The Stacks project

64.7 Properties of Spaces defined by properties of schemes

Any étale local property of schemes gives rise to a corresponding property of algebraic spaces via the following lemma.

Lemma 64.7.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{P}$ be a property of schemes which is local in the étale topology, see Descent, Definition 35.12.1. The following are equivalent

  1. for some scheme $U$ and surjective étale morphism $U \to X$ the scheme $U$ has property $\mathcal{P}$, and

  2. for every scheme $U$ and every étale morphism $U \to X$ the scheme $U$ has property $\mathcal{P}$.

If $X$ is representable this is equivalent to $\mathcal{P}(X)$.

Proof. The implication (2) $\Rightarrow $ (1) is immediate. For the converse, choose a surjective étale morphism $U \to X$ with $U$ a scheme that has $\mathcal{P}$ and let $V$ be an étale $X$-scheme. Then $U \times _ X V \rightarrow V$ is an étale surjection of schemes, so $V$ inherits $\mathcal{P}$ from $U \times _ X V$, which in turn inherits $\mathcal{P}$ from $U$ (see discussion following Descent, Definition 35.12.1). The last claim is clear from (1) and Descent, Definition 35.12.1. $\square$

Definition 64.7.2. Let $\mathcal{P}$ be a property of schemes which is local in the étale topology. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. We say $X$ has property $\mathcal{P}$ if any of the equivalent conditions of Lemma 64.7.1 hold.

Remark 64.7.3. Here is a list of properties which are local for the étale topology (keep in mind that the fpqc, fppf, syntomic, and smooth topologies are stronger than the étale topology):

  1. locally Noetherian, see Descent, Lemma 35.13.1,

  2. Jacobson, see Descent, Lemma 35.13.2,

  3. locally Noetherian and $(S_ k)$, see Descent, Lemma 35.14.1,

  4. Cohen-Macaulay, see Descent, Lemma 35.14.2,

  5. Gorenstein, see Duality for Schemes, Lemma 48.24.6,

  6. reduced, see Descent, Lemma 35.15.1,

  7. normal, see Descent, Lemma 35.15.2,

  8. locally Noetherian and $(R_ k)$, see Descent, Lemma 35.15.3,

  9. regular, see Descent, Lemma 35.15.4,

  10. Nagata, see Descent, Lemma 35.15.5.

Any étale local property of germs of schemes gives rise to a corresponding property of algebraic spaces. Here is the obligatory lemma.

Lemma 64.7.4. Let $\mathcal{P}$ be a property of germs of schemes which is étale local, see Descent, Definition 35.18.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$ be a point of $X$. Consider étale morphisms $a : U \to X$ where $U$ is a scheme. The following are equivalent

  1. for any $U \to X$ as above and $u \in U$ with $a(u) = x$ we have $\mathcal{P}(U, u)$, and

  2. for some $U \to X$ as above and $u \in U$ with $a(u) = x$ we have $\mathcal{P}(U, u)$.

If $X$ is representable, then this is equivalent to $\mathcal{P}(X, x)$.

Proof. Omitted. $\square$

Definition 64.7.5. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$. Let $\mathcal{P}$ be a property of germs of schemes which is étale local. We say $X$ has property $\mathcal{P}$ at $x$ if any of the equivalent conditions of Lemma 64.7.4 hold.

Remark 64.7.6. Let $P$ be a property of local rings. Assume that for any étale ring map $A \to B$ and $\mathfrak q$ is a prime of $B$ lying over the prime $\mathfrak p$ of $A$, then $P(A_\mathfrak p) \Leftrightarrow P(B_\mathfrak q)$. Then we obtain an étale local property of germs $(U, u)$ of schemes by setting $\mathcal{P}(U, u) = P(\mathcal{O}_{U, u})$. In this situation we will use the terminology “the local ring of $X$ at $x$ has $P$” to mean $X$ has property $\mathcal{P}$ at $x$. Here is a list of such properties $P$:

  1. Noetherian, see More on Algebra, Lemma 15.44.1,

  2. dimension $d$, see More on Algebra, Lemma 15.44.2,

  3. regular, see More on Algebra, Lemma 15.44.3,

  4. discrete valuation ring, follows from (2), (3), and Algebra, Lemma 10.119.7,

  5. reduced, see More on Algebra, Lemma 15.45.4,

  6. normal, see More on Algebra, Lemma 15.45.6,

  7. Noetherian and depth $k$, see More on Algebra, Lemma 15.45.8,

  8. Noetherian and Cohen-Macaulay, see More on Algebra, Lemma 15.45.9,

  9. Noetherian and Gorenstein, see Dualizing Complexes, Lemma 47.21.8.

There are more properties for which this holds, for example G-ring and Nagata. If we every need these we will add them here as well as references to detailed proofs of the corresponding algebra facts.


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