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The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third module, and also for a pair of a right-module and a left-module over any ring, with result an abelian group.
[4] Tor R 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 ...
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 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 ...
Ab, the category of abelian groups, with the direct sum of abelian groups as monoidal product and the trivial group as unit. More generally, the category R-Mod of (left) modules over a ring R (commutative or not) becomes a cocartesian monoidal category with the direct sum of modules as tensor product and the trivial module as unit.
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 ...
For example it is common to take A to be Z/2Z, so that coefficients are modulo 2. This becomes straightforward in the absence of 2- torsion in the homology. Quite generally, the result indicates the relationship that holds between the Betti numbers b i of X and the Betti numbers b i , F with coefficients in a field F .
Roughly speaking, the tensor product is the metric space completion of the ordinary tensor product. This is an example of a topological tensor product. The tensor product allows Hilbert spaces to be collected into a symmetric monoidal category. [1]