I am trying to figure out something concerning the index of lattices.
The question came about after reading the paper of W.Fulton and B.Sturmfels,   ([*"Intersection theorey on toric varieties"*][1]). To paraphrase one of their constructions: in their paper they define the intersection between rational subspaces $V,W \subset \mathbb{R}^n$ and say that the multiplicity between these
should be $$ [\mathbb{Z}^n:M+ M']$$
where $M=V\cap\mathbb{Z}^n$ and $M'=W\cap\mathbb{Z}^n$. Here, the notation means the index of the sublattice $M+M'$ in $\mathbb{Z}^n$.

Now, let $L=V^{\perp}\cap\mathbb{Z}^n$ and $L'=W^{\perp}\cap\mathbb{Z}^n$ where $V^{\perp}$ denotes the orthogonal complement of $V$ in $\mathbb{R}^n$.


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**Question:**
Is it true that $$ [\mathbb{Z}^n:M+ M']=[\mathbb{Z}^n:L+ L']?$$

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This is true if $n=2$, and all the examples I have considered in $n=3$ satisfy this as well. I have very little familliarity with lattice theory, so I would really appreciate any advice, or where to look.

Thank you very much for your consideration!

(ps. I posted this question for $n=3$ at [math.stackexchange.com][2] a couple of days ago, but there were no takers, so I hope it is ok to post it here as well.)


  [1]: http://arxiv.org/abs/alg-geom/9403002BlockquoteBlockquote
  [2]: http://math.stackexchange.com/questions/353852/index-of-the-sum-of-two-lattices-determined-by-rational-planes