijective correspondence between isomorphism classes of
321:
179:
10871:, Algebraic K-theory I, Lecture Notes in Math., vol. 341, Springer-Verlag, 1973, pp. 266â292.
10434:
Lluis-Puebla, Emilio; Loday, Jean-Louis; Gillet, Henri; Soulé, Christophe; Snaith, Victor (1992),
8558:) have recently been determined, but whether the latter groups are cyclic, and whether the groups
8197:
of regular rings is finite, i.e. any finitely generated module has a finite projective resolution
6953:
6597:
6499:
6036:{\displaystyle K_{2}(A)\rightarrow K_{2}(A/I)\rightarrow K_{1}(A,I)\rightarrow K_{1}(A)\cdots \ .}
2864:-theory or topological cyclic homology is possible, then many other "nearby" calculations follow.
10468:, Encyclopedia of Mathematics and its Applications, vol. 87 (corrected paperback ed.),
8607:
7835:
7548:-construction are functorial by definition. This fact is not automatic in the plus-construction.
668:
and Hirzebruch quickly transported Grothendieck's construction to topology and used it to define
135:
8050:
8708:
8276:-theory has provided deep insight into various aspects of algebraic geometry and topology, the
5062:
4966:
2623:-theory. Quillen's proposed spectral sequence would start from the Ă©tale cohomology of a ring
888:
390:
80:
6776:
2776:-theory with the foundations of intersection theory described in volume six of Grothendieck's
2011:
975:
could be fit together into an exact sequence similar to the relative homology exact sequence.
215:
is the ring of integers in a number field, this generalizes the classical construction of the
10756:
10669:
7830:
7015:
3439:
1320:
884:
779:(roughly "main conjecture"). The fact that triangulations were stable under subdivision led
279:
119:
10204:
Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1
2153:
and in particular is the obstruction to the uniqueness of a homotopy between a manifold and
11316:
11093:
10973:
10657:
10633:
10582:
10420:
10388:
10280:
10232:
10211:
10192:
10142:
10119:
10091:
10060:
10014:
9981:
9890:
9863:
9771:
9729:
9666:
9533:
8737:
8257:
8128:
8023:
7264:
6293:
4786:
4225:
3512:
2722:
892:
851:
784:
669:
661:
378:
370:
362:
333:
106:. Intersection theory is still a motivating force in the development of (higher) algebraic
10590:
10513:
10457:
10340:
10288:
10150:
10068:
10022:
9950:
9920:
9822:
9541:
9386:
8:
11205:
The Obstruction to Finding a Boundary for an Open Manifold of Dimension Greater than Five
11061:
8746:
8732:
8622:
8281:
8261:
7806:
5703:
5271:
5038:
For a non-commutative ring, the determinant cannot in general be defined, but the map GL(
4236:) generated by elementary matrices equals the commutator subgroup . Indeed, the group GL(
4213:
3542:
3384:
2811:
2310:
2264:
1677:-construction builds a category, not an abelian group, and unlike Segal's Î-objects, the
1626:
157:
84:
58:
11097:
10095:
7842:
7708:
7598:
6999:
2710:
2612:
445:, and so it is an abelian group. If the basis element corresponding to a vector bundle
11429:
11401:
11373:
11279:
11229:
11181:
10773:
10686:
10667:; Murthy, M.P. (1967), "Grothendieck groups and Picard groups of abelian group rings",
10494:, Modern BirkhÀuser Classics (Paperback reprint of the 1996 2nd ed.), Boston, MA:
9630:
9374:
8716:
8677:
7976:
7872:
7621:
7578:
7554:
7011:
6542:
5696:
2913:
2885:
2853:
2461:
1883:
788:
787:. A simple homotopy equivalence is defined in terms of adding simplices or cells to a
438:
259:
111:
10595:
10351:
10293:
4159:
11302:
11195:
11176:
10940:
10916:
10810:
10568:
10499:
10473:
10443:
10374:
10326:
10304:
10266:
10178:
10107:
10046:
10000:
9936:
9906:
9849:
9757:
9654:
9521:
9366:
9208:
8724:
8672:
8099:
7280:
7095:
7059:
7003:
5177:
4221:
4130:{\displaystyle \operatorname {GL} (A)=\operatorname {colim} \operatorname {GL} (n,A)}
3572:
3489:
2920:
2830:-groups. Bokstedt's version of the Dennis trace map was a transformation of spectra
2330:
1651:-theory which was to prove enormously successful. This new definition began with an
1519:
1454:
337:
88:
10720:
10701:
4860:{\displaystyle 1\to \operatorname {SL} (A)\to \operatorname {GL} (A)\to A^{*}\to 1.}
2814:. This was based around the existence of the Dennis trace map, a homomorphism from
1647:
In the spring of 1972, Quillen found another approach to the construction of higher
769:
11449:
11421:
11393:
11365:
11294:
11221:
11190:
11101:
11044:
11026:
11013:
10935:
10893:
10879:
10828:
10765:
10715:
10678:
10586:
10560:
10509:
10453:
10408:
10366:
10336:
10322:
10284:
10258:
10170:
10146:
10099:
10064:
10038:
10018:
9969:
9946:
9916:
9870:
9841:
9829:
9818:
9775:. See also Lecture IV and the references in (Friedlander & Weibel
9749:
9646:
9537:
9511:
9497:
9485:
9382:
9358:
8601:
8434:
8194:
5282:
5267:
5196:
4598:
4389:{\displaystyle K_{1}(A,I)=\ker \left({K_{1}(D(A,I))\rightarrow K_{1}(A)}\right)\ .}
4062:{\displaystyle K_{1}(A)=\operatorname {GL} (A)^{\mbox{ab}}=\operatorname {GL} (A)/}
3884:
3854:
3806:{\displaystyle K_{0}(A,I)=\ker \left({K_{0}(D(A,I))\rightarrow K_{0}(A)}\right)\ .}
2788:
2726:
1882:
is homotopy equivalent to a finite complex if and only if the invariant vanishes.
1686:
1450:
1158:
869:
780:
765:
313:
183:
62:
4897:) splits as the direct sum of the group of units and the special Whitehead group:
2296:
11312:
11298:
10969:
10605:
10578:
10556:
10546:
10536:
10521:
10439:
10416:
10384:
10362:
10276:
10254:
10244:
10228:
10207:
10188:
10166:
10138:
10115:
10056:
10034:
10010:
9977:
9932:
9886:
9859:
9837:
9767:
9745:
9662:
9529:
9204:
8179:
7276:
6300:
6157:
5692:
5659:
5448:
5028:
3450:
2845:
2803:
1717:(taking the loop space corrects the indexing). Quillen additionally proved his "
1485:
544:
329:
9500:(1969), "Sur les sous-groupes arithmétiques des groupes semi-simples déployés",
7514:) discrete, this definition doesn't differ in higher degrees and also holds for
3488:
An algebro-geometric variant of this construction is applied to the category of
1398:
Inspired in part by Matsumoto's theorem, Milnor made a definition of the higher
800:. The torsion of a homotopy equivalence takes values in a group now called the
11453:
11293:, Progr. Math., vol. 88, Boston, MA: BirkhĂ€user Boston, pp. 247â435,
11018:
11001:
10629:
10601:
10412:
10396:
10347:
10219:
Quillen, Daniel (1974), "Higher K-theory for categories with exact sequences",
10199:
10158:
9874:
9346:
8147:
7572:
7531:
7252:
6289:
5122:
4610:
4400:
3934:
3876:
2840:. This transformation factored through the fixed points of a circle action on
2468:
2044:-cobordism is a cylinder if and only if the Whitehead torsion of the inclusion
1714:
1656:
1652:
1446:
1297:
775:
665:
434:
295:
271:
263:
231:
131:
11384:
Whitehead, J.H.C. (1941), "On incidence matrices, nuclei and homotopy types",
10859:, Lecture Notes in Mathematics, vol. 657, SpringerâVerlag, pp. 40â84
10564:
10262:
10042:
9733:
9650:
2259:-theory, Waldhausen made significant technical advances in the foundations of
2160:. Consideration of these questions led Waldhausen to introduce his algebraic
1434:-theory. Additionally, Thomason discovered that there is no analog of Milnor
954:) is the subgroup of elementary matrices. They also provided a definition of
11475:
10955:
10744:
10649:
10221:
New developments in topology (Proc. Sympos. Algebraic Topology, Oxford, 1972)
10111:
9988:
9658:
9525:
9370:
8296:
The first and one of the most important calculations of the higher algebraic
7485:{\displaystyle K_{n}(R)=\pi _{n}(B\operatorname {GL} (R)^{+}\times K_{0}(R))}
3527:
3302:
2604:
2388:
1969:
1849:
1526:-theory since the work of Grothendieck, and so Quillen was led to define the
1522:. The Adams operations had been known to be related to Chern classes and to
854:, an invariant related to Whitehead torsion, to disprove the Hauptvermutung.
374:
50:
10960:
10495:
10370:
9817:, Mathematics Lecture Note Series, New York-Amsterdam: W.A. Benjamin, Inc.,
1518:), the map became a homotopy equivalence. This modification was called the
660:
also became defined for non-commutative rings, where it had applications to
53:. Geometric, algebraic, and arithmetic objects are assigned objects called
11160:-theory I, Lecture Notes in Math., vol. 341, Springer Verlag, 1973, 85â147.
10806:
9957:
9707:"ag.algebraic geometry - Quillen's motivation of higher algebraic K-theory"
8449:
8438:
8410:
8301:
8182:
7702:
7260:
5440:
5073:
4153:
4141:
3892:
3477:
3391:
2705:", an analog of topological Chern classes which took elements of algebraic
2582:{\displaystyle H^{p}(X,{\mathcal {K}}_{p})\cong \operatorname {CH} ^{p}(X)}
2074:
1792:
to be the free abelian group on isomorphism classes of coherent sheaves on
1571:
is connected, so Quillen's definition failed to give the correct value for
1091:) of elementary matrices, the universal central extension is now written St
792:
204:
123:
102:-group, but even this single group has plenty of applications, such as the
9845:
7098:
of spaces and the long exact sequence for relative K-groups arises as the
5256:{\displaystyle \varphi \colon \operatorname {St} (A)\to \mathrm {GL} (A),}
3922:
provided this definition, which generalizes the group of units of a ring:
2697:-vector spaces, and he found an analog of the Bott element in topological
11057:
10997:
10126:
10075:
8223:
7698:
6538:
5651:
5166:
4785:
which is a quotient of the usual split short exact sequence defining the
3179:
3160:{\displaystyle \dim _{\mathfrak {p}}:K_{0}\left(A\right)\to \mathbf {Z} }
2795:-theory to derived categories, Thomason was able to prove that algebraic
2702:
2180:) which is defined so that it plays essentially the same role for higher
1914:
1301:
1106:
1060:
847:
345:
317:
216:
127:
46:
11062:"Differential topology from the point of view of simple homotopy theory"
11049:
11030:
9516:
8020:
moves the homotopy groups up one degree, hence the shift in degrees for
5031:. For Dedekind domains with all quotients by maximal ideals finite, SK
3500:-group of the category of locally free sheaves (or coherent sheaves) on
361:. In the mid-20th century, the RiemannâRoch theorem was generalized by
11433:
11405:
11377:
11233:
11105:
10897:
10833:
10814:
10777:
10728:
10697:
10690:
10664:
10174:
10103:
9973:
9901:, Cambridge Studies in Advanced Mathematics, vol. 101, Cambridge:
9806:
9753:
9378:
8618:
7283:. He originally found this idea while studying the group cohomology of
3919:
3417:
2325:-theory spectra would, to each open subset of a variety, associate the
2073:-cobordisms is their uniqueness. The natural notion of equivalence is
1710:
865:
839:
548:
536:
139:
115:
8444:(a finite extension of the rationals), then the algebraic K-groups of
1339:-theory were proposed. Swan and Gersten both produced definitions of
1070:
The next major development in the subject came with the definition of
603:. This gives two ways of determining an element in the Chow group of
11466:
11031:"Sur les sous-groupes aritmetiques des groupes semi-simples deployes"
10875:
7103:
4961:
is nonzero was given by Ischebeck in 1980 and by Grayson in 1981. If
2078:
1913:-theoretic way. This reinterpretation happened through the study of
1697:
but whose morphisms are defined in terms of short exact sequences in
11440:
Whitehead, J.H.C. (1939), "Simplicial spaces, nuclei and m-groups",
11425:
11397:
11369:
11225:
11145:, Proc. ICM Nice 1970, vol. 2, Gauthier-Villars, Paris, 1971, 47â52.
10769:
10682:
10438:, Lecture Notes in Mathematics, vol. 1491, Berlin, Heidelberg:
10352:"Algebraic K-theory of rings of integers in local and global fields"
9639:
Institut des Hautes Ătudes Scientifiques. Publications MathĂ©matiques
9362:
3888:
3170:
is the map sending every (class of a) finitely generated projective
118:. The subject also includes classical number-theoretic topics like
8700:
449:
is denoted , then for each short exact sequence of vector bundles:
42:
38:
9744:, Lecture Notes in Mathematics, vol. 1126, Berlin, New York:
6278:{\displaystyle K_{*}^{M}(k):=T^{*}(k^{\times })/(a\otimes (1-a)),}
1909:
Whitehead torsion was eventually reinterpreted in a more directly
1044:. By applying this description recursively, he produced negative
684:(satisfying some mild technical constraints) a sequence of groups
243:
is a field, it is exactly the group of units. For a number field
134:, as well as more modern concerns like the construction of higher
11356:
Wall, C. T. C. (1965), "Finiteness conditions for CW-complexes",
10958:; Wagoner, John (1973), "Pseudo-isotopies of compact manifolds",
7728:
are analogous to the definitions of morphisms in the category of
7140:
Quillen gave two constructions, the "plus-construction" and the "
6923:
satisfying the defining relations of the Milnor K-group. Hence
1968:
are homotopy equivalences (in the categories TOP, PL, or DIFF).
1874:
is the fundamental group of the space. This invariant is called
1835:
1335:
In the late 1960s and early 1970s, several definitions of higher
70:
4244:) was first defined and studied by Whitehead, and is called the
2799:-theory had all the expected properties of a cohomology theory.
2737:-theory. For varieties defined over the complex numbers, Ă©tale
2395:-groups, proved that on a regular surface, the cohomology group
1735:
and led to simpler proofs, but still did not yield any negative
278:-groups of algebraic varieties were not known until the work of
8300:-groups of a ring were made by Quillen himself for the case of
7235:{\displaystyle K_{n}(R)=\pi _{n}(B\operatorname {GL} (R)^{+}),}
5907:
There is also an extension of the exact sequence for relative K
5121:) is trivial, and this may be extended to square-free degree.
3816:
where the map is induced by projection along the first factor.
2923:, regarded as a monoid under direct sum. Any ring homomorphism
2010:(in TOP, PL, or DIFF as appropriate). This theorem proved the
1560:-theory to the Adams operations allowed Quillen to compute the
1030:. Bass recognized that this theorem provided a description of
623:, or one can first apply the Chern character and Todd class of
10133:, Annals of Mathematics Studies, vol. 72, Princeton, NJ:
7006:) cohomology of the field and Milnor K-theory modulo 2 is the
1402:-groups of a field. He referred to his definition as "purely
1300:, which expresses the solvability of quadratic equations over
365:
to all algebraic varieties. In Hirzebruch's formulation, the
11252:, Proc. Sympos. Pure Math., vol. XVII, 1970, pp. 88â123.
10654:
On the Structure and Classification of Differential Manifolds
10206:, Montreal, Quebec: Canad. Math. Congress, pp. 171â176,
5421:
2207:. In particular, Waldhausen showed that there is a map from
1681:-construction works directly with short exact sequences. If
1430:-theory of a field is the highest weight-graded piece of the
619:-theory and then apply the Chern character and Todd class of
11350:
Algebraic and geometric topology (New Brunswick, N.J., 1983)
11241:
Generateurs, relations et revetements de groupes algebriques
10433:
7383: > 0 so one often defines the higher algebraic
6763:{\displaystyle \partial :k^{*}\rightarrow H^{1}(k,\mu _{m})}
2753:-theory with finite coefficients became isomorphic to Ă©tale
2627:
and, in high enough degrees and after completing at a prime
2293:-theory from the need to invoke analogs of exact sequences.
2285:
is for Segal) defined in terms of chains of cofibrations in
1816:-theory, he could prove that for a regular ring or variety,
1666:-construction has its roots in Grothendieck's definition of
10403:, Contemporary Mathematics, vol. 243, Providence, RI:
6437:
The tensor product on the tensor algebra induces a product
5650:
Matsumoto's original theorem is even more general: For any
4938:) vanishes, and the determinant map is an isomorphism from
11352:, Lecture Notes in Mathematics, vol. 1126 (1985), 318â419.
10165:, Lecture Notes in Math, vol. 341, Berlin, New York:
8420:
8417:) reproved Quillen's computation using different methods.
8260:. It applies to categories with cofibrations (also called
6371:{\displaystyle \left\{a\otimes (1-a):\ a\neq 0,1\right\}.}
5076:
provides a generalisation of the determinant giving a map
4870:
The determinant is split by including the group of units
3883:
into a commutative ring with the class as identity. The
1728:-theory agreed with each other. This yielded the correct
1705:-groups of the exact category are the homotopy groups of Ω
10321:, Chapman and Hall Mathematics Series, London, New York:
8264:). This is a more general concept than exact categories.
4775:{\displaystyle 1\to SK_{1}(A)\to K_{1}(A)\to A^{*}\to 1,}
3673:{\displaystyle D(A,I)=\{(x,y)\in A\times A:x-y\in I\}\ .}
933:) is the infinite general linear group (the union of all
653:). Upon replacing vector bundles by projective modules,
401:
be a smooth algebraic variety. To each vector bundle on
10301:
Beilinson's Conjectures on Special Values of L-Functions
11267:, 4e serie (1985), 437â552; erratum 22 (1989), 675â677.
9349:(1950), "On the commutator group of a simple algebra",
8083:
This definition coincides with the above definition of
4927:
is a Euclidean domain (e.g. a field, or the integers) S
2654:
suggested to Browder that there should be a variant of
2297:
Algebraic topology and algebraic geometry in algebraic
1391:
and are related to homotopy-invariant modifications of
1143:
further extended some of the exact sequences known for
8287:
8226:, with =Σ ± . This isomorphism extends to the higher
7687:{\displaystyle M'\longleftarrow N\longrightarrow M'',}
5478:
is finite for the ring of integers of a number field.
3990:
1356:-groups. Karoubi and Villamayor defined well-behaved
1157:, and it had striking applications to number theory.
10223:, London Math. Soc. Lecture Note Ser., vol. 11,
8208:, and a simple argument shows that the canonical map
8053:
8026:
7999:
7979:
7963:{\displaystyle K_{i}(P)=\pi _{i+1}(\mathrm {BQ} P,0)}
7898:
7875:
7845:
7809:
7773:
7738:
7711:
7655:
7624:
7601:
7581:
7557:
7396:
7337:
7289:
7165:
6956:
6929:
6817:
6779:
6708:
6645:
6600:
6554:
6502:
6443:
6312:
6186:
6097:
5924:
5714:
5529:
5309:
5208:
5007:. For a Dedekind domain, this is the case: indeed, K
4798:
4699:
4411:
4279:
4162:
4081:
3950:
3696:
3591:
3245:
3216:
3187:
3112:
3008:
2806:
discovered an entirely novel technique for computing
2701:-theory. Soule used this theory to construct "Ă©tale
2607:
of a number field could be expressed in terms of the
2517:
2401:
2339:
2069:
An obvious question associated with the existence of
1804:-theory of its category of vector bundles, while its
1371:
was sometimes a proper quotient of the BassâSchanuel
1185:
887:
of the space. All such vector bundles come from the
458:
37:
is a subject area in mathematics with connections to
10986:
Comptes Rendus de l'Académie des Sciences, Série A-B
9960:(1993), "The K-theory of finite fields, revisited",
9885:, World Sci. Publ., River Edge, NJ, pp. 1â119,
6489:{\displaystyle K_{m}\times K_{n}\rightarrow K_{m+n}}
5675: > 1) and, in the limit, stable second
2460:. Inspired by this, Gersten conjectured that for a
352:
determines subspaces of these vector spaces, and if
91:. In the modern language, Grothendieck defined only
11412:Whitehead, J.H.C. (1950), "Simple homotopy types",
11132:
Annales Scientifiques de l'Ăcole Normale SupĂ©rieure
11036:
Annales Scientifiques de l'Ăcole Normale SupĂ©rieure
9503:
Annales Scientifiques de l'Ăcole Normale SupĂ©rieure
8576:about the class groups of cyclotomic integers. See
7331:and noted some of his calculations were related to
6173:led Milnor to the following definition of "higher"
5292:
is zero for any finite field. The computation of K
3849:as a ring without identity. The independence from
2603:Lichtenbaum conjectured that special values of the
2381:-group of the total space. This is now called the
397:, his generalization of Hirzebruch's theorem. Let
389:-theory takes its name from a 1957 construction of
11212:Smale, S (1962), "On the structure of manifolds",
9828:
8583:
8072:
8039:
8012:
7985:
7962:
7881:
7854:
7821:
7791:
7756:
7720:
7686:
7630:
7610:
7587:
7563:
7484:
7368:
7323:
7234:
6983:
6942:
6912:
6804:-th roots of unity in some separable extension of
6792:
6762:
6683:
6627:
6586:
6529:
6488:
6389:⧠3 they differ in general. For example, we have
6370:
6277:
6132:
6035:
5883:
5639:
5409:
5255:
4859:
4774:
4577:
4388:
4204:
4129:
4061:
3805:
3672:
3278:
3231:
3202:
3159:
3092:
2581:
2440:
2356:
2081:proved that for simply connected smooth manifolds
1673:. Unlike Grothendieck's definition, however, the
1285:
982:-theory from this period culminated in Bass' book
555:. Additionally, he proved that a proper morphism
501:
10996:
10607:The K-book: an introduction to Algebraic K-theory
10523:The K-book: An introduction to algebraic K-theory
10401:The development of algebraic K-theory before 1980
10236:(relation of Q-construction to plus-construction)
9995:, Graduate Studies in Mathematics, vol. 67,
9869:
9789:
9776:
8598:non-commutative main conjecture of Iwasawa theory
6587:{\displaystyle a_{1}\otimes \cdots \otimes a_{n}}
6385:= 0,1,2 these coincide with those below, but for
2658:-theory with finite coefficients. He introduced
2066:-cobordisms and elements of the Whitehead group.
1414:-theory is actually a direct summand of the true
730:to be defined only for rings, not for varieties.
11473:
11288:
11272:Le principe de sciendage et l'inexistence d'une
10949:Classes de fasiceaux et theoreme de RiemannâRoch
10628:
8475:) modulo torsion. For example, for the integers
8146:-groups of (the exact category of) locally free
7271:for the size of the matrix tending to infinity,
7014:. The analogous statement for odd primes is the
4965:is a Dedekind domain whose quotient field is an
2508:contains a field, and using this he proved that
2321:-theory would provide an example. The sheaf of
2309:that it might be possible to create a theory of
2032:are not assumed to be simply connected, then an
270:-groups of rings was a difficult achievement of
11002:"K-theorie algebrique et K-theorie topologique"
7732:, where morphisms are given as correspondences
4588:
3571:and define the "double" to be a subring of the
3492:; it associates with a given algebraic variety
2778:Séminaire de Géométrie Algébrique du Bois Marie
2252:and whose homotopy fiber is a homology theory.
2110:-cobordism theorem is the classifying space of
1000:. This is a four-term exact sequence relating
274:, and many of the basic facts about the higher
10954:
10033:, Springer Monographs in Mathematics, Berlin:
9931:, Springer Monographs in Mathematics, Berlin:
8796:Whitehead 1939, Whitehead 1941, Whitehead 1950
8659:-groups of the category of finitely generated
4999:in GL. When this fails, one can ask whether K
3279:{\displaystyle {\tilde {K}}_{0}\left(A\right)}
2504:. Soon Quillen proved that this is true when
1578:. Additionally, it did not give any negative
1453:in topology, he had constructed maps from the
1319:) is essentially structured around the law of
10820:Bulletin de la Société Mathématique de France
10707:Bulletin of the American Mathematical Society
9899:Central simple algebras and Galois cohomology
9896:
9203:, Classics in mathematics, Berlin, New York:
5431:For non-Archimedean local fields, the group K
3416:Finitely generated projective modules over a
2988:is commutative, we can define a subgroup of
2391:, influenced by Gersten's work on sheaves of
10696:
7152:One possible definition of higher algebraic
6678:
6646:
5631:
5589:
3661:
3613:
3345:) to be the kernel of the corresponding map
2733:-theory for the Ă©tale topology called Ă©tale
1492:. This map is acyclic, and after modifying
1277:
1274:
1262:
1223:
676:-theory was one of the first examples of an
441:on isomorphism classes of vector bundles on
405:, Grothendieck associates an invariant, its
301:
79:-theory was discovered in the late 1950s by
11249:Nonabelian homological algebra and K-theory
10980:Karoubi, Max (1968), "Foncteurs derives et
10702:"The homotopy theory of projective modules"
10028:
9993:Introduction to Quadratic Forms over Fields
9929:Class field theory. From theory to practice
7993:. Note the classifying space of a groupoid
4969:(a finite extension of the rationals) then
3895:embeds as a subgroup of the group of units
2441:{\displaystyle H^{2}(X,{\mathcal {K}}_{2})}
961:of a homomorphism of rings and proved that
615:, one can first compute the pushforward in
10805:
10663:
10031:Reciprocity laws. From Euler to Eisenstein
9728:
7792:{\displaystyle X\leftarrow Z\rightarrow Y}
7618:is defined, objects of which are those of
6288:thus as graded parts of a quotient of the
5056:
5053:) is a generalisation of the determinant.
4222:elementary matrices used in linear algebra
3378:Grothendieck group § Further examples
764:for group rings was earlier introduced by
680:: It associates to each topological space
11439:
11411:
11383:
11194:
11048:
11025:
11017:
10939:
10832:
10790:Bokstedt, M., Hsiang, W. C., Madsen, I.,
10719:
10640:, Proc. Sympos. Pure Math., vol. 3,
10541:
10530:
10316:
10239:
9897:Gille, Philippe; Szamuely, TamĂĄs (2006),
9629:
9515:
9496:
9463:
9461:
7353:
7308:
7275:is the classifying space construction of
7018:, proved by Voevodsky, Rost, and others.
2953:) by mapping (the class of) a projective
2916:of the set of isomorphism classes of its
2848:. In the course of proving an algebraic
516:. These generators and relations define
502:{\displaystyle 0\to V'\to V\to V''\to 0,}
10622:
10485:
10161:(1973), "Higher algebraic K-theory. I",
9406:
9404:
9339:
9293:
9291:
9164:
9162:
8546:), and the orders of the finite groups K
8106:, this definition agrees with the above
7324:{\displaystyle GL_{n}(\mathbb {F} _{q})}
4621:) and thus descends to a map det :
3423:are free and so in this case once again
2333:converging from the sheaf cohomology of
2126:) is a space that classifies bundles of
2036:-cobordism need not be a cylinder. The
1948:whose boundary is the disjoint union of
178:and is closely related to the notion of
11291:The Grothendieck Festschrift Volume III
10979:
10914:
10869:-theory as generalized sheaf cohomology
10840:
10548:Algebraic K-theory and its applications
10246:Algebraic K-theory and its applications
10218:
10198:
10157:
9956:
9788:(Friedlander & Weibel
9672:
9431:
9332:
9330:
9198:
9176:
9174:
8414:
7697:where the first arrow is an admissible
7369:{\displaystyle K_{1}(\mathbb {F} _{q})}
7035:
6684:{\displaystyle \{a_{1},\ldots ,a_{n}\}}
5682:
5501:is divisible by 4, and otherwise zero.
4601:, one can define a determinant det: GL(
2650:The necessity of localizing at a prime
1693:is a category with the same objects as
1296:This relation is also satisfied by the
1161:'s 1968 thesis showed that for a field
1120:) to be the kernel of the homomorphism
369:, the theorem became a statement about
14:
11474:
11208:, Thesis, Princeton University (1965).
11079:
10648:
10600:
10537:Higher Algebraic K-Theory: an overview
10463:
10436:Higher algebraic K-theory: an overview
10395:
10346:
10298:
10125:
10074:
9690:
9681:
9599:
9565:
9556:
9553:Rosenberg (1994) Theorem 4.3.15, p.214
9547:
9470:
9458:
9440:
9422:
9413:
9392:
9230:
8617:Another fundamental conjecture due to
8098:is the category of finitely generated
7701:and the second arrow is an admissible
7646:âł are isomorphism classes of diagrams
7156:-theory of rings was given by Quillen
5504:
4970:
2844:, which suggested a relationship with
2760:Throughout the 1970s and early 1980s,
1844:The earliest application of algebraic
1593:, which was classically the source of
1449:'s. As part of Quillen's work on the
11211:
11174:
11056:
10951:, mimeographed notes, Princeton 1957.
10743:
10638:Vector bundles and homogeneous spaces
10466:An algebraic introduction to K-theory
10202:(1975), "Higher algebraic K-theory",
9614:
9605:
9574:
9449:
9401:
9309:
9300:
9288:
9279:
9270:
9261:
9239:
9221:
9183:
9159:
8479:, Borel proved that (modulo torsion)
8233:
7521:
7147:
6144:. The map is not always surjective.
5420:and remarked that the proof followed
5285:: this leads to Matsumoto's theorem.
2860:-theory, so that if a calculation in
2741:-theory is isomorphic to topological
1724:theorem" that his two definitions of
1629:whose homotopy groups are the higher
1422:-groups have a filtration called the
1418:-theory of the field. Specifically,
883:is defined using vector bundles on a
11355:
11177:"Categories and cohomology theories"
11069:Publications MathĂ©matiques de l'IHĂS
10885:Publications MathĂ©matiques de l'IHĂS
10874:
10727:
9926:
9805:
9583:
9345:
9327:
9318:
9285:Rosenberg (1994) Theorem 2.3.2, p.74
9171:
4270:is defined in terms of the "double"
3687:is defined in terms of the "double"
2271:he introduced a simplicial category
2085:of dimension at least 5, isotopy of
1976:-cobordism theorem asserted that if
1944:-dimensional manifold with boundary
1063:gave another definition of negative
10792:The cyclotomic trace and algebraic
10555:, vol. 147, Berlin, New York:
10253:, vol. 147, Berlin, New York:
9987:
8778:Grothendieck 1957, BorelâSerre 1958
8592:-groups are used in conjectures on
7030:The accepted definitions of higher
4991:can be interpreted as saying that K
3305:, we can extend the definition of K
3223:
3194:
3119:
3081:
3049:
3043:
2617:AtiyahâHirzebruch spectral sequence
24:
10519:
10427:
10319:Introduction to algebraic K-theory
10131:Introduction to algebraic K-theory
9258:Amer. J. Math., 72 (1950) pp. 1â57
8005:
7944:
7941:
7757:{\displaystyle Z\subset X\times Y}
7021:
6931:
6819:
6709:
6147:
5237:
5234:
4882:) into the general linear group GL
4152: + 1) as the upper left
3071:
3068:
3065:
2540:
2424:
2357:{\displaystyle {\mathcal {K}}_{n}}
2343:
2164:-theory of spaces. The algebraic
1852:in 1963. Wall found that a space
1505:) slightly to produce a new space
1445:-theory to be widely accepted was
1326:
512:Grothendieck imposed the relation
25:
11498:
11460:
9633:(2003), "Motivic cohomology with
9620:Gille & Szamuely (2006) p.108
9611:Gille & Szamuely (2006) p.184
8683:Fundamental theorem of algebraic
5102:) may be defined as the kernel.
3232:{\displaystyle M_{\mathfrak {p}}}
3203:{\displaystyle A_{\mathfrak {p}}}
2867:
2305:Quillen suggested to his student
2289:. This freed the foundations of
1259:
993:fundamental theorem of algebraic
857:The first adequate definition of
733:
395:GrothendieckâRiemannâRoch theorem
104:GrothendieckâRiemannâRoch theorem
9832:; Grayson, Daniel, eds. (2005),
9336:Gille & Szamuely (2006) p.48
9324:Gille & Szamuely (2006) p.47
5835:
5809:
5763:
5737:
5568:
5439:) is the direct sum of a finite
5382:
5338:
5324:
5011:is generated by the images of GL
3875:is a commutative ring, then the
3303:ring without an identity element
3153:
2448:is isomorphic to the Chow group
2267:, and for a Waldhausen category
2263:-theory. Waldhausen introduced
2054:vanishes. This generalizes the
1956:and for which the inclusions of
1662:." Like Segal's Î-objects, the
1488:acting on the classifying space
1202:
373:: The Euler characteristic of a
320:proved what is now known as the
10984:-theorie. Categories filtres",
10909:-theory and Hochschild homology
10785:Topological Hochschild homology
10721:10.1090/s0002-9904-1962-10826-x
10612:Graduate Studies in Mathematics
9782:
9722:
9699:
9623:
9592:
9490:
9479:
9248:
9192:
9156:DundasâGoodwillieâMcCarthy 2012
9150:
9141:
9132:
9123:
9114:
9105:
9096:
9087:
9078:
9069:
9060:
9051:
9042:
9033:
9024:
9015:
9006:
8997:
8988:
8979:
8970:
8961:
8952:
8943:
8934:
8925:
8916:
8907:
8898:
8889:
8880:
8871:
8862:
8853:
8844:
8625:) says that all of the groups
8584:Applications and open questions
8013:{\displaystyle B{\mathcal {G}}}
7379:This definition only holds for
5003:is generated by the image of GL
4995:is generated by the image of GL
4667:). This map splits via the map
3860:
2383:BrownâGersten spectral sequence
1441:The first definition of higher
1438:-theory for a general variety.
678:extraordinary cohomology theory
367:HirzebruchâRiemannâRoch theorem
110:-theory through its links with
11263:, Ann. Scient. Ec. Norm. Sup.
11119:, Princeton Univ. Press, 1971.
11084:-theory and Quadratic Forms",
10911:, unpublished preprint (1976).
10303:, Boston, MA: Academic Press,
10082:-theory and quadratic forms",
9734:"Algebraic K-theory of spaces"
8835:
8826:
8817:
8808:
8799:
8790:
8781:
8772:
8763:
8636:) are finitely generated when
8610:concerns the higher algebraic
8578:QuillenâLichtenbaum conjecture
8193:-theory coincide. Indeed, the
7957:
7937:
7915:
7909:
7783:
7777:
7670:
7664:
7479:
7476:
7470:
7448:
7441:
7429:
7413:
7407:
7363:
7348:
7318:
7303:
7226:
7217:
7210:
7198:
7182:
7176:
6978:
6972:
6860:
6757:
6738:
6725:
6622:
6616:
6524:
6518:
6467:
6336:
6324:
6269:
6266:
6254:
6245:
6237:
6224:
6208:
6202:
6133:{\displaystyle xyx^{-1}y^{-1}}
6021:
6015:
6002:
5999:
5987:
5974:
5971:
5957:
5944:
5941:
5935:
5900:runs over all prime ideals of
5872:
5856:
5840:
5800:
5784:
5768:
5728:
5610:
5598:
5546:
5540:
5395:
5378:
5351:
5334:
5328:
5320:
5300:) is complicated: Tate proved
5247:
5241:
5230:
5227:
5221:
5195:It can also be defined as the
5169:found the right definition of
4973:, corollary 16.3) shows that S
4851:
4838:
4835:
4829:
4820:
4817:
4811:
4802:
4763:
4750:
4747:
4741:
4728:
4725:
4719:
4703:
4566:
4552:
4539:
4536:
4530:
4517:
4514:
4502:
4489:
4486:
4472:
4459:
4456:
4450:
4437:
4434:
4422:
4372:
4366:
4353:
4350:
4347:
4335:
4329:
4302:
4290:
4255:
4199:
4196:
4190:
4178:
4172:
4163:
4124:
4112:
4094:
4088:
4056:
4053:
4047:
4035:
4029:
4020:
4012:
4006:
3986:
3979:
3967:
3961:
3789:
3783:
3770:
3767:
3764:
3752:
3746:
3719:
3707:
3628:
3616:
3607:
3595:
3552:
3253:
3149:
3016:
2576:
2570:
2551:
2528:
2435:
2412:
2003:is isomorphic to the cylinder
1936:-cobordant if there exists an
1542:). Not only did this recover
1244:
1232:
490:
479:
473:
462:
174:is isomorphic to the integers
13:
1:
11331:-theory of topological spaces
11080:Milnor, J (1970), "Algebraic
10815:"Le theoreme de RiemannâRoch"
10642:American Mathematical Society
10553:Graduate Texts in Mathematics
10405:American Mathematical Society
10251:Graduate Texts in Mathematics
9997:American Mathematical Society
9799:
8594:special values of L-functions
8267:
7829:, which is defined to be the
6943:{\displaystyle \partial ^{n}}
3939:infinite general linear group
2456:) of codimension 2 cycles on
1876:Wall's finiteness obstruction
409:. The set of all classes on
11299:10.1007/978-0-8176-4576-2_10
11261:-theory and Ă©tale cohomology
11196:10.1016/0040-9383(74)90022-6
11122:Nobile, A., Villamayor, O.,
10941:10.1016/0021-8693(71)90030-5
10787:. Preprint, Bielefeld, 1986.
9678:Rosenberg (1994) pp. 245â246
9276:Rosenberg (1994) 2.5.4, p.95
9267:Rosenberg (1994) 2.5.1, p.92
9236:Rosenberg (1994) 1.5.3, p.27
9227:Rosenberg (1994) 1.5.1, p.27
8692:Basic theorems in algebraic
8425:-groups of rings of integers
8272:While the Quillen algebraic
8161:-groups are usually written
7100:long exact homotopy sequence
6984:{\displaystyle K_{n}^{M}(k)}
6950:may be regarded as a map on
6628:{\displaystyle K_{n}^{M}(k)}
6530:{\displaystyle K_{*}^{M}(F)}
6059:. Given commuting matrices
5426:Law of Quadratic Reciprocity
4589:Commutative rings and fields
3383:(Projective) modules over a
2219:) which generalizes the map
1800:-theory of a variety is the
1079:universal central extensions
230:) is closely related to the
7:
10317:Silvester, John R. (1981),
10029:Lemmermeyer, Franz (2000),
9147:BokstedtâHsiangâMadsen 1993
8666:
6091:as images. The commutator
4644:), one can also define the
3371:
2729:then invented an analog of
2106:The proper context for the
1999:are simply connected, then
1469:) to the homotopy fiber of
783:to introduce the notion of
757:A group closely related to
27:Subject area in mathematics
10:
11503:
11276:-theorie de Milnor globale
11113:Introduction to Algebraic
11019:10.7146/math.scand.a-11024
10850:-theory with coefficients
10470:Cambridge University Press
10225:Cambridge University Press
10215:(Quillen's Q-construction)
10135:Princeton University Press
9903:Cambridge University Press
9120:Thomason and Trobaugh 1990
8534:The torsion subgroups of K
8435:ring of algebraic integers
8241:
8073:{\displaystyle \pi _{i+1}}
7529:
6155:
6046:
5162:Steinberg group (K-theory)
5159:
5019:in GL. The subgroup of SK
4688:split short exact sequence
3375:
3056: prime ideal of
2745:-theory. Moreover, Ă©tale
2255:In order to fully develop
1836:Applications of algebraic
1564:-groups of finite fields.
1534:as the homotopy groups of
1364:, but their equivalent of
1105:. In the spring of 1967,
573:determines a homomorphism
285:
11467:K theory preprint archive
11000:; Villamayor, O. (1971),
10947:Grothendieck, Alexander,
10565:10.1007/978-1-4612-4314-4
10464:Magurn, Bruce A. (2009),
10263:10.1007/978-1-4612-4314-4
10043:10.1007/978-3-662-12893-0
9879:An overview of algebraic
9651:10.1007/s10240-003-0010-6
9201:K-Theory: an Introduction
8596:and the formulation of a
8448:are finitely generated.
8316:is the finite field with
8244:Waldhausen S-construction
7973:with a fixed zero-object
7544:-groups, defined via the
6162:The above expression for
3545:real-valued functions on
3506:compact topological space
1077:. Steinberg studied the
1059:). In independent work,
699:EilenbergâSteenrod axioms
11454:10.1112/plms/s2-45.1.243
10863:Brown, K., Gersten, S.,
10084:Inventiones Mathematicae
9467:Lemmermeyer (2000) p.385
9021:Hatcher and Wagoner 1973
8756:
8640:is a finitely generated
8292:-groups of finite fields
8248:A third construction of
8127:. More generally, for a
7705:. Note the morphisms in
7503:) is path connected and
7279:, and the is Quillen's
6793:{\displaystyle \mu _{m}}
5512:states that for a field
2639:-adic completion of the
2118:is a CAT manifold, then
697:) which satisfy all the
607:from a vector bundle on
294:-theory was detailed by
11442:Proc. London Math. Soc.
11285:, no. 3, 1992, 571â588.
11138:, no. 3, 1968, 581â616.
10700:; Schanuel, S. (1962),
10371:10.1007/3-540-27855-9_5
9958:Jardine, John Frederick
9410:Lemmermeyer (2000) p.66
8895:KaroubiâVillamayor 1971
8600:and in construction of
8493:)/tors.=0 for positive
8452:used this to calculate
8429:Quillen proved that if
6548:The images of elements
6051:There is a pairing on K
5110:has prime degree then S
5057:Central simple algebras
4646:special Whitehead group
3288:reduced zeroth K-theory
1924:-dimensional manifolds
1824:-theory, and therefore
1382:-groups are now called
1308:was able to prove that
1026:, and the localization
344:form vector spaces. A
302:The Grothendieck group
11175:Segal, Graeme (1974),
10754:of algebraic cycles",
10531:Pedagogical references
10413:10.1090/conm/243/03695
9927:Gras, Georges (2003),
9696:Rosenberg (1994) p.289
9687:Rosenberg (1994) p.246
9571:Rosenberg (1994) p.200
9476:Silvester (1981) p.228
9102:DwyerâFriedlander 1982
8922:NesterenkoâSuslin 1990
8886:NobileâVillamayor 1968
8787:AtiyahâHirzebruch 1961
8644:-algebra. (The groups
8572:) vanish depends upon
8256:-construction, due to
8252:-theory groups is the
8142:are defined to be the
8074:
8041:
8014:
7987:
7964:
7883:
7869:of the exact category
7856:
7823:
7800:
7793:
7758:
7722:
7688:
7632:
7612:
7589:
7565:
7486:
7370:
7325:
7236:
7094:) are functors into a
7034:-groups were given by
6985:
6944:
6914:
6794:
6764:
6685:
6629:
6588:
6531:
6490:
6372:
6279:
6134:
6037:
5885:
5641:
5424:'s first proof of the
5411:
5288:One can compute that K
5257:
5063:central simple algebra
4967:algebraic number field
4861:
4776:
4617:, which vanishes on E(
4579:
4390:
4228:states that the group
4206:
4148:), which embeds in GL(
4131:
4063:
3853:is an analogue of the
3807:
3674:
3280:
3233:
3204:
3161:
3094:
2852:-theory analog of the
2709:-theory to classes in
2583:
2442:
2358:
1769:-theory (or sometimes
1567:The classifying space
1287:
1022:, the polynomial ring
889:clutching construction
864:of a ring was made by
503:
393:which appeared in the
391:Alexander Grothendieck
379:characteristic classes
180:vector space dimension
138:and special values of
81:Alexander Grothendieck
11386:Annals of Mathematics
11358:Annals of Mathematics
10757:Annals of Mathematics
10670:Annals of Mathematics
10634:Hirzebruch, Friedrich
10623:Historical references
10486:Srinivas, V. (2008),
9846:10.1007/3-540-27855-9
9730:Waldhausen, Friedhelm
9315:Rosenberg (1994) p.78
9306:Rosenberg (1994) p.81
9297:Rosenberg (1994) p.75
9256:Simple homotopy types
9199:Karoubi, Max (2008),
9168:Rosenberg (1994) p.30
8712:-theory of a category
8574:Vandiver's conjecture
8398: − 1)
8262:Waldhausen categories
8075:
8042:
8040:{\displaystyle K_{i}}
8015:
7988:
7965:
7884:
7857:
7831:geometric realisation
7824:
7794:
7766:
7759:
7723:
7689:
7633:
7613:
7590:
7566:
7487:
7371:
7326:
7265:general linear groups
7237:
7016:Bloch-Kato conjecture
6998:The relation between
6986:
6945:
6915:
6800:denotes the group of
6795:
6765:
6686:
6630:
6589:
6532:
6491:
6373:
6280:
6135:
6038:
5886:
5642:
5412:
5258:
5150:
4862:
4777:
4580:
4391:
4207:
4132:
4064:
3909:
3808:
3675:
3281:
3234:
3205:
3162:
3095:
2981:a covariant functor.
2891:
2596:. This is known as
2584:
2443:
2359:
2265:Waldhausen categories
1757:-theory of a variety
1715:geometric realization
1321:quadratic reciprocity
1288:
1176:) was isomorphic to:
1037:entirely in terms of
662:group representations
504:
382:class is the degree.
371:Euler characteristics
334:meromorphic functions
312:In the 19th century,
120:quadratic reciprocity
11143:Cohomology of groups
10802:(3) (1993), 465â539.
10658:Cambridge University
10407:, pp. 211â238,
10365:, pp. 139â190,
10361:, Berlin, New York:
10359:Handbook of K-theory
10127:Milnor, John Willard
10076:Milnor, John Willard
9871:Friedlander, Eric M.
9836:, Berlin, New York:
9834:Handbook of K-Theory
9748:, pp. 318â419,
9218:, see Theorem I.6.18
8608:Parshin's conjecture
8051:
8024:
7997:
7977:
7896:
7873:
7843:
7807:
7771:
7736:
7709:
7653:
7622:
7599:
7579:
7555:
7394:
7335:
7287:
7163:
6954:
6927:
6815:
6777:
6706:
6643:
6598:
6552:
6500:
6441:
6310:
6294:multiplicative group
6184:
6095:
5922:
5712:
5683:Long exact sequences
5527:
5474:/2, and in general K
5307:
5206:
5035:is a torsion group.
4796:
4787:special linear group
4697:
4409:
4277:
4160:
4079:
3948:
3887:similarly induces a
3694:
3589:
3321:be the extension of
3243:
3214:
3185:
3110:
3006:
2713:. Unlike algebraic
2515:
2399:
2337:
1183:
893:general linear group
852:Reidemeister torsion
785:simple homotopy type
670:topological K-theory
569:to a smooth variety
456:
363:Friedrich Hirzebruch
322:RiemannâRoch theorem
57:-groups. These are
11202:Siebenmann, Larry,
11128:-theorie algebrique
11098:1970InMat...9..318M
11050:10.24033/asens.1174
10915:Gersten, S (1971),
10543:Rosenberg, Jonathan
10241:Rosenberg, Jonathan
10227:, pp. 95â103,
10169:, pp. 85â147,
10096:1970InMat...9..318M
10078:(1970), "Algebraic
9631:Voevodsky, Vladimir
9562:Milnor (1971) p.123
9517:10.24033/asens.1174
9446:Milnor (1971) p.175
9428:Milnor (1971) p.102
9419:Milnor (1971) p.101
8913:Milnor 1970, p. 319
8733:Redshift conjecture
8282:K-groups of a field
7889:is then defined as
7822:{\displaystyle BQP}
7638:and morphisms from
6971:
6901:
6808:. This extends to
6615:
6517:
6430:is nonzero for odd
6303:, generated by the
6201:
5704:long exact sequence
5520:-group is given by
5510:Matsumoto's theorem
5505:Matsumoto's theorem
5272:elementary matrices
4987:The vanishing of SK
4399:There is a natural
4214:commutator subgroup
3834:) is isomorphic to
3496:the Grothendieck's
3490:algebraic varieties
3434:) is isomorphic to
3405:) is isomorphic to
3178:to the rank of the
2812:Hochschild homology
2012:Poincaré conjecture
1898:-cobordism between
1840:-theory in topology
1761:and an open subset
1633:-groups (including
332:, then the sets of
89:algebraic varieties
85:intersection theory
11487:Algebraic geometry
11482:Algebraic K-theory
11106:10.1007/bf01425486
10898:10.1007/BF02684687
10834:10.24033/bsmf.1500
10811:Serre, Jean-Pierre
10630:Atiyah, Michael F.
10175:10.1007/BFb0067053
10104:10.1007/BF01425486
9974:10.1007/BF00961219
9875:Weibel, Charles W.
9754:10.1007/BFb0074449
9637:/2-coefficients",
9589:Milnor (1971) p.69
9580:Milnor (1971) p.63
9455:Milnor (1971) p.81
9254:J.H.C. Whitehead,
9245:Milnor (1971) p.15
9189:Milnor (1971) p.14
9048:BrownâGersten 1973
8805:BassâSchanuel 1962
8580:for more details.
8070:
8037:
8010:
7983:
7960:
7879:
7855:{\displaystyle QP}
7852:
7819:
7789:
7754:
7721:{\displaystyle QP}
7718:
7684:
7628:
7611:{\displaystyle QP}
7608:
7585:
7561:
7482:
7366:
7321:
7232:
7148:The +-construction
7060:classifying spaces
7012:Vladimir Voevodsky
6981:
6957:
6940:
6910:
6884:
6790:
6760:
6681:
6625:
6601:
6584:
6543:graded-commutative
6527:
6503:
6486:
6368:
6275:
6187:
6140:is an element of K
6130:
6033:
5881:
5697:field of fractions
5637:
5407:
5377:
5253:
5147:) is non-trivial.
5125:also showed that S
5027:may be studied by
4857:
4772:
4575:
4386:
4202:
4127:
4059:
3994:
3803:
3670:
3276:
3229:
3200:
3157:
3090:
3063:
2921:projective modules
2918:finitely generated
2914:Grothendieck group
2854:Novikov conjecture
2579:
2462:regular local ring
2438:
2354:
1884:Laurent Siebenmann
1773:′-theory).
1556:, the relation of
1455:classifying spaces
1304:. In particular,
1283:
872:. In topological
789:simplicial complex
642:) is now known as
499:
439:free abelian group
425:. By definition,
421:) from the German
338:differential forms
260:class field theory
203:is related to the
122:and embeddings of
112:motivic cohomology
18:Algebraic K theory
11346:-theory of spaces
11308:978-0-8176-3487-2
11270:Thomason, R. W.,
11255:Thomason, R. W.,
11165:Higher algebraic
11163:Quillen, Daniel,
11150:Higher algebraic
11148:Quillen, Daniel,
11141:Quillen, Daniel,
11027:Matsumoto, Hideya
10905:Higher algebraic
10798:. Invent. Math.,
10796:-theory of spaces
10614:, vol. 145,
10574:978-0-387-94248-3
10505:978-0-8176-4736-0
10479:978-0-521-10658-0
10449:978-3-540-55007-5
10380:978-3-540-23019-9
10332:978-0-412-22700-4
10310:978-0-12-581120-0
10272:978-0-387-94248-3
10184:978-3-540-06434-3
10052:978-3-540-66957-9
10006:978-0-8218-1095-8
9942:978-3-540-44133-5
9912:978-0-521-86103-8
9855:978-3-540-30436-4
9830:Friedlander, Eric
9763:978-3-540-15235-4
9742:-theory of spaces
9506:, 4 (in French),
9498:Matsumoto, Hideya
9437:Gras (2003) p.205
9214:978-3-540-79889-7
9180:Milnor (1971) p.5
8720:-group of a field
8673:Additive K-theory
8602:higher regulators
7986:{\displaystyle 0}
7882:{\displaystyle P}
7862:. Then, the i-th
7631:{\displaystyle P}
7588:{\displaystyle P}
7564:{\displaystyle P}
7281:plus construction
7096:homotopy category
7008:Milnor conjecture
6909:
6344:
6029:
5880:
5481:We further have K
5406:
5374:
5363:
5283:Steinberg symbols
5281:is determined by
5061:In the case of a
4571:
4382:
4226:Whitehead's lemma
4218:elementary matrix
3993:
3799:
3666:
3573:Cartesian product
3309:as follows. Let
3256:
3057:
3042:
3019:
2810:-theory based on
2331:spectral sequence
2114:-cobordisms. If
1520:plus construction
1426:, and the Milnor
1424:weight filtration
1101:) and called the
114:and specifically
16:(Redirected from
11494:
11456:
11436:
11408:
11392:(5): 1197â1239,
11380:
11340:Waldhausen, F.,
11322:Waldhausen, F.,
11319:
11236:
11199:
11198:
11108:
11076:
11066:
11053:
11052:
11022:
11021:
10993:
10976:
10944:
10943:
10917:"On the functor
10900:
10860:
10842:Browder, William
10837:
10836:
10780:
10740:
10724:
10723:
10693:
10660:
10645:
10618:
10593:
10526:
10516:
10482:
10460:
10423:
10392:(survey article)
10391:
10356:
10343:
10323:Chapman and Hall
10313:
10291:
10235:
10214:
10195:
10154:(lower K-groups)
10153:
10122:
10071:
10025:
9984:
9953:
9923:
9893:
9866:
9825:
9793:
9786:
9780:
9774:
9726:
9720:
9719:
9718:
9717:
9703:
9697:
9694:
9688:
9685:
9679:
9676:
9670:
9669:
9627:
9621:
9618:
9612:
9609:
9603:
9602:), cf. Lemma 1.8
9596:
9590:
9587:
9581:
9578:
9572:
9569:
9563:
9560:
9554:
9551:
9545:
9544:
9519:
9494:
9488:
9486:Hideya Matsumoto
9483:
9477:
9474:
9468:
9465:
9456:
9453:
9447:
9444:
9438:
9435:
9429:
9426:
9420:
9417:
9411:
9408:
9399:
9398:Lam (2005) p.139
9396:
9390:
9389:
9343:
9337:
9334:
9325:
9322:
9316:
9313:
9307:
9304:
9298:
9295:
9286:
9283:
9277:
9274:
9268:
9265:
9259:
9252:
9246:
9243:
9237:
9234:
9228:
9225:
9219:
9217:
9196:
9190:
9187:
9181:
9178:
9169:
9166:
9157:
9154:
9148:
9145:
9139:
9136:
9130:
9127:
9121:
9118:
9112:
9109:
9103:
9100:
9094:
9091:
9085:
9082:
9076:
9073:
9067:
9064:
9058:
9055:
9049:
9046:
9040:
9037:
9031:
9028:
9022:
9019:
9013:
9010:
9004:
9001:
8995:
8992:
8986:
8983:
8977:
8974:
8968:
8965:
8959:
8956:
8950:
8947:
8941:
8938:
8932:
8929:
8923:
8920:
8914:
8911:
8905:
8902:
8896:
8893:
8887:
8884:
8878:
8875:
8869:
8866:
8860:
8857:
8851:
8848:
8842:
8839:
8833:
8830:
8824:
8823:BassâMurthy 1967
8821:
8815:
8812:
8806:
8803:
8797:
8794:
8788:
8785:
8779:
8776:
8770:
8767:
8728:-theory spectrum
8623:Bass' conjecture
8437:in an algebraic
8411:Rick Jardine
8320:elements, then:
8195:global dimension
8148:coherent sheaves
8079:
8077:
8076:
8071:
8069:
8068:
8046:
8044:
8043:
8038:
8036:
8035:
8019:
8017:
8016:
8011:
8009:
8008:
7992:
7990:
7989:
7984:
7969:
7967:
7966:
7961:
7947:
7936:
7935:
7908:
7907:
7888:
7886:
7885:
7880:
7861:
7859:
7858:
7853:
7828:
7826:
7825:
7820:
7798:
7796:
7795:
7790:
7763:
7761:
7760:
7755:
7727:
7725:
7724:
7719:
7693:
7691:
7690:
7685:
7680:
7663:
7637:
7635:
7634:
7629:
7617:
7615:
7614:
7609:
7594:
7592:
7591:
7586:
7575:; associated to
7570:
7568:
7567:
7562:
7518: = 0.
7491:
7489:
7488:
7483:
7469:
7468:
7456:
7455:
7428:
7427:
7406:
7405:
7375:
7373:
7372:
7367:
7362:
7361:
7356:
7347:
7346:
7330:
7328:
7327:
7322:
7317:
7316:
7311:
7302:
7301:
7241:
7239:
7238:
7233:
7225:
7224:
7197:
7196:
7175:
7174:
6990:
6988:
6987:
6982:
6970:
6965:
6949:
6947:
6946:
6941:
6939:
6938:
6919:
6917:
6916:
6911:
6907:
6906:
6902:
6900:
6892:
6872:
6871:
6859:
6858:
6840:
6839:
6827:
6826:
6799:
6797:
6796:
6791:
6789:
6788:
6769:
6767:
6766:
6761:
6756:
6755:
6737:
6736:
6724:
6723:
6690:
6688:
6687:
6682:
6677:
6676:
6658:
6657:
6634:
6632:
6631:
6626:
6614:
6609:
6593:
6591:
6590:
6585:
6583:
6582:
6564:
6563:
6536:
6534:
6533:
6528:
6516:
6511:
6495:
6493:
6492:
6487:
6485:
6484:
6466:
6465:
6453:
6452:
6402:
6401:
6377:
6375:
6374:
6369:
6364:
6360:
6342:
6284:
6282:
6281:
6276:
6244:
6236:
6235:
6223:
6222:
6200:
6195:
6139:
6137:
6136:
6131:
6129:
6128:
6116:
6115:
6071:, take elements
6055:with values in K
6042:
6040:
6039:
6034:
6027:
6014:
6013:
5986:
5985:
5967:
5956:
5955:
5934:
5933:
5890:
5888:
5887:
5882:
5878:
5868:
5867:
5852:
5851:
5839:
5838:
5832:
5824:
5823:
5814:
5813:
5812:
5796:
5795:
5780:
5779:
5767:
5766:
5760:
5752:
5751:
5742:
5741:
5740:
5724:
5723:
5702:then there is a
5646:
5644:
5643:
5638:
5588:
5583:
5582:
5573:
5572:
5571:
5561:
5560:
5539:
5538:
5416:
5414:
5413:
5408:
5404:
5403:
5402:
5390:
5385:
5376:
5375:
5372:
5359:
5358:
5346:
5341:
5327:
5319:
5318:
5270:of the group of
5268:Schur multiplier
5262:
5260:
5259:
5254:
5240:
5029:Mennicke symbols
4866:
4864:
4863:
4858:
4850:
4849:
4781:
4779:
4778:
4773:
4762:
4761:
4740:
4739:
4718:
4717:
4599:commutative ring
4584:
4582:
4581:
4576:
4569:
4562:
4551:
4550:
4529:
4528:
4501:
4500:
4482:
4471:
4470:
4449:
4448:
4421:
4420:
4395:
4393:
4392:
4387:
4380:
4379:
4375:
4365:
4364:
4328:
4327:
4289:
4288:
4268:relative K-group
4211:
4209:
4208:
4205:{\displaystyle }
4203:
4136:
4134:
4133:
4128:
4068:
4066:
4065:
4060:
4019:
3996:
3995:
3991:
3960:
3959:
3885:exterior product
3855:Excision theorem
3812:
3810:
3809:
3804:
3797:
3796:
3792:
3782:
3781:
3745:
3744:
3706:
3705:
3685:relative K-group
3679:
3677:
3676:
3671:
3664:
3286:is known as the
3285:
3283:
3282:
3277:
3275:
3264:
3263:
3258:
3257:
3249:
3238:
3236:
3235:
3230:
3228:
3227:
3226:
3209:
3207:
3206:
3201:
3199:
3198:
3197:
3166:
3164:
3163:
3158:
3156:
3148:
3137:
3136:
3124:
3123:
3122:
3099:
3097:
3096:
3091:
3086:
3085:
3084:
3074:
3062:
3058:
3055:
3053:
3052:
3038:
3027:
3026:
3021:
3020:
3012:
2839:
2789:derived category
2727:Eric Friedlander
2723:William G. Dwyer
2711:Ă©tale cohomology
2693:
2682:
2653:
2638:
2630:
2613:Ă©tale cohomology
2588:
2586:
2585:
2580:
2566:
2565:
2550:
2549:
2544:
2543:
2527:
2526:
2447:
2445:
2444:
2439:
2434:
2433:
2428:
2427:
2411:
2410:
2363:
2361:
2360:
2355:
2353:
2352:
2347:
2346:
2251:
2215:) to a space Wh(
2159:
2053:
2020:
2009:
1987:is compact, and
1982:
1943:
1820:-theory equaled
1788:) for a variety
1723:
1687:abelian category
1582:-groups. Since
1475:
1451:Adams conjecture
1360:-groups for all
1292:
1290:
1289:
1284:
1222:
1217:
1216:
1207:
1206:
1205:
1195:
1194:
1159:Hideya Matsumoto
1135:
924:
870:Stephen Schanuel
838:is the integral
781:J.H.C. Whitehead
766:J.H.C. Whitehead
611:: Starting from
598:
568:
543:coming from the
515:
508:
506:
505:
500:
489:
472:
336:and meromorphic
316:and his student
314:Bernhard Riemann
258:) is related to
238:
202:
184:commutative ring
173:
83:in his study of
63:abstract algebra
61:in the sense of
21:
11502:
11501:
11497:
11496:
11495:
11493:
11492:
11491:
11472:
11471:
11463:
11426:10.2307/2372133
11398:10.2307/1970465
11370:10.2307/1970382
11309:
11246:Swan, Richard,
11239:Steinberg, R.,
11226:10.2307/2372978
11064:
10923:
10903:Dennis, R. K.,
10770:10.2307/1970902
10753:
10683:10.2307/1970360
10644:, pp. 7â38
10625:
10602:Weibel, Charles
10575:
10557:Springer-Verlag
10533:
10506:
10480:
10450:
10440:Springer-Verlag
10430:
10428:Further reading
10397:Weibel, Charles
10381:
10363:Springer-Verlag
10354:
10348:Weibel, Charles
10333:
10311:
10273:
10255:Springer-Verlag
10200:Quillen, Daniel
10185:
10167:Springer-Verlag
10159:Quillen, Daniel
10053:
10035:Springer-Verlag
10007:
9943:
9933:Springer-Verlag
9913:
9856:
9838:Springer-Verlag
9802:
9797:
9796:
9787:
9783:
9764:
9746:Springer-Verlag
9727:
9723:
9715:
9713:
9705:
9704:
9700:
9695:
9691:
9686:
9682:
9677:
9673:
9628:
9624:
9619:
9615:
9610:
9606:
9597:
9593:
9588:
9584:
9579:
9575:
9570:
9566:
9561:
9557:
9552:
9548:
9495:
9491:
9484:
9480:
9475:
9471:
9466:
9459:
9454:
9450:
9445:
9441:
9436:
9432:
9427:
9423:
9418:
9414:
9409:
9402:
9397:
9393:
9363:10.2307/2372036
9347:Wang, Shianghaw
9344:
9340:
9335:
9328:
9323:
9319:
9314:
9310:
9305:
9301:
9296:
9289:
9284:
9280:
9275:
9271:
9266:
9262:
9253:
9249:
9244:
9240:
9235:
9231:
9226:
9222:
9215:
9205:Springer-Verlag
9197:
9193:
9188:
9184:
9179:
9172:
9167:
9160:
9155:
9151:
9146:
9142:
9137:
9133:
9128:
9124:
9119:
9115:
9110:
9106:
9101:
9097:
9092:
9088:
9083:
9079:
9074:
9070:
9065:
9061:
9056:
9052:
9047:
9043:
9039:Waldhausen 1985
9038:
9034:
9030:Waldhausen 1978
9029:
9025:
9020:
9016:
9011:
9007:
9002:
8998:
8993:
8989:
8984:
8980:
8976:Siebenmann 1965
8975:
8971:
8966:
8962:
8957:
8953:
8948:
8944:
8939:
8935:
8930:
8926:
8921:
8917:
8912:
8908:
8903:
8899:
8894:
8890:
8885:
8881:
8876:
8872:
8867:
8863:
8858:
8854:
8849:
8845:
8840:
8836:
8831:
8827:
8822:
8818:
8813:
8809:
8804:
8800:
8795:
8791:
8786:
8782:
8777:
8773:
8768:
8764:
8759:
8678:Bloch's formula
8669:
8649:
8630:
8586:
8567:
8553:
8541:
8517:
8488:
8470:
8460:
8427:
8406: â„ 1.
8389:
8380:
8363:
8354:
8338:
8329:
8315:
8294:
8270:
8246:
8240:
8221:
8214:
8203:
8169:
8118:
8089:
8058:
8054:
8052:
8049:
8048:
8031:
8027:
8025:
8022:
8021:
8004:
8003:
7998:
7995:
7994:
7978:
7975:
7974:
7940:
7925:
7921:
7903:
7899:
7897:
7894:
7893:
7874:
7871:
7870:
7844:
7841:
7840:
7808:
7805:
7804:
7772:
7769:
7768:
7737:
7734:
7733:
7710:
7707:
7706:
7673:
7656:
7654:
7651:
7650:
7623:
7620:
7619:
7600:
7597:
7596:
7595:a new category
7580:
7577:
7576:
7556:
7553:
7552:
7534:
7528:
7509:
7464:
7460:
7451:
7447:
7423:
7419:
7401:
7397:
7395:
7392:
7391:
7357:
7352:
7351:
7342:
7338:
7336:
7333:
7332:
7312:
7307:
7306:
7297:
7293:
7288:
7285:
7284:
7277:homotopy theory
7250:
7220:
7216:
7192:
7188:
7170:
7166:
7164:
7161:
7160:
7150:
7028:
6966:
6961:
6955:
6952:
6951:
6934:
6930:
6928:
6925:
6924:
6893:
6888:
6877:
6873:
6867:
6863:
6854:
6850:
6835:
6831:
6822:
6818:
6816:
6813:
6812:
6784:
6780:
6778:
6775:
6774:
6751:
6747:
6732:
6728:
6719:
6715:
6707:
6704:
6703:
6699:there is a map
6691:. For integer
6672:
6668:
6653:
6649:
6644:
6641:
6640:
6610:
6605:
6599:
6596:
6595:
6578:
6574:
6559:
6555:
6553:
6550:
6549:
6512:
6507:
6501:
6498:
6497:
6474:
6470:
6461:
6457:
6448:
6444:
6442:
6439:
6438:
6428:
6422:
6410:
6400:
6395:
6394:
6393:
6317:
6313:
6311:
6308:
6307:
6301:two-sided ideal
6240:
6231:
6227:
6218:
6214:
6196:
6191:
6185:
6182:
6181:
6168:
6160:
6158:Milnor K-theory
6154:
6143:
6121:
6117:
6108:
6104:
6096:
6093:
6092:
6081:Steinberg group
6058:
6054:
6049:
6009:
6005:
5981:
5977:
5963:
5951:
5947:
5929:
5925:
5923:
5920:
5919:
5914:
5910:
5863:
5859:
5847:
5843:
5834:
5833:
5828:
5819:
5815:
5808:
5807:
5803:
5791:
5787:
5775:
5771:
5762:
5761:
5756:
5747:
5743:
5736:
5735:
5731:
5719:
5715:
5713:
5710:
5709:
5693:Dedekind domain
5685:
5670:
5660:Chevalley group
5584:
5578:
5574:
5567:
5566:
5562:
5556:
5552:
5534:
5530:
5528:
5525:
5524:
5507:
5484:
5477:
5465:
5454:
5449:divisible group
5434:
5398:
5394:
5386:
5381:
5373: odd prime
5371:
5367:
5354:
5350:
5342:
5337:
5323:
5314:
5310:
5308:
5305:
5304:
5295:
5291:
5280:
5233:
5207:
5204:
5203:
5182:Steinberg group
5175:
5164:
5158:
5156:
5142:
5131:
5116:
5106:states that if
5097:
5082:
5059:
5048:
5034:
5026:
5023:generated by SL
5022:
5018:
5014:
5010:
5006:
5002:
4998:
4994:
4990:
4979:
4960:
4944:
4933:
4915:
4903:
4892:
4877:
4845:
4841:
4797:
4794:
4793:
4757:
4753:
4735:
4731:
4713:
4709:
4698:
4695:
4694:
4681:
4654:
4627:
4591:
4558:
4546:
4542:
4524:
4520:
4496:
4492:
4478:
4466:
4462:
4444:
4440:
4416:
4412:
4410:
4407:
4406:
4360:
4356:
4323:
4319:
4318:
4314:
4284:
4280:
4278:
4275:
4274:
4264:
4262:
4246:Whitehead group
4161:
4158:
4157:
4080:
4077:
4076:
4015:
3989:
3985:
3955:
3951:
3949:
3946:
3945:
3928:
3917:
3915:
3901:
3891:structure. The
3882:
3869:
3866:
3840:
3825:
3777:
3773:
3740:
3736:
3735:
3731:
3701:
3697:
3695:
3692:
3691:
3590:
3587:
3586:
3567:be an ideal of
3561:
3559:
3541:of the ring of
3539:
3534:coincides with
3459:
3451:Dedekind domain
3429:
3400:
3380:
3374:
3359:
3351:
3340:
3337:and we define K
3308:
3265:
3259:
3248:
3247:
3246:
3244:
3241:
3240:
3222:
3221:
3217:
3215:
3212:
3211:
3193:
3192:
3188:
3186:
3183:
3182:
3152:
3138:
3132:
3128:
3118:
3117:
3113:
3111:
3108:
3107:
3080:
3079:
3075:
3064:
3054:
3048:
3047:
3046:
3028:
3022:
3011:
3010:
3009:
3007:
3004:
3003:
2994:
2980:
2970:
2948:
2937:
2907:
2899:
2897:
2874:
2846:cyclic homology
2831:
2804:R. Keith Dennis
2786:
2691:
2680:
2670:
2662:-theory groups
2651:
2636:
2628:
2619:in topological
2598:Bloch's formula
2561:
2557:
2545:
2539:
2538:
2537:
2522:
2518:
2516:
2513:
2512:
2495:
2487:) injects into
2482:
2429:
2423:
2422:
2421:
2406:
2402:
2400:
2397:
2396:
2372:
2364:, the sheaf of
2348:
2342:
2341:
2340:
2338:
2335:
2334:
2303:
2277:
2245:
2234:
2226:
2220:
2198:
2190:
2154:
2144:
2130:-cobordisms on
2095:
2045:
2015:
2004:
1977:
1937:
1862:
1842:
1808:-theory is the
1783:
1734:
1718:
1672:
1639:
1624:
1610:
1599:
1588:
1577:
1555:
1548:
1517:
1504:
1486:Adams operation
1470:
1468:
1390:
1377:
1370:
1347:
1333:
1314:
1218:
1212:
1208:
1201:
1200:
1196:
1190:
1186:
1184:
1181:
1180:
1171:
1156:
1149:
1142:
1121:
1115:
1103:Steinberg group
1096:
1086:
1076:
1054:
1043:
1036:
1017:
1006:
974:
967:
960:
941:
907:
901:
882:
863:
823:
802:Whitehead group
763:
755:
753:
746:
739:
729:
716:
707:
692:
672:. Topological
659:
648:
580:
574:
556:
545:Chern character
513:
482:
465:
457:
454:
453:
385:The subject of
330:Riemann surface
310:
308:
290:The history of
288:
280:Robert Thomason
253:
234:
225:
196:
190:
167:
161:
132:complex numbers
97:
69:-groups of the
28:
23:
22:
15:
12:
11:
5:
11500:
11490:
11489:
11484:
11470:
11469:
11462:
11461:External links
11459:
11458:
11457:
11437:
11414:Amer. J. Math.
11409:
11381:
11353:
11338:
11320:
11307:
11286:
11268:
11253:
11244:
11237:
11220:(3): 387â399,
11214:Amer. J. Math.
11209:
11200:
11189:(3): 293â312,
11172:
11161:
11146:
11139:
11120:
11109:
11092:(4): 318â344,
11077:
11054:
11023:
10994:
10977:
10956:Hatcher, Allen
10952:
10945:
10934:(2): 212â237,
10921:
10912:
10901:
10872:
10861:
10838:
10803:
10788:
10783:Bokstedt, M.,
10781:
10764:(2): 349â379,
10751:
10745:Bloch, Spencer
10741:
10725:
10714:(4): 425â428,
10694:
10661:
10650:Barden, Dennis
10646:
10624:
10621:
10620:
10619:
10598:
10573:
10539:
10532:
10529:
10528:
10527:
10517:
10504:
10483:
10478:
10461:
10448:
10429:
10426:
10425:
10424:
10393:
10379:
10344:
10331:
10314:
10309:
10296:
10271:
10237:
10216:
10196:
10183:
10155:
10123:
10090:(4): 318â344,
10072:
10051:
10026:
10005:
9989:Lam, Tsit-Yuen
9985:
9968:(6): 579â595,
9954:
9941:
9924:
9911:
9894:
9867:
9854:
9826:
9801:
9798:
9795:
9794:
9781:
9762:
9721:
9698:
9689:
9680:
9671:
9622:
9613:
9604:
9591:
9582:
9573:
9564:
9555:
9546:
9489:
9478:
9469:
9457:
9448:
9439:
9430:
9421:
9412:
9400:
9391:
9357:(2): 323â334,
9338:
9326:
9317:
9308:
9299:
9287:
9278:
9269:
9260:
9247:
9238:
9229:
9220:
9213:
9191:
9182:
9170:
9158:
9149:
9140:
9131:
9122:
9113:
9104:
9095:
9086:
9077:
9068:
9059:
9050:
9041:
9032:
9023:
9014:
9005:
8996:
8987:
8978:
8969:
8960:
8951:
8942:
8933:
8924:
8915:
8906:
8897:
8888:
8879:
8870:
8861:
8859:Matsumoto 1969
8852:
8843:
8841:Steinberg 1962
8834:
8825:
8816:
8807:
8798:
8789:
8780:
8771:
8761:
8760:
8758:
8755:
8754:
8753:
8744:
8735:
8730:
8722:
8714:
8706:
8698:
8689:
8680:
8675:
8668:
8665:
8647:
8628:
8585:
8582:
8562:
8547:
8535:
8532:
8531:
8511:
8506:
8486:
8466:
8456:
8426:
8419:
8408:
8407:
8385:
8374:
8369:
8359:
8349:
8344:
8334:
8327:
8311:
8293:
8286:
8269:
8266:
8242:Main article:
8239:
8232:
8230:-groups, too.
8219:
8212:
8201:
8165:
8114:
8110:definition of
8087:
8067:
8064:
8061:
8057:
8034:
8030:
8007:
8002:
7982:
7971:
7970:
7959:
7956:
7953:
7950:
7946:
7943:
7939:
7934:
7931:
7928:
7924:
7920:
7917:
7914:
7911:
7906:
7902:
7878:
7851:
7848:
7818:
7815:
7812:
7788:
7785:
7782:
7779:
7776:
7753:
7750:
7747:
7744:
7741:
7717:
7714:
7695:
7694:
7683:
7679:
7676:
7672:
7669:
7666:
7662:
7659:
7627:
7607:
7604:
7584:
7573:exact category
7560:
7532:Q-construction
7530:Main article:
7527:
7520:
7507:
7493:
7492:
7481:
7478:
7475:
7472:
7467:
7463:
7459:
7454:
7450:
7446:
7443:
7440:
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7434:
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7426:
7422:
7418:
7415:
7412:
7409:
7404:
7400:
7365:
7360:
7355:
7350:
7345:
7341:
7320:
7315:
7310:
7305:
7300:
7296:
7292:
7253:homotopy group
7246:
7243:
7242:
7231:
7228:
7223:
7219:
7215:
7212:
7209:
7206:
7203:
7200:
7195:
7191:
7187:
7184:
7181:
7178:
7173:
7169:
7149:
7146:
7125:) â
7117:) â
7058:) in terms of
7036:Quillen (1973)
7027:
7020:
6980:
6977:
6974:
6969:
6964:
6960:
6937:
6933:
6921:
6920:
6905:
6899:
6896:
6891:
6887:
6883:
6880:
6876:
6870:
6866:
6862:
6857:
6853:
6849:
6846:
6843:
6838:
6834:
6830:
6825:
6821:
6787:
6783:
6771:
6770:
6759:
6754:
6750:
6746:
6743:
6740:
6735:
6731:
6727:
6722:
6718:
6714:
6711:
6695:invertible in
6680:
6675:
6671:
6667:
6664:
6661:
6656:
6652:
6648:
6624:
6621:
6618:
6613:
6608:
6604:
6581:
6577:
6573:
6570:
6567:
6562:
6558:
6526:
6523:
6520:
6515:
6510:
6506:
6483:
6480:
6477:
6473:
6469:
6464:
6460:
6456:
6451:
6447:
6426:
6420:
6408:
6396:
6379:
6378:
6367:
6363:
6359:
6356:
6353:
6350:
6347:
6341:
6338:
6335:
6332:
6329:
6326:
6323:
6320:
6316:
6290:tensor algebra
6286:
6285:
6274:
6271:
6268:
6265:
6262:
6259:
6256:
6253:
6250:
6247:
6243:
6239:
6234:
6230:
6226:
6221:
6217:
6213:
6210:
6207:
6204:
6199:
6194:
6190:
6166:
6156:Main article:
6153:
6146:
6141:
6127:
6124:
6120:
6114:
6111:
6107:
6103:
6100:
6056:
6052:
6048:
6045:
6044:
6043:
6032:
6026:
6023:
6020:
6017:
6012:
6008:
6004:
6001:
5998:
5995:
5992:
5989:
5984:
5980:
5976:
5973:
5970:
5966:
5962:
5959:
5954:
5950:
5946:
5943:
5940:
5937:
5932:
5928:
5912:
5908:
5892:
5891:
5877:
5874:
5871:
5866:
5862:
5858:
5855:
5850:
5846:
5842:
5837:
5831:
5827:
5822:
5818:
5811:
5806:
5802:
5799:
5794:
5790:
5786:
5783:
5778:
5774:
5770:
5765:
5759:
5755:
5750:
5746:
5739:
5734:
5730:
5727:
5722:
5718:
5684:
5681:
5666:
5648:
5647:
5636:
5633:
5630:
5627:
5624:
5621:
5618:
5615:
5612:
5609:
5606:
5603:
5600:
5597:
5594:
5591:
5587:
5581:
5577:
5570:
5565:
5559:
5555:
5551:
5548:
5545:
5542:
5537:
5533:
5506:
5503:
5482:
5475:
5463:
5452:
5432:
5418:
5417:
5401:
5397:
5393:
5389:
5384:
5380:
5370:
5366:
5362:
5357:
5353:
5349:
5345:
5340:
5336:
5333:
5330:
5326:
5322:
5317:
5313:
5293:
5289:
5278:
5277:For a field, K
5264:
5263:
5252:
5249:
5246:
5243:
5239:
5236:
5232:
5229:
5226:
5223:
5220:
5217:
5214:
5211:
5173:
5157:
5154:
5149:
5140:
5129:
5114:
5104:Wang's theorem
5095:
5080:
5058:
5055:
5046:
5032:
5024:
5020:
5016:
5012:
5008:
5004:
5000:
4996:
4992:
4988:
4977:
4958:
4942:
4931:
4913:
4901:
4890:
4875:
4868:
4867:
4856:
4853:
4848:
4844:
4840:
4837:
4834:
4831:
4828:
4825:
4822:
4819:
4816:
4813:
4810:
4807:
4804:
4801:
4783:
4782:
4771:
4768:
4765:
4760:
4756:
4752:
4749:
4746:
4743:
4738:
4734:
4730:
4727:
4724:
4721:
4716:
4712:
4708:
4705:
4702:
4679:
4652:
4625:
4611:group of units
4590:
4587:
4586:
4585:
4574:
4568:
4565:
4561:
4557:
4554:
4549:
4545:
4541:
4538:
4535:
4532:
4527:
4523:
4519:
4516:
4513:
4510:
4507:
4504:
4499:
4495:
4491:
4488:
4485:
4481:
4477:
4474:
4469:
4465:
4461:
4458:
4455:
4452:
4447:
4443:
4439:
4436:
4433:
4430:
4427:
4424:
4419:
4415:
4401:exact sequence
4397:
4396:
4385:
4378:
4374:
4371:
4368:
4363:
4359:
4355:
4352:
4349:
4346:
4343:
4340:
4337:
4334:
4331:
4326:
4322:
4317:
4313:
4310:
4307:
4304:
4301:
4298:
4295:
4292:
4287:
4283:
4263:
4260:
4254:
4201:
4198:
4195:
4192:
4189:
4186:
4183:
4180:
4177:
4174:
4171:
4168:
4165:
4138:
4137:
4126:
4123:
4120:
4117:
4114:
4111:
4108:
4105:
4102:
4099:
4096:
4093:
4090:
4087:
4084:
4070:
4069:
4058:
4055:
4052:
4049:
4046:
4043:
4040:
4037:
4034:
4031:
4028:
4025:
4022:
4018:
4014:
4011:
4008:
4005:
4002:
3999:
3988:
3984:
3981:
3978:
3975:
3972:
3969:
3966:
3963:
3958:
3954:
3935:abelianization
3926:
3916:
3913:
3908:
3899:
3880:
3877:tensor product
3868:
3864:
3859:
3838:
3823:
3814:
3813:
3802:
3795:
3791:
3788:
3785:
3780:
3776:
3772:
3769:
3766:
3763:
3760:
3757:
3754:
3751:
3748:
3743:
3739:
3734:
3730:
3727:
3724:
3721:
3718:
3715:
3712:
3709:
3704:
3700:
3681:
3680:
3669:
3663:
3660:
3657:
3654:
3651:
3648:
3645:
3642:
3639:
3636:
3633:
3630:
3627:
3624:
3621:
3618:
3615:
3612:
3609:
3606:
3603:
3600:
3597:
3594:
3560:
3557:
3551:
3537:
3528:vector bundles
3486:
3485:
3457:
3443:
3427:
3414:
3398:
3373:
3370:
3357:
3349:
3338:
3306:
3274:
3271:
3268:
3262:
3255:
3252:
3225:
3220:
3196:
3191:
3168:
3167:
3155:
3151:
3147:
3144:
3141:
3135:
3131:
3127:
3121:
3116:
3101:
3100:
3089:
3083:
3078:
3073:
3070:
3067:
3061:
3051:
3045:
3041:
3037:
3034:
3031:
3025:
3018:
3015:
2992:
2978:
2966:
2946:
2935:
2905:
2898:
2895:
2890:
2873:
2866:
2784:
2666:
2635:, abut to the
2631:invertible in
2590:
2589:
2578:
2575:
2572:
2569:
2564:
2560:
2556:
2553:
2548:
2542:
2536:
2533:
2530:
2525:
2521:
2491:
2478:
2469:fraction field
2437:
2432:
2426:
2420:
2417:
2414:
2409:
2405:
2368:
2351:
2345:
2302:
2295:
2275:
2243:
2232:
2224:
2196:
2188:
2142:
2093:
1860:
1841:
1834:
1781:
1732:
1670:
1653:exact category
1637:
1622:
1608:
1597:
1586:
1575:
1553:
1546:
1513:
1500:
1464:
1447:Daniel Quillen
1386:
1375:
1368:
1343:
1332:
1325:
1312:
1298:Hilbert symbol
1294:
1293:
1282:
1279:
1276:
1273:
1270:
1267:
1264:
1261:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1221:
1215:
1211:
1204:
1199:
1193:
1189:
1169:
1154:
1147:
1140:
1113:
1092:
1082:
1074:
1052:
1041:
1034:
1015:
1004:
972:
965:
958:
937:
899:
880:
861:
821:
776:Hauptvermutung
770:Henri Poincaré
761:
754:
751:
744:
737:
732:
725:
712:
705:
688:
657:
646:
578:
510:
509:
498:
495:
492:
488:
485:
481:
478:
475:
471:
468:
464:
461:
309:
306:
300:
296:Charles Weibel
287:
284:
272:Daniel Quillen
264:Hilbert symbol
251:
232:group of units
223:
194:
165:
95:
26:
9:
6:
4:
3:
2:
11499:
11488:
11485:
11483:
11480:
11479:
11477:
11468:
11465:
11464:
11455:
11451:
11447:
11443:
11438:
11435:
11431:
11427:
11423:
11419:
11415:
11410:
11407:
11403:
11399:
11395:
11391:
11387:
11382:
11379:
11375:
11371:
11367:
11363:
11359:
11354:
11351:
11347:
11343:
11339:
11336:
11333:
11332:
11327:
11326:
11321:
11318:
11314:
11310:
11304:
11300:
11296:
11292:
11287:
11284:
11281:
11277:
11273:
11269:
11266:
11262:
11258:
11254:
11251:
11250:
11245:
11242:
11238:
11235:
11231:
11227:
11223:
11219:
11215:
11210:
11207:
11206:
11201:
11197:
11192:
11188:
11184:
11183:
11178:
11173:
11170:
11166:
11162:
11159:
11155:
11151:
11147:
11144:
11140:
11137:
11133:
11129:
11125:
11121:
11118:
11114:
11110:
11107:
11103:
11099:
11095:
11091:
11087:
11086:Invent. Math.
11083:
11078:
11074:
11070:
11063:
11059:
11055:
11051:
11046:
11042:
11038:
11037:
11032:
11028:
11024:
11020:
11015:
11011:
11007:
11003:
10999:
10995:
10991:
10987:
10983:
10978:
10975:
10971:
10967:
10963:
10962:
10957:
10953:
10950:
10946:
10942:
10937:
10933:
10929:
10925:
10920:
10913:
10910:
10906:
10902:
10899:
10895:
10891:
10887:
10886:
10881:
10877:
10873:
10870:
10866:
10862:
10858:
10855:
10853:
10847:
10843:
10839:
10835:
10830:
10826:
10822:
10821:
10816:
10812:
10808:
10807:Borel, Armand
10804:
10801:
10797:
10793:
10789:
10786:
10782:
10779:
10775:
10771:
10767:
10763:
10759:
10758:
10750:
10746:
10742:
10738:
10734:
10730:
10726:
10722:
10717:
10713:
10709:
10708:
10703:
10699:
10695:
10692:
10688:
10684:
10680:
10676:
10672:
10671:
10666:
10662:
10659:
10655:
10651:
10647:
10643:
10639:
10635:
10631:
10627:
10626:
10617:
10613:
10609:
10608:
10603:
10599:
10597:
10592:
10588:
10584:
10580:
10576:
10570:
10566:
10562:
10558:
10554:
10550:
10549:
10544:
10540:
10538:
10535:
10534:
10525:
10524:
10518:
10515:
10511:
10507:
10501:
10497:
10493:
10489:
10484:
10481:
10475:
10471:
10467:
10462:
10459:
10455:
10451:
10445:
10441:
10437:
10432:
10431:
10422:
10418:
10414:
10410:
10406:
10402:
10398:
10394:
10390:
10386:
10382:
10376:
10372:
10368:
10364:
10360:
10353:
10349:
10345:
10342:
10338:
10334:
10328:
10324:
10320:
10315:
10312:
10306:
10302:
10297:
10295:
10290:
10286:
10282:
10278:
10274:
10268:
10264:
10260:
10256:
10252:
10248:
10247:
10242:
10238:
10234:
10230:
10226:
10222:
10217:
10213:
10209:
10205:
10201:
10197:
10194:
10190:
10186:
10180:
10176:
10172:
10168:
10164:
10160:
10156:
10152:
10148:
10144:
10140:
10136:
10132:
10128:
10124:
10121:
10117:
10113:
10109:
10105:
10101:
10097:
10093:
10089:
10085:
10081:
10077:
10073:
10070:
10066:
10062:
10058:
10054:
10048:
10044:
10040:
10036:
10032:
10027:
10024:
10020:
10016:
10012:
10008:
10002:
9998:
9994:
9990:
9986:
9983:
9979:
9975:
9971:
9967:
9963:
9959:
9955:
9952:
9948:
9944:
9938:
9934:
9930:
9925:
9922:
9918:
9914:
9908:
9904:
9900:
9895:
9892:
9888:
9884:
9880:
9876:
9872:
9868:
9865:
9861:
9857:
9851:
9847:
9843:
9839:
9835:
9831:
9827:
9824:
9820:
9816:
9812:
9808:
9804:
9803:
9792:), Lecture VI
9791:
9785:
9778:
9773:
9769:
9765:
9759:
9755:
9751:
9747:
9743:
9739:
9735:
9731:
9725:
9712:
9708:
9702:
9693:
9684:
9675:
9668:
9664:
9660:
9656:
9652:
9648:
9645:(1): 59â104,
9644:
9640:
9636:
9632:
9626:
9617:
9608:
9601:
9598:(Weibel
9595:
9586:
9577:
9568:
9559:
9550:
9543:
9539:
9535:
9531:
9527:
9523:
9518:
9513:
9509:
9505:
9504:
9499:
9493:
9487:
9482:
9473:
9464:
9462:
9452:
9443:
9434:
9425:
9416:
9407:
9405:
9395:
9388:
9384:
9380:
9376:
9372:
9368:
9364:
9360:
9356:
9352:
9348:
9342:
9333:
9331:
9321:
9312:
9303:
9294:
9292:
9282:
9273:
9264:
9257:
9251:
9242:
9233:
9224:
9216:
9210:
9206:
9202:
9195:
9186:
9177:
9175:
9165:
9163:
9153:
9144:
9138:Bokstedt 1986
9135:
9126:
9117:
9111:Thomason 1985
9108:
9099:
9090:
9081:
9072:
9063:
9054:
9045:
9036:
9027:
9018:
9009:
9000:
8991:
8982:
8973:
8964:
8955:
8946:
8940:Thomason 1992
8937:
8928:
8919:
8910:
8901:
8892:
8883:
8874:
8865:
8856:
8847:
8838:
8829:
8820:
8811:
8802:
8793:
8784:
8775:
8766:
8762:
8752:
8750:
8745:
8743:
8741:
8736:
8734:
8731:
8729:
8727:
8723:
8721:
8719:
8715:
8713:
8711:
8707:
8705:
8703:
8699:
8697:
8695:
8690:
8688:
8686:
8681:
8679:
8676:
8674:
8671:
8670:
8664:
8662:
8658:
8654:
8650:
8643:
8639:
8635:
8631:
8624:
8620:
8615:
8613:
8609:
8605:
8603:
8599:
8595:
8591:
8581:
8579:
8575:
8571:
8566:
8561:
8557:
8551:
8545:
8539:
8529:
8526:for positive
8525:
8521:
8515:
8510:
8507:
8504:
8500:
8496:
8492:
8485:
8482:
8481:
8480:
8478:
8474:
8469:
8464:
8459:
8455:
8451:
8447:
8443:
8440:
8436:
8432:
8424:
8418:
8416:
8412:
8405:
8401:
8397:
8393:
8388:
8384:
8378:
8373:
8370:
8367:
8362:
8358:
8353:
8348:
8345:
8342:
8337:
8333:
8326:
8323:
8322:
8321:
8319:
8314:
8310:
8305:
8303:
8302:finite fields
8299:
8291:
8285:
8283:
8279:
8275:
8265:
8263:
8259:
8255:
8251:
8245:
8238:-construction
8237:
8231:
8229:
8225:
8218:
8211:
8207:
8200:
8196:
8192:
8188:
8184:
8181:
8177:
8173:
8168:
8164:
8160:
8155:
8153:
8149:
8145:
8141:
8137:
8134:, the higher
8133:
8130:
8126:
8122:
8117:
8113:
8109:
8105:
8103:
8097:
8093:
8086:
8081:
8065:
8062:
8059:
8055:
8032:
8028:
8000:
7980:
7954:
7951:
7948:
7932:
7929:
7926:
7922:
7918:
7912:
7904:
7900:
7892:
7891:
7890:
7876:
7868:
7866:
7849:
7846:
7838:
7837:
7832:
7816:
7813:
7810:
7799:
7786:
7780:
7774:
7765:
7751:
7748:
7745:
7742:
7739:
7731:
7715:
7712:
7704:
7700:
7681:
7677:
7674:
7667:
7660:
7657:
7649:
7648:
7647:
7645:
7641:
7625:
7605:
7602:
7582:
7574:
7558:
7549:
7547:
7543:
7539:
7533:
7526:-construction
7525:
7519:
7517:
7513:
7506:
7502:
7498:
7473:
7465:
7461:
7457:
7452:
7444:
7438:
7435:
7432:
7424:
7420:
7416:
7410:
7402:
7398:
7390:
7389:
7388:
7386:
7382:
7377:
7358:
7343:
7339:
7313:
7298:
7294:
7290:
7282:
7278:
7274:
7270:
7266:
7262:
7258:
7254:
7249:
7229:
7221:
7213:
7207:
7204:
7201:
7193:
7189:
7185:
7179:
7171:
7167:
7159:
7158:
7157:
7155:
7145:
7143:
7138:
7136:
7132:
7128:
7124:
7120:
7116:
7112:
7108:
7105:
7101:
7097:
7093:
7089:
7085:
7081:
7077:
7073:
7069:
7065:
7061:
7057:
7053:
7049:
7045:
7041:
7037:
7033:
7025:
7019:
7017:
7013:
7009:
7005:
7001:
6996:
6994:
6993:Galois symbol
6991:, called the
6975:
6967:
6962:
6958:
6935:
6903:
6897:
6894:
6889:
6885:
6881:
6878:
6874:
6868:
6864:
6855:
6851:
6847:
6844:
6841:
6836:
6832:
6828:
6823:
6811:
6810:
6809:
6807:
6803:
6785:
6781:
6752:
6748:
6744:
6741:
6733:
6729:
6720:
6716:
6712:
6702:
6701:
6700:
6698:
6694:
6673:
6669:
6665:
6662:
6659:
6654:
6650:
6638:
6619:
6611:
6606:
6602:
6579:
6575:
6571:
6568:
6565:
6560:
6556:
6546:
6544:
6540:
6521:
6513:
6508:
6504:
6481:
6478:
6475:
6471:
6462:
6458:
6454:
6449:
6445:
6435:
6434:(see below).
6433:
6429:
6425:
6416:
6412:
6407:
6399:
6392:
6388:
6384:
6365:
6361:
6357:
6354:
6351:
6348:
6345:
6339:
6333:
6330:
6327:
6321:
6318:
6314:
6306:
6305:
6304:
6302:
6298:
6295:
6291:
6272:
6263:
6260:
6257:
6251:
6248:
6241:
6232:
6228:
6219:
6215:
6211:
6205:
6197:
6192:
6188:
6180:
6179:
6178:
6176:
6172:
6165:
6159:
6151:
6145:
6125:
6122:
6118:
6112:
6109:
6105:
6101:
6098:
6090:
6086:
6082:
6078:
6074:
6070:
6066:
6062:
6030:
6024:
6018:
6010:
6006:
5996:
5993:
5990:
5982:
5978:
5968:
5964:
5960:
5952:
5948:
5938:
5930:
5926:
5918:
5917:
5916:
5905:
5903:
5899:
5898:
5875:
5869:
5864:
5860:
5853:
5848:
5844:
5829:
5825:
5820:
5816:
5804:
5797:
5792:
5788:
5781:
5776:
5772:
5757:
5753:
5748:
5744:
5732:
5725:
5720:
5716:
5708:
5707:
5706:
5705:
5701:
5698:
5694:
5690:
5680:
5678:
5674:
5669:
5665:
5661:
5657:
5653:
5634:
5628:
5625:
5622:
5619:
5616:
5613:
5607:
5604:
5601:
5595:
5592:
5585:
5579:
5575:
5563:
5557:
5553:
5549:
5543:
5535:
5531:
5523:
5522:
5521:
5519:
5516:, the second
5515:
5511:
5502:
5500:
5496:
5492:
5488:
5479:
5473:
5469:
5460:
5458:
5450:
5447:, say, and a
5446:
5442:
5438:
5429:
5427:
5423:
5399:
5391:
5387:
5368:
5364:
5360:
5355:
5347:
5343:
5331:
5315:
5311:
5303:
5302:
5301:
5299:
5286:
5284:
5275:
5273:
5269:
5250:
5244:
5224:
5218:
5215:
5212:
5209:
5202:
5201:
5200:
5198:
5193:
5191:
5187:
5183:
5179:
5172:
5168:
5163:
5153:
5148:
5146:
5139:
5135:
5128:
5124:
5120:
5113:
5109:
5105:
5101:
5094:
5090:
5086:
5079:
5075:
5071:
5068:over a field
5067:
5064:
5054:
5052:
5045:
5041:
5036:
5030:
4985:
4983:
4976:
4972:
4968:
4964:
4956:
4952:
4948:
4941:
4937:
4930:
4926:
4921:
4919:
4911:
4907:
4900:
4896:
4889:
4885:
4881:
4873:
4854:
4846:
4842:
4832:
4826:
4823:
4814:
4808:
4805:
4799:
4792:
4791:
4790:
4788:
4769:
4766:
4758:
4754:
4744:
4736:
4732:
4722:
4714:
4710:
4706:
4700:
4693:
4692:
4691:
4689:
4685:
4678:
4674:
4670:
4666:
4662:
4659:) := SL(
4658:
4651:
4647:
4643:
4639:
4635:
4631:
4624:
4620:
4616:
4612:
4608:
4604:
4600:
4596:
4572:
4563:
4559:
4555:
4547:
4543:
4533:
4525:
4521:
4511:
4508:
4505:
4497:
4493:
4483:
4479:
4475:
4467:
4463:
4453:
4445:
4441:
4431:
4428:
4425:
4417:
4413:
4405:
4404:
4403:
4402:
4383:
4376:
4369:
4361:
4357:
4344:
4341:
4338:
4332:
4324:
4320:
4315:
4311:
4308:
4305:
4299:
4296:
4293:
4285:
4281:
4273:
4272:
4271:
4269:
4259:
4253:
4251:
4247:
4243:
4239:
4235:
4231:
4227:
4223:
4219:
4215:
4193:
4187:
4184:
4181:
4175:
4169:
4166:
4155:
4151:
4147:
4143:
4121:
4118:
4115:
4109:
4106:
4103:
4100:
4097:
4091:
4085:
4082:
4075:
4074:
4073:
4050:
4044:
4041:
4038:
4032:
4026:
4023:
4016:
4009:
4003:
4000:
3997:
3982:
3976:
3973:
3970:
3964:
3956:
3952:
3944:
3943:
3942:
3940:
3936:
3932:
3925:
3921:
3912:
3907:
3905:
3898:
3894:
3890:
3886:
3878:
3874:
3863:
3858:
3857:in homology.
3856:
3852:
3848:
3845:), regarding
3844:
3837:
3833:
3829:
3822:
3819:The relative
3817:
3800:
3793:
3786:
3778:
3774:
3761:
3758:
3755:
3749:
3741:
3737:
3732:
3728:
3725:
3722:
3716:
3713:
3710:
3702:
3698:
3690:
3689:
3688:
3686:
3667:
3658:
3655:
3652:
3649:
3646:
3643:
3640:
3637:
3634:
3631:
3625:
3622:
3619:
3610:
3604:
3601:
3598:
3592:
3585:
3584:
3583:
3581:
3577:
3574:
3570:
3566:
3556:
3550:
3548:
3544:
3540:
3533:
3529:
3525:
3521:
3518:
3516:
3510:
3507:
3503:
3499:
3495:
3491:
3483:
3479:
3475:
3471:
3467:
3463:
3456:
3452:
3448:
3444:
3441:
3437:
3433:
3426:
3422:
3419:
3415:
3412:
3408:
3404:
3397:
3393:
3392:vector spaces
3389:
3386:
3382:
3381:
3379:
3369:
3367:
3363:
3355:
3348:
3344:
3336:
3332:
3328:
3324:
3320:
3316:
3312:
3304:
3300:
3295:
3293:
3289:
3272:
3269:
3266:
3260:
3250:
3218:
3189:
3181:
3177:
3173:
3145:
3142:
3139:
3133:
3129:
3125:
3114:
3106:
3105:
3104:
3103:where :
3087:
3076:
3059:
3039:
3035:
3032:
3029:
3023:
3013:
3002:
3001:
3000:
2999:) as the set
2998:
2991:
2987:
2982:
2977:
2973:
2969:
2964:
2960:
2956:
2952:
2945:
2941:
2934:
2930:
2926:
2922:
2919:
2915:
2911:
2908:takes a ring
2904:
2894:
2889:
2887:
2883:
2879:
2871:
2865:
2863:
2859:
2855:
2851:
2847:
2843:
2838:
2834:
2829:
2825:
2821:
2817:
2813:
2809:
2805:
2800:
2798:
2794:
2790:
2783:
2779:
2775:
2771:
2767:
2763:
2758:
2756:
2752:
2748:
2744:
2740:
2736:
2732:
2728:
2724:
2720:
2716:
2712:
2708:
2704:
2703:Chern classes
2700:
2696:
2689:
2686:) which were
2685:
2678:
2674:
2669:
2665:
2661:
2657:
2648:
2646:
2642:
2634:
2626:
2622:
2618:
2614:
2610:
2606:
2605:zeta function
2601:
2599:
2595:
2573:
2567:
2562:
2558:
2554:
2546:
2534:
2531:
2523:
2519:
2511:
2510:
2509:
2507:
2503:
2499:
2494:
2490:
2486:
2481:
2477:
2473:
2470:
2466:
2463:
2459:
2455:
2451:
2430:
2418:
2415:
2407:
2403:
2394:
2390:
2389:Spencer Bloch
2386:
2384:
2380:
2376:
2371:
2367:
2349:
2332:
2328:
2324:
2320:
2316:
2312:
2308:
2307:Kenneth Brown
2300:
2294:
2292:
2288:
2284:
2280:
2274:
2270:
2266:
2262:
2258:
2253:
2249:
2242:
2238:
2230:
2223:
2218:
2214:
2210:
2206:
2202:
2194:
2187:
2183:
2179:
2175:
2171:
2167:
2163:
2157:
2152:
2148:
2141:
2137:
2133:
2129:
2125:
2121:
2117:
2113:
2109:
2104:
2102:
2099:
2092:
2088:
2084:
2080:
2076:
2072:
2067:
2065:
2061:
2057:
2052:
2048:
2043:
2039:
2035:
2031:
2027:
2022:
2018:
2013:
2007:
2002:
1998:
1994:
1990:
1986:
1980:
1975:
1971:
1970:Stephen Smale
1967:
1963:
1959:
1955:
1951:
1947:
1941:
1935:
1931:
1927:
1923:
1919:
1917:
1912:
1907:
1905:
1901:
1897:
1893:
1889:
1885:
1881:
1877:
1873:
1869:
1866:
1859:
1855:
1851:
1850:C. T. C. Wall
1847:
1839:
1833:
1831:
1827:
1823:
1819:
1815:
1811:
1807:
1803:
1799:
1795:
1791:
1787:
1780:
1776:
1772:
1768:
1764:
1760:
1756:
1752:
1748:
1746:
1740:
1738:
1731:
1727:
1722:
1716:
1712:
1708:
1704:
1700:
1696:
1692:
1688:
1684:
1680:
1676:
1669:
1665:
1661:
1660:-construction
1659:
1654:
1650:
1645:
1643:
1636:
1632:
1628:
1621:
1617:
1612:
1607:
1603:
1596:
1592:
1585:
1581:
1574:
1570:
1565:
1563:
1559:
1552:
1545:
1541:
1537:
1533:
1529:
1525:
1521:
1516:
1512:
1508:
1503:
1499:
1495:
1491:
1487:
1483:
1479:
1473:
1467:
1463:
1459:
1456:
1452:
1448:
1444:
1439:
1437:
1433:
1429:
1425:
1421:
1417:
1413:
1409:
1405:
1401:
1396:
1394:
1389:
1385:
1381:
1374:
1367:
1363:
1359:
1355:
1351:
1346:
1342:
1338:
1330:
1324:
1322:
1318:
1311:
1307:
1303:
1299:
1280:
1271:
1268:
1265:
1256:
1253:
1250:
1247:
1241:
1238:
1235:
1229:
1226:
1219:
1213:
1209:
1197:
1191:
1187:
1179:
1178:
1177:
1175:
1168:
1164:
1160:
1153:
1146:
1139:
1136:. The group
1133:
1129:
1125:
1119:
1112:
1108:
1104:
1100:
1095:
1090:
1085:
1080:
1073:
1068:
1066:
1062:
1058:
1051:
1047:
1040:
1033:
1029:
1025:
1021:
1014:
1010:
1003:
999:
998:
996:
989:
985:
981:
976:
971:
964:
957:
953:
949:
945:
940:
936:
932:
928:
922:
918:
914:
910:
905:
898:
894:
890:
886:
879:
875:
871:
867:
860:
855:
853:
849:
845:
841:
837:
834:
830:
827:
820:
815:
811:
807:
803:
799:
794:
790:
786:
782:
778:
777:
771:
767:
760:
750:
743:
736:
731:
728:
724:
720:
715:
711:
704:
700:
696:
691:
687:
683:
679:
675:
671:
667:
663:
656:
652:
645:
641:
637:
632:
630:
626:
622:
618:
614:
610:
606:
602:
596:
592:
588:
584:
577:
572:
567:
563:
559:
554:
550:
546:
542:
538:
534:
530:
525:
523:
519:
496:
493:
486:
483:
476:
469:
466:
459:
452:
451:
450:
448:
444:
440:
436:
432:
428:
424:
420:
416:
412:
408:
404:
400:
396:
392:
388:
383:
380:
376:
375:vector bundle
372:
368:
364:
360:
355:
351:
347:
343:
339:
335:
331:
327:
323:
319:
315:
305:
299:
297:
293:
283:
281:
277:
273:
269:
265:
261:
257:
250:
246:
242:
237:
233:
229:
222:
219:. The group
218:
214:
210:
206:
200:
193:
188:
185:
181:
177:
171:
164:
159:
155:
151:
146:
144:
142:
137:
133:
129:
125:
124:number fields
121:
117:
113:
109:
105:
101:
98:, the zeroth
94:
90:
86:
82:
78:
74:
72:
68:
64:
60:
56:
52:
51:number theory
48:
44:
40:
36:
34:
19:
11445:
11441:
11417:
11413:
11389:
11385:
11364:(1): 56â69,
11361:
11357:
11349:
11345:
11341:
11334:
11330:
11323:
11290:
11282:
11275:
11271:
11264:
11260:
11256:
11247:
11240:
11217:
11213:
11203:
11186:
11180:
11168:
11164:
11157:
11156:, Algebraic
11153:
11149:
11142:
11135:
11134:, 4e serie,
11127:
11123:
11116:
11112:
11111:Milnor, J.,
11089:
11085:
11081:
11072:
11068:
11058:Mazur, Barry
11040:
11034:
11009:
11006:Math. Scand.
11005:
10998:Karoubi, Max
10989:
10985:
10981:
10965:
10959:
10948:
10931:
10927:
10918:
10908:
10904:
10889:
10883:
10868:
10864:
10857:
10851:
10849:
10845:
10824:
10818:
10799:
10795:
10791:
10784:
10761:
10755:
10748:
10736:
10732:
10711:
10705:
10677:(1): 16â73,
10674:
10668:
10653:
10637:
10606:
10547:
10522:
10520:Weibel, C.,
10491:
10487:
10465:
10435:
10400:
10358:
10318:
10300:
10245:
10220:
10203:
10162:
10130:
10087:
10083:
10079:
10030:
9992:
9965:
9961:
9928:
9898:
9882:
9878:
9833:
9814:
9810:
9784:
9741:
9737:
9724:
9714:, retrieved
9711:MathOverflow
9710:
9701:
9692:
9683:
9674:
9642:
9638:
9634:
9625:
9616:
9607:
9594:
9585:
9576:
9567:
9558:
9549:
9507:
9501:
9492:
9481:
9472:
9451:
9442:
9433:
9424:
9415:
9394:
9354:
9351:Am. J. Math.
9350:
9341:
9320:
9311:
9302:
9281:
9272:
9263:
9255:
9250:
9241:
9232:
9223:
9200:
9194:
9185:
9152:
9143:
9134:
9125:
9116:
9107:
9098:
9089:
9084:Browder 1976
9080:
9075:Quillen 1975
9071:
9066:Quillen 1973
9062:
9053:
9044:
9035:
9026:
9017:
9008:
8999:
8990:
8981:
8972:
8963:
8954:
8949:Quillen 1971
8945:
8936:
8927:
8918:
8909:
8900:
8891:
8882:
8877:Gersten 1969
8873:
8864:
8855:
8846:
8837:
8832:Karoubi 1968
8828:
8819:
8810:
8801:
8792:
8783:
8774:
8765:
8748:
8739:
8738:Topological
8725:
8717:
8709:
8701:
8693:
8684:
8660:
8656:
8652:
8645:
8641:
8637:
8633:
8626:
8616:
8611:
8606:
8589:
8587:
8569:
8564:
8559:
8555:
8549:
8543:
8537:
8533:
8527:
8523:
8519:
8513:
8508:
8502:
8498:
8494:
8490:
8483:
8476:
8472:
8467:
8462:
8457:
8453:
8450:Armand Borel
8445:
8441:
8439:number field
8430:
8428:
8422:
8409:
8403:
8399:
8395:
8391:
8386:
8382:
8376:
8371:
8365:
8360:
8356:
8351:
8346:
8340:
8335:
8331:
8324:
8317:
8312:
8308:
8306:
8297:
8295:
8289:
8277:
8273:
8271:
8253:
8249:
8247:
8235:
8227:
8216:
8209:
8205:
8198:
8190:
8186:
8183:regular ring
8175:
8171:
8166:
8162:
8158:
8156:
8151:
8143:
8139:
8135:
8131:
8124:
8120:
8115:
8111:
8107:
8101:
8095:
8091:
8084:
8082:
8080:of a space.
7972:
7864:
7863:
7834:
7801:
7767:
7703:monomorphism
7696:
7643:
7639:
7550:
7545:
7541:
7537:
7535:
7523:
7515:
7511:
7504:
7500:
7496:
7494:
7387:-theory via
7384:
7380:
7378:
7272:
7268:
7261:direct limit
7256:
7247:
7244:
7153:
7151:
7141:
7139:
7134:
7130:
7126:
7122:
7118:
7114:
7110:
7106:
7091:
7087:
7083:
7079:
7075:
7071:
7067:
7063:
7055:
7051:
7047:
7043:
7039:
7031:
7029:
7023:
7010:, proven by
6997:
6992:
6922:
6805:
6801:
6772:
6696:
6692:
6636:
6547:
6436:
6431:
6423:
6418:
6414:
6405:
6403:
6397:
6390:
6386:
6382:
6380:
6296:
6287:
6174:
6170:
6163:
6161:
6149:
6088:
6084:
6076:
6072:
6068:
6064:
6060:
6050:
5906:
5901:
5896:
5895:
5893:
5699:
5688:
5686:
5676:
5672:
5667:
5663:
5655:
5649:
5517:
5513:
5509:
5508:
5498:
5494:
5490:
5486:
5480:
5471:
5467:
5461:
5456:
5444:
5441:cyclic group
5436:
5430:
5419:
5297:
5287:
5276:
5265:
5194:
5189:
5185:
5176:: it is the
5170:
5165:
5151:
5144:
5137:
5133:
5126:
5118:
5111:
5107:
5103:
5099:
5092:
5088:
5084:
5077:
5074:reduced norm
5069:
5065:
5060:
5050:
5043:
5039:
5037:
4986:
4984:) vanishes.
4981:
4974:
4971:Milnor (1971
4962:
4954:
4950:
4946:
4939:
4935:
4928:
4924:
4922:
4917:
4909:
4905:
4898:
4894:
4887:
4883:
4879:
4871:
4869:
4784:
4683:
4676:
4672:
4668:
4664:
4660:
4656:
4649:
4645:
4641:
4637:
4633:
4629:
4622:
4618:
4614:
4606:
4602:
4594:
4592:
4398:
4267:
4265:
4257:
4249:
4248:of the ring
4245:
4241:
4237:
4233:
4229:
4217:
4216:. Define an
4154:block matrix
4149:
4145:
4142:direct limit
4139:
4071:
3930:
3923:
3918:
3910:
3903:
3896:
3893:Picard group
3872:
3870:
3861:
3850:
3846:
3842:
3835:
3831:
3827:
3820:
3818:
3815:
3684:
3682:
3579:
3575:
3568:
3564:
3562:
3554:
3546:
3535:
3531:
3526:) of (real)
3523:
3519:
3514:
3513:topological
3508:
3501:
3497:
3493:
3487:
3481:
3478:Picard group
3473:
3472:, where Pic(
3469:
3465:
3461:
3454:
3446:
3435:
3431:
3424:
3420:
3406:
3402:
3395:
3387:
3365:
3361:
3353:
3346:
3342:
3334:
3330:
3326:
3322:
3318:
3314:
3310:
3298:
3296:
3291:
3287:
3175:
3171:
3169:
3102:
2996:
2989:
2985:
2984:If the ring
2983:
2975:
2971:
2967:
2962:
2958:
2954:
2950:
2943:
2939:
2932:
2931:gives a map
2928:
2924:
2909:
2902:
2901:The functor
2900:
2892:
2881:
2877:
2875:
2869:
2861:
2857:
2849:
2841:
2836:
2832:
2827:
2823:
2819:
2815:
2807:
2801:
2796:
2792:
2781:
2773:
2769:
2765:
2761:
2759:
2754:
2750:
2746:
2742:
2738:
2734:
2730:
2718:
2714:
2706:
2698:
2694:
2687:
2683:
2676:
2672:
2667:
2663:
2659:
2655:
2649:
2644:
2640:
2632:
2624:
2620:
2608:
2602:
2597:
2593:
2591:
2505:
2501:
2497:
2492:
2488:
2484:
2479:
2475:
2471:
2464:
2457:
2453:
2449:
2392:
2387:
2378:
2374:
2369:
2365:
2326:
2322:
2318:
2304:
2298:
2290:
2286:
2282:
2278:
2272:
2268:
2260:
2256:
2254:
2247:
2240:
2236:
2228:
2221:
2216:
2212:
2208:
2204:
2203:)) does for
2200:
2192:
2185:
2181:
2177:
2173:
2169:
2165:
2161:
2155:
2150:
2146:
2139:
2135:
2131:
2127:
2123:
2119:
2115:
2111:
2107:
2105:
2100:
2097:
2090:
2086:
2082:
2070:
2068:
2063:
2059:
2055:
2050:
2046:
2041:
2037:
2033:
2029:
2025:
2023:
2016:
2005:
2000:
1996:
1992:
1988:
1984:
1978:
1973:
1965:
1961:
1957:
1953:
1949:
1945:
1939:
1933:
1929:
1925:
1921:
1915:
1910:
1908:
1903:
1899:
1895:
1891:
1887:
1879:
1875:
1871:
1867:
1864:
1857:
1853:
1845:
1843:
1837:
1829:
1825:
1821:
1817:
1813:
1809:
1805:
1801:
1797:
1793:
1789:
1785:
1778:
1774:
1770:
1766:
1762:
1758:
1754:
1750:
1749:
1744:
1741:
1736:
1729:
1725:
1720:
1706:
1702:
1698:
1694:
1690:
1682:
1678:
1674:
1667:
1663:
1657:
1648:
1646:
1641:
1634:
1630:
1619:
1615:
1613:
1605:
1601:
1594:
1590:
1583:
1579:
1572:
1568:
1566:
1561:
1557:
1550:
1543:
1539:
1535:
1531:
1527:
1523:
1514:
1510:
1506:
1501:
1497:
1493:
1489:
1481:
1477:
1471:
1465:
1461:
1457:
1442:
1440:
1435:
1431:
1427:
1423:
1419:
1415:
1411:
1407:
1403:
1399:
1397:
1392:
1387:
1383:
1379:
1372:
1365:
1361:
1357:
1353:
1349:
1344:
1340:
1336:
1334:
1328:
1316:
1309:
1302:local fields
1295:
1173:
1166:
1162:
1151:
1144:
1137:
1131:
1127:
1123:
1117:
1110:
1102:
1098:
1093:
1088:
1083:
1071:
1069:
1064:
1056:
1049:
1045:
1038:
1031:
1027:
1023:
1019:
1012:
1008:
1001:
994:
991:
987:
983:
979:
977:
969:
962:
955:
951:
947:
943:
938:
934:
930:
926:
920:
916:
912:
908:
903:
896:
877:
873:
858:
856:
843:
835:
832:
828:
825:
818:
813:
809:
805:
804:and denoted
801:
797:
793:cell complex
774:
758:
756:
748:
741:
734:
726:
722:
718:
713:
709:
702:
694:
689:
685:
681:
673:
654:
650:
643:
639:
635:
633:
628:
624:
620:
616:
612:
608:
604:
600:
594:
590:
586:
582:
575:
570:
565:
561:
557:
552:
540:
532:
528:
526:
521:
517:
511:
446:
442:
430:
426:
422:
418:
414:
410:
406:
402:
398:
386:
384:
358:
353:
349:
341:
325:
311:
303:
291:
289:
275:
267:
255:
248:
247:, the group
244:
240:
235:
227:
220:
212:
208:
205:Picard group
198:
191:
189:, the group
186:
175:
169:
162:
153:
149:
147:
140:
128:real numbers
107:
99:
92:
76:
75:
66:
54:
32:
30:
29:
11448:: 243â327,
11420:(1): 1â57,
11012:: 265â307,
10992:: A328âA331
10729:Bass, Hyman
10698:Bass, Hyman
10665:Bass, Hyman
9807:Bass, Hyman
9510:(2): 1â62,
9129:Dennis 1976
9003:Barden 1963
8931:Totaro 1992
8904:Milnor 1970
8850:Milnor 1971
8769:Weibel 1999
8663:-modules)
8224:isomorphism
8138:-groups of
8100:projective
7699:epimorphism
6635:are termed
6539:graded ring
6177:-groups by
6169:of a field
5652:root system
5199:of the map
5167:John Milnor
2643:-theory of
2373:-groups on
2184:-groups as
2172:is a space
2168:-theory of
1918:-cobordisms
1600:. Because
1530:-theory of
1107:John Milnor
1061:Max Karoubi
848:John Milnor
601:pushforward
599:called the
537:Chow groups
413:was called
346:line bundle
318:Gustav Roch
217:class group
211:, and when
116:Chow groups
47:ring theory
11476:Categories
11342:Algebraic
11325:Algebraic
11257:Algebraic
10961:Astérisque
10928:J. Algebra
10876:Cerf, Jean
10865:Algebraic
10846:Algebraic
10827:: 97â136,
10739:, Benjamin
10733:Algebraic
10656:(Thesis),
10591:0801.19001
10514:1125.19300
10496:BirkhÀuser
10488:Algebraic
10458:0746.19001
10341:0468.18006
10289:0801.19001
10151:0237.18005
10069:0949.11002
10023:1068.11023
9951:1019.11032
9921:1137.12001
9823:0174.30302
9811:Algebraic
9800:References
9738:Algebraic
9716:2021-03-26
9542:0261.20025
9387:0040.30302
9093:Soulé 1979
9057:Bloch 1974
8994:Mazur 1963
8985:Smale 1962
8958:Segal 1974
8747:Rigidity (
8655:) are the
8619:Hyman Bass
8588:Algebraic
8421:Algebraic
8364:) = 0 for
8288:Algebraic
8258:Waldhausen
8222:(R) is an
8180:noetherian
8123:) for all
6639:, denoted
5266:or as the
5160:See also:
4953:. This is
4144:of the GL(
3920:Hyman Bass
3543:continuous
3504:. Given a
3418:local ring
3376:See also:
2876:The lower
2780:. There,
2721:-theory.
2500:) for all
1711:loop space
1007:of a ring
984:Algebraic
902:of a ring
885:suspension
866:Hyman Bass
840:group ring
634:The group
549:Todd class
148:The lower
143:-functions
136:regulators
31:Algebraic
11154:-theory I
10892:: 5â173,
10747:(1974), "
10112:0020-9910
9659:0073-8301
9526:0012-9593
9371:0002-9327
9012:Cerf 1970
8967:Wall 1965
8868:Swan 1968
8814:Bass 1968
8522:)/tors.=
8056:π
7923:π
7784:→
7778:←
7764:such that
7749:×
7743:⊂
7671:⟶
7665:⟵
7458:×
7439:
7421:π
7259:) is the
7208:
7190:π
7104:fibration
7062:so that
6932:∂
6895:⊗
6886:μ
6861:→
6856:∗
6848:×
6845:⋯
6842:×
6837:∗
6820:∂
6782:μ
6749:μ
6726:→
6721:∗
6710:∂
6663:…
6572:⊗
6569:⋯
6566:⊗
6541:which is
6509:∗
6468:→
6455:×
6349:≠
6331:−
6322:⊗
6261:−
6252:⊗
6233:×
6220:∗
6193:∗
6123:−
6110:−
6025:⋯
6003:→
5975:→
5945:→
5873:→
5857:→
5841:→
5805:⊕
5801:→
5785:→
5769:→
5733:⊕
5729:→
5679:-groups.
5632:⟩
5614:∣
5605:−
5596:⊗
5590:⟨
5580:×
5564:⊗
5558:×
5462:We have K
5443:of order
5400:∗
5365:∏
5361:×
5356:∗
5231:→
5219:
5213::
5210:φ
4852:→
4847:∗
4839:→
4827:
4821:→
4809:
4803:→
4789:, namely
4764:→
4759:∗
4751:→
4729:→
4704:→
4540:→
4518:→
4490:→
4460:→
4438:→
4354:→
4312:
4256:Relative
4188:
4170:
4110:
4104:
4086:
4045:
4027:
4004:
3977:
3933:) is the
3867:as a ring
3771:→
3729:
3656:∈
3650:−
3638:×
3632:∈
3553:Relative
3476:) is the
3411:dimension
3254:~
3150:→
3044:⋂
3017:~
2974:, making
2802:In 1976,
2757:-theory.
2568:
2555:≅
2377:, to the
2317:of which
2079:Jean Cerf
1870:), where
1832:-theory.
1739:-groups.
1474:− 1
1395:-theory.
1378:. Their
1306:John Tate
1278:⟩
1260:∖
1254:∈
1248::
1239:−
1230:⊗
1224:⟨
1214:×
1198:⊗
1192:×
876:-theory,
846:. Later
831:), where
812:), where
535:) to the
491:→
480:→
474:→
463:→
239:, and if
182:. For a
126:into the
11280:Topology
11182:Topology
11060:(1963),
11043:: 1â62,
11029:(1969),
10878:(1970),
10844:(1978),
10813:(1958),
10731:(1968),
10652:(1964),
10636:(1961),
10604:(2013),
10545:(1994),
10399:(1999),
10350:(2005),
10243:(1994),
10129:(1971),
9991:(2005),
9962:K-Theory
9877:(1999),
9809:(1968),
9732:(1985),
8751:-theory)
8667:See also
8505:positive
8268:Examples
8174:). When
8104:-modules
7678:″
7661:′
7551:Suppose
6417:⧠2 but
5620:≠
4671:â GL(1,
4636:. As E(
4224:). Then
3464:) = Pic(
3372:Examples
3210:-module
3174:-module
2957:-module
2592:for all
2239:)) â Wh(
1878:because
1627:spectrum
1476:, where
1348:for all
1109:defined
1053:−n
1048:-groups
978:Work in
925:, where
581: :
560: :
487:″
470:′
435:quotient
71:integers
43:topology
39:geometry
11434:2372133
11406:1970465
11378:1970382
11317:1106918
11234:2372978
11169:-theory
11124:Sur la
11117:-theory
11094:Bibcode
10974:0353337
10778:1970902
10737:-theory
10691:1970360
10583:1282290
10492:-theory
10421:1732049
10389:2181823
10281:1282290
10233:0335604
10212:0422392
10193:0338129
10143:0349811
10120:0260844
10092:Bibcode
10061:1761696
10015:2104929
9982:1268594
9891:1715873
9883:-theory
9864:2182598
9815:-theory
9772:0802796
9667:2031199
9534:0240214
9379:2372036
8742:-theory
8704:-theory
8696:-theory
8687:-theory
8497:unless
8465:) and K
8433:is the
8413: (
8185:, then
7833:of the
7730:motives
7263:of the
7074:) and (
7026:-theory
7022:Higher
6637:symbols
6496:making
6299:by the
6292:of the
6152:-theory
6148:Milnor
6079:in the
6047:Pairing
5180:of the
4640:) â
SL(
4609:to the
4212:is its
4140:is the
3937:of the
3517:-theory
2912:to the
2872:-groups
2315:spectra
2311:sheaves
2301:-theory
2276:⋅
2158:×
2134:. The
2075:isotopy
2008:×
1920:. Two
1713:of the
1701:. The
1689:, then
1480:is the
1331:-groups
1327:Higher
997:-theory
988:-theory
946:)) and
798:torsion
437:of the
433:) is a
286:History
160:, then
35:-theory
11432:
11404:
11376:
11315:
11305:
11232:
11075:: 5â93
10972:
10776:
10689:
10596:Errata
10589:
10581:
10571:
10512:
10502:
10476:
10456:
10446:
10419:
10387:
10377:
10339:
10329:
10307:
10294:Errata
10287:
10279:
10269:
10231:
10210:
10191:
10181:
10149:
10141:
10118:
10110:
10067:
10059:
10049:
10021:
10013:
10003:
9980:
9949:
9939:
9919:
9909:
9889:
9862:
9852:
9821:
9770:
9760:
9665:
9657:
9540:
9532:
9524:
9385:
9377:
9369:
9211:
8499:i=4k+1
8215:(R) â
8189:- and
8129:scheme
8094:). If
8047:being
7867:-group
7571:is an
7495:Since
7245:Here Ï
7046:) and
7004:Galois
6908:
6773:where
6343:
6028:
5894:where
5879:
5497:/2 if
5405:
5197:kernel
5178:center
5072:, the
5015:and SL
4886:, so
4570:
4381:
4156:, and
3889:λ-ring
3798:
3665:
3511:, the
2868:Lower
1995:, and
1709:, the
1685:is an
1404:ad hoc
747:, and
666:Atiyah
423:Klasse
324:. If
262:, the
59:groups
49:, and
11430:JSTOR
11402:JSTOR
11374:JSTOR
11348:, in
11230:JSTOR
11065:(PDF)
10774:JSTOR
10687:JSTOR
10355:(PDF)
9375:JSTOR
8757:Notes
8501:with
8178:is a
7836:nerve
7642:âČ to
7267:over
7255:, GL(
7251:is a
7102:of a
7000:Ă©tale
6995:map.
6411:) = 0
6083:with
6067:over
5911:and K
5695:with
5691:is a
5422:Gauss
5188:) of
5091:and S
4955:false
4949:) to
4923:When
4101:colim
4072:Here
3530:over
3438:, by
3409:, by
3385:field
3356:) â K
3301:is a
2884:be a
2467:with
2281:(the
1964:into
850:used
514:= +
407:class
328:is a
158:field
156:is a
11303:ISBN
10569:ISBN
10500:ISBN
10474:ISBN
10444:ISBN
10375:ISBN
10327:ISBN
10305:ISBN
10267:ISBN
10179:ISBN
10108:ISSN
10047:ISBN
10001:ISBN
9937:ISBN
9907:ISBN
9850:ISBN
9790:1999
9777:1999
9758:ISBN
9655:ISSN
9600:2005
9522:ISSN
9367:ISSN
9209:ISBN
8415:1993
8402:for
8390:) =
8339:) =
8307:If
8234:The
7536:The
7522:The
7082:) â
7002:(or
6413:for
6381:For
6075:and
6063:and
5493:) =
5470:) =
5123:Wang
5087:) â
5042:) â
4912:â SK
4908:) â
4874:= GL
4675:) â
4663:)/E(
4632:) â
4605:) â
4593:For
4266:The
4240:)/E(
3683:The
3563:Let
3468:) â
3445:For
3440:rank
3394:and
3390:are
3364:) =
3180:free
2942:) â
2886:ring
2725:and
2028:and
2014:for
1960:and
1952:and
1942:+ 1)
1932:are
1928:and
1902:and
1890:and
1719:+ =
1549:and
1150:and
1126:) â
968:and
915:) /
868:and
589:) â
547:and
130:and
11450:doi
11422:doi
11394:doi
11366:doi
11335:. I
11295:doi
11222:doi
11191:doi
11130:,
11102:doi
11045:doi
11014:doi
10990:267
10936:doi
10894:doi
10829:doi
10800:111
10766:doi
10716:doi
10679:doi
10616:AMS
10594:.
10587:Zbl
10561:doi
10510:Zbl
10454:Zbl
10409:doi
10367:doi
10337:Zbl
10292:.
10285:Zbl
10259:doi
10171:doi
10147:Zbl
10100:doi
10065:Zbl
10039:doi
10019:Zbl
9970:doi
9947:Zbl
9917:Zbl
9842:doi
9819:Zbl
9750:doi
9647:doi
9538:Zbl
9512:doi
9383:Zbl
9359:doi
8368:â„1,
8284:.)
8150:on
8108:BGL
7839:of
7497:BGL
7137:).
6594:in
5687:If
5459:).
5184:St(
4920:).
4884:(A)
4613:of
4309:ker
3906:).
3871:If
3726:ker
3480:of
3297:If
3290:of
3115:dim
3077:dim
2961:to
2842:THH
2837:THH
2313:of
2103:).
2077:.
2024:If
2019:â„ 5
1981:â„ 5
1972:'s
1707:BQC
1569:BGL
1536:BGL
1507:BGL
1494:BGL
1484:th
1458:BGL
1122:St(
1018:of
1011:to
906:is
842:of
791:or
664:.
551:of
539:of
348:on
340:on
207:of
87:on
11478::
11446:45
11444:,
11428:,
11418:72
11416:,
11400:,
11390:42
11388:,
11372:,
11362:81
11360:,
11313:MR
11311:,
11301:,
11283:31
11278:,
11265:18
11228:,
11218:84
11216:,
11187:13
11185:,
11179:,
11100:,
11088:,
11073:15
11071:,
11067:,
11039:,
11033:,
11010:28
11008:,
11004:,
10988:,
10970:MR
10968:,
10964:,
10932:17
10930:,
10926:,
10890:39
10888:,
10882:,
10825:86
10823:,
10817:,
10809:;
10772:,
10762:99
10760:,
10712:68
10710:,
10704:,
10685:,
10675:86
10673:,
10632:;
10610:,
10585:,
10579:MR
10577:,
10567:,
10559:,
10551:,
10508:,
10498:,
10472:,
10452:,
10442:,
10417:MR
10415:,
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10383:,
10373:,
10357:,
10335:,
10325:,
10283:,
10277:MR
10275:,
10265:,
10257:,
10249:,
10229:MR
10208:MR
10189:MR
10187:,
10177:,
10145:,
10139:MR
10137:,
10116:MR
10114:,
10106:,
10098:,
10086:,
10063:,
10057:MR
10055:,
10045:,
10037:,
10017:,
10011:MR
10009:,
9999:,
9978:MR
9976:,
9964:,
9945:,
9935:,
9915:,
9905:,
9887:MR
9873:;
9860:MR
9858:,
9848:,
9840:,
9768:MR
9766:,
9756:,
9736:,
9709:,
9663:MR
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9653:,
9643:98
9641:,
9536:,
9530:MR
9528:,
9520:,
9460:^
9403:^
9381:,
9373:,
9365:,
9355:72
9353:,
9329:^
9290:^
9207:,
9173:^
9161:^
8604:.
8552:+2
8540:+1
8516:+1
8394:/(
8379:â1
8304::
8204:â
8154:.
7436:GL
7376:.
7205:GL
7066:â
6545:.
6537:a
6212::=
5915::
5904:.
5428:.
5274:.
5216:St
5192:.
4910:A*
4872:A*
4855:1.
4824:GL
4806:SL
4690::
4669:A*
4634:A*
4607:A*
4597:a
4252:.
4185:GL
4167:GL
4107:GL
4083:GL
4042:GL
4024:GL
4001:GL
3992:ab
3974:GL
3941::
3582::
3549:.
3453:,
3449:a
3368:.
3333:â
3329:â
3313:=
3294:.
2927:â
2888:.
2835:â
2675:;
2559:CH
2474:,
2450:CH
2385:.
2250:))
2049:â
2021:.
1991:,
1983:,
1906:.
1691:QC
1611:.
1602:GL
1591:GL
1490:BU
1384:KV
1323:.
1165:,
935:GL
927:GL
909:GL
806:Wh
768:.
740:,
564:â
298:.
282:.
145:.
73:.
45:,
41:,
11452::
11424::
11396::
11368::
11344:K
11329:K
11297::
11274:K
11259:K
11224::
11193::
11167:K
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11136:1
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11096::
11090:9
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11041:2
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8648:n
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