--- title: Buchsbaum Simplicial Affine Semigroups url: https://www.emergentmind.com/topics/buchsbaum-simplicial-affine-semigroups type: topic --- # Buchsbaum Simplicial Affine Semigroups Buchsbaum simplicial affine semigroups are affine semigroups whose rational polyhedral cone is simplicial and whose semigroup ring is Buchsbaum. In the contemporary literature, the subject is organized around several complementary viewpoints: decomposition of simplicial semigroup rings into shifted monomial ideals over the subring generated by extremal rays, two-dimensional geometric characterizations for convex-body semigroups, and gap-theoretic classifications for simplicial semigroups with finite complement in their integer cone. Across these approaches, the simplicial hypothesis makes Apéry sets, extremal rays, and pseudo-Frobenius elements particularly effective invariants for deciding Buchsbaumness and relating it to Cohen–Macaulay, Gorenstein, seminormal, and projective-closure phenomena [1206.1735] [1402.2597] [2507.09267]. ## 1. Semigroup-theoretic framework A positive affine semigroup \(B \subset \mathbb{Z}^d\) is a finitely generated subsemigroup of \(\mathbb{N}^m\) with no invertible elements except \(0\). Its rational polyhedral cone is denoted \(C(B)\), and the abelian group generated by \(B\) is \(G(B)\). The associated semigroup ring over a field \(K\) is \[ K[B] = \bigoplus_{b \in B} K\cdot t^b \subset K[t_1,\dots,t_m], \] with \(\deg(t^b)=b\). The semigroup \(B\), equivalently \(C(B)\), is simplicial if the cone \(C(B)\) is generated by \(d\) linearly independent extremal rays. If \(e_1,\dots,e_d \in B\) are minimal generators of these extremal rays and \(A=\langle e_1,\dots,e_d\rangle\), then \(C(A)=C(B)\), so \(K[A]\subseteq K[B]\) is a finite extension and \(G(B)/G(A)\) is finite [1206.1735]. In dimension \(2\), the simplicial condition means that the cone has exactly two extremal rays, usually denoted \(\tau_1\) and \(\tau_2\). For an affine simplicial semigroup \(S \subset \mathbb{N}^r\), one also considers the saturation-like set \[ \overline{S} := \{a\in \mathbb{N}^r \mid a+n_i\in S,\ \forall i\}, \] where \(\{n_1,\dots,n_m\}\) is the minimal generating set of \(S\). A fundamental theorem used throughout the geometric literature states that \(S\) is Buchsbaum if and only if \(\overline{S}\) is Cohen–Macaulay. In this sense, Buchsbaumness for simplicial affine semigroups is often studied by translating it into a Cohen–Macaulay problem for a closely related semigroup [1402.2597]. The Buchsbaum property is ring-theoretic: a semigroup \(S\) is called Buchsbaum when \(k[S]\) is a Buchsbaum ring. The decomposition and classification results considered below are characteristic-free in the stated simplicial settings, and several papers emphasize that Buchsbaum, Cohen–Macaulay, Gorenstein, normal, and seminormal properties are independent of the field \(K\) in the simplicial case [1206.1735]. ## 2. Decomposition-theoretic characterization in the simplicial case For a finite extension \(K[A]\subseteq K[B]\) with \(A\) generated by extremal rays, the semigroup ring \(K[B]\) admits a canonical decomposition as a \(\mathbb{Z}^m\)-graded \(K[A]\)-module: \[ K[B] = \bigoplus_{g \in G} I_g(-h_g), \qquad G:=G(B)/G(A), \] where each \(I_g\subseteq K[A]\) is a monomial ideal and each \(h_g\in G(B)\) is a degree shift. In the simplicial setting, the construction is controlled by the Apéry set \[ B_A := \{x\in B \mid x\notin B+(A\setminus\{0\})\} = \operatorname{Ap}(B;A). \] For each coset \(g\in G\), one sets \(\Gamma_g:=\{x\in B_A\mid x\in g\}\), writes each \(v\in \Gamma_g\) uniquely as \(v=\sum_{k=1}^d \lambda_k e_k\) with \(\lambda_k\in \mathbb{Q}\), and defines \[ h_g=\sum_{k=1}^d \bigl(\min_{v\in \Gamma_g}\lambda_k\bigr)e_k, \qquad I_g=K\cdot\{t^{v-h_g}\mid v\in \Gamma_g\}\cdot K[A]. \] Because the extremal generators \(e_1,\dots,e_d\) are linearly independent, the shifts \(h_g\) and the resulting decomposition are uniquely determined by \(A\) [1206.1735]. In this language, the Buchsbaum property has an explicit simplicial criterion. The ring \(K[B]\) is Buchsbaum if and only if, for every \(g\in G\), the monomial ideal \(I_g\) is either \(K[A]\) or the homogeneous maximal ideal \(K[A]_+\), and whenever \(I_g=K[A]_+\), one has \(h_g+b\in B\) for every \(b\in \operatorname{Hilb}(B)\). Equivalently, if \[ H:=\{h_g\mid I_g=K[A]_+\}, \qquad C:=\operatorname{Hilb}(B)\setminus\{0,e_1,\dots,e_d\}, \] then \(K[B]\) is Buchsbaum if and only if \[ (H+C)\cap H = \varnothing. \] This yields Algorithm 2 of the paper: compute \(A\) from the extremal rays, compute \(B_A\), the \(\Gamma_g\), the decomposition data \((I_g,h_g)\), reject as soon as some \(I_g\notin\{K[A],K[A]_+\}\), and otherwise test \((H+C)\cap H=\varnothing\) [1206.1735]. The same decomposition is also used to compute Castelnuovo–Mumford regularity in the homogeneous case: \[ \operatorname{reg} K[B] = \max_{g\in G}\{\operatorname{reg} I_g + \deg(h_g)\}. \] Since each \(I_g\) has projective dimension at most \(d-1\), this route is typically much faster than resolving \(K[B]\) directly. The decomposition paper uses this framework to test Buchsbaum, Cohen–Macaulay, Gorenstein, normal, and seminormal properties, all of which imply the toric Eisenbud–Goto bound \(\operatorname{reg} K[B]\le \deg K[B]-\operatorname{codim} K[B]\) in the cases treated there [1206.1735]. A representative example is \[ B=\{(4,0,0),(0,4,0),(0,0,4),(1,0,3),(0,2,2),(3,0,1),(1,2,1)\}\subset \mathbb{N}^3, \] with \(A=\{(4,0,0),(0,4,0),(0,0,4)\}\). The decomposition computed in the paper is \[ K[B]=K[A]\oplus K[A](-1)^4\oplus K[A](-2)^2\oplus (x_1,x_2,x_3)(-1). \] From this, the authors obtain \(\operatorname{depth} K[B]=1\), so \(K[B]\) is not Cohen–Macaulay and hence not normal; the seminormality test returns false, the Buchsbaum test returns true, and \(\operatorname{reg} K[B]=2\) while \(\deg K[B]-\operatorname{codim} K[B]=4\), so the Eisenbud–Goto bound holds in this example [1206.1735]. ## 3. Two-dimensional geometric families A distinct line of work studies Buchsbaum simplicial affine semigroups arising from convex bodies in \(\mathbb{R}^2_{\ge 0}\). For an affine circle semigroup \(S\subset \mathbb{N}^2\), the criterion is especially clean: \(S\) is Buchsbaum if and only if \(\operatorname{int}(C)=\operatorname{int}(\overline{S})\) and, for \(j=1,2\), the semigroup \(\overline{S}\cap \tau_j\) is generated by only one element. Here \(C\) is the integer cone, \(\tau_1,\tau_2\) are its extremal rays, and the criterion is obtained by combining the Rosales–Sánchez reduction “\(S\) Buchsbaum iff \(\overline{S}\) Cohen–Macaulay” with the two-dimensional Cohen–Macaulay test that every hole \(a\in C\setminus S\) must fail to remain inside after translation by at least one extremal-ray generator [1402.2597]. For simplicial affine convex polygonal semigroups \(P\subset \mathbb{N}^2\), the criterion has two cases. If \(\operatorname{int}(C)=\operatorname{int}(\overline{P})\), then \(P\) is Buchsbaum if and only if \(\overline{P}\cap \tau_j\) is generated by only one element for \(j=1,2\). If \(\operatorname{int}(C)\neq \operatorname{int}(\overline{P})\), the criterion is expressed in terms of finite controlling regions constructed from the polygon and its extremal rays: \(P\) is Buchsbaum if and only if \(\Upsilon'=\varnothing\) and \(\Upsilon\subset \overline{P}\). The paper also proves that every affine convex polygonal semigroup associated to a triangle with rational vertices is Buchsbaum [1402.2597]. These criteria were designed to avoid prohibitively large Apéry-set computations. In the polygonal setting, the cone is decomposed into strips and finite regions \(\mathcal{B}_1,\mathcal{B}_2,\Upsilon_1,\Upsilon_2,\Upsilon\), and Buchsbaumness reduces to finite membership checks inside these explicitly constructed sets. The same paper emphasizes the practicality of this approach: PolySGTools implements routines such as PolygonalSG, BelongToSG, and PSGIsBuchsbaumQ, and the reported polygonal example is decided in approximately \(0.73\) seconds, whereas a direct application of an earlier Apéry-set criterion would require checking membership for about \(2\times 7{,}771{,}556{,}800{,}000\) elements [1402.2597]. A second geometric family is given by proportionally modular affine semigroups \[ S=\{x\in \mathbb{N}^p \mid f(x)\bmod b \le g(x)\}, \] with \(f,g:\mathbb{Q}^p\to \mathbb{Q}\) linear forms with integer coefficients after clearing denominators. In dimension \(p=2\), every nontrivial proportionally modular affine semigroup is simplicial. When \(g_1g_2\le 0\), the paper proves that such a semigroup is Cohen–Macaulay and Buchsbaum; the argument uses a translation lemma along the unique generator \(u\) of the solutions of \(g=0\) and \(f\bmod b=0\), from which one gets \(\overline{S}=S\), and then applies the theorem that \(T\) is Buchsbaum if and only if \(\overline{T}\) is Cohen–Macaulay. In the same setting, Gorensteinness is decided by the existence of a unique maximal element in the finite set \[ \{h\in H \mid h-u\notin S,\ h-\widetilde{u}\notin S\}, \] where \(H\) is the finite strip used in the generator algorithm [1512.01513]. ## 4. Gaps, pseudo-Frobenius elements, and finite-complement classifications For simplicial affine semigroups with finite complement in their integer cone, Buchsbaumness admits a sharp gap-theoretic description. If \(S\subseteq \mathbb{N}^d\) is a simplicial \(\mathcal{C}\)-semigroup, meaning that \(\mathcal{C}=\operatorname{cone}(S)\cap \mathbb{N}^d\) and \(\mathcal{C}\setminus S\) is finite, the gap set is \[ \mathcal{H}(S)=\mathcal{C}\setminus S, \] the genus is \(g(S)=|\mathcal{H}(S)|\), and the pseudo-Frobenius set is \[ PF(S)=\{h\in \mathcal{H}(S)\mid h+(S\setminus\{0\})\subseteq S\}. \] The main theorem of the 2025 paper states that, for \(d>1\), \(S\) is Buchsbaum if and only if \(\mathcal{H}(S)=PF(S)\). Equivalently, every gap is pseudo-Frobenius, or every \(h\in \mathcal{H}(S)\) satisfies \(h+(S\setminus\{0\})\subseteq S\) [2507.09267]. This criterion is further converted into a complete structural description. Let \(\mathcal{M}(S)\) denote the minimal elements of \(S\setminus\{0\}\) with respect to the cone order \(\le_{\mathcal{C}}\), and let a “multset” of \(\mathcal{C}\) mean an antichain under \(\le_{\mathcal{C}}\) containing a set of extremal rays. Then, for a simplicial \(\mathcal{C}\)-semigroup with \(d>1\), the following are equivalent: \(S\) is Buchsbaum; \(S\setminus\{0\}\) is an ideal of \(\mathcal{C}\), that is, \((S\setminus\{0\})+\mathcal{C}\subseteq S\setminus\{0\}\); and \[ S=(\mathcal{M}(S)+\mathcal{C})\cup \{0\} \] for some multset \(\mathcal{M}(S)\) containing the extremal rays. Thus Buchsbaum simplicial \(\mathcal{C}\)-semigroups are precisely the \(\mathcal{C}\)-ideals generated by their minimal elements [2507.09267]. The maximal embedding dimension case is also described explicitly. If \[ \operatorname{msg}(S)=\{a_1,\dots,a_d\}\sqcup \{a_{d+1},\dots,a_{d+m}\}, \] with \(\{a_1,\dots,a_d\}\) the extremal rays, then for a Buchsbaum simplicial \(\mathcal{C}\)-semigroup of maximal embedding dimension one has \[ \mu(I_S)=\frac12\bigl[m(m+1)+d(d-1)g(S)\bigr]. \] This formula separates the contribution from pairs of non-extremal minimal generators and the contribution from the genus times the extremal-ray structure [2507.09267]. A closely related but more specialized criterion appears in the theory of simplicial affine semigroups of maximal projective dimension. If \(S\) is simplicial with extremal rays \(a_1,\dots,a_d\), define \[ \mathcal{D}(S)=\{a\in G(S)\setminus S \mid a+2a_i,\ a+2a_j\in S\ \text{for some } i\neq j\}. \] For simplicial MPD-semigroups in \(\mathbb{N}^d\), \(d>1\), the semigroup ring \(k[S]\) is Buchsbaum if and only if \(\mathcal{D}(S)=PF(S)\). The broader simplicial Buchsbaum criterion used there is \[ \{a\in G(S)\mid a+2a_i,\ a+2a_j\in S\ \text{for some } i\neq j\}+\operatorname{Hilb}(S)\subseteq S, \] and the MPD specialization identifies the “double-extremal” holes with pseudo-Frobenius elements [2304.14806]. ## 5. Homological viewpoints: depth, local cohomology, and projective closure The local-cohomological characterization of a Buchsbaum ring is the condition \[ \mathfrak{m}\cdot H^i_{\mathfrak{m}}(R)=0 \quad \text{for all } i<\dim R. \] This criterion is part of the standard background, but the decomposition algorithms for simplicial semigroup rings discussed above do not use it directly; instead, they exploit the decomposition \(K[B]=\bigoplus_g I_g(-h_g)\) and explicit combinatorial criteria on the summands \(I_g\) and shifts \(h_g\) [1206.1735]. Apéry sets nevertheless encode a large part of the relevant homological structure. For a simplicial affine semigroup \(S\subseteq \mathbb{N}^d\) with extremal rays \(E=\{a_1,\dots,a_d\}\), the paper on depth proves that \(k[S]\) has depth one if and only if \(\operatorname{Ap}(S;b)\) has a maximal element for some, equivalently every, \(b\in S\). It also recalls that \(S\) is Cohen–Macaulay if and only if for all \(a,b\in \operatorname{Ap}(S;E)\) with \(b-a\in \bigoplus_{i=1}^d \mathbb{Z}a_i\), one has \(a=b\). In dimension \(3\), depth two is characterized by the absence of maximal elements in \(\operatorname{Ap}(S;b)\) together with the existence of a maximal element in \(\operatorname{Ap}(S;a_i)\cap \operatorname{Ap}(S;a_j)\) for some pair of extremal rays. In dimension \(4\), depth two is described by analogous two-ray Apéry intersections plus explicit translation conditions involving a permutation of four extremal rays [2307.16501]. These depth characterizations are directly relevant to Buchsbaumness but do not themselves constitute a Buchsbaum criterion. The depth paper explicitly presents them as a route toward Buchsbaum verification: one still has to compute the local cohomology modules \(H^i_{\mathfrak{m}}(k[S])\) and check that \(\mathfrak{m}\) annihilates them for all \(i