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href="http://dbpedia.org/resource/Algebraic_K-theory">Algebraic K-theory</a> </h1> </div> </div> <div class="row"> <div class="col"> <div class="text-muted"> An Entity of Type: <a href="http://dbpedia.org/ontology/Place">place</a>, from Named Graph: <a href="http://dbpedia.org">http://dbpedia.org</a>, within Data Space: <a href="http://dbpedia.org">dbpedia.org</a> </div> </div> </div> <div class="row pt-2"> <div class="col-xs-9 col-sm-10"> 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. </div> </div> </div> </section>   <section> <div class="container-xl"> <div class="row"> <div class="table-responsive"> <table class="table table-hover table-sm table-light"> <thead> <tr> <th class="col-xs-3 ">Property</th> <th class="col-xs-9 px-3">Value</th> </tr> </thead> <tbody> <tr class="odd"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/abstract">dbo:abstract</a> </td><td class="col-10 text-break"><ul> <li>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 discovered 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, the group 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, the group 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. (en)</li> <li style="display:none;">Das mathematische Teilgebiet der Algebraischen K-Theorie beschäftigt sich mit dem Studium von Ringen bzw. Vektorbündeln auf Schemata. sei stets ein unitärer Ring. Die algebraischen K-Gruppen sind eine Folge abelscher Gruppen , die dem Ring zugeordnet sein sollen und Informationen über diesen kodieren. Es gibt in der Mathematik verschiedene Arten von K-Theorien. Mit "algebraischer K-Theorie" ist in aller Regel die auf Quillen zurückgehende Definition gemeint. Milnors K-Theorie stimmt mit dieser im Allgemeinen nur für überein. Die Entwicklung der algebraischen K-Theorie wurde unter anderem von der topologischen K-Theorie motiviert, sie hängt aber nicht unmittelbar mit dieser zusammen. (de)</li> <li style="display:none;">En mathématiques, la K-théorie algébrique est une branche importante de l'algèbre homologique. Son objet est de définir et d'appliquer une suite de foncteurs Kn de la catégorie des anneaux dans celle des groupes abéliens. Pour des raisons historiques, K0 et K1 sont conçus en des termes un peu différents des Kn pour n ≥ 2. Ces deux K-groupes sont en effet plus accessibles et ont plus d'applications que ceux d'indices supérieurs. La théorie de ces derniers est bien plus profonde et ils sont beaucoup plus difficiles à calculer, ne serait-ce que pour l'anneau des entiers. Le groupe abélien K0(A) généralise la construction du groupe des classes d'idéaux d'un anneau A en utilisant les A-modules projectifs. Il a été développé dans les années 1960 et 1970 — au cours desquelles la « conjecture de Serre » sur les modules projectifs est devenue le théorème de Quillen-Suslin — et a été relié à beaucoup d'autres problèmes algébriques classiques. De même, le groupe K1(A) est une modification du groupe des unités, en utilisant les matrices élémentaires ; il est important en topologie, en particulier lorsque A est un anneau de groupe, parce qu'un groupe quotient, le (en), contient la (en), utilisée en théorie du type simple d'homotopie et de la chirurgie. Le groupe K0(A) contient aussi d'autres invariants, comme l'invariant de finitude[Quoi ?]. Depuis les années 1980, la K-théorie algébrique a eu de plus en plus d'applications en géométrie algébrique. Par exemple, la cohomologie motivique lui est intimement liée. (fr)</li> <li style="display:none;">数学では、代数的K-理論(algebraic K-theory)は、ある非負な整数 n に対して環からアーベル群への函手の系列を定義して適用することに関係したホモロジー代数の重要な一部である。歴史的理由により、 K0 と K1 は、n ≥ 2 に対する Kn とはいくらか異なった項と考えられている。実際、高次の群よりも低次の群は受け入れやすく、より多くの応用を持っている。高次の群の理論は、（ R が整数の環であるときでさえ）非常に深く、計算することが確かに困難である。群 K0(R) は、射影加群を使い、環のイデアル類群の構成を一般化したことになる。1960年代、1970年代の発展は、現在は(Quillen–Suslin theorem)となっている射影加群についてのジャン＝ピエール・セール(Jean-Pierre Serre)の予想を解こうとした努力に関係していた。キレン・サスリンの定理は、この分野で発見された古典的代数の他の問題に多く関連している。同じように、K1(R) は、行列の基本変形を使った環の可逆元の群の変形である。群 K1(R) はトポロジー、特に、R が群環のときに重要である。なぜなら、その商である(Whitehead group)が、単純ホモトピー論(simple homotopy theory)や(surgery theory)の理論における問題を研究するためのホワイトヘッドの捩れを含んでいるからである。群 K0(R) もたとえば有限性不変量のような他の不変量を含んでいる。1980年代以降、代数的K-理論は、ますます代数幾何学へ多くの応用が増加している。たとえば、モチーヴィックコホモロジー(motivic cohomology)は密接に代数的K-理論に関係している。 (ja)</li> <li style="display:none;">수학에서 대수적 K이론(代數的K理論, 영어: algebraic K-theory)은 환의 가군들을 다루는 K이론의 한 종류다. (ko)</li> <li style="display:none;">In de homologische algebra, een deelgebied van de wiskunde,is de algebraïsche K-theorie de studie van ringen respectievelijk vectorbundels op schema's. De algebraische K-groepen vormen een rij abelse groepen die behoren bij de unitaire ring en daarover informatie coderen. (nl)</li> <li style="display:none;">K-teoria algébrica é uma parte importante da álgebra homológica, preocupada com definição e aplicação de uma seqüência Kn(R) de funtores dos anéis para grupos abelianos, para todos inteiros n. A K-teoria é uma maneira sistemática de tentar lidar com invariantes abelianos da teoria das matrizes, chamando-se-lhe, por vezes, álgebra linear estável. A ideia é a de que, se tivermos duas matrizes grandes, a matriz A e a matriz B, que não comutam, elas passarão a comutar se colocadas em posições ortogonais em blocos diferentes. (pt)</li> <li style="display:none;">Inom matematiken är algebraisk K-teori ett viktigt delområde av homologisk algebra och rör definitionen och tillämpningar av en följd Kn(R) av funktorer från ringar till abelska grupper, för alla naturliga tal n. Av historiska skäl behandlar man ofta de lägre K-grupperna K0 och K1 något annorlunda än de högre K-grupperna Kn för n ≥ 2. De lägre grupperna är enklare att behandla och har hittills haft större tillämpning än de högre grupperna. De högre K-grupperna är också betydligt svårare att beräkna (till och med när R är ringen av heltal). Den nu använda definitionen av Quillen täcker dock samtliga K-grupper. Gruppen K0(R) generaliserar konstruktionen av av en ring genom att använda . Dess utveckling på 1960- och 1970-talet var relaterat till försök att bevisa en förmodan av Serre om projektiva moduler numera känd som ; flera andra samband med klassiska algebraiska problem upptäcktes under denna period. Analogt är K1(R) en modifiering av i en ring, genom att använda elementär matristeori. Gruppen K1(R) är viktig inom topologin, speciellt då R är en , eftersom dess kvot Whiteheadgruppen innehåller som används till att undersöka problem i enkel homotopiteori; gruppen K0(R) innehåller även andra invarianter såsom ändlighetsinvarianten. Sedan 1980-talet har algebraisk K-teori använts i ökande takt inom algebraisk geometri. Exempelvis är nära relaterad till algebraisk K-teori. 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class="odd"><td class="col-2"><a class="uri" href="http://www.w3.org/2000/01/rdf-schema#comment">rdfs:comment</a> </td><td class="col-10 text-break"><ul> <li style="display:none;">수학에서 대수적 K이론(代數的K理論, 영어: algebraic K-theory)은 환의 가군들을 다루는 K이론의 한 종류다. (ko)</li> <li style="display:none;">In de homologische algebra, een deelgebied van de wiskunde,is de algebraïsche K-theorie de studie van ringen respectievelijk vectorbundels op schema's. De algebraische K-groepen vormen een rij abelse groepen die behoren bij de unitaire ring en daarover informatie coderen. (nl)</li> <li style="display:none;">K-teoria algébrica é uma parte importante da álgebra homológica, preocupada com definição e aplicação de uma seqüência Kn(R) de funtores dos anéis para grupos abelianos, para todos inteiros n. A K-teoria é uma maneira sistemática de tentar lidar com invariantes abelianos da teoria das matrizes, chamando-se-lhe, por vezes, álgebra linear estável. A ideia é a de que, se tivermos duas matrizes grandes, a matriz A e a matriz B, que não comutam, elas passarão a comutar se colocadas em posições ortogonais em blocos diferentes. (pt)</li> <li>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. (en)</li> <li style="display:none;">Das mathematische Teilgebiet der Algebraischen K-Theorie beschäftigt sich mit dem Studium von Ringen bzw. Vektorbündeln auf Schemata. sei stets ein unitärer Ring. Die algebraischen K-Gruppen sind eine Folge abelscher Gruppen , die dem Ring zugeordnet sein sollen und Informationen über diesen kodieren. Es gibt in der Mathematik verschiedene Arten von K-Theorien. Mit "algebraischer K-Theorie" ist in aller Regel die auf Quillen zurückgehende Definition gemeint. Milnors K-Theorie stimmt mit dieser im Allgemeinen nur für überein. (de)</li> <li style="display:none;">En mathématiques, la K-théorie algébrique est une branche importante de l'algèbre homologique. Son objet est de définir et d'appliquer une suite de foncteurs Kn de la catégorie des anneaux dans celle des groupes abéliens. Pour des raisons historiques, K0 et K1 sont conçus en des termes un peu différents des Kn pour n ≥ 2. Ces deux K-groupes sont en effet plus accessibles et ont plus d'applications que ceux d'indices supérieurs. La théorie de ces derniers est bien plus profonde et ils sont beaucoup plus difficiles à calculer, ne serait-ce que pour l'anneau des entiers. (fr)</li> <li style="display:none;">数学では、代数的K-理論(algebraic K-theory)は、ある非負な整数 n に対して環からアーベル群への函手の系列を定義して適用することに関係したホモロジー代数の重要な一部である。歴史的理由により、 K0 と K1 は、n ≥ 2 に対する Kn とはいくらか異なった項と考えられている。実際、高次の群よりも低次の群は受け入れやすく、より多くの応用を持っている。高次の群の理論は、（ R が整数の環であるときでさえ）非常に深く、計算することが確かに困難である。 (ja)</li> <li style="display:none;">Inom matematiken är algebraisk K-teori ett viktigt delområde av homologisk algebra och rör definitionen och tillämpningar av en följd Kn(R) av funktorer från ringar till abelska grupper, för alla naturliga tal n. Av historiska skäl behandlar man ofta de lägre K-grupperna K0 och K1 något annorlunda än de högre K-grupperna Kn för n ≥ 2. De lägre grupperna är enklare att behandla och har hittills haft större tillämpning än de högre grupperna. De högre K-grupperna är också betydligt svårare att beräkna (till och med när R är ringen av heltal). Den nu använda definitionen av Quillen täcker dock samtliga K-grupper. 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