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 .
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 ...
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 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.
an element of a tensor product of a module E with its dual maps canonically into End R (E), and thence onto its trace (Bourbaki II.4.3) In the noncommutative case, an element of a tensor product of a module E with its dual maps canonically into End R (E), but no trace is defined; the map E ∗ ⊗ R F ∗ → Bilin Z (E, F; R) should be canonical
Let k be a field, A an associative k-algebra, and M an A-bimodule.The enveloping algebra of A is the tensor product = of A with its opposite algebra.Bimodules over A are essentially the same as modules over the enveloping algebra of A, so in particular A and M can be considered as A e-modules.