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 .
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 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.
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 ...
For any commutative ring R, the category of R-algebras is monoidal with the tensor product of algebras as the product and R as the unit. The category of pointed spaces (restricted to compactly generated spaces for example) is monoidal with the smash product serving as the product and the pointed 0-sphere (a two-point discrete space) serving as ...
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 ...
In mathematics, an invertible sheaf is a sheaf on a ringed space that has an inverse with respect to tensor product of sheaves of modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle. Due to their interactions with Cartier divisors, they play a central role in the study of algebraic varieties.