论文标题
主要理想的整体关闭并不总是主要的
The integral closure of a primary ideal is not always primary
论文作者
论文摘要
1936年，克鲁尔（Krull）询问主要理想的整体闭合是否仍然是主要的。五十年后，霍纳克（Huneke）通过给出一个主要的多项式理想，部分地回答了这个问题，在常规的特征特征$ p = 2 $中，其积分闭合并不是主要的。我们对Krull关于具有任何特征的多项式环的问题进行反例。我们还发现，多项式$ f = x^6 + y^6 + x^4 z t + z^3 $由Briançon和Speder在1975年给出的jacobian理想$ j $是克鲁尔问题的反例。令$ v_1 $为由$ f = 0 $定义的超曲面，而$ v_2 $是其单数基因座。 Briançon和Speder证明，惠特尼的平等性并不意味着Zariski Equistionality，表明这对$（v_1 \ setminus v_2，\ v_2）$满足了whitney围绕起源的条件，但失败了Zariski的Equisingular条件。我们发现，这对$（v_1 \ setMinus v_2，\ v_2）$在整体关闭$ \ bar {j} $的嵌入式元素下使惠特尼的条件失败，这意味着$ v_1 $不是whitney of $ v_2 $。此外，我们还表明，该超表面的惠特尼分层与Hauenstein和Wampler给出的同骨组的分层不同，这与Thom-Boardman Singularity有关。
In 1936, Krull asked if the integral closure of a primary ideal is still primary. Fifty years later, Huneke partially answered this question by giving a primary polynomial ideal whose integral closure is not primary in a regular local ring of characteristic $p=2$. We provide counterexamples to Krull's question regarding polynomial rings with any characteristics. We also find that the Jacobian ideal $J$ of the polynomial $f = x^6 + y^6 + x^4 z t + z^3$ given by Briançon and Speder in 1975 is a counterexample to Krull's question. Let $V_1$ be the hypersurface defined by $f = 0$ and $V_2$ be its singular locus. Briançon and Speder proved that Whitney equisingularity does not imply Zariski equisingularity by showing that the pair $(V_1 \setminus V_2,\ V_2)$ satisfies Whitney's conditions around the origin but fails Zariski's equisingular conditions. We discover that the pair $(V_1 \setminus V_2,\ V_2)$ fails Whitney's conditions at the variety of the embedded prime of the integral closure $\bar{J}$, which means that $V_1$ is not Whitney regular along $V_2$. Moreover, we also show that Whitney stratification of this hypersurface is different from the stratification of isosingular sets given by Hauenstein and Wampler, which is related to Thom-Boardman singularity.
