30.10 The singular locus of a morphism

Let $f : X \to S$ be a finite type morphism of schemes. The set $U$ of points where $f$ is smooth is an open of $X$ (by Morphisms, Definition 28.32.1). In many situations it is useful to have a canonical closed subscheme $\text{Sing}(f) \subset X$ whose complement is $U$ and whose formation commutes with arbitrary change of base.

If $f$ is of finite presentation, then one choice would be to consider the closed subscheme $Z$ cut out by functions which are affine locally “strictly standard” in the sense of Smoothing Ring Maps, Definition 16.2.3. It follows from Smoothing Ring Maps, Lemma 16.2.7 that if $f' : X' \to S'$ is the base change of $f$ by a morphism $S' \to S$, then $Z' \subset S' \times _ S Z$ where $Z'$ is the closed subscheme of $X'$ cut out by functions which are affine locally strictly standard. However, equality isn't clear. The notion of a strictly standard element was useful in the chapter on Popescu's theorem. The closed subscheme defined by these elements is (as far as we know) not used in the literature1.

If $f$ is flat, of finite presentation, and the fibres of $f$ all are equidimensional of dimension $d$, then the $d$th fitting ideal of $\Omega _{X/S}$ is used to get a good closed subscheme. For any morphism of finite type the closed subschemes of $X$ defined by the fitting ideals of $\Omega _{X/S}$ define a stratification of $X$ in terms of the rank of $\Omega _{X/S}$ whose formation commutes with base change. This can be helpful; it is related to embedding dimensions of fibres, see Varieties, Section 32.45.

Lemma 30.10.1. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $X = Z_{-1} \supset Z_0 \supset Z_1 \supset \ldots$ be the closed subschemes defined by the fitting ideals of $\Omega _{X/S}$. Then the formation of $Z_ i$ commutes with arbitrary base change.

Proof. Observe that $\Omega _{X/S}$ is a finite type quasi-coherent $\mathcal{O}_ X$-module (Morphisms, Lemma 28.31.12) hence the fitting ideals are defined. If $f' : X' \to S'$ is the base change of $f$ by $g : S' \to S$, then $\Omega _{X'/S'} = (g')^*\Omega _{X/S}$ where $g' : X' \to X$ is the projection (Morphisms, Lemma 28.31.10). Hence $(g')^{-1}\text{Fit}_ i(\Omega _{X/S}) \cdot \mathcal{O}_{X'} = \text{Fit}_ i(\Omega _{X'/S'})$. This means that

$Z'_ i = (g')^{-1}(Z_ i) = Z_ i \times _ X X'$

scheme theoretically and this is the meaning of the statement of the lemma. $\square$

The $0$th fitting ideal of $\Omega$ cuts out the “ramified locus” of the morphism.

Lemma 30.10.2. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. The closed subscheme $Z \subset X$ cut out by the $0$th fitting ideal of $\Omega _{X/S}$ is exactly the set of points where $f$ is not unramified.

Proof. By Lemma 30.9.3 the complement of $Z$ is exactly the locus where $\Omega _{X/S}$ is zero. This is exactly the set of points where $f$ is unramified by Morphisms, Lemma 28.33.2. $\square$

Lemma 30.10.3. Let $f : X \to S$ be a morphism of schemes. Let $d \geq 0$ be an integer. Assume

1. $f$ is flat,

2. $f$ is locally of finite presentation, and

3. every nonempty fibre of $f$ is equidimensional of dimension $d$.

Let $Z \subset X$ be the closed subscheme cut out by the $d$th fitting ideal of $\Omega _{X/S}$. Then $Z$ is exactly the set of points where $f$ is not smooth.

Proof. By Lemma 30.9.6 the complement of $Z$ is exactly the locus where $\Omega _{X/S}$ can be generated by at most $d$ elements. Hence the lemma follows from Morphisms, Lemma 28.32.14. $\square$

 If $f$ is a local complete intersection morphism (More on Morphisms, Definition 36.54.2) then the closed subscheme cut out by the locally strictly standard elements is the correct thing to look at.

Comment #2115 by David Hansen on

In Lemma 30.10.1, the morphism f should be assumed locally of finite presentation, since otherwise the Fitting ideals aren't necessarily defined!

Comment #2143 by on

@David: If you read the first line of the proof, then you will see that we payed attention to this. Fitting ideals of finite type quasi-coherent modules are defined in Section 30.9.

Comment #2145 by David Hansen on

Amazing!

Comment #3852 by Matthieu Romagny on

Typo in third sentence of this section: it is useful to have a canonical closed subscheme

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