Constructible Commutative Semigroup Rings

• Constructible commutative semigroup rings are semigroup rings k[S] over a field, where the affine semigroup S is constructed via algebraic operations like gluing and join.
• They facilitate the explicit construction of projective closures and one-dimensional local rings with controlled Hilbert functions and strongly indispensable minimal free resolutions.
• Utilizing techniques such as Gröbner bases and tensor products, these rings reveal deep combinatorial and homological insights, addressing open problems in commutative algebra.

A constructible commutative semigroup ring is a semigroup ring $k[S]$ over a field $k$, where the affine semigroup $S \subset \mathbb{N}^d$ is obtained via algebraic constructions such as gluing and join. These operations furnish large, explicit families of semigroup rings with prescribed homological and algebraic properties—most notably, projective closures exhibiting the Cohen–Macaulay (CM) or Gorenstein property, infinite families of one-dimensional Gorenstein local rings with controlled Hilbert functions, and classes of semigroup rings with strongly indispensable minimal free resolutions. The study of such rings is tightly interwoven with the structure of the underlying semigroups and their defining lattice ideals, presenting a rich interplay between combinatorial and homological algebraic features (Singh et al., 2023).

1. Affine Semigroups and Their Semigroup Rings

An affine semigroup $S \subset \mathbb{N}^d$ is a finitely generated submonoid of $\mathbb{N}^d$. The associated semigroup ring is defined as $$k[S] := k[t^{a_1}, \ldots, t^{a_n}] \subset k[t_1, \ldots, t_d]$$ for generators $a_1, \ldots, a_n$ of $S$, where $t^{a_i} := t_1^{a_{i1}} \cdots t_d^{a_{id}}$. Equivalently, $k[S] \cong R / I(S)$ with $R = k[x_1, \ldots, x_n]$ and the prime lattice ideal $I(S) = \ker(\varphi)$, $\varphi(x_i) = t^{a_i}$, generated by all binomials $x^u - x^v$ such that $\sum u_i a_i = \sum v_i a_i$ (Singh et al., 2023).

2. Gluing of Numerical Semigroups and Its Ring-Theoretic Consequences

Given numerical semigroups $S_1 = \langle m_1, \ldots, m_\ell \rangle$ and $S_2 = \langle n_1, \ldots, n_k \rangle$ (submonoids of $\mathbb{N}$ with $\gcd$ of generators $1$), gluing constructs a new numerical semigroup $$S = S_1 \#_{p, q} S_2 := \langle q m_1, \ldots, q m_\ell, p n_1, \ldots, p n_k \rangle$$ for suitable integers $p \in S_1$ and $q \in S_2$ satisfying $\gcd(p,q) = 1$ and natural exclusion conditions on $p$ and $q$. The defining ideal $I(S)$ of the glued semigroup can be explicitly computed as

$$I(S) = (f_1, \ldots, f_a, \ g_1, \ldots, g_b, \ x_1^{b_1} \cdots x_\ell^{b_\ell} - y_1^{a_1} \cdots y_k^{a_k})$$

for generators $f_i$ of $I(S_1)$ and $g_j$ of $I(S_2)$ (Singh et al., 2023).

Table: Key Conditions in Numerical-Semigroup Gluing

Parameter	Description	Constraints
$p$	Linear combination in $S_1$	$\gcd(p,q) = 1$, $p \notin \{n_j\}$
$q$	Linear combination in $S_2$	$q \notin \{m_i\}$, set-intersection empty
$S$	Glued semigroup	$S = S_1 \#_{p,q} S_2$ as above

These glued semigroups serve as the basis for constructing projective closures and local rings with desired homological properties.

3. Homological Properties: Cohen–Macaulayness and the Gorenstein Condition

For "nice gluing" $S_1 \#_{p, q} S_2$, defined by $p = b_1 m_1 + \cdots + b_\ell m_\ell$, $q = a_1 n_1$ with $b_1 + \cdots + b_\ell \geq a_1$, the projective closure $C(S) \subset \mathbb{P}^e$, with homogeneous coordinate ring $k[x_0, x_1, \ldots, x_e]/I_h(S)$, exhibits the following property ([(Singh et al., 2023), Theorem 2.10]):

• If the largest generator of $S$ is $p n_k$ (from $S_2$), $C(S)$ is arithmetically Cohen–Macaulay (aCM) or Gorenstein if $C(S_1), C(S_2)$ are so.
• If the largest generator is $q m_\ell$ (from $S_1$), $C(S)$ fails to be aCM.

The proof utilizes a Gröbner basis construction, showing that the union of homogenized Gröbner bases for $I(S_1)$ and $I(S_2)$, together with the binomial corresponding to the gluing, remains a Gröbner basis for $I(S)$. The Gorenstein property for one-dimensional local rings follows from the symmetry of the glued semigroup under nice gluing.

Under "star gluing" (with $q = \sum a_j n_j$, $\sum a_j < \sum b_i$), if the tangent cones of $C(S_1)$, $C(S_2)$ are Cohen–Macaulay, then so is $C(S)$. These constructions yield infinite families of symmetric semigroups whose corresponding local Gorenstein rings have Cohen–Macaulay tangent cones and thus non-decreasing Hilbert functions (positive answers to Rossi's question) (Singh et al., 2023).

4. Joins of Affine Semigroups and Strongly Indispensable Resolutions

For affine semigroups $S_1, S_2 \subset \mathbb{N}^d$ with disjoint and $\mathbb{Q}$-linearly independent sets of extremal rays $E_1, E_2$, the join $S = S_1 \cup S_2 = \langle G_1 \cup G_2 \rangle$ produces a new simplicial semigroup. The semigroup ring $k[S]$ possesses a minimal graded free resolution that is strongly indispensable (SIFR) if, at each homological position, the difference of two distinct $T$-degrees does not lie in $S$ (Singh et al., 2023).

Theorem (3.4): $k[S]$ has a strongly indispensable minimal free resolution if and only if both $k[S_1]$ and $k[S_2]$ do. The tensor product of minimal free resolutions $F_\bullet(S_1) \otimes_k F_\bullet(S_2)$ yields a minimal free resolution of $k[S]$, and the Cauchy-product formula for the Betti numbers holds:

$$\beta_{p, m}(S) = \sum_{i + i' = p} \sum_{j + j' = m} \beta_{i, j}(S_1) \cdot \beta_{i', j'}(S_2).$$

The differentials act as in the tensor product of complexes.

5. Explicit Constructions and Illustrative Examples

Several concrete families and explicit computations underscore the power of the gluing and join processes:

• Cohen–Macaulay Gluing: $S_1 = \langle 3, 5 \rangle$, $S_2 = \langle 7, 8 \rangle$; with $p=11$, $q=15$, the glued semigroup $S$ yields projective closure $C(S) \subset \mathbb{P}^3$ that is arithmetically Cohen–Macaulay.
• Gorenstein Monomial Curves and Hilbert Functions: For $S_1 = \langle 3,5,7 \rangle$, $S_2 = \langle 9, 11 \rangle$, star gluing with $p=28$, $q=29$ gives $S = \langle 87, 145, 203, 252, 308 \rangle$. The Hilbert function $H(n)$ computed as $\dim_k m^n/m^{n+1}$ is non-decreasing: $H(0) = 1, H(1) = 5, H(2) = 10, H(3) = 15, \ldots$.
• Join Example and SIFR: For $$S_1 = \langle (1,0), (1,2) \rangle \subset \mathbb{N}^2$$ and $$S_2 = \langle (0,1), (2,1) \rangle \subset \mathbb{N}^2$$, the join $S$ with generators $\{(1,0), (1,2), (0,1), (2,1)\}$ has $I(S) = (x_1^2 - x_2, y_1^2 - y_2^3)$ in $k[x_1, x_2, y_1, y_2]$. The tensor product of the two Koszul-type resolutions yields a 4-step minimal resolution, and since both factors are strongly indispensable, so is the join (Singh et al., 2023).

6. Broader Algebraic and Combinatorial Relevance

The gluing and join constructions address open questions in the theory of semigroup rings and local algebra. Infinitely many new examples of Gorenstein local rings with non-decreasing Hilbert functions (affirming Rossi's question) are produced via star gluing. The join operation provides a systematic way to construct semigroup rings with strongly indispensable minimal free resolutions, contributing examples relevant to the question posed by Charalambous and Thoma on lattice ideals. The explicit description of generators and relations also aids computational approaches in commutative algebra and leads to effective criteria for key algebraic properties (Singh et al., 2023).