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Associated Graded Ring
In mathematics, the associated graded ring of a ring ''R'' with respect to a proper ideal ''I'' is the graded ring: :\operatorname_I R = \oplus_^\infty I^n/I^. Similarly, if ''M'' is a left ''R''-module, then the associated graded module is the graded module over \operatorname_I R: :\operatorname_I M = \oplus_^\infty I^n M/ I^ M. Basic definitions and properties For a ring ''R'' and ideal ''I'', multiplication in \operatorname_IR is defined as follows: First, consider homogeneous elements a \in I^i/I^ and b \in I^j/I^ and suppose a' \in I^i is a representative of ''a'' and b' \in I^j is a representative of ''b''. Then define ab to be the equivalence class of a'b' in I^/I^. Note that this is well-defined modulo I^. Multiplication of inhomogeneous elements is defined by using the distributive property. A ring or module may be related to its associated graded ring or module through the initial form map. Let ''M'' be an ''R''-module and ''I'' an ideal of ''R''. Given f \in M, the in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and mathematical analysis, analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of mathematical object, abstract objects and the use of pure reason to proof (mathematics), prove them. These objects consist of either abstraction (mathematics), abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of inference rule, deductive rules to already established results. These results include previously proved theorems, axioms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Enveloping Algebra
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal enveloping algebras are used in the representation theory of Lie groups and Lie algebras. For example, Verma modules can be constructed as quotients of the universal enveloping algebra. In addition, the enveloping algebra gives a precise definition for the Casimir operators. Because Casimir operators commute with all elements of a Lie algebra, they can be used to classify representations. The precise definition also allows the importation of Casimir operators into other areas of mathematics, specifically, those that have a differential algebra. They also play a central role in some recent developments in mathematics. In particular, their dual provides a commutative example of the objects studied in non-commutative geometry, the quantum groups. This dual can be shown, by the Gelfa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rees Algebra
In commutative algebra, the Rees algebra of an ideal ''I'' in a commutative ring ''R'' is defined to be R t\bigoplus_^ I^n t^n\subseteq R The extended Rees algebra of ''I'' (which some authors refer to as the Rees algebra of ''I'') is defined asR t,t^\bigoplus_^I^nt^n\subseteq R ,t^This construction has special interest in algebraic geometry since the projective scheme defined by the Rees algebra of an ideal in a ring is the blowing-up of the spectrum of the ring along the subscheme defined by the ideal.Eisenbud-Harris, ''The geometry of schemes''. Springer-Verlag, 197, 2000 Properties * Assume ''R'' is Noetherian; then ''R t' is also Noetherian. The Krull dimension of the Rees algebra is \dim R t\dim R+1 if ''I'' is not contained in any prime ideal ''P'' with \dim(R/P)=\dim R; otherwise \dim R t\dim R. The Krull dimension of the extended Rees algebra is \dim R t, t^\dim R+1. * If J\subseteq I are ideals in a Noetherian ring ''R'', then the ring extension R tsubseteq R t/math> ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graded (mathematics)
In mathematics, the term “graded” has a number of meanings, mostly related: In abstract algebra, it refers to a family of concepts: * An algebraic structure X is said to be I-graded for an index set I if it has a gradation or grading, i.e. a decomposition into a direct sum X = \bigoplus_ X_i of structures; the elements of X_i are said to be "homogeneous of degree ''i'' ". ** The index set I is most commonly \N or \Z, and may be required to have extra structure depending on the type of X. ** Grading by \Z_2 (i.e. \Z/2\Z) is also important; see e.g. signed set (the \Z_2-graded sets). ** The trivial (\Z- or \N-) gradation has X_0 = X, X_i = 0 for i \neq 0 and a suitable trivial structure 0. ** An algebraic structure is said to be doubly graded if the index set is a direct product of sets; the pairs may be called "bidegrees" (e.g. see Spectral sequence). * A I-graded vector space or graded linear space is thus a vector space with a decomposition into a direct sum V = \b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Filtered Ring
In mathematics, a filtered algebra is a generalization of the notion of a graded algebra. Examples appear in many branches of mathematics, especially in homological algebra and representation theory. A filtered algebra over the field k is an algebra (A,\cdot) over k that has an increasing sequence \ \subseteq F_0 \subseteq F_1 \subseteq \cdots \subseteq F_i \subseteq \cdots \subseteq A of subspaces of A such that :A=\bigcup_ F_ and that is compatible with the multiplication in the following sense: : \forall m,n \in \mathbb,\quad F_m\cdot F_n\subseteq F_. Associated graded algebra In general there is the following construction that produces a graded algebra out of a filtered algebra. If A is a filtered algebra then the ''associated graded algebra'' \mathcal(A) is defined as follows: The multiplication is well-defined and endows \mathcal(A) with the structure of a graded algebra, with gradation \_. Furthermore if A is associative then so is \mathcal(A). Also if A is un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Filtration (mathematics)
In mathematics, a filtration \mathcal is an indexed family (S_i)_ of subobjects of a given algebraic structure S, with the index i running over some totally ordered index set I, subject to the condition that ::if i\leq j in I, then S_i\subseteq S_j. If the index i is the time parameter of some stochastic process, then the filtration can be interpreted as representing all historical but not future information available about the stochastic process, with the algebraic structure S_i gaining in complexity with time. Hence, a process that is adapted to a filtration \mathcal is also called non-anticipating, because it cannot "see into the future". Sometimes, as in a filtered algebra, there is instead the requirement that the S_i be subalgebras with respect to some operations (say, vector addition), but not with respect to other operations (say, multiplication) that satisfy only S_i \cdot S_j \subseteq S_, where the index set is the natural numbers; this is by analogy with a graded ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exterior Algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues. The exterior product of two vectors u and v, denoted by u \wedge v, is called a bivector and lives in a space called the ''exterior square'', a vector space that is distinct from the original space of vectors. The magnitude of u \wedge v can be interpreted as the area of the parallelogram with sides u and v, which in three dimensions can also be computed using the cross product of the two vectors. More generally, all parallel plane surfaces with the same orientation and area have the same bivector as a measure of their oriented area. Like the cross product, the exterior product is anticommutative, meaning ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degeneration (algebraic Geometry)
In algebraic geometry, a degeneration (or specialization) is the act of taking a limit of a family of varieties. Precisely, given a morphism :\pi: \mathcal \to C, of a variety (or a scheme) to a curve ''C'' with origin 0 (e.g., affine or projective line), the fibers :\pi^(t) form a family of varieties over ''C''. Then the fiber \pi^(0) may be thought of as the limit of \pi^(t) as t \to 0. One then says the family \pi^(t), t \ne 0 ''degenerates'' to the ''special'' fiber \pi^(0). The limiting process behaves nicely when \pi is a flat morphism and, in that case, the degeneration is called a flat degeneration. Many authors assume degenerations to be flat. When the family \pi^(t) is trivial away from a special fiber; i.e., \pi^(t) is independent of t \ne 0 up to (coherent) isomorphisms, \pi^(t), t \ne 0 is called a general fiber. Degenerations of curves In the study of moduli of curves, the important point is to understand the boundaries of the moduli, which amounts to understand deg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clifford Algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing. They are named after the English mathematician William Kingdon Clifford. The most familiar Clifford algebras, the orthogonal Clifford algebras, are also referred to as (''pseudo-'')''Riemannian Clifford algebras'', as distinct from ''symplectic Clifford algebras''.see for ex. Introduction and basic properties A Clifford algebra is a unital associative algebra that contains and is generated by a vector space over a field , where is equipped with a qua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coordinate Ring
In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime ideal. If the condition of generating a prime ideal is removed, such a set is called an (affine) algebraic set. A Zariski open subvariety of an affine variety is called a quasi-affine variety. Some texts do not require a prime ideal, and call ''irreducible'' an algebraic variety defined by a prime ideal. This article refers to zero-loci of not necessarily prime ideals as affine algebraic sets. In some contexts, it is useful to distinguish the field in which the coefficients are considered, from the algebraically closed field (containing ) over which the zero-locus is considered (that is, the points of the affine variety are in ). In this case, the variety is said ''defined over'' , and the points of the variety that belong to are said ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré–Birkhoff–Witt Theorem
In mathematics, more specifically in the theory of Lie algebras, the Poincaré–Birkhoff–Witt theorem (or PBW theorem) is a result giving an explicit description of the universal enveloping algebra of a Lie algebra. It is named after Henri Poincaré, Garrett Birkhoff, and Ernst Witt. The terms ''PBW type theorem'' and ''PBW theorem'' may also refer to various analogues of the original theorem, comparing a filtered algebra to its associated graded algebra, in particular in the area of quantum groups. Statement of the theorem Recall that any vector space ''V'' over a field has a basis; this is a set ''S'' such that any element of ''V'' is a unique (finite) linear combination of elements of ''S''. In the formulation of Poincaré–Birkhoff–Witt theorem we consider bases of which the elements are totally ordered by some relation which we denote ≤. If ''L'' is a Lie algebra over a field K, let ''h'' denote the canonical K-linear map from ''L'' into the universal envelo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Law
In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self-dual condition, ;Modular law: implies where are arbitrary elements in the lattice, ≤ is the partial order, and ∨ and ∧ (called join and meet respectively) are the operations of the lattice. This phrasing emphasizes an interpretation in terms of projection onto the sublattice , a fact known as the diamond isomorphism theorem. An alternative but equivalent condition stated as an equation (see below) emphasizes that modular lattices form a variety in the sense of universal algebra. Modular lattices arise naturally in algebra and in many other areas of mathematics. In these scenarios, modularity is an abstraction of the 2nd Isomorphism Theorem. For example, the subspaces of a vector space (and more generally the submodules of a module over a ring) form a modular lattice. In a not necessarily modular lattice, there may sti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
