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 .
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.
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 ...
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 ...
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 ...
A module is called flat if taking the tensor product of it with any exact sequence of R-modules preserves exactness. Torsionless A module is called torsionless if it embeds into its algebraic dual. Simple A simple module S is a module that is not {0} and whose only submodules are {0} and S. Simple modules are sometimes called irreducible. [5 ...
The ordinary tensor product makes vector spaces, abelian groups, R-modules, or R-algebras into monoidal categories. Monoidal categories can be seen as a generalization of these and other examples. Monoidal categories can be seen as a generalization of these and other examples.
Let R be a commutative ring and let A and B be R-algebras.Since A and B may both be regarded as R-modules, their tensor product. is also an R-module.The tensor product can be given the structure of a ring by defining the product on elements of the form a ⊗ b by [1] [2]