Betti splittings for powers of sums of ideals

2016/05/31 by Hop D. Nguyen, Nguyen, Hop D. · 1 citation
Computer Science · Mathematics · #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #Rings and Algebras (math.RA) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.1605.09621

openalex publication_date 2016/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let A and B be standard graded polynomial rings over a field k and I and J be non-zero, proper homogeneous ideals contained in A and B, respectively. Denote by P the sum of I and J in R=A⊗_k B. Under reasonable conditions on k, I and J, we provide exact formulas and describe the asymptotic behavior of the depth and the regularity of the powers of P in terms of the data of I and J. Thereby, we strengthen previous work of H.T. Hà, N.V. Trung and T.N. Trung. Our main technical result says that, under the aforementioned conditions, for all s≥ 0 and all n≥ 1, the simple decomposition I^sPⁿ=I^s+1P^n-1+I^sJⁿ yields a Betti splitting for I^sPⁿ. A decomposition of an ideal L as a sum of two subideals is called a Betti splitting if the minimal free resolution of L is completely determined by those of the summands and their intersection.

Citations

• THE EVENTUAL STABILITY OF DEPTH, ASSOCIATED PRIMES AND COHOMOLOGY OF A GRADED MODULE

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• Nonincreasing depth functions of monomial ideals

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