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searching for Category of modules 33 found (67 total)

Quasitriangular Hopf algebra (771 words) [view diff] exact match in snippet view article find links to article

construction. If the Hopf algebra H is quasitriangular, then the category of modules over H is braided with braiding c U , V ( u ⊗ v ) = T ( R ⋅ ( u ⊗

Module spectrum (103 words) [view diff] exact match in snippet view article find links to article

be considered as either analog or generalization of the derived category of modules over a ring. Lurie defines the K-theory of a ring spectrum R to be

Quasi-bialgebra (1,449 words) [view diff] exact match in snippet view article find links to article

non-coassociativity. One of their key properties is that the corresponding category of modules forms a tensor category. A quasi-bialgebra B A = ( A , Δ , ε , Φ

Highly structured ring spectrum (2,290 words) [view diff] exact match in snippet view article find links to article

category of modules over an A ∞ {\displaystyle A_{\infty }} -operad and the category of monoids are Quillen equivalent and likewise the category of modules

Schur functor (1,251 words) [view diff] exact match in snippet view article find links to article

functors (named after Issai Schur) are certain functors from the category of modules over a fixed commutative ring to itself. They generalize the constructions

Topological half-exact functor (172 words) [view diff] exact match in snippet view article find links to article

(most frequently in applications, category of abelian groups or category of modules over a fixed ring) that has a following property: for each sequence

Injective module (3,919 words) [view diff] exact match in snippet view article find links to article

cogenerators are injective modules that faithfully represent the entire category of modules. Injective resolutions measure how far from injective a module is

Five lemma (867 words) [view diff] exact match in snippet view article find links to article

four lemmas. To perform diagram chasing, we assume that we are in a category of modules over some ring, so that we may speak of elements of the objects in

Lie algebra representation (4,312 words) [view diff] exact match in snippet view article find links to article

category of representations of a Lie algebra is the same as the category of modules over its enveloping algebra. Let g {\displaystyle {\mathfrak {g}}}

Derived category (4,503 words) [view diff] exact match in snippet view article find links to article

{\displaystyle {\mathcal {A}}} be an abelian category. (Examples include the category of modules over a ring and the category of sheaves of abelian groups on a topological

Free presentation (563 words) [view diff] exact match in snippet view article find links to article

Proposition—Let F, G be left-exact contravariant functors from the category of modules over a commutative ring R to abelian groups and θ a natural transformation

Weak Hopf algebra (1,313 words) [view diff] exact match in snippet view article find links to article

Nikshych and Ostrik that any fusion category is equivalent to a category of modules over a weak Hopf algebra. A weak bialgebra ( H , μ , η , Δ , ε )

Homological algebra (3,859 words) [view diff] exact match in snippet view article find links to article

far" F is from being exact. Let R be a ring and let ModR be the category of modules over R. Let B be in ModR and set T(B) = HomR(A,B), for fixed A in

Hopfian object (842 words) [view diff] exact match in snippet view article find links to article

prove the second statement. Here are several basic results in the category of modules. It is especially important to remember that RR being hopfian or

Zero element (1,108 words) [view diff] exact match in snippet view article find links to article

zero module, containing only the identity (a zero object in the category of modules over a ring) A zero morphism in a category is a generalised absorbing

Direct sum (2,858 words) [view diff] exact match in snippet view article find links to article

An additive category is an abstraction of the properties of the category of modules. In such a category, finite products and coproducts agree, and the

Generator (category theory) (320 words) [view diff] exact match in snippet view article

sets, any set with at least two elements is a cogenerator. In the category of modules over a ring R, a generator in a finite direct sum with itself contains

Weak equivalence (homotopy theory) (868 words) [view diff] exact match in snippet view article

complexes. Let A be a Grothendieck abelian category, for example the category of modules over a ring or the category of sheaves of abelian groups on a topological

Injective hull (1,052 words) [view diff] exact match in snippet view article find links to article

has an injective hull (these three conditions are satisfied by the category of modules over a ring). Every object in a Grothendieck category has an injective

Warehouse management system (1,657 words) [view diff] exact match in snippet view article find links to article

cloud. A WMS may be a standalone product, or can be a module or category of modules within a larger Enterprise Resource Planning (ERP) system, Shipping

Uniform module (1,349 words) [view diff] exact match in snippet view article find links to article

1016/0022-4049(84)90075-6 Hanna, A.; Shamsuddin, A. (1984), Duality in the category of modules: Applications, Reinhard Fischer, ISBN 978-3889270177 Miyashita, Y

Derived functor (3,092 words) [view diff] exact match in snippet view article find links to article

→ A b {\displaystyle A\otimes _{R}-:R{\text{-Mod}}\to Ab} ; The category of modules has enough projectives so that left derived functors always exists

Representation theory of Hopf algebras (1,181 words) [view diff] exact match in snippet view article find links to article

an antipode, and H is a Hopf algebra. The desire for a monoidal category of modules with functorial tensor products and dual representations is therefore

Claus Michael Ringel (630 words) [view diff] exact match in snippet view article find links to article

doi:10.1007/978-3-0348-8658-1_6 Ringel, Claus Michael (1991). "The category of modules with good filtrations over a quasi-hereditary algebra has almost

Ext functor (3,876 words) [view diff] exact match in snippet view article find links to article

R} be a ring and let R -Mod {\displaystyle R{\text{-Mod}}} be the category of modules over R {\displaystyle R} . (One can take this to mean either left

Spectral sequence (10,712 words) [view diff] exact match in snippet view article find links to article

spectral sequence by various tricks. Fix an abelian category, such as a category of modules over a ring, and a nonnegative integer r 0 {\displaystyle r_{0}}

Local system (2,681 words) [view diff] exact match in snippet view article find links to article

{Mod}}(R)} from the fundamental groupoid of X {\displaystyle X} to the category of modules over a commutative ring R {\displaystyle R} , where typically R =

Clifford algebra (9,191 words) [view diff] exact match in snippet view article find links to article

algebra (its representation theory: the equivalence class of the category of modules over it) depends on only the signature (p − q) mod 8. This is an

Coherent sheaf (6,934 words) [view diff] exact match in snippet view article find links to article

quasi-coherent sheaves on an affine scheme are equivalent to the category of modules over the underlying ring, and the adjunction comes from taking global

Quotient of an abelian category (1,640 words) [view diff] exact match in snippet view article find links to article

K_{0}({\mathcal {A/B}})\to 0} Section 12.10 The Stacks Project "109.76 The category of modules modulo torsion modules". The Stacks Project. Gabriel, Pierre, Des

Triangulated category (5,798 words) [view diff] exact match in snippet view article find links to article

"large" triangulated categories are compactly generated: The derived category of modules over a ring R is compactly generated by one object, the R-module

Algebraic K-theory (10,647 words) [view diff] exact match in snippet view article find links to article

similar to, but slightly weaker than, the properties satisfied by a category of modules or vector bundles. From this he constructed an auxiliary category

Vertex operator algebra (8,906 words) [view diff] exact match in snippet view article find links to article

\mathbb {Z} )} -invariant up to a constant. Huang showed that the category of modules of a regular VOA is a modular tensor category, and its fusion rules