Fluctuations in depth and associated primes of powers of ideals

Roswitha Rissner, Irena Swanson
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Abstract

We count the numbers of associated primes of powers of ideals as defined in $\href{https://dx.doi.org/10.1007/s11512-013-0184-1}{[2]}$. We generalize those ideals to monomial ideals $\operatorname{BHH}(m, r, s)$ for $r \geq 2, m, s \geq1$; we establish partially the associated primes of powers of these ideals, and we establish completely the depth function of quotients by powers of these ideals: the depth function is periodic of period $r$ repeated $m$ times on the initial interval before settling to a constant value. The number of needed variables for these depth functions are lower than those from general constructions in $\href{https://doi.org/10.1090/proc/15083}{[6]}$.
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理想幂的深度波动和相关素数
我们计算了 $\href{https://dx.doi.org/10.1007/s11512-013-0184-1}{[2]}$ 中定义的幂的相关素数。我们把这些理想归纳为 $r \geq 2, m, s \geq1$ 的单项式理想 $operatorname{BHH}(m,r,s)$;我们部分地建立了这些理想的幂的相关素数,并完全建立了这些理想的幂的商的深度函数:深度函数是周期为 $r$ 的周期性函数,在初始区间上重复 $m$ 次,然后稳定为一个常值。这些深度函数所需的变量数目低于 $\href{https://doi.org/10.1090/proc/15083}{[6]}$ 中一般构造的变量数目。
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