en>EdwardGoldobin: many small improvements throughout the text

2014-02-02T23:47:54Z

many small improvements throughout the text

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{{disambiguation}}
In [[mathematics]], the term “'''graded'''” has a number of related meanings:
* An [[algebraic structure]] <math>X</math> 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 <math>X = \oplus_{i \in I} X_i</math> of structures; the elements of <math>X_i</math> are said to be “homogenous of degree ''i''”.
** The index set I is most commonly <math>\mathbb{N}</math> or <math>\mathbb{Z}</math>, and may be required to have extra structure depending on the type of <math>X</math>.
** The '''trivial''' (<math>\mathbb{Z}</math>- or <math>\mathbb{N}</math>-) gradation has <math>X_0 = X</math>, <math>X_i = 0</math> for <math>i ≠ 0</math> and a suitable trivial structure <math>0</math>.
** An algebraic structure is said to be [[doubly graded]] if the index set is a direct product; the pairs may be called as “bidegrees” (e.g. see [[spectral sequence]]).
* A I-[[graded vector space]] or '''graded linear space''' for a set I is thus a vector space with a decomposition into a direct sum <math>V = \oplus_{i \in I} V_i</math> of spaces.
** A [[graded linear map]] is a map between graded vector spaces respecting their gradations.
* A [[graded ring]] is a ring that is a direct sum of abelian groups <math>R_i</math> such that <math>R_i R_j \subseteq R_{i+j}</math>, with <math>i</math> taken from some monoid, usually <math>\mathbb{N}</math> or <math>\mathbb{Z}</math>, or semigroup (for a ring without identity).
** The [[associated graded ring]] of a commutative ring <math>R</math> with respect to a proper ideal <math>I</math> is <math>\operatorname{gr}_I R = \oplus_{n \in \mathbb{N}} I^n/I^{n+1}</math>.
* A [[graded module]] is left module <math>M</math> over a graded ring which is a direct sum <math>\oplus_{i \in I} M_i</math> of modules satisfying <math>R_i M_j \subseteq M_{i+j}</math>.
** The [[associated graded module]] of an <math>R</math>-module <math>M</math> with respect to a proper ideal <math>I</math> is <math>\operatorname{gr}_I M = \oplus_{n \in \mathbb{N}} I^n M/ I^{n+1} M</math>.
** A [[differential graded module]], '''differential graded <math>\mathbb{Z}</math>-module''' or '''DG-module''' is a graded module <math>M</math> with a '''differential''' <math>d\colon M \to M \colon M_i \to M_{i+1}</math> making <math>M</math> a '''chain complex''', i.e. <math>d \circ d=0</math> .
* A [[graded algebra]] is an algebra <math>A</math> over a ring <math>R</math> that is graded as a ring; if <math>R</math> is graded we also require <math>A_iR_j \subseteq A_{i+j} ⊇ R_iA_j</math>.
** The [[graded Leibniz rule]] for a map <math>d\colon A \to A</math> on a graded algebra <math>A</math> specifies that <math>d(a \cdot b) = (da) \cdot b + (-1)^{|a|}a \cdot (db)</math> .
** A [[differential graded algebra]], '''DG-algebra''' or '''DGAlgebra''' is a graded algebra which is a differential graded module whose differential obeys the graded Leibniz rule.
** A '''DGA''' is an augmented DG-algebra, or '''[[differential graded augmented algebra]]''', (see [[differential graded algebra]]).
** A [[superalgebra]] is a a '''Z'''<sub>2</sub>-graded algebra.
*** A [[graded-commutative]] superalgebra satisfies the “supercommutative” law <math>yx = (-1)^{|x| |y|}xy.\,</math> for homogenous ''x'',''y'', where <math>|a|</math> represents the “parity” of <math>a</math>, i.e. 0 or 1 depending on the component it lies in.
** '''CDGA''' may refer to the category of augmented differential graded commutative algebras.
* A [[graded Lie algebra]] is a Lie algebra which is graded as a vector space by a gradation compatible with its Lie bracket.
** A [[graded Lie superalgebra]] is a graded Lie algebra with the requirement for anticommutativity of its Lie bracket relaxed.
** A [[supergraded Lie superalgebra]] is a graded Lie superalgebra with an additional super '''Z'''/2'''Z'''-gradation.
** A [[Differential graded Lie algebra]] is a graded vector space over a field of characteristic zero together with a bilinear map <math>[,]: L_i \otimes L_j \to L_{i+j}</math> and a differential <math>d: L_i \to L_{i-1}</math> satisfying <math>[x,y] = (-1)^{|x||y|+1}[y,x],</math> for any homogeneous elements ''x'', ''y'' in ''L'', the “graded Jacobi identity” and the graded Leibniz rule.
* The '''Graded Brauer group''' is a synonym for the [[Brauer–Wall group]] <math>BW(F)</math> classifying finite-dimensional graded central division algebras over the field ''F''.
* An <math>\mathcal{A}</math>-[[graded category]] for a category <math>\mathcal{A}</math> is a category <math>\mathcal{C}</math> together with a functor <math>F:\mathcal{C} \rightarrow \mathcal{A}</math>.
** A [[differential graded category]] or ''' DG category''' is a category whose morphism sets form differential graded <math>\mathbb{Z}</math>-modules.
* [[Graded manifold]] – extension of the manifold concept based on ideas coming from supersymmetry and supercommutative algebra, including sections on
** [[Graded manifold#Graded functions|Graded function]]
** [[Graded manifold#Graded vector fields|Graded vector fields]]
** [[Graded manifold#Graded exterior forms|Graded exterior forms]]
** [[Graded manifold#Graded differential geometry|Graded differential geometry]]
** [[Graded manifold#Graded differential calculus|Graded differential calculus]]
* [[Graded derivation]] , ''but see also the section [[Derivation (abstract algebra)#Graded derivations|Graded derivations]] in [[Derivation (abstract algebra)]]''
* [[Functionally graded element]]s are elements used in finite element analysis.
* A [[graded poset]] is a poset ''P'' with a '''rank function''' <math>ρ\colon P \to N</math> compatible with the ordering (so ρ(''x'')<ρ(''y'') ⇐ ''x'' < ''y'') such that ''y'' covers ''x'' ⇒ <math>\rho(y)=\rho(x)+1</math> .

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en>EdwardGoldobin: many small improvements throughout the text