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I want to do an exercise in the book Toric Varieties (by David Cox)

Exercise 3.3.5. Let $\overline{\phi}:N \rightarrow N'$ be a surjective $\mathbb{Z}$-linear mapping and let $\widehat{\sigma}$ and $\sigma'$ be cones in $N_{\mathbb{R}}$ and $N'_{\mathbb{R}}$ respectively with the property that $\overline{\phi}_{\mathbb{R}}$ maps $\widehat{\sigma}$ bijectively onto $\sigma'$. Prove that $\overline{\phi}$ has a splitting $\overline{\nu}:N' \rightarrow N$ such that $\overline{\nu}$ maps $\sigma'$ to $\widehat{\sigma}$.

My approach: we consider the following exact sequence of lattices:

$0 \longrightarrow N_0 = ker(\overline{\phi})\longrightarrow N \xrightarrow{ \ \overline{\phi } \ } N' \longrightarrow 0$

Because $N'$ is free over $\mathbb{Z}$, the exact sequence above splits exact. That is, there exists $\overline{\nu} : N' \rightarrow N$ such that $\overline{\phi} \circ \overline{\nu} = id_{N'}$. That is, I want find $\overline{\nu}$ such that $\overline{\nu}(\sigma') \subseteq \widehat{\sigma}$. Recall that the existence of $\overline{\nu}$ is not unique:

Because $\overline{\phi}$ is onto, if $N' \simeq \mathbb{Z}^r$, then $e_j = \overline{\phi}(u_j)$ with $u_j \in N$ for each $j \in \lbrace 1,2,\dots,r \rbrace$. Since $N'$ is free with $\mathbb{Z}$-basis $\lbrace e_1,\dots,e_r\rbrace$, for fixed $\lbrace u_j \rbrace$ we can define $\overline{\nu}$ by extending $e_j \longmapsto u_j$.

Now, let us consider the case of that $\widehat{\sigma}$ is a strongli convex rational polyhedral cone, i.e., we can write $\widehat{\sigma} = \lbrace\widehat{v_1},\dots,\widehat{v_k} \rbrace$, where $\widehat{v_1},\dots,\widehat{v_k}\in N \subseteq N_{\mathbb{R}}$ and $\widehat{v_1},\dots,\widehat{v_k}$ are linearly independent over $\mathbb{R}$. Note that for each $\widehat{v_i}$, $\widehat{v_i} = \sum_{j=1}^r a_{ij}e_j$ with $a_{ij} \in \mathbb{R}$ for $ i =1,\dots,k$ and $j = 1,\dots,r$. Now, $e_j = \overline{\phi}(\widehat{w_j})$ for each j. We hope $\overline{\nu}(\sum_{j=1}^r a_{ij}e_j) = \widehat{v_i}$.

My main question:

(1) What shall I do in next step?

(2) Where should I use the hypothesis that $\overline{\phi}_{\mathbb{R}}$ maps $\widehat{\sigma}$ bijectively onto $\sigma'$?

(3) Is this statement always true for any arbitrary cone?

share|improve this question
    
Should I use $\overline{\phi}_{\mathbb{R}}$ maps $\widehat{\sigma}$ bijectively onto $\sigma'$ to show that ${ \overline{\phi}_{\mathbb{R}}(\widehat{v_i}) = \overline{\phi}(\widehat{v_i}) }$ are linearly independent over $\mathbb{R}$, and we can extend them to a basis in $N'_{\mathbb{R}} \simeq \mathbb{R}^r$. Moreover, we can choose all of them are lattice points. Linear independents over $\mathbb{R}$ implies linear independents over $\mathbb{Z}$ ... –  Peter Hu May 12 '12 at 12:10
    
But $rank_{\mathbb{Z}}(N) \geq rank_{\mathbb{Z}}(N')$, I think the method above is useless... –  Peter Hu May 12 '12 at 12:22
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