Rees Algebras of Squarefree Monomial Ideals

• Rees algebras of squarefree monomial ideals are blowup algebras encoding the asymptotic behavior of all powers of ideals generated by distinct-variable monomials.
• They are characterized by explicit combinatorial constructions using generator graphs and polyhedral methods to determine properties like normality, Cohen–Macaulayness, and Koszulness.
• Research explores how combinatorial structures, from vertex decomposable complexes to Ferrers ideals, influence algebraic invariants such as depth, regularity, and symbolic defects.

A Rees algebra of a squarefree monomial ideal is a blowup algebra that encodes the asymptotic behavior of all powers (or symbolic powers) of an ideal generated by monomials with support on distinct subsets of variables. The subject forms a nexus between commutative algebra, combinatorics (particularly graph theory and hypergraph theory), polyhedral geometry, and optimization. Key themes include the explicit description of defining equations, the normality, Cohen–Macaulayness, and Koszulness of the algebras, and the profound impact of combinatorial properties of the underlying objects (graphs, clutters, simplicial complexes) on the algebraic invariants and homological properties.

1. Combinatorial Construction and Defining Equations

For a squarefree monomial ideal $I \subset S = K[x_1, \dots, x_n]$, generated by monomials $f_1, \dots, f_m$, one constructs the Rees algebra via a surjection

$$\varphi: S[y_1, \dots, y_m] \to S[It], \quad y_j \mapsto f_j t,$$

with kernel $J$ giving the defining equations. For ideals generated in fixed degree, $J$ has an explicit structure: it is generated by binomials of the form

$$T_{\alpha, \beta} = \left( T_{\alpha} \frac{f_{\beta}}{\gcd(f_{\alpha}, f_{\beta})} \right) - \left( T_{\beta} \frac{f_{\alpha}}{\gcd(f_{\alpha}, f_{\beta})} \right),$$

where $\alpha, \beta$ index disjoint subsets of generators (Fouli et al., 2012). For ideals of linear type, all generators of $J$ are linear forms; more generally, the relation type $\operatorname{rt}(I)$ (maximum degree among minimal generators of $J$) is sharply bounded above by $n - 2$ for $n \leq 5$, with extremal examples constructed for $n > 4$.

The combinatorial structure underlying these relations is captured by the generator graph $\mathcal{G}(I)$, which has vertices for the generators and edges encoding nontrivial gcds. Nonlinear relations correspond to even closed walks in this graph (Alilooee et al., 2013).

2. Normality, Cohen–Macaulayness, and Koszul Property

Normality and Cohen–Macaulayness are crucial for the behavior of the Rees algebra and for its geometric interpretation in blowing up a variety along a monomial subscheme. For squarefree monomial ideals attached to vertex decomposable simplicial complexes, the associated Rees algebra is always a normal Cohen–Macaulay domain, and the ideal enjoys strong persistence (the set of associated primes of powers stabilizes monotonically) (Moradi, 2023). This extends to squarefree weakly polymatroidal ideals, cover ideals of vertex decomposable graphs (including chordal and Cameron–Walker graphs), and to facet ideals of pure forests.

Koszulness is tethered to the existence of a quadratic (often squarefree) Gröbner basis. For certain principal strongly stable ideals and multi-Rees algebras constructed from them, explicit combinatorial “sorting” techniques produce quadratic Gröbner bases, thereby ensuring Koszulness, normality, and Cohen–Macaulayness (Sosa, 2014). For Rees algebras associated to trees and unicyclic graphs with cycles of length $3$ or $4$—in the context of complementary edge ideals—a quadratic Gröbner basis is present (Ficarra et al., 22 Sep 2025), guaranteeing Koszulness.

3. Symbolic Powers, Polyhedral Constructions, and Optimization

The symbolic Rees algebra $$\mathcal{R}^s(I) = \bigoplus_{k=0}^\infty I^{(k)} t^k$$ records the increasing sequence of symbolic powers. Its structure is closely tied to the combinatorics of the underlying clutter or graph: generators correspond to irreducible $b$-covers (exponent vectors $a$ such that the sum over any edge is at least $b$) (Dupont, 2010).

Polyhedral geometry provides analytic tools via the covering polyhedron:

$Q(C) = \{ x \in \mathbb{R}_+^s : xC \ge 1 \}$

where $C$ is an incidence matrix. The Waldschmidt constant and ic-resurgence are then formulated as solutions to linear or linear-fractional programming problems over $Q(C)$, with explicit algorithms for their computation (Grisalde et al., 2021). Normality and integral closure are linked to equality of Newton polyhedra and irreducible polyhedra.

Connections to combinatorial optimization arise, especially through the Conforti–Cornuéjols/packing conjecture (Montaño et al., 2018, Francisco et al., 2013): equality of ordinary and symbolic powers $I^{(n)} = I^n$ for all $n$ corresponds to the max-flow min-cut property for clutters and can be algorithmically checked using linear programming duality.

4. Depth, Regularity, Analytic Spread, and Homological Invariants

The eventual depth of powers of $I$,

$\lim_{k \to \infty} \operatorname{depth} S/I^k,$

is controlled by analytic spread $\ell(I)$, as in

$\operatorname{depth} S/I = n - \ell(I),$

for ideals with Cohen–Macaulay Rees algebra (Herzog et al., 2012). For complementary edge ideals, the limiting depth equals the number of bipartite connected components $b(G)$ of $G$ (Ficarra et al., 22 Sep 2025). The index of depth stability is bounded above by $n - 2$, and sharp for path graphs.

Regularity bounds for linearly presented squarefree ideals are now established: for an ideal $I$ generated in degree $d$ with linear syzygies,

$$\operatorname{reg}(I) \le \max\left\{ 3d, \frac{(d-1)n}{d+1} + 1 \right\},$$

and for those satisfying Serre’s $S_2$ condition, the projective/cohomological dimension is similarly bounded (Dao et al., 15 Jun 2024).

For ideals with linear powers, explicit formulas for Betti numbers as functions of $k$ are derived, especially for ideals generated by all squarefree monomials of degree $d$. The fiber type property and quadratic generation of the Rees ideal for polymatroidal ideals relate directly to White’s conjecture on matroids (Nicklasson, 2019).

5. Special Classes: Tame Ideals, Trees, Ferrers Ideals, and Vertex Decomposable Complexes

Tame ideals are precisely those whose blowup yields a regular scheme. For squarefree monomial ideals, tameness equates to the associated clutter being a union of isolated vertices and a complete $d$-partite $d$-uniform clutter, or equivalently to the facets of the underlying simplicial complex having pairwise disjoint complements (Nejad et al., 2016). These ideals are always of fiber type.

For three-dimensional Ferrers ideals satisfying the projection property, the Rees algebra is Koszul and of fiber type with a presentation ideal generated by quadratic $2$-minors (Lin et al., 2017).

Vertex decomposable complexes afford a systematic mechanism for constructing squarefree monomial ideals with normal, Cohen–Macaulay Rees algebras. The Biermann–Van Tuyl construction creates new families of such ideals via coloring techniques (Moradi, 2023).

6. Lefschetz Properties, Symbolic Defects, and Mixed Multiplicities

The Lefschetz properties (WLP/SLP) for artinian algebras defined by squarefree monomial ideals are accessed via multiplication maps encoded either by the log-matrix or incidence matrices of the underlying facet ideals (Holleben, 18 Apr 2024). Analytic spread equals the rank of the log-matrix, connecting algebraic and combinatorial spread.

Symbolic powers and their defects are quantified in terms of the $f$-vector of the simplicial complex. The symbolic defect polynomial records the number of failures of symbolic and ordinary powers to coincide at each graded piece.

Mixed multiplicities generalize classical Eulerian numbers: for pure $d$-dimensional simplicial complexes, positivity of mixed Eulerian numbers for sequences of facet ideals is proven, linking Hilbert polynomials of Rees algebras to combinatorial volumes (Holleben, 18 Apr 2024).

7. Synthesis and Future Directions

The paper of Rees algebras of squarefree monomial ideals reveals an intricate web of connections: algebraic invariants (syzygies, regularity, depth, normality) are determined by geometric, combinatorial, and optimization-theoretic data. The generator graph, covering polyhedron, and Simis cone transfer combinatorial and polyhedral properties into algebraic relations and homological features. Classes such as trees and vertex decomposable complexes ensure tractable and “nice” properties, while more general clutters and ideals furnish examples exhibiting extremality, fiber type, or homological complexity.

Open research directions include:

• Full classification of linear type and fiber type ideals via their generator graphs or facet complexes
• Quantitative behavior of regularity and Betti numbers in higher degree and non-uniform settings
• Algorithmic advances in correlating combinatorial optimization problems and Rees algebra invariants
• Generalizations of Lefschetz property criteria in terms of Rees algebra data
• Exploration of symbolic defect polynomials and their geometric/topological significance

These themes collectively underscore that Rees algebras of squarefree monomial ideals are central objects in modern combinatorial commutative algebra—providing both explicit, computable invariants and a deep theoretical bridge to geometry and optimization.