The Stacks project

76.28 Slicing Cohen-Macaulay morphisms

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $f_1, \ldots , f_ r \in \Gamma (X, \mathcal{O}_ X)$. In this case we denote $V(f_1, \ldots , f_ r)$ the closed subspace of $X$ cut out by $f_1, \ldots , f_ r$. More precisely, we can define $V(f_1, \ldots , f_ r)$ as the closed subspace of $X$ corresponding to the quasi-coherent sheaf of ideals generated by $f_1, \ldots , f_ r$, see Morphisms of Spaces, Lemma 67.13.1. Alternatively, we can choose a presentation $X = U/R$ and consider the closed subscheme $Z \subset U$ cut out by $f_1|U, \ldots , f_ r|_ U$. It is clear that $Z$ is an $R$-invariant (see Groupoids, Definition 39.19.1) closed subscheme and we may set $V(f_1, \ldots , f_ r) = Z/R_ Z$.

Lemma 76.28.1. Let $S$ be a scheme. Consider a cartesian diagram

\[ \xymatrix{ X \ar[d] & F \ar[l]^ p \ar[d] \\ Y & \mathop{\mathrm{Spec}}(k) \ar[l] } \]

where $X \to Y$ is a morphism of algebraic spaces over $S$ which is flat and locally of finite presentation, and where $k$ is a field over $S$. Let $f_1, \ldots , f_ r \in \Gamma (X, \mathcal{O}_ X)$ and $z \in |F|$ such that $f_1, \ldots , f_ r$ map to a regular sequence in the local ring $\mathcal{O}_{F, \overline{z}}$. Then, after replacing $X$ by an open subspace containing $p(z)$, the morphism

\[ V(f_1, \ldots , f_ r) \longrightarrow Y \]

is flat and locally of finite presentation.

Proof. Set $Z = V(f_1, \ldots , f_ r)$. It is clear that $Z \to X$ is locally of finite presentation, hence the composition $Z \to Y$ is locally of finite presentation, see Morphisms of Spaces, Lemma 67.28.2. Hence it suffices to show that $Z \to Y$ is flat in a neighbourhood of $p(z)$. Let $k'/k$ be an extension field. Then $F' = F \times _{\mathop{\mathrm{Spec}}(k)} \mathop{\mathrm{Spec}}(k')$ is surjective and flat over $F$, hence we can find a point $z' \in |F'|$ mapping to $z$ and the local ring map $\mathcal{O}_{F, \overline{z}} \to \mathcal{O}_{F', \overline{z}'}$ is flat, see Morphisms of Spaces, Lemma 67.30.8. Hence the image of $f_1, \ldots , f_ r$ in $\mathcal{O}_{F', \overline{z}'}$ is a regular sequence too, see Algebra, Lemma 10.68.5. Thus, during the proof we may replace $k$ by an extension field. In particular, we may assume that $z \in |F|$ comes from a section $z : \mathop{\mathrm{Spec}}(k) \to F$ of the structure morphism $F \to \mathop{\mathrm{Spec}}(k)$.

Choose a scheme $V$ and a surjective étale morphism $V \to Y$. Choose a scheme $U$ and a surjective étale morphism $U \to X \times _ Y V$. After possibly enlarging $k$ once more we may assume that $\mathop{\mathrm{Spec}}(k) \to F \to X$ factors through $U$ (as $U \to X$ is surjective). Let $u : \mathop{\mathrm{Spec}}(k) \to U$ be such a factorization and denote $v \in V$ the image of $u$. Note that the morphisms

\[ U_ v \times _{\mathop{\mathrm{Spec}}(\kappa (v))} \mathop{\mathrm{Spec}}(k) = U \times _ V \mathop{\mathrm{Spec}}(k) \to U \times _ Y \mathop{\mathrm{Spec}}(k) \to F \]

are étale (the first as the base change of $V \to V \times _ Y V$ and the second as the base change of $U \to X$). Moreover, by construction the point $u : \mathop{\mathrm{Spec}}(k) \to U$ gives a point of the left most space which maps to $z$ on the right. Hence the elements $f_1, \ldots , f_ r$ map to a regular sequence in the local ring on the right of the following map

\[ \mathcal{O}_{U_ v, u} \longrightarrow \mathcal{O}_{U_ v \times _{\mathop{\mathrm{Spec}}(\kappa (v)} \mathop{\mathrm{Spec}}(k), \overline{u}} = \mathcal{O}_{U \times _ V \mathop{\mathrm{Spec}}(k), \overline{u}}. \]

But since the displayed arrow is flat (combine More on Flatness, Lemma 38.2.5 and Morphisms of Spaces, Lemma 67.30.8) we see from Algebra, Lemma 10.68.5 that $f_1, \ldots , f_ r$ maps to a regular sequence in $\mathcal{O}_{U_ v, u}$. By More on Morphisms, Lemma 37.23.2 we conclude that the morphism of schemes

\[ V(f_1, \ldots , f_ r) \times _ X U = V(f_1|_ U, \ldots , f_ r|_ U) \to V \]

is flat in an open neighbourhood $U'$ of $u$. Let $X' \subset X$ be the open subspace corresponding to the image of $|U'| \to |X|$ (see Properties of Spaces, Lemmas 66.4.6 and 66.4.8). We conclude that $V(f_1, \ldots , f_ r) \cap X' \to Y$ is flat (see Morphisms of Spaces, Definition 67.30.1) as we have the commutative diagram

\[ \xymatrix{ V(f_1, \ldots , f_ r) \times _ X U' \ar[d]_ a \ar[r] & V \ar[d]^ b \\ V(f_1, \ldots , f_ r) \cap X' \ar[r] & Y } \]

with $a, b$ étale and $a$ surjective. $\square$

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