The homological shift algebra of a monomial ideal

2024/12/30 by Antonino Ficarra, Ficarra, Antonino, Ayesha Asloob Qureshı +1 · 3 citations
Computer Science · Mathematics · #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2412.21031

openalex publication_date 2024/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let S=K[x₁,…,x_n] be the polynomial ring over a field K, and let I⊂ S be a monomial ideal. In this paper, we introduce the ith homological shift algebras HS_i(R(I))=\bigoplus_k≥1HS_i(I^k) of I. If I has linear powers, these algebras have the structure of a finitely generated bigraded module over the Rees algebra R(I) of I. Hence, many invariants of HS_i(I^k), such as depth, associated primes, regularity, and the v-number, exhibit well behaved asymptotic behavior. We determine several families of monomial ideals I for which HS_i(I^k) has linear resolution for all k≫0. Finally, we show that HS_i(I^k) is Golod for all monomial ideals I⊂ S with linear powers and all k≫0.

Cited by

• Homological shifts of powers a complementary edge ideal
• Polymatroidal ideals and their asymptotic syzygies
• On the homological shifts of cover ideals of Cohen-Macaulay graphs

Related

• Polymatroidal ideals and their asymptotic syzygies
• Edge ideals and their asymptotic syzygies
• Monomial ideals whose powers have a linear resolution
• Linearization of monomial ideals
• Finding all monomials in a polynomial ideal