Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Tensor Products and Maps Factoring Through

  1. May 23, 2014 #1


    User Avatar
    Science Advisor
    Gold Member

    Hi, I understand the tensor product of modules as a new module in which every bilinear map becomes a linear map.

    But now I am trying to see the Tensor product of modules from the perspective of maps
    factoring through, i.e., from properties that allow a commutative triangle of maps. As a concrete example of what I mean:

    For homomorphisms f: A-->B and g: A-->C , a condition of the kernels allow
    the existence of a commutative triangle , i.e., the conditions allow the existence of a map h
    with the necessary properties so that hog=f. More specifically:

    If f: A -> B is a homomorphism of groups, and g: A -> C is a
    surjective homomorphism of groups, we have that f "descends to a homomorphism"
    of groups h: B -> C iff the kernel of g is contained in the kernel of f. In other words,
    there is an h with hog=f, which allows f to "factor through" and the associated triangle is commutative .
    Sorry, I don't know how to draw triangles here in ASCII.

    ** Now ** I'm curious as to how moding out the free module on a product VxW of R-modules
    by the standard necessary relations on the tensor product:

    i ) (a+a',b)=(a,b)+(a',b) and:

    ii) (a,b+b')=(a,b)+(a,b')

    allows for the existence of the linear map L that completes the commutative triangle, so, my question is:

    what specific algebraic result/theorem are we using to guarantee that imposing the above relations allow the existence of a linear map L : V(x)W --> Z , for an R-module Z, to factor thru the maps:

    (x): V x W --> V(x)W , and

    B: V x W --> Z

    i.e., L is the linear map defined on the tensor product V(x) W that represents the bilinear map B defined on the product V xW into Z ?

    WWGD: What Would Gauss Do?
    Last edited: May 23, 2014
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted