Search results
Results from the WOW.Com Content Network
In particular, any tensor product of R-modules can be constructed, if so desired, as a quotient of a tensor product of abelian groups by imposing the R-balanced product property. More category-theoretically, let σ be the given right action of R on M ; i.e., σ( m , r ) = m · r and τ the left action of R of N .
In particular, () is the usual tensor product of modules M and N over R. Geometrically, the derived tensor product corresponds to the intersection product (of derived schemes ). Example : Let R be a simplicial commutative ring , Q ( R ) → R be a cofibrant replacement, and Ω Q ( R ) 1 {\displaystyle \Omega _{Q(R)}^{1}} be the module of ...
The tensor product of two vector spaces is a vector space that is defined up to an isomorphism.There are several equivalent ways to define it. Most consist of defining explicitly a vector space that is called a tensor product, and, generally, the equivalence proof results almost immediately from the basic properties of the vector spaces that are so defined.
Moreover, for a fixed ring R, Tor is a functor in each variable (from R-modules to abelian groups). For a commutative ring R and R-modules A and B, Tor R i (A, B) is an R-module (using that A ⊗ R B is an R-module in this case). For a non-commutative ring R, Tor R i (A, B) is only an abelian group, in general.
The tensor product of commutative algebras is of frequent use in algebraic geometry. For affine schemes X , Y , Z with morphisms from X and Z to Y , so X = Spec( A ), Y = Spec( R ), and Z = Spec( B ) for some commutative rings A , R , B , the fiber product scheme is the affine scheme corresponding to the tensor product of algebras:
The tensor product is the direct product of objects, and any terminal object (empty product) is the unit object. The category of bimodules over a ring R is monoidal (using the ordinary tensor product of modules), but not necessarily symmetric. If R is commutative, the category of left R-modules is symmetric monoidal. The latter example class ...
Mitchell's embedding theorem states every abelian category arises as a full subcategory of the category of modules over some ring. Projective limits and inductive limits exist in the categories of left and right modules. [4] Over a commutative ring, together with the tensor product of modules ⊗, the category of modules is a symmetric monoidal ...
In this interpretation, the category End(R) = Bimod(R, R) is exactly the monoidal category of R-R-bimodules with the usual tensor product over R the tensor product of the category. In particular, if R is a commutative ring, every left or right R-module is canonically an R-R-bimodule, which gives a monoidal embedding of the category R-Mod into ...