I am working some problems that can potentially be on a qualifying exam about tensor algebras and have come across some questions about field of fractions which is something I have not seen for a while and so I have not really been able to work through all the appropriate definitions to arrive at the correct proof. Any help would be greatly appreciated.

Let $I$ be an integral domain and let $Frac(I)$ denote its field of fractions http://en.wikipedia.org/wiki/Field_of_fractions.

let $\wedge^k M$ be the $k$-th exterior power of $M$ that is $T^k(V)/A^k(V)$ where $A(M)$ is the ideal generated by all $m \otimes m$ for $m \in M$ and $T^k(M) = M \otimes M \otimes \ldots \otimes M$ is tensor product of $k$ modules.

Consider an $I$-module $M \subset Frac(I)$.

How do we show $\wedge^k M$ is a torsion module for $k \geq 2$?

I think it is clear that the exterior power should be zero for $k <2$ but I am still not sure this is trivial after putting all the definitions together.

  • 1
    $\begingroup$ If M = Frac(I) is a one-dimensional vector space, then the 0th and 1st exterior powers of M are isomorphic to M, I believe. $\endgroup$ – Jack Schmidt Oct 19 '11 at 23:57

Let K be the field of fractions of the integral domain I, and let IMK. Since tensor product is associative, K ⊗ ⋀(M) = ⋀(KM) = ⋀(K) = KK, which is zero for k ≥ 2.

An I-module N such that KN = 0 is (by definition in some areas) a torsion I-module. If in = 0 and i ≠ 0, then 1 ⊗ n = 1/iin = 0. The other inclusion follows from the flatness of K.

  • $\begingroup$ Could you explain the first two lines of this answer, please? $\endgroup$ – yaa09d Nov 17 '11 at 2:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.