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 $Y$.
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.
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
$X$ is J-2,
there exists an open covering of $X$ all of whose members are J-2 schemes,
for every affine open $\mathop{\mathrm{Spec}}(R) = U \subset X$ the ring $R$ is J-2, and
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.
Any scheme locally of finite type over a field.
Any scheme locally of finite type over a Noetherian complete local ring.
Any scheme locally of finite type over $\mathbf{Z}$.
Any scheme locally of finite type over a Noetherian local ring of dimension $1$.
Any scheme locally of finite type over a Nagata ring of dimension $1$.
Any scheme locally of finite type over a Dedekind ring of characteristic zero.
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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