Betti splittings for powers of sums of ideals

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 $R=A{\otimes }_{k}B$. 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 $s\ge 0$ and all $n\ge 1$, the simple decomposition $I^s{P}^n=I^{s+1}{P}^{n-1}+I^s{J}^n$ yields a Betti splitting for $I^s{P}^n$. 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.

arXiv commentThis paper contains no error to the author's knowledge. The author has integrated it to arXiv:1607.07380 [math.AC] and does not intend to publish it on its own

Cited by in corpus (1)

• Nonincreasing depth functions of monomial ideals