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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] |
Algebraic K-theory
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] |
Suspension (topology)
In topology, a branch of mathematics, the suspension of a topological space ''X'' is intuitively obtained by stretching ''X'' into a cylinder and then collapsing both end faces to points. One views ''X'' as "suspended" between these end points. The suspension of ''X'' is denoted by ''SX'' or susp(''X''). There is a variation of the suspension for pointed space, which is called the reduced suspension and denoted by Σ''X''. The "usual" suspension ''SX'' is sometimes called the unreduced suspension, unbased suspension, or free suspension of ''X'', to distinguish it from Σ''X.'' Free suspension The (free) suspension SX of a topological space X can be defined in several ways. 1. SX is the quotient space (X \times ,1/(X\times \, X\times \). In other words, it can be constructed as follows: * Construct the cylinder X \times ,1/math>. * Consider the entire set X\times \ as a single point ("glue" all its points together). * Consider the entire set X\times \ as a single point ("g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |