Admissible matchings and the Castelnuovo–Mumford regularity of square-free powers

Abstract

Square-free powers of square-free monomial ideals relate homological properties of monomial ideals with combinatorial matching structures. Let

be any square-free monomial ideal, and let

H I $ \mathcal {H}_I $ script upper H Subscript upper I

denote its associated hypergraph. We refine the concept of

k $ k $ k

-admissible matching, originally introduced by Erey and Hibi for graphs, by defining a generalized version for arbitrary hypergraphs. This new notion allows us to establish a sharp lower bound for the Castelnuovo–Mumford regularity of the

k th $ k\text {th} $ k th

square-free power

I [ k ] $ I^{[k]} $ upper I Superscript left bracket k right bracket

. When

I $ I $ upper I

is equigenerated, the bound is given in terms of a new combinatorial invariant, the

k $ k $ k

-admissible matching number

aim ( H I , k ) $ \mathrm {aim}(\mathcal {H}_I, k) $ aim left parenthesis script upper H Subscript upper I Baseline comma k right parenthesis

, provided

I [ k ] ≠ 0 $ I^{[k]} \neq 0 $ upper I Superscript left bracket k right bracket Baseline not equals 0

. Surprisingly, for edge ideals

I ( G ) $ I(G) $ upper I left parenthesis upper G right parenthesis

of graphs, this gives the first general lower bound on the regularity of

I ( G ) [ k ] $ I(G)^{[k]}$ upper I left parenthesis upper G right parenthesis Superscript left bracket k right bracket

. When

G $ G $ upper G

is a forest, our invariant coincides with the

k $ k $ k

-admissible matching number of Erey and Hibi. Moreover, we prove that for any block graph

G $ G $ upper G

, the bound becomes an exact formula, thereby generalizing the known regularity formula for forests. We conclude by proposing a conjecture on the regularity of square-free powers of edge ideals of chordal graphs.