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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 ...
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.
Download as PDF; Printable version; In other projects ... It is the tensor product of the r many ... as graded R-modules. Also, by the definition of a tensor product ...
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 ...
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 ...
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]