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 is said to be -graded for an index set if it has a gradation or grading, i.e. a decomposition into a direct sum of structures; the elements of are said to be "homogeneous of degree i ". The index set is most commonly or , and may be required to have extra structure depending on the type of . Grading by (i.e. ) is also important; see e.g. signed set (the -graded sets). The trivial (- or -) gradation has for and a suitable trivial structure . 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 -graded vector space or graded linear space is thus a vector space with a decomposition into a direct sum 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 additive abelian groups such that , with taken from some monoid, usually or , or semigroup (for a ring without identity). The associated graded ring of a commutative ring with respect to a proper ideal is . A graded module is left module over a graded ring that is a direct sum of modules satisfying . The associated graded module of an -module with respect to a proper ideal is . A differential graded module, differential graded -module or DG-module is a graded module with a differential making a chain complex, i.e. . A graded algebra is an algebra over a ring that is graded as a ring; if is graded we also require . The graded Leibniz rule for a map on a graded algebra specifies that . A differential graded algebra, DG-algebra or DGAlgebra is a graded algebra that is a differential graded module whose differential obeys the graded Leibniz rule. A homogeneous derivation on a graded algebra A is a homogeneous linear map of grade d = |D| on A such that acting on homogeneous elements of A.