reduced homologies of shifted complexes

Question

I am trying to understand dimension of reduced homologies of shifted complexes. Lemma 2.15 of the book Combinatorial commutative algebra by Miller and Sturmfels explains it. But the last part of the proof confuses me. It may have some misprints, I suppose. Is there any other reference for this statement? (In the proof, is it true in general that $\tilde{C_i}(k*\Delta)=\tilde{C_i}(\Gamma)$ and hence $dim_K\tilde{H}_{i+1}(k*\Delta, \Gamma)=\#\{F\in k*\Delta: dim\ F=i+1, F\notin \Gamma\}$? If so, then by the discussion given in the proof the latter set is equal to $\{k\cup F: \text{$F$ an $i$-facet of $\Delta$}, k\cup \Delta\notin \Gamma\}$. Hence the lemma is proved and there is no need for the discussion in the last paragraph (about minimal near-cone etc.). What is wrong in the above argument? Lemma 2.15: Fix a shifted simplicial complex $\Gamma$ on $1,\ldots,k$, and let $\Delta\subseteq \Gamma$ consists of the faces of $\Gamma$ not having $k$ as a vertex. Then $\dim_K\tilde{H}_i(\Gamma;K)$ equals the number of dimension $i$ facets $F$ of $\Delta$ such that $F\cup k$ is not a face of $\Gamma$.