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The Stacks project

Lemma 31.22.7. Let \varphi : X \to S be a flat morphism which is locally of finite presentation. Let T \subset X be a closed subscheme. Let x \in T with image s \in S.

  1. If T_ s \subset X_ s is a quasi-regular immersion in a neighbourhood of x, then there exists an open U \subset X and a relative quasi-regular immersion Z \subset U such that Z_ s = T_ s \cap U_ s and T \cap U \subset Z.

  2. If T_ s \subset X_ s is a quasi-regular immersion in a neighbourhood of x, the morphism T \to X is of finite presentation, and T \to S is flat at x, then we can choose U and Z as in (1) such that T \cap U = Z.

  3. If T_ s \subset X_ s is a quasi-regular immersion in a neighbourhood of x, and T is cut out by c equations in a neighbourhood of x, where c = \dim _ x(X_ s) - \dim _ x(T_ s), then we can choose U and Z as in (1) such that T \cap U = Z.

In each case Z \to U is a regular immersion by Lemma 31.22.4. In particular, if T \to S is locally of finite presentation and flat and all fibres T_ s \subset X_ s are quasi-regular immersions, then T \to X is a relative quasi-regular immersion.

Proof. Choose affine open neighbourhoods \mathop{\mathrm{Spec}}(A) of s and \mathop{\mathrm{Spec}}(B) of x such that \varphi (\mathop{\mathrm{Spec}}(B)) \subset \mathop{\mathrm{Spec}}(A). Let \mathfrak p \subset A be the prime ideal corresponding to s. Let \mathfrak q \subset B be the prime ideal corresponding to x. Let I \subset B be the ideal corresponding to T. By the initial assumption of the lemma we know that A \to B is flat and of finite presentation. The assumption in (1) means that, after shrinking \mathop{\mathrm{Spec}}(B), we may assume I(B \otimes _ A \kappa (\mathfrak p)) is generated by a quasi-regular sequence of elements. After possibly localizing B at some g \in B, g \not\in \mathfrak q we may assume there exist f_1, \ldots , f_ r \in I which map to a quasi-regular sequence in B \otimes _ A \kappa (\mathfrak p) which generates I(B \otimes _ A \kappa (\mathfrak p)). By Algebra, Lemmas 10.69.6 and 10.68.6 we may assume after another localization that f_1, \ldots , f_ r \in I form a regular sequence in B \otimes _ A \kappa (\mathfrak p). By Lemma 31.18.9 it follows that Z_1 = V(f_1) \subset \mathop{\mathrm{Spec}}(B) is a relative effective Cartier divisor, again after possibly localizing B. Applying the same lemma again, but now to Z_2 = V(f_1, f_2) \subset Z_1 we see that Z_2 \subset Z_1 is a relative effective Cartier divisor. And so on until one reaches Z = Z_ n = V(f_1, \ldots , f_ n). Then Z \to \mathop{\mathrm{Spec}}(B) is a regular immersion and Z is flat over S, in particular Z \to \mathop{\mathrm{Spec}}(B) is a relative quasi-regular immersion over \mathop{\mathrm{Spec}}(A). This proves (1).

To see (2) consider the closed immersion Z \to D. The surjective ring map u : \mathcal{O}_{D, x} \to \mathcal{O}_{Z, x} is a map of flat local \mathcal{O}_{S, s}-algebras which are essentially of finite presentation, and which becomes an isomorphisms after dividing by \mathfrak m_ s. Hence it is an isomorphism, see Algebra, Lemma 10.128.4. It follows that Z \to D is an isomorphism in a neighbourhood of x, see Algebra, Lemma 10.126.6.

To see (3), after possibly shrinking U we may assume that the ideal of Z is generated by a regular sequence f_1, \ldots , f_ r (see our construction of Z above) and the ideal of T is generated by g_1, \ldots , g_ c. We claim that c = r. Namely,

\begin{align*} \dim _ x(X_ s) & = \dim (\mathcal{O}_{X_ s, x}) + \text{trdeg}_{\kappa (s)}(\kappa (x)), \\ \dim _ x(T_ s) & = \dim (\mathcal{O}_{T_ s, x}) + \text{trdeg}_{\kappa (s)}(\kappa (x)), \\ \dim (\mathcal{O}_{X_ s, x}) & = \dim (\mathcal{O}_{T_ s, x}) + r \end{align*}

the first two equalities by Algebra, Lemma 10.116.3 and the second by r times applying Algebra, Lemma 10.60.13. As T \subset Z we see that f_ i = \sum b_{ij} g_ j. But the ideals of Z and T cut out the same quasi-regular closed subscheme of X_ s in a neighbourhood of x. Hence the matrix (b_{ij}) \bmod \mathfrak m_ x is invertible (some details omitted). Hence (b_{ij}) is invertible in an open neighbourhood of x. In other words, T \cap U = Z after shrinking U.

The final statements of the lemma follow immediately from part (2), combined with the fact that Z \to S is locally of finite presentation if and only if Z \to X is of finite presentation, see Morphisms, Lemmas 29.21.3 and 29.21.11. \square


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