线性代数 | excellent algebraic space

注：本文为 “excellent algebraic space” 相关讨论。
英文引文，机翻未校。
如有内容异常，请看原文。

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What is an excellent algebraic space?

什么是优秀代数空间？（What is an excellent algebraic space?）

What does it mean to say that an algebraic space $S$ is excellent? One knows that excellence of a Noetherian ring is not a property that is etale local (that is, excellence cannot be checked over an etale cover; there are counterexamples in EGA). Thus I wonder: what does excellence actually mean in the context of algebraic spaces?
当我们说一个代数空间 $S$ 是“优秀的（excellent）”时，其含义是什么？我们知道，诺特环（Noetherian ring）的优秀性并非平展局部（etale local）性质——也就是说，无法通过平展覆盖（etale cover）来验证优秀性，在《代数几何基础》（EGA）中存在相关反例。因此我想知道：在代数空间的背景下，优秀性的实际含义究竟是什么？

The question comes from looking at this paper http://imrn.oxfordjournals.org/content/2006/75273 of Max Lieblich. He uses the phrase “excellent algebraic space” five times in the paper without discussing its meaning (as far as I can tell), so I presume the notion is standard. I would be very grateful if someone could explain what it means.
这个问题源于阅读马克斯·利布利希（Max Lieblich）的一篇论文（链接：http://imrn.oxfordjournals.org/content/2006/75273）。据我观察，他在论文中五次使用了“优秀代数空间（excellent algebraic space）”这一表述，却未对其含义进行说明，因此我推测这是一个标准概念。若有人能解释其含义，我将不胜感激。

asked May 1, 2015 at 16:12
O-Ren Ishii

A variant of your question would be: Is being excellent an etale local property of schemes? An affirmative answer to this would give an unambiguous meaning to ‘excellent algebraic space’. Excellent is made of three conditions - G-ring + J2 + universally catenary. (see stacks.math.columbia.edu/tag/07QS) Unfortunately, the property of being universally catenary does not seem to be an etale local property (see stacks.math.columbia.edu/tag/0355).
你的问题可衍生出另一个版本：优秀性是否为概形（schemes）的平展局部性质？若该问题能得到肯定回答，“优秀代数空间”的含义便会清晰明确。优秀性由三个条件构成，即 G-环（G-ring）、J2 条件（J2 condition）与泛链条件（universally catenary）。遗憾的是，泛链性质似乎并非平展局部性质。

Amit H
Commented May 2, 2015 at 10:45

Let $X$ be a Noetherian algebraic space.
设 $X$ 为一个诺特代数空间（Noetherian algebraic space）。

We say $X$ is quasi-excellent if the following equivalent conditions hold: (1) for every scheme $U$ and etale morphism $U\to X$ the scheme $U$ is quasi-excellent, and (2) for some scheme $U$ and surjective etale morphism $U\to X$ the scheme $U$ is quasi-excellent.
若满足以下两个等价条件，则称 $X$ 为“拟优秀的（quasi-excellent）”：（1）对于任意概形 $U$ 及平展态射 $U\to X$，概形 $U$ 是拟优秀的；（2）存在某个概形 $U$ 及满平展态射 $U\to X$，使得概形 $U$ 是拟优秀的。

We say $X$ is universally catenary if for every quasi-separated morphism $Y\to X$ locally of finite type, the topological space $|Y|$ of $Y$ is catenary. (Note that the space $|Y|$ is a sober locally Noetherian topological space.)
若对于任意拟分离（quasi-separated）且局部有限型（locally of finite type）的态射 $Y\to X$，$Y$ 的拓扑空间 $|Y|$ 是链状的（catenary），则称 $X$ 满足“泛链条件（universally catenary）”。（注：空间 $|Y|$ 是一个清醒的（sober）局部诺特拓扑空间。）

We say $X$ is excellent if it is quasi-excellent and universally catenary.
若 $X$ 既为拟优秀的，又满足泛链条件，则称 $X$ 是“优秀的（excellent）”。

Discussion: The problem with this definition is that it is hard to check (so it is not clear that there exist excellent algebraic spaces, besides the ones we know about, namely algebraic spaces of finite type over an excellent base scheme). On the other hand, everywhere in Max’s paper you can replace excellent by quasi-excellent. In fact, in order to use Artin’s results on representability all you need is to work over a base where the local rings are G-rings. Moreover, to establish a stack is algebraic, you may work etale locally on the base (if I understand correctly, this is how the algebraic space you are talking about arises in his paper), hence you can always assume the base is a scheme.
讨论：该定义存在一个问题，即难以验证——除了我们已知的、在优秀基概形（excellent base scheme）上有限型的代数空间外，目前尚不清楚是否存在其他优秀代数空间。不过，在马克斯的论文中，所有“优秀的”表述实际上都可替换为“拟优秀的”。事实上，若要应用阿廷（Artin）关于可表性的结论，只需在局部环为 G-环的基上进行研究即可。此外，要证明一个层（stack）是代数的，可在基上进行平展局部研究（若我理解无误，你所提及的代数空间在其论文中正是通过这种方式构造的），因此始终可假设基为一个概形。

answered May 8, 2015 at 12:08
answered May 8, 2015 at 12:08
Dracula

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via：

• ag.algebraic geometry - What is an excellent algebraic space? - MathOverflow
  https://mathoverflow.net/questions/204458/what-is-an-excellent-algebraic-space
• Section 15.52 (07QS): Excellent rings—The Stacks project
  https://stacks.math.columbia.edu/tag/07QS