Matches in DBpedia 2015-10 for { <http://dbpedia.org/resource/Algebraic_K-theory> ?p ?o }
- Algebraic_K-theory abstract "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-theory of the integers.K-theory was discovered in the late 1950s by Alexander Grothendieck in his study of intersection theory on algebraic varieties. Intersection theory is still a motivating force in the development of 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 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 higher K-groups of algebraic varieties were not known until the work of Robert Thomason.".
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- Algebraic_K-theory wikiPageWikiLink Abelianization.
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- Algebraic_K-theory wikiPageWikiLink American_Mathematical_Society.
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- Algebraic_K-theory wikiPageWikiLink Atiyah–Hirzebruch_spectral_sequence.
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- Algebraic_K-theory wikiPageWikiLink Category:Algebraic_K-theory.
- Algebraic_K-theory wikiPageWikiLink Category:Algebraic_geometry.
- Algebraic_K-theory wikiPageWikiLink Cell_complex.
- Algebraic_K-theory wikiPageWikiLink Center_(group_theory).
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- Algebraic_K-theory wikiPageWikiLink Class_field_theory.
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- Algebraic_K-theory wikiPageWikiLink Exterior_algebra.
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- Algebraic_K-theory wikiPageWikiLink Extraordinary_cohomology_theory.
- Algebraic_K-theory wikiPageWikiLink Fibration.
- Algebraic_K-theory wikiPageWikiLink Field_(mathematics).
- Algebraic_K-theory wikiPageWikiLink Field_of_fractions.
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- Algebraic_K-theory wikiPageWikiLink Geometric_realisation.
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- Algebraic_K-theory wikiPageWikiLink Geometry.
- Algebraic_K-theory wikiPageWikiLink Global_dimension.