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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, 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 classi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exact Category
In mathematics, specifically in category theory, an exact category is a category equipped with short exact sequences. The concept is due to Daniel Quillen and 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphism-closed Subcategory
In category theory, a branch of mathematics, a subcategory \mathcal of a category \mathcal is said to be isomorphism closed or replete if every \mathcal-isomorphism h:A\to B with A\in\mathcal belongs to \mathcal. https://www.cs.cornell.edu/courses/cs6117/2018sp/Lectures/Subcategories.pdf This implies that both B and h^:B\to A belong to \mathcal as well. A subcategory that is isomorphism closed and full is called strictly full. In the case of full subcategories it is sufficient to check that every \mathcal-object that is isomorphic to an \mathcal-object is also an \mathcal-object. This condition is very natural. For example, in the category of topological spaces one usually studies properties that are invariant under homeomorphisms—so-called topological properties In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is 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 system, a netw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cylinder Functor
A cylinder () 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 a solid ball versus sphere surface)—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 A. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
