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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"). 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 $\mathbb{Z}^n \cap affine(M+M')$, where $affine(M)$ is the affine hull of the lattice $M$.

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 $\mathbb{R}^n$.

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

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.

Example: Here is a simple example for $n=3$ which hopefully clarifies the question. Let $V$ be the line determined by the vector $(0,0,1)$ (i.e. $V=\mathbb{R} \cdot (0,0,1)$) and let $W$ be the line determined by $(k,k,1)$, for $k \in \mathbb{N}$. Then, with the notation above, $M + M'=\mathbb{Z}< (0,0,1)> + \mathbb{Z} <(k,k,1)> $ and $$ [\mathbb{Z}^3:M+M']=[\mathbb{Z}^3 \cap affine(M+M'):M+M' ])=k,$$ since a basis for $\mathbb{Z}^3 \cap affine(M+M')$ is given by $(0,0,1)$ and $(1,1,0)$.

On the other hand $L+L'=\mathbb{Z} <(1,0,0),(0,1,0)> + \mathbb{Z} <(0,1,-k),(1,-1,0) > =$ $ =\mathbb{Z}<(1,0,0),(0,1,0),(0,0,-k)>$ and thus $$ [\mathbb{Z}^n:L+L']=k,$$ which tells us that the answer to the question is yes in this case.

Thank you very much for your consideration!

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

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If $V,W$ are $1-$dimensional (linearly independant) and $n\geq 3$ then the index of $M+M'$ (being or rank $2$) is infinite while $L+L'$ has finite index. I guess you want perhaps also $V+W$ and $V^\perp+W^\perp$ to be of full dimension $n$. – Roland Bacher Apr 9 '13 at 12:52
I think they are working, in the article mentioned, with the definition that if $L$ is not a full rank sublattice of $\mathbb{Z}^n$ then the index $[\mathbb{Z}^n:L]$ is by definition equal to $[affine(L) \cap \mathbb{Z}^n:L]$. However, I might be wrong since they do not explicitly state this, but I think this is the only thing that makes sense from the toric-goemetry perspective. – edwold Apr 9 '13 at 13:52

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