29.19 The singular locus, reprise

We look for a criterion that implies openness of the regular locus for any scheme locally of finite type over the base. Here is the definition.

Definition 29.19.1. Let $X$ be a locally Noetherian scheme. We say $X$ is J-2 if for every morphism $Y \to X$ which is locally of finite type the regular locus $\text{Reg}(Y)$ is open in $Y$.

This is the analogue of the corresponding notion for Noetherian rings, see More on Algebra, Definition 15.47.1.

Lemma 29.19.2. Let $X$ be a locally Noetherian scheme. The following are equivalent

1. $X$ is J-2,

2. there exists an open covering of $X$ all of whose members are J-2 schemes,

3. for every affine open $\mathop{\mathrm{Spec}}(R) = U \subset X$ the ring $R$ is J-2, and

4. there exists an affine open covering $S = \bigcup U_ i$ such that each $\mathcal{O}(U_ i)$ is J-2 for all $i$.

Moreover, in this case any scheme locally of finite type over $X$ is J-2 as well.

Proof. By Lemma 29.15.5 an open immersion is locally of finite type. A composition of morphisms locally of finite type is locally of finite type (Lemma 29.15.3). Thus it is clear that if $X$ is J-2 then any open and any scheme locally of finite type over $X$ is J-2 as well. This proves the final statement of the lemma.

If $\mathop{\mathrm{Spec}}(R)$ is J-2, then for every finite type $R$-algebra $A$ the regular locus of the scheme $\mathop{\mathrm{Spec}}(A)$ is open. Hence $R$ is J-2, by definition (see More on Algebra, Definition 15.47.1). Combined with the remarks above we conclude that (1) implies (3), and (2) implies (4). Of course (1) $\Rightarrow$ (2) and (3) $\Rightarrow$ (4) trivially.

To finish the proof we show that (4) implies (1). Assume (4) and let $Y \to X$ be a morphism locally of finite type. We can find an affine open covering $Y = \bigcup V_ j$ such that each $V_ j \to X$ maps into one of the $U_ i$. By Lemma 29.15.2 the induced ring map $\mathcal{O}(U_ i) \to \mathcal{O}(V_ j)$ is of finite type. Hence the regular locus of $V_ j = \mathop{\mathrm{Spec}}(\mathcal{O}(V_ j))$ is open. Since $\text{Reg}(Y) \cap V_ j = \text{Reg}(V_ j)$ we conclude that $\text{Reg}(Y)$ is open as desired. $\square$

Lemma 29.19.3. The following types of schemes are J-2.

1. Any scheme locally of finite type over a field.

2. Any scheme locally of finite type over a Noetherian complete local ring.

3. Any scheme locally of finite type over $\mathbf{Z}$.

4. Any scheme locally of finite type over a Noetherian local ring of dimension $1$.

5. Any scheme locally of finite type over a Nagata ring of dimension $1$.

6. Any scheme locally of finite type over a Dedekind ring of characteristic zero.

7. And so on.

Proof. By Lemma 29.19.2 we only need to show that the rings mentioned above are J-2. For this see More on Algebra, Proposition 15.48.7. $\square$

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