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 ...
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 ...
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.
0 (A, B) ≅ A ⊗ R B for any right R-module A and left R-module B. Tor R i (A, B) = 0 for all i > 0 if either A or B is flat (for example, free) as an R-module. In fact, one can compute Tor using a flat resolution of either A or B; this is more general than a projective (or free) resolution. [5] There are converses to the previous statement ...
Monoidal functors are the functors between monoidal categories that preserve the tensor product and monoidal natural transformations are the natural transformations, between those functors, which are "compatible" with the tensor product. Every monoidal category can be seen as the category B(∗, ∗) of a bicategory B with only one object ...
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 ...
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: