For monomial ideals with linear powers, the homological shift algebra is a finitely generated Rees module, making regularity, depth, associated primes, v-number, and Golodness of homological shift ideals eventually linear or stable.
Bayati, A Quasi-additive Property of Homological Shift Ideals
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The homological shift algebra of a monomial ideal
For monomial ideals with linear powers, the homological shift algebra is a finitely generated Rees module, making regularity, depth, associated primes, v-number, and Golodness of homological shift ideals eventually linear or stable.