Search results
Results from the WOW.Com Content Network
In this setup, for example, one can define a tensor field on a smooth manifold M as a (global or local) section of the tensor product (called tensor bundle) () where O is the sheaf of rings of smooth functions on M and the bundles , are viewed as locally free sheaves on M.
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 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:
The monoidal product is given by the tensor product of modules and the internal Hom is given by the space of R-linear maps (,) with its natural R-module structure. In particular, the category of vector spaces over a field K {\displaystyle K} is a symmetric, closed monoidal category.
In many circumstances conditions are imposed on the modules E i resolving the given module M. For example, a free resolution of a module M is a left resolution in which all the modules E i are free R-modules. Likewise, projective and flat resolutions are left resolutions such that all the E i are projective and flat R-modules, respectively.
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 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.