2025/08/14 by Ficarra, Antonino, Moradi, Somayeh · 6 citations
#Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.2508.10354
Let S = K[x1, …, xn] be the standard graded polynomial ring over a field K. In this paper, we address and completely solve two fundamental open questions in Commutative Algebra: (i) For which degrees d, does there exist a uniform combinatorial characterization of all squarefree monomial ideals in S having d-linear resolutions? (ii) For which degrees d, does having a linear resolution coincide with having linear powers for all squarefree monomial ideals of S generated in degree d? Let In,d(K) denote the class of squarefree monomial ideals of S having a d-linear resolution. Our main result establishes the equivalence of the following conditions: (a) Any squarefree monomial ideal I in S generated in degree d has a linear resolution, if and only if, I has linear powers. (b) In,d(K) is independent of the base field K. (c) d∈\0,1,2,n-2,n-1,n\. In each of these degrees, we show that a squarefree monomial ideal has a linear resolution if and only if all of its powers admit linear quotients, and we combinatorially classify such ideals. In contrast, for each degree 3≤ d≤ n-3, we construct fully-supported squarefree monomial ideals I and J in S generated in degree d such that the linear resolution property of I depends on the choice of the base field, J has a linear resolution and J2 does not have a linear resolution.