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K-theory Spectrum
In mathematics, given a ring ''R'', the ''K''-theory spectrum of ''R'' is an Ω-spectrum K_R whose ''n''th term is given by, writing \Sigma R for the suspension Suspension or suspended may refer to: Science and engineering * Suspension (topology), in mathematics * Suspension (dynamical systems), in mathematics * Suspension of a ring, in mathematics * Suspension (chemistry), small solid particles suspend ... of ''R'', :(K_R)_n = K_0(\Sigma^n R) \times BGL(\Sigma^n R)^+, where "+" means the Quillen's + construction.Dominique Arlettaz, Algebraic ''K''-theory of rings from a topological view poin/ref> By definition, K_i(R) = \pi_i(K_R). References Algebraic K-theory {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring (mathematics)
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ''ring'' is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. Formally, a ''ring'' is an abelian group whose operation is called ''addition'', with a second binary operation called ''multiplication'' that is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors use the term " " with a missing i to refer to the more general structure that omits this last requirement; see .) Whether a ring is commutative (that is, whether the order in which two elements are multiplied might change the result) has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ω-spectrum
In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. Every such cohomology theory is representable, as follows from Brown's representability theorem. This means that, given a cohomology theory\mathcal^*:\text^ \to \text,there exist spaces E^k such that evaluating the cohomology theory in degree k on a space X is equivalent to computing the homotopy classes of maps to the space E^k, that is\mathcal^k(X) \cong \left , E^k\right/math>.Note there are several different categories of spectra leading to many technical difficulties, but they all determine the same homotopy category, known as the stable homotopy category. This is one of the key points for introducing spectra because they form a natural home for stable homotopy theory. The definition of a spectrum There are many variations of the definition: in general, a ''spectrum'' is any sequence X_n of pointed topological spaces or pointed simplicial sets together with the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Suspension Of A Ring
In algebra, more specifically in algebraic K-theory, the suspension \Sigma R of a ring ''R'' is given byWeibel, III, Ex. 1.15 \Sigma(R) = C(R)/M(R) where C(R) is the ring of all infinite matrices with coefficients in ''R'' having only finitely many nonzero elements in each row or column and M(R) is its ideal of matrices having only finitely many nonzero elements. It is an analog of suspension Suspension or suspended may refer to: Science and engineering * Suspension (topology), in mathematics * Suspension (dynamical systems), in mathematics * Suspension of a ring, in mathematics * Suspension (chemistry), small solid particles suspend ... in topology. One then has: K_i(R) \simeq K_(\Sigma R). References * C. WeibelThe K-book: An introduction to algebraic K-theory Algebra {{algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
+ Construction
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 ''K''0, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
