Algebraic K-theory

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 Γ-objects could be used to define the 356:

is projective, then these subspaces are finite dimensional. The Riemann–Roch 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: 11351:, in Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 1, pp. 35–60, 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 Bass–Schanuel 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 Riemann–Roch 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 Grothendieck–Riemann–Roch theorem specializes to Hirzebruch's theorem.

6989: 6633: 6535: 11143: 11045: 9503: 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 11217: 10825: 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. 11262: 11342: 11338: 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}},} 10710: 2717:-theory, étale cohomology is highly computable, so étale Chern classes provided an effective tool for detecting the existence of elements in 10891: 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: 10301:

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 11320: 10574: 10505: 10479: 10449: 10380: 10332: 10310: 10272: 10184: 10052: 10006: 9942: 9912: 9855: 9763: 9707: 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 Grothendieck–Riemann–Roch 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 10544: 10242: 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 10887:"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. 10848: 10613: 8280:-groups have proved particularly difficult to compute except in a few isolated but interesting cases. (See also: 2398: 677: 11303:

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 Γ-objects. Segal's approach is a homotopy analog of Grothendieck's construction of 10643: 10617: 10554: 10406: 10252: 9998: 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. 11504: 11499: 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 10471: 10226: 10136: 9904: 6080: 5181: 5161: 4687: 3213: 3184: 2764:-theory on singular varieties still lacked adequate foundations. While it was believed that Quillen's 11257:, ́Colloq. Theorie des Groupes Algebriques, Gauthier-Villars, Paris, 1962, pp. 113–127. (French) 10165:

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

11183:, Proc. Intern. Congress Math., Vancouver, 1974, vol. I, Canad. Math. Soc., 1975, pp. 171–176. 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: 10878:, Algebraic K-theory I, Lecture Notes in Math., vol. 341, Springer-Verlag, 1973, pp. 266–292. 10436:

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. 10470:, 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 étale 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

10761: 10672: 7830: 7015: 3439: 1320: 884: 779:(roughly "main conjecture"). The fact that triangulations were stable under subdivision led 279: 119: 10206:

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

11330: 11104: 10982: 10659: 10635: 10584: 10422: 10390: 10282: 10234: 10213: 10194: 10144: 10121: 10093: 10062: 10016: 9983: 9892: 9865: 9773: 9731: 9667: 9534: 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 10592: 10515: 10459: 10342: 10290: 10152: 10070: 10024: 9952: 9922: 9824: 9542: 9386: 8: 11218:

The Obstruction to Finding a Boundary for an Open Manifold of Dimension Greater than Five

11072: 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 Γ-objects, the 1626: 157: 84: 58: 11108: 10097: 7842: 7708: 7598: 6999: 2710: 2612: 445:, and so it is an abelian group. If the basis element corresponding to a vector bundle 11445: 11416: 11387: 11293: 11242: 11193: 10778: 10689: 10670:; Murthy, M.P. (1967). "Grothendieck groups and Picard groups of abelian group rings". 10496:, Modern Birkhäuser Classics (Paperback reprint of the 1996 2nd ed.), Boston, MA: 9631: 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: 10597: 10353: 10295: 4159: 11316: 11207: 11188: 10948: 10924: 10816: 10570: 10501: 10475: 10445: 10376: 10328: 10306: 10268: 10180: 10109: 10048: 10002: 9938: 9908: 9851: 9759: 9655: 9522: 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: 10724: 10705: 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: 11466: 11437: 11408: 11379: 11308: 11234: 11202: 11112: 11054: 11036: 11022: 10943: 10900: 10886: 10834: 10770: 10719: 10681: 10588: 10562: 10511: 10455: 10410: 10368: 10338: 10324: 10286: 10260: 10172: 10148: 10101: 10066: 10040: 10020: 9971: 9948: 9918: 9872: 9843: 9831: 9820: 9777:. See also Lecture IV and the references in (Friedlander & Weibel 9751: 9647: 9538: 9512: 9498: 9486: 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: 11326: 11312: 10978: 10607: 10580: 10558: 10548: 10538: 10523: 10441: 10418: 10386: 10364: 10278: 10256: 10246: 10230: 10209: 10190: 10168: 10140: 10117: 10058: 10036: 10012: 9979: 9934: 9888: 9861: 9839: 9769: 9747: 9663: 9530: 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: 9501:(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 11470: 11307:, Progr. Math., vol. 88, Boston, MA: Birkhäuser Boston, pp. 247–435, 11027: 11010: 10631: 10603: 10414: 10398: 10349: 10221:

Quillen, Daniel (1974), "Higher K-theory for categories with exact sequences",

10201: 10160: 9876: 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: 11399:

Whitehead, J.H.C. (1941). "On incidence matrices, nuclei and homotopy types".

10866:, Lecture Notes in Mathematics, vol. 657, Springer–Verlag, pp. 40–84 10566: 10264: 10044: 9735: 9651: 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 11493: 10964: 10749: 10651: 10223:

New developments in topology (Proc. Sympos. Algebraic Topology, Oxford, 1972)

10113: 9990: 9659: 9526: 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: 10969: 10497: 10372: 9819:, 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 11172:-theory I, Lecture Notes in Math., vol. 341, Springer Verlag, 1973, 85–147. 10812: 9959: 9708:"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 9847: 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 11068: 11006: 10128: 10077: 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: 11073:"Differential topology from the point of view of simple homotopy theory" 11059: 11040: 9517: 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 Riemann–Roch theorem was generalized by 11449: 11420: 11391: 11246: 11116: 10904: 10839: 10820: 10782: 10733: 10701: 10693: 10667: 10176: 10105: 9975: 9903:, Cambridge Studies in Advanced Mathematics, vol. 101, Cambridge: 9808: 9755: 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 11484: 11041:"Sur les sous-groupes aritmetiques des groupes semi-simples deployes" 10882: 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

11457:

Whitehead, J.H.C. (1939). "Simplicial spaces, nuclei and m-groups".

11441: 11412: 11383: 11238: 11157:, Proc. ICM Nice 1970, vol. 2, Gauthier-Villars, Paris, 1971, 47–52. 10774: 10685: 10440:, Lecture Notes in Mathematics, vol. 1491, Berlin, Heidelberg: 10354:"Algebraic K-theory of rings of integers in local and global fields" 9640:

Institut des Hautes Études Scientifiques. Publications Mathématiques

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: 9746:, 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 11370:

Wall, C. T. C. (1965). "Finiteness conditions for CW-complexes".

10967:; 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, étale 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 10135:, 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

11266:, Proc. Sympos. Pure Math., vol. XVII, 1970, pp. 88–123. 10656:

On the Structure and Classification of Differential Manifolds

10208:, Montreal, Quebec: Canad. Math. Congress, pp. 171–176, 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 11364:

Algebraic and geometric topology (New Brunswick, N.J., 1983)

11255:

Generateurs, relations et revetements de groupes algebriques

10435: 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 étale 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 10405:, 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 11366:, Lecture Notes in Mathematics, vol. 1126 (1985), 318–419. 10167:, 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 10323:, 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

10303:

Beilinson's Conjectures on Special Values of L-Functions

11281:, 4e serie (1985), 437–552; erratum 22 (1989), 675–677. 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. 10225:, 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 "étale 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 Bass–Schanuel

1185: 887:

of the space. All such vector bundles come from the

458: 37:

is a subject area in mathematics with connections to

10995:

Comptes Rendus de l'Académie des Sciences, Série A-B

9962:(1993), "The K-theory of finite fields, revisited", 9887:, World Sci. Publ., River Edge, NJ, pp. 1–119, 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 11428:Whitehead, J.H.C. (1950). "Simple homotopy types". 11144:

Annales Scientifiques de l'École Normale Supérieure

11046:

Annales Scientifiques de l'École Normale Supérieure

9504:

Annales Scientifiques de l'École Normale Supérieure

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 11225:Smale, S (1962). "On the structure of manifolds". 9830: 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: 11005: 10609:The K-book: an introduction to Algebraic K-theory 10525:The K-book: An introduction to algebraic K-theory 10403:The development of algebraic K-theory before 1980 10238:(relation of Q-construction to plus-construction) 9997:, Graduate Studies in Mathematics, vol. 67, 9871: 9791: 9778: 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. 11491: 11302: 11286:Le principe de sciendage et l'inexistence d'une 10958:Classes de fasiceaux et theoreme de Riemann–Roch 10630: 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 11011:"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 10963: 10035:, Springer Monographs in Mathematics, Berlin: 9933:, 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 10826:Bulletin de la Société Mathématique de France 10711:Bulletin of the American Mathematical Society 9901:Central simple algebras and Galois cohomology 9898: 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 10700: 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 étale topology called étale 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 11263:Nonabelian homological algebra and K-theory 10989:Karoubi, Max (1968). "Foncteurs derives et 10706:"The homotopy theory of projective modules" 10030: 9995:Introduction to Quadratic Forms over Fields 9931: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 10811: 10666: 10033:Reciprocity laws. From Euler to Eisenstein 9730: 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 11456: 11427: 11398: 11206: 11058: 11035: 11026: 10947: 10838: 10796:Bokstedt, M., Hsiang, W. C., Madsen, I., 10723: 10642:, Proc. Sympos. Pure Math., vol. 3, 10543: 10532: 10318: 10241: 9899:Gille, Philippe; Szamuely, Tamás (2006), 9630: 9516: 9497: 9464: 9462: 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,} 10624: 10487: 10163:(1973), "Higher algebraic K-theory. I", 9407: 9405: 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 11305:The Grothendieck Festschrift Volume III 10988: 10922: 10876:-theory as generalized sheaf cohomology 10847: 10550:Algebraic K-theory and its applications 10248:Algebraic K-theory and its applications 10220: 10200: 10159: 9958: 9790:(Friedlander & Weibel 9673: 9432: 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: 11492: 11221:, Thesis, Princeton University (1965). 11090: 10650: 10602: 10539:Higher Algebraic K-Theory: an overview 10465: 10438:Higher algebraic K-theory: an overview 10397: 10348: 10300: 10127: 10076: 9691: 9682: 9600: 9566: 9557: 9554:Rosenberg (1994) Theorem 4.3.15, p.214 9548: 9471: 9459: 9441: 9423: 9414: 9393: 9230: 8617:Another fundamental conjecture due to 8098:is the category of finitely generated 7701:and the second arrow is an admissible 7646:″ 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 11224: 11186: 11067: 10960:, mimeographed notes, Princeton 1957. 10748: 10640:Vector bundles and homogeneous spaces 10468:An algebraic introduction to K-theory 10204:(1975), "Higher algebraic K-theory", 9615: 9606: 9575: 9450: 9402: 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 11369: 11189:"Categories and cohomology theories" 11080:Publications Mathématiques de l'IHÉS 10892:Publications Mathématiques de l'IHÉS 10881: 10732: 9928: 9807: 9584: 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 10798:The cyclotomic trace and algebraic 10557:, vol. 147, Berlin, New York: 10255:, vol. 147, Berlin, New York: 9989: 8778:Grothendieck 1957, Borel–Serre 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:Atiyah–Hirzebruch spectral sequence 24: 10521: 10429: 10321:Introduction to algebraic K-theory 10133:Introduction to algebraic K-theory 9258:Amer. J. Math., 72 (1950) pp. 1–57 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: 11516: 11478: 9634:(2003), "Motivic cohomology with 9621:Gille & Szamuely (2006) p.108 9612: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:Grothendieck–Riemann–Roch theorem 104:Grothendieck–Riemann–Roch theorem 9834:; 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 Γ-objects, the 1488:acting on the classifying space 1202: 373:: The Euler characteristic of a 320:proved what is now known as the 10993:-theorie. Categories filtres". 10917:-theory and Hochschild homology 10791:Topological Hochschild homology 10725:10.1090/s0002-9904-1962-10826-x 10614:Graduate Studies in Mathematics 9784: 9724: 9700: 9624: 9593: 9491: 9480: 9248: 9192: 9156:Dundas–Goodwillie–McCarthy 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:Brown–Gersten spectral sequence 1441:The first definition of higher 1438:-theory for a general variety. 678:extraordinary cohomology theory 367:Hirzebruch–Riemann–Roch theorem 110:-theory through its links with 11277:, Ann. Scient. Ec. Norm. Sup. 11131:, Princeton Univ. Press, 1971. 11095:-theory and Quadratic Forms". 10919:, unpublished preprint (1976). 10305:, Boston, MA: Academic Press, 10084:-theory and quadratic forms", 9736:"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:Quillen–Lichtenbaum 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: 11345:-theory of topological spaces 11091:Milnor, J (1970). "Algebraic 10821:"Le theoreme de Riemann–Roch" 10644:American Mathematical Society 10555:Graduate Texts in Mathematics 10407:American Mathematical Society 10253:Graduate Texts in Mathematics 9999:American Mathematical Society 9801: 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 11313:10.1007/978-0-8176-4576-2_10 11275:-theory and étale cohomology 11208:10.1016/0040-9383(74)90022-6 11134:Nobile, A., Villamayor, O., 10949:10.1016/0021-8693(71)90030-5 10793:. Preprint, Bielefeld, 1986. 9679:Rosenberg (1994) pp. 245–246 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: 10319:Silvester, John R. (1981), 10031:Lemmermeyer, Franz (2000), 9147:Bokstedt–Hsiang–Madsen 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: 11521: 11290:-theorie de Milnor globale 11125:Introduction to Algebraic 11028:10.7146/math.scand.a-11024 10857:-theory with coefficients 10472:Cambridge University Press 10227:Cambridge University Press 10217:(Quillen's Q-construction) 10137:Princeton University Press 9905: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, étale 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: 11485:K theory preprint archive 11009:; Villamayor, O. (1971). 10956:Grothendieck, Alexander, 10567:10.1007/978-1-4612-4314-4 10466:Magurn, Bruce A. (2009), 10265:10.1007/978-1-4612-4314-4 10045:10.1007/978-3-662-12893-0 9881:An overview of algebraic 9652:10.1007/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:Eilenberg–Steenrod axioms 11471:10.1112/plms/s2-45.1.243 10870:Brown, K., Gersten, S., 10086:Inventiones Mathematicae 9468: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 11299:, no. 3, 1992, 571–588. 11150:, no. 3, 1968, 581–616. 10704:; Schanuel, S. (1962). 10373:10.1007/3-540-27855-9_5 9960:Jardine, John Frederick 9411:Lemmermeyer (2000) p.66 8895:Karoubi–Villamayor 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 11459:Proc. London Math. Soc 11187:Segal, Graeme (1974). 10759:of algebraic cycles". 10533:Pedagogical references 10415:10.1090/conm/243/03695 9929:Gras, Georges (2003), 9697:Rosenberg (1994) p.289 9688:Rosenberg (1994) p.246 9572:Rosenberg (1994) p.200 9477:Silvester (1981) p.228 9102:Dwyer–Friedlander 1982 8922:Nesterenko–Suslin 1990 8886:Nobile–Villamayor 1968 8787:Atiyah–Hirzebruch 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 11401:Annals of Mathematics 11372:Annals of Mathematics 10762:Annals of Mathematics 10673:Annals of Mathematics 10636:Hirzebruch, Friedrich 10625:Historical references 10488:Srinivas, V. (2008), 9848:10.1007/3-540-27855-9 9732: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 11155:Cohomology of groups 10808:(3) (1993), 465–539. 10660:Cambridge University 10409:, pp. 211–238, 10367:, pp. 139–190, 10363:, Berlin, New York: 10361:Handbook of K-theory 10129:Milnor, John Willard 10078:Milnor, John Willard 9873:Friedlander, Eric M. 9838:, Berlin, New York: 9836:Handbook of K-Theory 9750:, pp. 318–419, 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:Riemann–Roch theorem 57:-groups. These are 11215:Siebenmann, Larry, 11140:-theorie algebrique 11109:1970InMat...9..318M 11060:10.24033/asens.1174 10923:Gersten, S (1971). 10545:Rosenberg, Jonathan 10243:Rosenberg, Jonathan 10229:, pp. 95–103, 10171:, pp. 85–147, 10098:1970InMat...9..318M 10080:(1970), "Algebraic 9632:Voevodsky, Vladimir 9563:Milnor (1971) p.123 9518:10.24033/asens.1174 9447:Milnor (1971) p.175 9429:Milnor (1971) p.102 9420: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 11505:Algebraic geometry 11500:Algebraic K-theory 11117:10.1007/bf01425486 10905:10.1007/BF02684687 10840:10.24033/bsmf.1500 10817:Serre, Jean-Pierre 10632:Atiyah, Michael F. 10177:10.1007/BFb0067053 10106:10.1007/BF01425486 9976:10.1007/BF00961219 9877:Weibel, Charles W. 9756:10.1007/BFb0074449 9638:/2-coefficients", 9590:Milnor (1971) p.69 9581:Milnor (1971) p.63 9456:Milnor (1971) p.81 9254:J.H.C. Whitehead, 9245:Milnor (1971) p.15 9189:Milnor (1971) p.14 9048:Brown–Gersten 1973 8805:Bass–Schanuel 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 11360:-theory of spaces 11322:978-0-8176-3487-2 11284:Thomason, R. W., 11269:Thomason, R. W., 11177:Higher algebraic 11175:Quillen, Daniel, 11162:Higher algebraic 11160:Quillen, Daniel, 11153:Quillen, Daniel, 11037:Matsumoto, Hideya 10913:Higher algebraic 10804:. Invent. Math., 10802:-theory of spaces 10616:, vol. 145, 10576:978-0-387-94248-3 10507:978-0-8176-4736-0 10481:978-0-521-10658-0 10451:978-3-540-55007-5 10382:978-3-540-23019-9 10334:978-0-412-22700-4 10312:978-0-12-581120-0 10274:978-0-387-94248-3 10186:978-3-540-06434-3 10054:978-3-540-66957-9 10008:978-0-8218-1095-8 9944:978-3-540-44133-5 9914:978-0-521-86103-8 9857:978-3-540-30436-4 9832:Friedlander, Eric 9765:978-3-540-15235-4 9744:-theory of spaces 9507:, 4 (in French), 9499:Matsumoto, Hideya 9438: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 11512: 11474: 11453: 11424: 11407:(5): 1197–1239. 11395: 11354:Waldhausen, F., 11336:Waldhausen, F., 11333: 11250: 11212: 11210: 11120: 11087: 11077: 11064: 11062: 11032: 11030: 11002: 10985: 10953: 10951: 10925:"On the functor 10908: 10867: 10849:Browder, William 10844: 10842: 10786: 10745: 10729: 10727: 10697: 10663: 10647: 10620: 10595: 10528: 10518: 10484: 10462: 10425: 10394:(survey article) 10393: 10358: 10345: 10325:Chapman and Hall 10315: 10293: 10237: 10216: 10197: 10156:(lower K-groups) 10155: 10124: 10073: 10027: 9986: 9955: 9925: 9895: 9868: 9827: 9795: 9788: 9782: 9776: 9728: 9722: 9721: 9719: 9718: 9704: 9698: 9695: 9689: 9686: 9680: 9677: 9671: 9670: 9628: 9622: 9619: 9613: 9610: 9604: 9603:), cf. Lemma 1.8 9597: 9591: 9588: 9582: 9579: 9573: 9570: 9564: 9561: 9555: 9552: 9546: 9545: 9520: 9495: 9489: 9487:Hideya Matsumoto 9484: 9478: 9475: 9469: 9466: 9457: 9454: 9448: 9445: 9439: 9436: 9430: 9427: 9421: 9418: 9412: 9409: 9400: 9399:Lam (2005) p.139 9397: 9391: 9390: 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:Bass–Murthy 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:étale cohomology 2693: 2682: 2653: 2638: 2630: 2613:étale 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: 11520: 11519: 11515: 11514: 11513: 11511: 11510: 11509: 11490: 11489: 11481: 11442:10.2307/2372133 11413:10.2307/1970465 11384:10.2307/1970382 11323: 11260:Swan, Richard, 11253:Steinberg, R., 11239:10.2307/2372978 11075: 10931: 10911:Dennis, R. K., 10775:10.2307/1970902 10758: 10686:10.2307/1970360 10646:, pp. 7–38 10627: 10604:Weibel, Charles 10577: 10559:Springer-Verlag 10535: 10508: 10482: 10452: 10442:Springer-Verlag 10432: 10430:Further reading 10399:Weibel, Charles 10383: 10365:Springer-Verlag 10356: 10350:Weibel, Charles 10335: 10313: 10275: 10257:Springer-Verlag 10202:Quillen, Daniel 10187: 10169:Springer-Verlag 10161:Quillen, Daniel 10055: 10037:Springer-Verlag 10009: 9945: 9935:Springer-Verlag 9915: 9858: 9840:Springer-Verlag 9804: 9799: 9798: 9789: 9785: 9766: 9748:Springer-Verlag 9729: 9725: 9716: 9714: 9706: 9705: 9701: 9696: 9692: 9687: 9683: 9678: 9674: 9629: 9625: 9620: 9616: 9611: 9607: 9598: 9594: 9589: 9585: 9580: 9576: 9571: 9567: 9562: 9558: 9553: 9549: 9496: 9492: 9485: 9481: 9476: 9472: 9467: 9460: 9455: 9451: 9446: 9442: 9437: 9433: 9428: 9424: 9419: 9415: 9410: 9403: 9398: 9394: 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: ≥ 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: 11518: 11508: 11507: 11502: 11488: 11487: 11480: 11479:External links 11477: 11476: 11475: 11454: 11425: 11396: 11367: 11352: 11334: 11321: 11300: 11282: 11267: 11258: 11251: 11233:(3): 387–399. 11222: 11213: 11201:(3): 293–312. 11184: 11173: 11158: 11151: 11132: 11121: 11103:(4): 318–344. 11088: 11065: 11033: 11003: 10986: 10965:Hatcher, Allen 10961: 10954: 10942:(2): 212–237. 10929: 10920: 10909: 10879: 10868: 10845: 10809: 10794: 10789:Bokstedt, M., 10787: 10769:(2): 349–379. 10756: 10750:Bloch, Spencer 10746: 10730: 10718:(4): 425–428. 10698: 10664: 10652:Barden, Dennis 10648: 10626: 10623: 10622: 10621: 10600: 10575: 10541: 10534: 10531: 10530: 10529: 10519: 10506: 10485: 10480: 10463: 10450: 10431: 10428: 10427: 10426: 10395: 10381: 10346: 10333: 10316: 10311: 10298: 10273: 10239: 10218: 10198: 10185: 10157: 10125: 10092:(4): 318–344, 10074: 10053: 10028: 10007: 9991:Lam, Tsit-Yuen 9987: 9970:(6): 579–595, 9956: 9943: 9926: 9913: 9896: 9869: 9856: 9828: 9803: 9800: 9797: 9796: 9783: 9764: 9723: 9699: 9690: 9681: 9672: 9623: 9614: 9605: 9592: 9583: 9574: 9565: 9556: 9547: 9490: 9479: 9470: 9458: 9449: 9440: 9431: 9422: 9413: 9401: 9392: 9357:(2): 323–334. 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: 7437: 7434: 7431: 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:) → 7117:) → 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: 11517: 11506: 11503: 11501: 11498: 11497: 11495: 11486: 11483: 11482: 11472: 11468: 11464: 11460: 11455: 11451: 11447: 11443: 11439: 11435: 11431: 11430:Amer. J. Math 11426: 11422: 11418: 11414: 11410: 11406: 11402: 11397: 11393: 11389: 11385: 11381: 11377: 11373: 11368: 11365: 11361: 11357: 11353: 11350: 11347: 11346: 11341: 11340: 11335: 11332: 11328: 11324: 11318: 11314: 11310: 11306: 11301: 11298: 11295: 11291: 11287: 11283: 11280: 11276: 11272: 11268: 11265: 11264: 11259: 11256: 11252: 11248: 11244: 11240: 11236: 11232: 11228: 11227:Amer. J. Math 11223: 11220: 11219: 11214: 11209: 11204: 11200: 11196: 11195: 11190: 11185: 11182: 11178: 11174: 11171: 11167: 11163: 11159: 11156: 11152: 11149: 11145: 11141: 11137: 11133: 11130: 11126: 11122: 11118: 11114: 11110: 11106: 11102: 11098: 11094: 11089: 11085: 11081: 11074: 11070: 11066: 11061: 11056: 11052: 11048: 11047: 11042: 11038: 11034: 11029: 11024: 11020: 11016: 11012: 11008: 11004: 11000: 10996: 10992: 10987: 10984: 10980: 10976: 10972: 10971: 10966: 10962: 10959: 10955: 10950: 10945: 10941: 10937: 10933: 10928: 10921: 10918: 10914: 10910: 10906: 10902: 10898: 10894: 10893: 10888: 10884: 10880: 10877: 10873: 10869: 10865: 10862: 10860: 10854: 10850: 10846: 10841: 10836: 10832: 10828: 10827: 10822: 10818: 10814: 10813:Borel, Armand 10810: 10807: 10803: 10799: 10795: 10792: 10788: 10784: 10780: 10776: 10772: 10768: 10764: 10763: 10755: 10751: 10747: 10743: 10739: 10735: 10731: 10726: 10721: 10717: 10713: 10712: 10707: 10703: 10699: 10695: 10691: 10687: 10683: 10679: 10675: 10674: 10669: 10665: 10661: 10657: 10653: 10649: 10645: 10641: 10637: 10633: 10629: 10628: 10619: 10615: 10611: 10610: 10605: 10601: 10599: 10594: 10590: 10586: 10582: 10578: 10572: 10568: 10564: 10560: 10556: 10552: 10551: 10546: 10542: 10540: 10537: 10536: 10527: 10526: 10520: 10517: 10513: 10509: 10503: 10499: 10495: 10491: 10486: 10483: 10477: 10473: 10469: 10464: 10461: 10457: 10453: 10447: 10443: 10439: 10434: 10433: 10424: 10420: 10416: 10412: 10408: 10404: 10400: 10396: 10392: 10388: 10384: 10378: 10374: 10370: 10366: 10362: 10355: 10351: 10347: 10344: 10340: 10336: 10330: 10326: 10322: 10317: 10314: 10308: 10304: 10299: 10297: 10292: 10288: 10284: 10280: 10276: 10270: 10266: 10262: 10258: 10254: 10250: 10249: 10244: 10240: 10236: 10232: 10228: 10224: 10219: 10215: 10211: 10207: 10203: 10199: 10196: 10192: 10188: 10182: 10178: 10174: 10170: 10166: 10162: 10158: 10154: 10150: 10146: 10142: 10138: 10134: 10130: 10126: 10123: 10119: 10115: 10111: 10107: 10103: 10099: 10095: 10091: 10087: 10083: 10079: 10075: 10072: 10068: 10064: 10060: 10056: 10050: 10046: 10042: 10038: 10034: 10029: 10026: 10022: 10018: 10014: 10010: 10004: 10000: 9996: 9992: 9988: 9985: 9981: 9977: 9973: 9969: 9965: 9961: 9957: 9954: 9950: 9946: 9940: 9936: 9932: 9927: 9924: 9920: 9916: 9910: 9906: 9902: 9897: 9894: 9890: 9886: 9882: 9878: 9874: 9870: 9867: 9863: 9859: 9853: 9849: 9845: 9841: 9837: 9833: 9829: 9826: 9822: 9818: 9814: 9810: 9806: 9805: 9794:), Lecture VI 9793: 9787: 9780: 9775: 9771: 9767: 9761: 9757: 9753: 9749: 9745: 9741: 9737: 9733: 9727: 9713: 9709: 9703: 9694: 9685: 9676: 9669: 9665: 9661: 9657: 9653: 9649: 9646:(1): 59–104, 9645: 9641: 9637: 9633: 9627: 9618: 9609: 9602: 9599:(Weibel 9596: 9587: 9578: 9569: 9560: 9551: 9544: 9540: 9536: 9532: 9528: 9524: 9519: 9514: 9510: 9506: 9505: 9500: 9494: 9488: 9483: 9474: 9465: 9463: 9453: 9444: 9435: 9426: 9417: 9408: 9406: 9396: 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: 11462: 11458: 11433: 11429: 11404: 11400: 11378:(1): 56–69. 11375: 11371: 11363: 11359: 11355: 11348: 11344: 11337: 11304: 11296: 11289: 11285: 11278: 11274: 11270: 11261: 11254: 11230: 11226: 11216: 11198: 11192: 11180: 11176: 11169: 11168:, Algebraic 11165: 11161: 11154: 11147: 11146:, 4e serie, 11139: 11135: 11128: 11124: 11123:Milnor, J., 11100: 11097:Invent. Math 11096: 11092: 11083: 11079: 11069:Mazur, Barry 11050: 11044: 11018: 11014: 11007:Karoubi, Max 11001:: A328–A331. 10998: 10994: 10990: 10974: 10968: 10957: 10939: 10935: 10926: 10916: 10912: 10896: 10890: 10875: 10871: 10864: 10858: 10856: 10852: 10830: 10824: 10805: 10801: 10797: 10790: 10766: 10760: 10753: 10741: 10737: 10715: 10709: 10680:(1): 16–73. 10677: 10671: 10655: 10639: 10608: 10549: 10524: 10522:Weibel, C., 10493: 10489: 10467: 10437: 10402: 10360: 10320: 10302: 10247: 10222: 10205: 10164: 10132: 10089: 10085: 10081: 10032: 9994: 9967: 9963: 9930: 9900: 9884: 9880: 9835: 9816: 9812: 9786: 9743: 9739: 9726: 9715:. Retrieved 9712:MathOverflow 9711: 9702: 9693: 9684: 9675: 9643: 9639: 9635: 9626: 9617: 9608: 9595: 9586: 9577: 9568: 9559: 9550: 9508: 9502: 9493: 9482: 9473: 9452: 9443: 9434: 9425: 9416: 9395: 9354: 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: 11465:: 243–327. 11436:(1): 1–57. 11021:: 265–307. 11015:Math. Scand 10744:. Benjamin. 10734:Bass, Hyman 10702:Bass, Hyman 10668:Bass, Hyman 9809:Bass, Hyman 9511:(2): 1–62, 9351:Am. J. Math 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 11494:Categories 11356:Algebraic 11339:Algebraic 11271:Algebraic 10970:Astérisque 10936:J. Algebra 10883:Cerf, Jean 10872:Algebraic 10853:Algebraic 10833:: 97–136. 10738:Algebraic 10658:(Thesis). 10593:0801.19001 10516:1125.19300 10498:Birkhäuser 10490:Algebraic 10460:0746.19001 10343:0468.18006 10291:0801.19001 10153:0237.18005 10071:0949.11002 10025:1068.11023 9953:1019.11032 9923:1137.12001 9825:0174.30302 9813:Algebraic 9802:References 9740:Algebraic 9717:2021-03-26 9543:0261.20025 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 11166:-theory I 10899:: 5–173. 10752:(1974). " 10114:0020-9910 9660:0073-8301 9527:0012-9593 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 11294:Topology 11194:Topology 11071:(1963). 11053:: 1–62. 11039:(1969). 10885:(1970). 10851:(1978), 10819:(1958). 10736:(1968). 10654:(1964). 10638:(1961), 10606:(2013), 10547:(1994), 10401:(1999), 10352:(2005), 10245:(1994), 10131:(1971), 9993:(2005), 9964:K-Theory 9879:(1999), 9811:(1968), 9734:(1985), 8751:-theory) 8667:See also 8505:positive 8268:Examples 8174:). When 8104:-modules 7678:″ 7661:′ 7551:Suppose 6417:≧ 2 but 5620:≠ 4671:→ GL(1, 4636:. As E( 4224:). Then 3464:) = Pic( 3372:Examples 3210:-module 3174:-module 2957:-module 2592:for all 2239:)) → 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 11450:2372133 11421:1970465 11392:1970382 11331:1106918 11247:2372978 11181:-theory 11136:Sur la 11129:-theory 11105:Bibcode 11086:: 5–93. 10983:0353337 10783:1970902 10742:-theory 10694:1970360 10585:1282290 10494:-theory 10423:1732049 10391:2181823 10283:1282290 10235:0335604 10214:0422392 10195:0338129 10145:0349811 10122:0260844 10094:Bibcode 10063:1761696 10017:2104929 9984:1268594 9893:1715873 9885:-theory 9866:2182598 9817:-theory 9774:0802796 9668:2031199 9535:0240214 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:) ◅ 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 11448: 11419: 11390: 11329: 11319: 11245: 10981: 10781: 10692: 10598:Errata 10591: 10583: 10573: 10514: 10504: 10478: 10458: 10448: 10421: 10389: 10379: 10341: 10331: 10309: 10296:Errata 10289: 10281: 10271: 10233: 10212: 10193: 10183: 10151: 10143: 10120: 10112: 10069: 10061: 10051: 10023: 10015: 10005: 9982: 9951: 9941: 9921: 9911: 9891: 9864: 9854: 9823: 9772: 9762: 9666: 9658: 9541: 9533: 9525: 9385: 9377: 9369: 9211: 8499:i=4k+1 8215:(R) → 8189:- and 8129:scheme 8094:). If 8047:being 7867:-group 7571:is an 7495:Since 7245:Here π 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 11446:JSTOR 11417:JSTOR 11388:JSTOR 11362:, in 11243:JSTOR 11076:(PDF) 10779:JSTOR 10690:JSTOR 10357:(PDF) 9375:JSTOR 8757:Notes 8501:with 8178:is a 7836:nerve 7642:′ to 7267:over 7255:, GL( 7251:is a 7102:of a 7000:étale 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:) → 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 11317:ISBN 10571:ISBN 10502:ISBN 10476:ISBN 10446:ISBN 10377:ISBN 10329:ISBN 10307:ISBN 10269:ISBN 10181:ISBN 10110:ISSN 10049:ISBN 10003:ISBN 9939:ISBN 9909:ISBN 9852:ISBN 9792:1999 9779:1999 9760:ISBN 9656:ISSN 9601:2005 9523:ISSN 9367:ISSN 9209:ISBN 8415:1993 8402:for 8390:) = 8339:) = 8307:If 8234:The 7536:The 7522:The 7082:) ⇒ 7002:(or 6413:for 6381:For 6075:and 6063:and 5493:) = 5470:) = 5123:Wang 5087:) → 5042:) → 4912:⊕ SK 4908:) ≅ 4874:= GL 4675:) → 4663:)/E( 4632:) → 4605:) → 4593:For 4266:The 4240:)/E( 3683:The 3563:Let 3468:) ⊕ 3445:For 3440:rank 3394:and 3390:are 3364:) = 3180:free 2942:) → 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:) → 968:and 915:) / 868:and 589:) → 547:and 130:and 11467:doi 11438:doi 11409:doi 11380:doi 11349:. I 11309:doi 11235:doi 11203:doi 11142:, 11113:doi 11055:doi 11023:doi 10999:267 10944:doi 10901:doi 10835:doi 10806:111 10771:doi 10720:doi 10682:doi 10618:AMS 10596:. 10589:Zbl 10563:doi 10512:Zbl 10456:Zbl 10411:doi 10369:doi 10339:Zbl 10294:. 10287:Zbl 10261:doi 10173:doi 10149:Zbl 10102:doi 10067:Zbl 10041:doi 10021:Zbl 9972:doi 9949:Zbl 9919:Zbl 9844:doi 9821:Zbl 9752:doi 9648:doi 9539:Zbl 9513:doi 9383:Zbl 9359:doi 8368:≥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:≥ 5 1981:≥ 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 11496:: 11463:45 11461:. 11444:. 11434:72 11432:. 11415:. 11405:42 11403:. 11386:. 11376:81 11374:. 11327:MR 11325:, 11315:, 11297:31 11292:, 11279:18 11241:. 11231:84 11229:. 11199:13 11197:. 11191:. 11111:. 11099:. 11084:15 11082:. 11078:. 11049:. 11043:. 11019:28 11017:. 11013:. 10997:. 10979:MR 10977:, 10973:, 10940:17 10938:. 10934:. 10897:39 10895:. 10889:. 10831:86 10829:. 10823:. 10815:; 10777:. 10767:99 10765:. 10716:68 10714:. 10708:. 10688:. 10678:86 10676:. 10634:; 10612:, 10587:, 10581:MR 10579:, 10569:, 10561:, 10553:, 10510:, 10500:, 10474:, 10454:, 10444:, 10419:MR 10417:, 10387:MR 10385:, 10375:, 10359:, 10337:, 10327:, 10285:, 10279:MR 10277:, 10267:, 10259:, 10251:, 10231:MR 10210:MR 10191:MR 10189:, 10179:, 10147:, 10141:MR 10139:, 10118:MR 10116:, 10108:, 10100:, 10088:, 10065:, 10059:MR 10057:, 10047:, 10039:, 10019:, 10013:MR 10011:, 10001:, 9980:MR 9978:, 9966:, 9947:, 9937:, 9917:, 9907:, 9889:MR 9875:; 9862:MR 9860:, 9850:, 9842:, 9770:MR 9768:, 9758:, 9738:, 9710:. 9664:MR 9662:, 9654:, 9644:98 9642:, 9537:, 9531:MR 9529:, 9521:, 9461:^ 9404:^ 9381:. 9373:. 9365:. 9355:72 9353:. 9329:^ 9290:^ 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Knowledge

Algebraic K-theory

Index