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arXiv 1702.01766math.ACmath.AG
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理想和的符号幂

Symbolic powers of sums of ideals

Huy Tai Ha, Dang Hop Nguyen, Ngo Viet Trung, Tran Nam Trung

备注：
22 pages, to appear in Math. Z.; This version does not contain the result on the depth function of powers of a homogeneous ideal, due to a recommendation of the referee
发表信息：
Mathematische Zeitschrift 294, 1499-1520 (2020)
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摘要：

设$I$和$J$是域$k$上两个Noether代数$A$和$B$中的非零理想.令$I+J$表示由$A\otimes_k B$中的$I$和$J$生成的理想。我们证明了以下扩展的符号权力：$$(I+J)^{(n)} = \sum_{i+j = n} I^{(i)} J^{(j)}.$$如果$A$和$B$是多项式环，如果CHARA $(k) = 0$或如果$I$和$J$是单项理想，我们给出了精确的公式的深度和Castelnuovo-Mumford正则性的$(I+J)^{(n)}$，这取决于之间的相互作用的符号权力的$I$和$J$。该证明涉及独立感兴趣的结果，该结果表明，在上述假设下，对于所有$i \ge 0$，$n \ge 0$，诱导映射Tor $_i^A(k,I^{(n)}) \to$ Tor $_i^A(k,I^{(n-1)})$为零。我们还研究了$(I+J)^{(n)}$的其他性质和不变量，如普通幂和符号幂之间的相等性，Waldschleman常数和Cohen-Macaulayness。

英文摘要：

Let $I$ and $J$ be nonzero ideals in two Noetherian algebras $A$ and $B$ over a field $k$. Let $I+J$ denote the ideal generated by $I$ and $J$ in $A\otimes_k B$. We prove the following expansion for the symbolic powers: $$(I+J)^{(n)} = \sum_{i+j = n} I^{(i)} J^{(j)}.$$ If $A$ and $B$ are polynomial rings and if chara$(k) = 0$ or if $I$ and $J$ are monomial ideals, we give exact formulas for the depth and the Castelnuovo-Mumford regularity of $(I+J)^{(n)}$, which depend on the interplay between the symbolic powers of $I$ and $J$. The proof involves a result of independent interest which states that under the above assumption, the induced map Tor$_i^A(k,I^{(n)}) \to$ Tor$_i^A(k,I^{(n-1)})$ is zero for all $i \ge 0$, $n \ge 0$. We also investigate other properties and invariants of $(I+J)^{(n)}$ such as the equality between ordinary and symbolic powers, the Waldschmidt constant and the Cohen-Macaulayness.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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