Algebraic K-theory | Wikipedia audio article

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00:02:23 1 History
00:02:37 1.1 The Grothendieck group iK/isub0/sub
00:08:33 1.2 iK/isub0/sub, iK/isub1/sub, and iK/isub2/sub
00:14:37 1.3 Higher iK/i-groups
00:22:16 1.4 Applications of algebraic iK/i-theory in topology
00:26:59 1.5 Algebraic topology and algebraic geometry in algebraic iK/i-theory
00:34:45 2 Lower K-groups
00:35:04 2.1 iK/isub0/sub
00:37:26 2.1.1 Examples
00:37:54 2.1.2 Relative Ksub0/sub
00:39:16 2.1.3 iK/isub0/sub as a ring
00:40:57 2.2 iK/isub1/sub
00:41:30 2.2.1 Relative iK/isub1/sub
00:43:31 2.2.2 Commutative rings and fields
00:45:07 2.2.3 Central simple algebras
00:48:19 2.3 iK/isub2/sub
00:49:03 2.3.1 Matsumoto's theorem
00:50:47 2.3.2 Long exact sequences
00:51:10 2.3.3 Pairing
00:52:49 3 Milnor iK/i-theory
00:54:50 4 Higher iK/i-theory
00:55:34 4.1 The +-construction
00:56:46 4.2 The Q-construction
01:00:20 4.3 The S-construction
01:01:18 5 Examples
01:03:02 5.1 Algebraic K-groups of finite fields
01:03:09 5.2 Algebraic K-groups of rings of integers
01:05:23 6 Applications and open questions
01:06:01 7 See also
01:06:10 8 Notes
01:06:28 9 References
01:06:38 10 Further reading
01:06:56 10.1 Pedagogical references
01:07:33 10.2 Historical references
01:07:44 11 External links
01:08:31 Z for positive k.The torsion subgroups of K2i+1(Z), and the orders of the finite groups K4k+2(Z) have recently been determined, but whether the latter groups are cyclic, and whether the groups K4k(Z) vanish depends upon Vandiver's conjecture about the class groups of cyclotomic integers. See Quillen–Lichtenbaum conjecture for more details.
01:08:57 Applications and open questions
01:09:50 See also



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SUMMARY
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Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called K-groups. These are groups in the sense of abstract algebra. They contain detailed information about the original object but are notoriously difficult to compute; for example, an important outstanding problem is to compute the K-groups of the integers.
K-theory was invented in the late 1950s by Alexander Grothendieck in his study of intersection theory on algebraic varieties. In the modern language, Grothendieck defined only K0, the zeroth K-group, but even this single group has plenty of applications, such as the Grothendieck–Riemann–Roch theorem. Intersection theory is still a motivating force in the development of (higher) algebraic K-theory through its links with motivic cohomology and specifically Chow groups. The subject also includes classical number-theoretic topics like quadratic reciprocity and embeddings of number fields into the real numbers and complex numbers, as well as more modern concerns like the construction of higher regulators and special values of L-functions.
The lower K-groups were discovered first, in the sense that adequate descriptions of these groups in terms of other algebraic structures were found. For example, if F is a field, then K0(F) is isomorphic to the integers Z and is closely related to the notion of vector space dimension. For a commutative ring R, K0(R) is related to the Picard group of R, and when R is the ring of integers in a number field, this generalizes the classical construction of the class group. The group K1(R) is closely related to the group of units R×, and if R is a field, it is exactly the group of units. For a number field F, K2(F) is related to class field theory, the Hilbert symbol, and the solvability of quadratic equations over completions. In contrast, finding the correct definition of the higher K-groups of rings was a difficult achievement of Daniel Quillen, and many of the basic facts about the h ...