论文标题
理想总和的整体关闭
Integral closures of powers of sums of ideals
论文作者
论文摘要
让$ k $为一个字段，让$ a $和$ b $为$ k $的多项式戒指，让$ s = a \ otimes_k b $。令$ i \ subseteq a $和$ j \ subseteq b $为单一理想。我们建立了$ i+j \ subseteq s $的二项式扩展，就$ i $和$ j $而言。特别是，对于积极的有理数$ u $，我们证明$（i+j）_u = \ sum_ {0 \ leω\ le u，\ le u，\ω\ in \ mathbb {q}}i_Ωj__{u-ω}，$，右侧的总和是有限的总和。使用$ i $和$ j $的合理权力的跳跃数量可以使这笔有限的总和更加精确。在$ i $和$ j $的情况下，我们进一步提供了足够的条件，以使该公式保持$ i+j $的整体关闭。在这些条件下，我们就$ i $和$ j $的权力方面提供了$ \ overline {（i+j）^k} $的深度和规律性的明确公式。
Let $k$ be a field, let $A$ and $B$ be polynomial rings over $k$, and let $S= A \otimes_k B$. Let $I \subseteq A$ and $J \subseteq B$ be monomial ideals. We establish a binomial expansion for rational powers of $I+J \subseteq S$ in terms of those of $I$ and $J$. Particularly, for a positive rational number $u$, we prove that $(I+J)_u = \sum_{0 \le ω\le u, \ ω\in \mathbb{Q}} I_ωJ_{u-ω},$ and that the sum on the right hand side is a finite sum. This finite sum can be made more precise using jumping numbers of rational powers of $I$ and $J$. We further give sufficient conditions for this formula to hold for the integral closures of powers of $I+J$ in terms of those of $I$ and $J$. Under these conditions, we provide explicit formulas for the depth and regularity of $\overline{(I+J)^k}$ in terms of those of powers of $I$ and $J$.
