differential graded module, or dg-module, is a Z {\displaystyle \mathbb {Z} } -graded module together with a differential; i.e., a square-zero graded...

gradation or grading. A graded module is defined similarly (see below for the precise definition). It generalizes graded vector spaces. A graded module that is...

}I^{n}M/I^{n+1}M} . A differential graded module, differential graded Z {\displaystyle \mathbb {Z} } -module or DG-module is a graded module M {\displaystyle...

the module Ω S / R {\displaystyle \Omega _{S/R}} of differentials in different, but equivalent ways. An R-linear derivation on S is an R-module homomorphism...

derivative and the interior product acting on differential forms. Graded derivations of superalgebras (i.e. Z2-graded algebras) are often called superderivations...

after finitely many steps. Graded A graded module is a module with a decomposition as a direct sum M = ⨁x Mx over a graded ring R = ⨁x Rx such that RxMy...

In mathematics, a D-module is a module over a ring D of differential operators. The major interest of such D-modules is as an approach to the theory of...

resolution of the differential graded algebra ( B ∙ , 0 ) {\displaystyle (B_{\bullet },0)} where B ∙ {\displaystyle B_{\bullet }} is the graded algebra with...

are endowed with the additional structure of a differential graded Z {\displaystyle \mathbb {Z} } -module. In detail, this means that Hom ⁡ ( A , B ) {\displaystyle...

Clifford algebra Cluster algebra Dendriform algebra Differential graded algebra Differential graded Lie algebra Exterior algebra F-algebra Filtered algebra...

Buchsbaum–Eisenbud acyclicity criterion Differential graded module "Graph complex". Bott, Raoul; Tu, Loring W. (1982), Differential Forms in Algebraic Topology,...

^{\bullet }{\mathfrak {g}}\otimes C^{\infty }(M)} considered as a differential graded module over the Lie algebra g {\displaystyle {\mathfrak {g}}} : K ∙ ...

groups of differential fields Differentially closed field Differential graded algebra – Algebraic structure in homological algebra D-module – Module over a...

defines an OX-module (called an associated sheaf) in a natural way. Similarly, if R is a graded ring and X is the Proj of R, then any graded module defines...

and is in fact a differential. In this setting, k {\displaystyle k} is viewed as a trivial g {\displaystyle {\mathfrak {g}}} -module while k ∼ Λ 0 g ∗...

naturally to the setting of graded commutative algebra, allowing for a natural foundation of calculus on supermanifolds, graded manifolds and associated...

geometry, graded manifolds are extensions of the concept of manifolds based on ideas coming from supersymmetry and supercommutative algebra. Both graded manifolds...

{\mathcal {J}}_{k,n}}} spans the space of differential k-forms in a manifold M of dimension n, when viewed as a module over the ring C∞(M) of smooth functions...

graded rings, differential graded rings). Algebras over the so-called commutative operad are commutative algebras, i.e., commutative (possibly graded...

of the module when r is the rank of the module. differential A differential graded module or dg-module is a graded module with a differential. direct...

R_{-1}\xrightarrow {s} R_{0}\xrightarrow {p} A\to 0} Then, by taking the differential graded module of derivations ( Der ( R ∙ ) , d ) {\displaystyle ({\text{Der}}(R_{\bullet...

In differential geometry, a Clifford module bundle, a bundle of Clifford modules or just Clifford module is a vector bundle whose fibers are Clifford...

exterior algebra can be defined for modules over a commutative ring. In particular, the algebra of differential forms in k {\displaystyle k} variables...

resolution in which the free modules Ei may be graded in such a way that the di and ε are graded linear maps. Among these graded free resolutions, the minimal...

differential graded category A differential graded category is a category whose Hom sets are equipped with structures of differential graded modules....

modules is injective. Every left injective module over a left Noetherian module can be decomposed as a direct sum of indecomposable injective modules...

mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. In other words, a graded Lie algebra...

{\displaystyle R_{+}} denote the ideal generated by positively graded elements. Then if M is a graded module over R for which M i = 0 {\displaystyle M_{i}=0} for...

Temperley-Lieb algebra. A differential graded algebra is an associative algebra together with a grading and a differential. For example, the de Rham algebra...

Localization of a module Completion (ring theory) Types Simple module, Semisimple module Indecomposable module Artinian module, Noetherian module Homological...