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

Short Exact Sequences and at Tensor Product

  1. Jul 14, 2014 #1

    WWGD

    User Avatar
    Science Advisor
    Gold Member

    Hi,let:

    0->A-> B -> 0

    ; A,B Z-modules, be a short exact sequence. It follows A is isomorphic with B.

    . We have that tensor product is

    right-exact , so that, for a ring R:

    0-> A(x)R-> B(x)R ->0

    is also exact. STILL: are A(x)R , B(x)R isomorphic?

    I suspect no, if R has torsion. Anyone have an example of

    A(x)R , B(x)R non-isomorphic, but A,B isomorphic? Thanks.
     
  2. jcsd
  3. Jul 14, 2014 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Yes, they are. You should look for a proof.
     
  4. Jul 14, 2014 #3

    WWGD

    User Avatar
    Science Advisor
    Gold Member

    Sorry, that is not what I meant to ask, instead , I am looking for an example where:

    A(x)R , B(x)R are isomorphic,

    but A,B are not isomorphic.

    Thanks.
     
  5. Jul 16, 2014 #4

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Oops, I never saw your reply. Weird. Anyway, consider

    [tex]\mathbb{Z}\otimes_\mathbb{Z} \mathbb{Z}_2\cong \mathbb{Z}_2 \cong \mathbb{Z}_2 \otimes_\mathbb{Z}\mathbb{Z}_2[/tex]

    In general, we have the relation

    [tex]\mathbb{Z}_n\otimes_\mathbb{Z} \mathbb{Z}_m \cong \mathbb{Z}_\text{gcd(m,n)}[/tex]

    This result will also give you a wealth of counterexamples (note that the above still holds if we define ##\mathbb{Z}_0 = \mathbb{Z}##)
     
  6. Jul 19, 2014 #5

    WWGD

    User Avatar
    Science Advisor
    Gold Member

    A, thanks, nice.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Short Exact Sequences and at Tensor Product
  1. Exact sequences (Replies: 5)

  2. Tensor product? (Replies: 7)

  3. Tensor product (Replies: 1)

  4. Short exact sequences (Replies: 2)

Loading...