构造非链状拟优局部预完备化
Constructing Noncatenary Quasi-Excellent Precompletions
AI总结：
本文研究局部环中两个素理想的粘合问题，给出存在拟优局部子环同时保持完备化与素谱性质的充分条件，并据此构造出素谱含非链状有限子集的拟优局部环。
摘要：
设 $T$ 为局部（诺特）环，并设 $Q_1$ 和 $Q_2$ 为 $T$ 的素理想。我们找到了存在 $T$ 的拟优局部子环 $B$ 满足以下条件的充分条件：（1）$B$ 在其极大理想处的完备化同构于 $T$ 在其极大理想处的完备化；（2）$B \cap Q_1 = B \cap Q_2$；（3）当视为偏序集时，$T/(Q_1 \cap Q_2)$ 的正高度素理想集合与 $B/(B \cap Q_1)$ 的正高度素理想集合相同；（4）对于 $i = 1$ 和 $i = 2$，在 $T_{Q_i}$ 的极小素理想与 $B_{B \cap Q_1}$ 的极小素理想之间存在一个保持余高度的双射。直观地说，这意味着 $T$ 包含一个拟优局部子环，其中 $Q_1$ 和 $Q_2$ 被“粘合在一起”，并且既保留了完备化，也保留了素谱的良好性质。我们利用这一结果证明某些完备局部环是一个拟优局部环的完备化，该拟优局部环的素谱作为偏序集包含有趣的非链状有限子集。
英文摘要：
Let $T$ be a local (Noetherian) ring and let $Q_1$ and $Q_2$ be prime ideals of $T$. We find sufficient conditions for there to exist a quasi-excellent local subring $B$ of $T$ satisfying the following conditions: (1) the completion of $B$ at its maximal ideal is isomorphic to the completion of $T$ at its maximal ideal, (2) $B \cap Q_1 = B \cap Q_2$, (3) the set of prime ideals of $T/(Q_1 \cap Q_2)$ of positive height is the same as the set of prime ideals of $B/(B \cap Q_1)$ of positive height when viewed as partially ordered sets, and (4) for $i = 1$ and for $i = 2$, there is a coheight preserving bijection between the minimal prime ideals of $T_{Q_i}$ and the minimal prime ideals of $B_{B \cap Q_1}$. Intuitively, this means that $T$ contains a quasi-excellent local subring in which $Q_1$ and $Q_2$ are "glued together" and such that both the completion and desirable properties of the prime spectrum are preserved. We use this result to show that certain complete local rings are the completion of a quasi-excellent local ring whose prime spectrum, when viewed as a partially ordered set, contains interesting noncatenary finite subsets.
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