If \(\mathbb R\), \(\mathbb Q\), \(\mathbb Z\), \(\mathbb N\) and \(\mathbb P\) have the usual meanings and if \(\Phi\) is an \(r\times n\) matrix over \(\mathbb Z\), then \(E = E^0\), where \(E^\alpha = \{\beta\in\mathbb N^n\mid \Phi\beta = \alpha\). If \(k\) is a field, \(\Lambda = kE\) is the monoid algebra of \(E\) over \(k\), realized as \(w: \Lambda \to k[x_1,\ldots, x_n]\), \(w(\beta)=w(x^\beta)=x_1^{\beta_1}\cdots x_n^{\beta_n}\), \(\beta = (\beta_1, \ldots, \beta_n)\). \(\Lambda^\alpha\) denotes the vector space with basis \(\{x^\beta \mid \beta\in E^\alpha\}\), turned into a \(\Lambda\)-module via \(x^\beta \cdot x^\gamma = x^{\beta+\gamma}\), \(\beta\in E\), \(\gamma\in E^\alpha\). \(\Lambda^\alpha\) is a f(initely) g(enerated) \(\Lambda\)-module which is (2.11) proven to be C(ohen)-M(acaulay) iff for all \(\gamma\in \bar E^\alpha\), \(\Gamma_\gamma = \emptyset\) or \(\Gamma_\gamma = \) acyclic, where \(\bar E^\alpha\) is the coset of \(\bar E\), the group generated by \(E\) in \(\mathbb Z^n\), containing \(E^\alpha\), and where \(\Gamma_\gamma = \cup \{\mathcal F^*\mid \text{supp-}\gamma \subset \text{supp-}\mathcal F\}\)). Here \(\mathcal F^*\) is a (dual) face on the dual polytope \(\mathcal P^*\) of the \(( n- r - 1)\)-dimensional convex polytope \(\mathcal P\) which is a non-degenerate cross-section of the \((n - r)\)-dimensional convex polyhedral cone \(C = \{\beta\in\mathbb R_+^n \mid \Phi\beta=0\}\), \(\text{supp }\beta = \{i\mid \beta_i > \mathcal P\}\), \(\text{supp-}\beta = \{i\mid \beta_i 0\) or \(i=t\) and \(\alpha > 0\). Furthermore \[ \text{depth }\Lambda^\alpha \begin{cases} = 0 \text{ if }\Lambda^\alpha = 0; \\ =s\text{ if }\beta\in\bar E^\alpha\) with \(\beta' < 0, \beta'' \ge 0; \\ = t\text{ if }\beta\in\bar E^\alpha\) with \(\beta' \ge 0, \beta'' < 0 \text{ and } \\ = s +t - 1 \text{ otherwise} \end{cases} \] (whence \(\Lambda^\alpha\) is C-M). It is noted (4.1) that \[ f(H^d(\Lambda^\alpha),x)= \sum_{\gamma\in E^\alpha, \Gamma_\gamma = \emptyset} x^\gamma, \] where \(d = \dim(\Lambda^\alpha)= \dim(\Lambda/\operatorname{Ann}\Lambda^\alpha) = \dim(\Lambda),\) and that (4.2) \(F(\Lambda^\alpha,x)_\infty\), the expansion at \(\infty\) of \(F(\Lambda^\alpha,x)\), equals \[ (-1)^d \sum_{\gamma\in E^\alpha, \Gamma_\gamma = \emptyset} x^\gamma + (-1)^{d-1}\sum_{\gamma\in E^\alpha, \Gamma_\gamma \ne\emptyset} \tilde\chi(\Gamma_\gamma) x^\gamma \] where \(\tilde\chi(\Gamma_\gamma)\) denotes the reduced Euler-characteristic of \(\Gamma_\gamma\). Hence (4.3) \[ F(\Lambda^\alpha,x)_\infty = (-1)^d \sum_{\Gamma_\gamma = \emptyset} x^\gamma \quad\text{iff } \tilde\chi(\Gamma_\gamma) = 0 \] whenever \(\Gamma_\gamma \ne \emptyset\), a result for which the converse is false. If \(\Omega(M) = \operatorname{Ext}_A^{c - d} (M,A)\), where \(A\) is G(orenstein), \(M\) is an f.g. \(A\)-module, \(c = \dim A\), \(d = \dim M\), then (4.5) \(E\cap \mathbb P^n \ne \emptyset\) implies that \(\Omega(\Lambda)\) is isomorphic to the ideal of \(\Lambda\) generated by all \(x^\beta\) in \(E\cap \mathbb P^n\). Also (4.6) \(H^d(\Lambda^\alpha\cong V^\alpha\), the \(k\)-vector space with basis \(\{x^\gamma\mid \Gamma_\gamma = \emptyset\}\) turned into \(\Lambda\)-module via \(x^\beta\cdot x^\gamma = x^{\beta+\gamma}\) if \(\Gamma_{\beta + \gamma} = \emptyset\) \(x^\beta\cdot x^\gamma = 0\), otherwise. Finally (5.2), there exist free (commutative) submonoids \(E_1,\ldots, E_t\) of \(E\), all of \(\text{rank }d= -\dim\Lambda\) and elements \(\delta_1,\ldots, \delta_t\) of \(E\), such that \(E= \cup (\delta_2 + E_i)\). \[ CF(E) = \{ \beta\in E\mid n\beta = \gamma + \delta,\ m\ge 1, \gamma, \delta\in E \Rightarrow \gamma = i\beta \text{ for some }0\le i\le m\}. \] If \(S\subseteq CF(E)\), then \(\mathcal F_S\) is the face of \(\mathcal P\) satisfying \(\text{supp }(\mathcal F_S = \cup_{\beta\in S} \text{supp }\beta\). If \(\gamma\in\mathbb Z^n\) \(\Lambda_S^\alpha)_\gamma\) denotes the \(\gamma\)-homogeneous part of \(\Lambda_S^\alpha\), the quotient of \(\Lambda^\alpha\) by the multiplicative set generated by \(S\). \(\Delta_\gamma\) is the simplicial complex with faces \(S\subset CF(E)\) such that \(\text{supp }\gamma \subset \cup_{\delta\in CF(E)\backslash S} (\text{supp }\delta) \). \[ \mathcal K(y^\infty,\Lambda^\alpha) = 0 \overset{\delta_0}{\rightarrow}\Lambda^\alpha \overset{\delta_1}{\rightarrow} \cup_i \Lambda_{y_i}^\alpha \overset{\delta_2}{\rightarrow} \cdots \overset{\delta_s}{\rightarrow} \Lambda_{y_1\cdots y_s}^\alpha \rightarrow 0, \] with \(\delta_{j + 1} (u) = \sum_{r=1}^{s-j} (-1)^{r - 1}(u)\), \(\varphi_{\ell_r}: M \to M_{y_{\ell_r}}\) the natural injection, \(M =\Lambda_{y_{i_1}\cdots y_{i_j}}^\alpha\), \(\{\ell_1 < \cdots \ell_{s-j}\}= \{1,\ldots, s\} \backslash \{i_1,\ldots, i_j\}\) and \(H^i(\Lambda^\alpha) = \ker \delta_{i+1}/\text{im }\delta_i\). (2.5) \( \mathcal K(y^\infty,\Lambda^\alpha)_\gamma\) is isomorphic to the augmented chain complex \(\tilde C(\Delta_\gamma)\) of \(\Delta_\gamma\), up to a shift in grading, with reduced homology groups \(\tilde H_i(\Delta_\gamma)\). Also from (2.8) it follows that if \(d = \dim \Lambda^\alpha\) and \(s = \vert CF(E)\vert\), then for all \(i\) \[ \tilde H_i(\Gamma_\gamma) \cong \tilde H_{s- d + i}(\Delta_\gamma), \] where \(\tilde H_i(\Gamma_\gamma)\) denotes reduced singular homology. In this way, this very interesting paper completes further a series of investigations scattered over a considerable collection of papers, both in time and place, to which the author has been the major contributor. For students of algebraic topology such beautiful applications are bound to be fascinating, while for the theory of linear diophantine equations these results are of great importance both as to content and as to the continued indication of a major direction of further development.