The Stacks project

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. $\square$

[1] 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.

Comments (5)

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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