Projective dimension of powers of cover ideal of Ferrers graphs

Abstract: Let $λ= (λ<em>1, \ldots, λ_n)$ be a partition with $λ_1 = m$. Denote by $J</em>λ$ the cover ideal in the polynomial ring ( S = k[x_1, \ldots, x_n, y_1, \ldots, y_m] ) associated to the Ferrers graph corresponding to $λ$. Let $d(λ)$ denote the number of distinct parts of $λ$. We prove that [ \operatorname{pd}(S/J_λ^t) = \min{t,\; d(λ)} + 1 ] for all $t \ge 1$.

• The paper proves that for every power t ≥ 1, the projective dimension is pd(S/Jλ^t) = 1 + min{t, d(λ)}, where d(λ) counts the distinct parts of the defining partition.
• The authors combine admissible-subgraph reductions, ordered matching numbers, regularity bounds, and induction to establish both upper and lower bounds for the formula.
• The result shows depth decreases by exactly one with each power until stabilizing after d(λ) steps, independently of the field characteristic, while extension to cochordal graphs remains open.

Overview and main result

This paper determines the projective dimension of all ordinary powers of the cover ideal of a Ferrers graph. Let $\lambda = (\lambda_1, \ldots, \lambda_n)$ be a partition with largest part $m$, let $G_\lambda$ be the associated Ferrers (bipartite) graph on vertex set $\{x_1,\ldots,x_n\} \cup \{y_1,\ldots,y_m\}$, and let $J_\lambda$ denote its cover ideal in $S = \kappa[x_1,\ldots,x_n,y_1,\ldots,y_m]$. Writing $d(\lambda)$ for the number of distinct parts of $\lambda$, the main theorem states that for every $t \ge 1$,

$\pd(S/J_\lambda^t) = 1 + \min\{t,\; d(\lambda)\}.$

Equivalently, via the Auslander–Buchsbaum formula, $m$0: the depth decreases by exactly one at each power until it stabilizes after $m$1 steps. This is a notably simple depth behavior. Explicit formulas for the depth function of powers are rare even for edge ideals of graphs, and for cover ideals of graphs the only prior explicit results concern paths and cycles—and those exhibit plateau regions before stabilization. The Ferrers case instead mirrors the behavior of powers of edge ideals of Cohen–Macaulay trees.

Algebraic framework: admissible subgraphs

The proof rests on a recent reduction due to Dung, Hang, and Vu, building on Hochster's formula. For a graph $m$2 with cover ideal $m$3, a subgraph $m$4 is $m$5-admissible if there is an exponent vector $m$6 such that an edge $m$7 lies in $m$8 precisely when it lies in $m$9 and $G_\lambda$0. The key identity is

$G_\lambda$1

For bipartite graphs, Herzog–Hibi–Trung's theorem gives $G_\lambda$2, so ordinary and symbolic powers coincide and this machinery applies directly to the problem at hand. Certificates of admissibility can then be written as pairs $G_\lambda$3, with $G_\lambda$4 if and only if $G_\lambda$5.

Ordered matchings and the number of distinct parts

The stabilizing value of the projective dimension is governed by the ordered matching number $G_\lambda$6, introduced by Constantinescu and Varbaro: Hoa–Kimura–Terai–Trung showed that $G_\lambda$7 is weakly increasing in $G_\lambda$8 with limit $G_\lambda$9. The paper proves the clean combinatorial identification

$\{x_1,\ldots,x_n\} \cup \{y_1,\ldots,y_m\}$0

The argument is short and explicit. If $\{x_1,\ldots,x_n\} \cup \{y_1,\ldots,y_m\}$1 are the row indices where $\{x_1,\ldots,x_n\} \cup \{y_1,\ldots,y_m\}$2 strictly decreases, then the edges $\{x_1,\ldots,x_n\} \cup \{y_1,\ldots,y_m\}$3 form an ordered matching, giving $\{x_1,\ldots,x_n\} \cup \{y_1,\ldots,y_m\}$4. Conversely, any ordered matching forces $\{x_1,\ldots,x_n\} \cup \{y_1,\ldots,y_m\}$5 along its rows, so $\{x_1,\ldots,x_n\} \cup \{y_1,\ldots,y_m\}$6. This lemma immediately implies both the eventual value $\{x_1,\ldots,x_n\} \cup \{y_1,\ldots,y_m\}$7 of the projective dimension and—since the formula holds for all $\{x_1,\ldots,x_n\} \cup \{y_1,\ldots,y_m\}$8—its characteristic-free nature.

Upper bound via decomposition into Ferrers subgraphs

The remaining task is to show that every $\{x_1,\ldots,x_n\} \cup \{y_1,\ldots,y_m\}$9-admissible subgraph $J_\lambda$0 of $J_\lambda$1 satisfies $J_\lambda$2. Given a certificate $J_\lambda$3, define for each $J_\lambda$4 the sets $J_\lambda$5 and $J_\lambda$6, and let $J_\lambda$7 be the induced subgraph on $J_\lambda$8. Then $J_\lambda$9 is the union of the $S = \kappa[x_1,\ldots,x_n,y_1,\ldots,y_m]$0, so $S = \kappa[x_1,\ldots,x_n,y_1,\ldots,y_m]$1. Each $S = \kappa[x_1,\ldots,x_n,y_1,\ldots,y_m]$2 is itself a Ferrers graph, and the Kalai–Meshulam theorem on intersections of Leray complexes yields the desired regularity bound. Combined with the ordered matching computation, this closes the upper side of the formula.

Lower bound by induction on distinct parts

The matching lower bound is constructive. By induction on $S = \kappa[x_1,\ldots,x_n,y_1,\ldots,y_m]$3: for $S = \kappa[x_1,\ldots,x_n,y_1,\ldots,y_m]$4, $S = \kappa[x_1,\ldots,x_n,y_1,\ldots,y_m]$5 itself is the only admissible subgraph and $S = \kappa[x_1,\ldots,x_n,y_1,\ldots,y_m]$6. For $S = \kappa[x_1,\ldots,x_n,y_1,\ldots,y_m]$7, let $S = \kappa[x_1,\ldots,x_n,y_1,\ldots,y_m]$8 be the first index where $S = \kappa[x_1,\ldots,x_n,y_1,\ldots,y_m]$9 and set $d(\lambda)$0, so $d(\lambda)$1. Applying the induction hypothesis to $d(\lambda)$2 and extending its certificate—setting $d(\lambda)$3 for $d(\lambda)$4, shifting all $d(\lambda)$5 up by one below $d(\lambda)$6, and zeroing them above—produces an admissible subgraph that is the disjoint union of the admissible subgraph for $d(\lambda)$7 and a complete bipartite graph. Regularity is additive over disjoint unions, giving $d(\lambda)$8. As the authors note, this induction effectively reduces the study of depths of powers of cover ideals of Ferrers graphs to the Cohen–Macaulay case.

Limitations and open questions

The result is specific to Ferrers graphs, and the paper is candid about the natural boundary of the method. Dung and Vu associate a type sequence to any cochordal graph, generalizing the partition attached to a Ferrers graph, and the authors pose as an open question whether the present results extend to symbolic powers of cover ideals of arbitrary cochordal graphs. The regularity bound $d(\lambda)$9 also relies on each layer $\lambda$0 being a Ferrers graph, so the decomposition argument does not transfer verbatim beyond this class. On the positive side, the formula shows the depth function here is independent of $\lambda$1, and all computations were verified in Macaulay2.

Conclusion

The paper settles the projective dimension of all powers of cover ideals of Ferrers graphs with the exact formula $\lambda$2, identifying the number of distinct parts of the partition as the sole invariant controlling the depth behavior. The proof combines the admissible-subgraph reduction of Dung–Hang–Vu, an exact computation of the ordered matching number, a Kalai–Meshulam-based regularity bound, and an inductive construction attaining it. The work adds Ferrers graphs to the short list of graph classes whose cover ideal depth functions are known explicitly, and leaves extension to cochordal graphs as the principal open direction.