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Basic Theorems In Algebraic K-theory
In mathematics, there are several theorems basic to algebraic ''K''-theory. Throughout, for simplicity, we assume when an exact category is a subcategory of another exact category, we mean it is strictly full subcategory (i.e., isomorphism-closed.) Theorems The localization theorem generalizes the localization theorem for abelian categories. Let C \subset D be exact categories. Then ''C'' is said to be cofinal in ''D'' if (i) it is closed under extension in ''D'' and if (ii) for each object ''M'' in ''D'' there is an ''N'' in ''D'' such that M \oplus N is in ''C''. The prototypical example is when ''C'' is the category of free modules and ''D'' is the category of projective module In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characterizati ...s. See also * Fundamental theor ... [...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] |
Exact Category
In mathematics, an exact category is a concept of category theory due to Daniel Quillen which is designed to encapsulate the properties of short exact sequences in abelian categories without requiring that morphisms actually possess kernels and cokernels, which is necessary for the usual definition of such a sequence. Definition An exact category E is an additive category possessing a class ''E'' of "short exact sequences": triples of objects connected by arrows : M' \to M \to M''\ satisfying the following axioms inspired by the properties of short exact sequences in an abelian category: * ''E'' is closed under isomorphisms and contains the canonical ("split exact") sequences: :: M' \to M' \oplus M''\to M''; * Suppose M \to M'' occurs as the second arrow of a sequence in ''E'' (it is an admissible epimorphism) and N \to M'' is any arrow in E. Then their pullback exists and its projection to N is also an admissible epimorphism. Dually, if M' \to M occurs as the first arrow of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Localization Theorem For Abelian Categories
Localization or localisation may refer to: Biology * Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence * Localization of sensation, ability to tell what part of the body is affected by touch or other sensation; see Allochiria * Neurologic localization, in neurology, the process of deducing the location of injury based on symptoms and neurological examination * Nuclear localization signal, an amino acid sequence on the surface of a protein which acts like a 'tag' to localize the protein in the cell * Sound localization, a listener's ability to identify the location or origin of a detected sound * Subcellular localization, organization of cellular components into different regions of a cell Engineering and technology * GSM localization, determining the location of an active cell phone or wireless transceiver * Robot localization, figuring out robot's position in an environment * Indoor positioning syste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cylinder Functor
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infinite curvilinear surface in various modern branches of geometry and topology. The shift in the basic meaning—solid versus surface (as in ball and sphere)—has created some ambiguity with terminology. The two concepts may be distinguished by referring to solid cylinders and cylindrical surfaces. In the literature the unadorned term cylinder could refer to either of these or to an even more specialized object, the ''right circular cylinder''. Types The definitions and results in this section are taken from the 1913 text ''Plane and Solid Geometry'' by George Wentworth and David Eugene Smith . A ' is a surface consisting of all the points on all the lines which are parallel to a given line and which pass through a fixed plane cur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Fibration
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy (, ; , ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. Formal definition Formally, a homotopy between two continuous functions ''f'' and ''g'' from a topological space ''X'' to a topological space ''Y'' is defined to be a continuous function H: X \times ,1\to Y from the product of the space ''X'' with the unit interval , 1to ''Y'' such that H(x,0) = f(x) and H(x,1) = g(x) for all x \in X. If we think of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dévissage
In algebraic geometry, dévissage is a technique introduced by Alexander Grothendieck for proving statements about coherent sheaves on noetherian schemes. Dévissage is an adaptation of a certain kind of noetherian induction. It has many applications, including the proof of generic flatness and the proof that higher direct images of coherent sheaves under proper morphisms are coherent. Laurent Gruson and Michel Raynaud extended this concept to the relative situation, that is, to the situation where the scheme under consideration is not necessarily noetherian, but instead admits a finitely presented morphism to another scheme. They did this by defining an object called a relative dévissage which is well-suited to certain kinds of inductive arguments. They used this technique to give a new criterion for a module to be flat. As a consequence, they were able to simplify and generalize the results of EGA IV 11 on descent of flatness. The word ''dévissage'' is French for ''unscrewing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quillen's Theorem A
In topology, a branch of mathematics, Quillen's Theorem A gives a sufficient condition for the classifying spaces of two categories to be homotopy equivalent. Quillen's Theorem B gives a sufficient condition for a square consisting of classifying spaces of categories to be homotopy Cartesian. The two theorems play central roles in Quillen's Q-construction In algebra, Quillen's Q-construction associates to an exact category (e.g., an abelian category) an algebraic K-theory. More precisely, given an exact category ''C'', the construction creates a topological space B^+C so that \pi_0 (B^+C) is the ... in algebraic K-theory and are named after Daniel Quillen. The precise statements of the theorems are as follows. In general, the homotopy fiber of Bf: BC \to BD is not naturally the classifying space of a category: there is no natural category Ff such that FBf = BFf. Theorem B constructs Ff in a case when f is especially nice. References * * * * Theorems in topology< ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cofinal Subcategory , a part of some Gregorian chants
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Cofinal may refer to: * Cofinal (mathematics) * Cofinality (mathematics) *Cofinal (music) A Gregorian mode (or church mode) is one of the eight systems of pitch organization used in Gregorian chant. History The name of Pope Gregory I was attached to the variety of chant that was to become the dominant variety in medieval western and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Module
In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist non-free modules. Given any set and ring , there is a free -module with basis , which is called the ''free module on'' or ''module of formal'' -''linear combinations'' of the elements of . A free abelian group is precisely a free module over the ring of integers. Definition For a ring R and an R-module M, the set E\subseteq M is a basis for M if: * E is a generating set for M; that is to say, every element of M is a finite sum of elements of E multiplied by coefficients in R; and * E is linearly independent, that is, for every subset \ of distinct elements of E, r_1 e_1 + r_2 e_2 + \cdots + r_n e_n = 0_M implies that r_1 = r_2 = \cdots = r_n = 0_R (where 0_M is the zero element of M and 0_R is t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Module
In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characterizations of these modules appear below. Every free module is a projective module, but the converse fails to hold over some rings, such as Dedekind rings that are not principal ideal domains. However, every projective module is a free module if the ring is a principal ideal domain such as the integers, or a polynomial ring (this is the Quillen–Suslin theorem). Projective modules were first introduced in 1956 in the influential book ''Homological Algebra'' by Henri Cartan and Samuel Eilenberg. Definitions Lifting property The usual category theoretical definition is in terms of the property of ''lifting'' that carries over from free to projective modules: a module ''P'' is projective if and only if for every surjective module homomor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Theorem Of Algebraic K-theory
In algebra, the fundamental theorem of algebraic ''K''-theory describes the effects of changing the ring of ''K''-groups from a ring ''R'' to R /math> or R , t^/math>. The theorem was first proved by Hyman Bass for K_0, K_1 and was later extended to higher ''K''-groups by Daniel Quillen. Description Let G_i(R) be the algebraic K-theory of the category of finitely generated modules over a noetherian ring ''R''; explicitly, we can take G_i(R) = \pi_i(B^+\text_R), where B^+ = \Omega BQ is given by Quillen's Q-construction. If ''R'' is a regular ring (i.e., has finite global dimension), then G_i(R) = K_i(R), the ''i''-th K-group of ''R''. This is an immediate consequence of the resolution theorem, which compares the K-theories of two different categories (with inclusion relation.) For a noetherian ring ''R'', the fundamental theorem states: *(i) G_i(R = G_i(R), \, i \ge 0. *(ii) G_i(R , t^ = G_i(R) \oplus G_(R), \, i \ge 0, \, G_(R) = 0. The proof of the theorem uses the Q-constr ... [...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] |
