标题： 保持维数的有限偏序集饱和嵌入到诺特唯一因子分解环谱中 显示英文标题

标题： Dimension-Preserving Saturated Embeddings of Finite Posets into the Spectra of Noetherian UFDs

摘要： 给定一个有限的偏序集$X$，我们找到存在局部Noether UFD$A$和偏序集的饱和嵌入$\phi : X \longrightarrow \mbox{Spec}(A)$的必要且充分条件，使得$\dim(X)=\dim(A)$。我们在$X$上施加的条件非常温和，表明存在一大类有限偏序集，可以在保持饱和链的情况下嵌入到与$X$维数相同的局部Noether UFD的谱中。 我们还证明了，给定任意有限偏序集 $Y$，存在一个半局部拟优秀环 $S$ 和一个饱和嵌入 $\psi: Y \longrightarrow \mbox{Spec}(S)$，使得若 $z$ 是 $Y$ 的最小元素，则 $\psi(z)$ 是 $S$ 的最小素理想，并且 $\psi(z)$ 的余高与 $Y$ 中从 $z$ 开始到 $Y$ 的最大元素结束的最长链的长度相同。 显示英文摘要

摘要： Given a finite poset $X$, we find necessary and sufficient conditions for there to exist a local Noetherian UFD $A$ and a saturated embedding of posets $\phi : X \longrightarrow \mbox{Spec}(A)$ such that $\dim(X)=\dim(A)$. The conditions imposed on $X$ in our characterization are remarkably mild, demonstrating that there is a large class of finite posets that can be embedded into the spectrum of a local Noetherian UFD of the same dimension as $X$ in a way that preserves saturated chains. We also show that given any finite poset $Y$, there exists a semi-local quasi-excellent ring $S$ and a saturated embedding $\psi: Y \longrightarrow \mbox{Spec}(S)$ such that if $z$ is a minimal element of $Y$, then $\psi(z)$ is a minimal prime ideal of $S$ and the coheight of $\psi(z)$ is the same as the length of the longest chain in $Y$ that starts at $z$ and ends at a maximal element of $Y$.