The Stacks project

37.21 Slicing Cohen-Macaulay morphisms

The results in this section eventually lead to the assertion that the fppf topology is the same as the “finitely presented, flat, quasi-finite” topology. The following lemma is very closely related to Divisors, Lemma 31.18.9.

Lemma 37.21.1. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ be a point with image $s \in S$. Let $h \in \mathfrak m_ x \subset \mathcal{O}_{X, x}$. Assume

  1. $f$ is locally of finite presentation,

  2. $f$ is flat at $x$, and

  3. the image $\overline{h}$ of $h$ in $\mathcal{O}_{X_ s, x} = \mathcal{O}_{X, x}/\mathfrak m_ s\mathcal{O}_{X, x}$ is a nonzerodivisor.

Then there exists an affine open neighbourhood $U \subset X$ of $x$ such that $h$ comes from $h \in \Gamma (U, \mathcal{O}_ U)$ and such that $D = V(h)$ is an effective Cartier divisor in $U$ with $x \in D$ and $D \to S$ flat and locally of finite presentation.

Proof. We are going to prove this by reducing to the Noetherian case. By openness of flatness (see Theorem 37.15.1) we may assume, after replacing $X$ by an open neighbourhood of $x$, that $X \to S$ is flat. We may also assume that $X$ and $S$ are affine. After possible shrinking $X$ a bit we may assume that there exists an $h \in \Gamma (X, \mathcal{O}_ X)$ which maps to our given $h$.

We may write $S = \mathop{\mathrm{Spec}}(A)$ and we may write $A = \mathop{\mathrm{colim}}\nolimits _ i A_ i$ as a directed colimit of finite type $\mathbf{Z}$ algebras. Then by Algebra, Lemma 10.166.1 or Limits, Lemmas 32.10.1, 32.8.2, and 32.10.1 we can find a cartesian diagram

\[ \xymatrix{ X \ar[r] \ar[d]_ f & X_0 \ar[d]^{f_0} \\ S \ar[r] & S_0 } \]

with $f_0$ flat and of finite presentation, $X_0$ affine, and $S_0$ affine and Noetherian. Let $x_0 \in X_0$, resp. $s_0 \in S_0$ be the image of $x$, resp. $s$. We may also assume there exists an element $h_0 \in \Gamma (X_0, \mathcal{O}_{X_0})$ which restricts to $h$ on $X$. (If you used the algebra reference above then this is clear; if you used the references to the chapter on limits then this follows from Limits, Lemma 32.10.1 by thinking of $h$ as a morphism $X \to \mathbf{A}^1_ S$.) Note that $\mathcal{O}_{X_ s, x}$ is a localization of $\mathcal{O}_{(X_0)_{s_0}, x_0} \otimes _{\kappa (s_0)} \kappa (s)$, so that $\mathcal{O}_{(X_0)_{s_0}, x_0} \to \mathcal{O}_{X_ s, x}$ is a flat local ring map, in particular faithfully flat. Hence the image $\overline{h}_0 \in \mathcal{O}_{(X_0)_{s_0}, x_0}$ is contained in $\mathfrak m_{(X_0)_{s_0}, x_0}$ and is a nonzerodivisor. We claim that after replacing $X_0$ by a principal open neighbourhood of $x_0$ the element $h_0$ is a nonzerodivisor in $B_0 = \Gamma (X_0, \mathcal{O}_{X_0})$ such that $B_0/h_0B_0$ is flat over $A_0 = \Gamma (S_0, \mathcal{O}_{S_0})$. If so then

\[ 0 \to B_0 \xrightarrow {h_0} B_0 \to B_0/h_0B_0 \to 0 \]

is a short exact sequence of flat $A_0$-modules. Hence this remains exact on tensoring with $A$ (by Algebra, Lemma 10.38.12) and the lemma follows.

It remains to prove the claim above. The corresponding algebra statement is the following (we drop the subscript ${}_0$ here): Let $A \to B$ be a flat, finite type ring map of Noetherian rings. Let $\mathfrak q \subset B$ be a prime lying over $\mathfrak p \subset A$. Assume $h \in \mathfrak q$ maps to a nonzerodivisor in $B_{\mathfrak q}/\mathfrak p B_{\mathfrak q}$. Goal: show that after possible replacing $B$ by $B_ g$ for some $g \in B$, $g \not\in \mathfrak q$ the element $h$ becomes a nonzerodivisor and $B/hB$ becomes flat over $A$. By Algebra, Lemma 10.98.2 we see that $h$ is a nonzerodivisor in $B_{\mathfrak q}$ and that $B_{\mathfrak q}/hB_{\mathfrak q}$ is flat over $A$. By openness of flatness, see Algebra, Theorem 10.128.4 or Theorem 37.15.1 we see that $B/hB$ is flat over $A$ after replacing $B$ by $B_ g$ for some $g \in B$, $g \not\in \mathfrak q$. Finally, let $I = \{ b \in B \mid hb = 0\} $ be the annihilator of $h$. Then $IB_{\mathfrak q} = 0$ as $h$ is a nonzerodivisor in $B_{\mathfrak q}$. Also $I$ is finitely generated as $B$ is Noetherian. Hence there exists a $g \in B$, $g \not\in \mathfrak q$ such that $IB_ g = 0$. After replacing $B$ by $B_ g$ we see that $h$ is a nonzerodivisor. $\square$

Lemma 37.21.2. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ be a point with image $s \in S$. Let $h_1, \ldots , h_ r \in \mathcal{O}_{X, x}$. Assume

  1. $f$ is locally of finite presentation,

  2. $f$ is flat at $x$, and

  3. the images of $h_1, \ldots , h_ r$ in $\mathcal{O}_{X_ s, x} = \mathcal{O}_{X, x}/\mathfrak m_ s\mathcal{O}_{X, x}$ form a regular sequence.

Then there exists an affine open neighbourhood $U \subset X$ of $x$ such that $h_1, \ldots , h_ r$ come from $h_1, \ldots , h_ r \in \Gamma (U, \mathcal{O}_ U)$ and such that $Z = V(h_1, \ldots , h_ r) \to U$ is a regular immersion with $x \in Z$ and $Z \to S$ flat and locally of finite presentation. Moreover, the base change $Z_{S'} \to U_{S'}$ is a regular immersion for any scheme $S'$ over $S$.

Proof. (Our conventions on regular sequences imply that $h_ i \in \mathfrak m_ x$ for each $i$.) The case $r = 1$ follows from Lemma 37.21.1 combined with Divisors, Lemma 31.18.1 to see that $V(h_1)$ remains an effective Cartier divisor after base change. The case $r > 1$ follows from a straightforward induction on $r$ (applying the result for $r = 1$ exactly $r$ times; details omitted).

Another way to prove the lemma is using the material from Divisors, Section 31.22. Namely, first by openness of flatness (see Theorem 37.15.1) we may assume, after replacing $X$ by an open neighbourhood of $x$, that $X \to S$ is flat. We may also assume that $X$ and $S$ are affine. After possible shrinking $X$ a bit we may assume that we have $h_1, \ldots , h_ r \in \Gamma (X, \mathcal{O}_ X)$. Set $Z = V(h_1, \ldots , h_ r)$. Note that $X_ s$ is a Noetherian scheme (because it is an algebraic $\kappa (s)$-scheme, see Varieties, Section 33.20) and that the topology on $X_ s$ is induced from the topology on $X$ (see Schemes, Lemma 26.18.5). Hence after shrinking $X$ a bit more we may assume that $Z_ s \subset X_ s$ is a regular immersion cut out by the $r$ elements $h_ i|_{X_ s}$, see Divisors, Lemma 31.20.8 and its proof. It is also clear that $r = \dim _ x(X_ s) - \dim _ x(Z_ s)$ because

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

the first two equalities by Algebra, Lemma 10.115.3 and the second by $r$ times applying Algebra, Lemma 10.59.12. Hence Divisors, Lemma 31.22.7 part (3) applies to show that (after Zariski shrinking $X$) the morphism $Z \to X$ is a regular immersion to which Divisors, Lemma 31.22.4 applies (which gives the flatness and the statement on base change). $\square$

Lemma 37.21.3. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ be a point with image $s \in S$. Assume

  1. $f$ is locally of finite presentation,

  2. $f$ is flat at $x$, and

  3. $\mathcal{O}_{X_ s, x}$ has $\text{depth} \geq 1$.

Then there exists an affine open neighbourhood $U \subset X$ of $x$ and an effective Cartier divisor $D \subset U$ containing $x$ such that $D \to S$ is flat and of finite presentation.

Proof. Pick any $h \in \mathfrak m_ x \subset \mathcal{O}_{X, x}$ which maps to a nonzerodivisor in $\mathcal{O}_{X_ s, x}$ and apply Lemma 37.21.1. $\square$

reference

Lemma 37.21.4. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ be a point with image $s \in S$. Assume

  1. $f$ is locally of finite presentation,

  2. $f$ is Cohen-Macaulay at $x$, and

  3. $x$ is a closed point of $X_ s$.

Then there exists a regular immersion $Z \to X$ containing $x$ such that

  1. $Z \to S$ is flat and locally of finite presentation,

  2. $Z \to S$ is locally quasi-finite, and

  3. $Z_ s = \{ x\} $ set theoretically.

Proof. We may and do replace $S$ by an affine open neighbourhood of $s$. We will prove the lemma for affine $S$ by induction on $d = \dim _ x(X_ s)$.

The case $d = 0$. In this case we show that we may take $Z$ to be an open neighbourhood of $x$. (Note that an open immersion is a regular immersion.) Namely, if $d = 0$, then $X \to S$ is quasi-finite at $x$, see Morphisms, Lemma 29.29.5. Hence there exists an affine open neighbourhood $U \subset X$ such that $U \to S$ is quasi-finite, see Morphisms, Lemma 29.55.2. Thus after replacing $X$ by $U$ we see that the fibre $X_ s$ is a finite discrete set. Hence after replacing $X$ by a further affine open neighbourhood of $X$ we see that $f^{-1}(\{ s\} ) = \{ x\} $ (because the topology on $X_ s$ is induced from the topology on $X$, see Schemes, Lemma 26.18.5). This proves the lemma in this case.

Next, assume $d > 0$. Note that because $x$ is a closed point of its fibre the extension $\kappa (s) \subset \kappa (x)$ is finite (by the Hilbert Nullstellensatz, see Morphisms, Lemma 29.20.3). Thus we see

\[ \text{depth}(\mathcal{O}_{X_ s, x}) = \dim (\mathcal{O}_{X_ s, x}) = d > 0 \]

the first equality as $\mathcal{O}_{X_ s, x}$ is Cohen-Macaulay and the second by Morphisms, Lemma 29.28.1. Thus we may apply Lemma 37.21.3 to find a diagram

\[ \xymatrix{ D \ar[r] \ar[rrd] & U \ar[r] \ar[rd] & X \ar[d] \\ & & S } \]

with $x \in D$. Note that $\mathcal{O}_{D_ s, x} = \mathcal{O}_{X_ s, x}/(\overline{h})$ for some nonzerodivisor $\overline{h}$, see Divisors, Lemma 31.18.1. Hence $\mathcal{O}_{D_ s, x}$ is Cohen-Macaulay of dimension one less than the dimension of $\mathcal{O}_{X_ s, x}$, see Algebra, Lemma 10.103.2 for example. Thus the morphism $D \to S$ is flat, locally of finite presentation, and Cohen-Macaulay at $x$ with $\dim _ x(D_ s) = \dim _ x(X_ s) - 1 = d - 1$. By induction hypothesis we can find a regular immersion $Z \to D$ having properties (a), (b), (c). As $Z \to D \to U$ are both regular immersions, we see that also $Z \to U$ is a regular immersion by Divisors, Lemma 31.21.7. This finishes the proof. $\square$

Lemma 37.21.5. Let $f : X \to S$ be a flat morphism of schemes which is locally of finite presentation. Let $s \in S$ be a point in the image of $f$. Then there exists a commutative diagram

\[ \xymatrix{ S' \ar[rr] \ar[rd]_ g & & X \ar[ld]^ f \\ & S } \]

where $g : S' \to S$ is flat, locally of finite presentation, locally quasi-finite, and $s \in g(S')$.

Proof. The fibre $X_ s$ is not empty by assumption. Hence there exists a closed point $x \in X_ s$ where $f$ is Cohen-Macaulay, see Lemma 37.20.7. Apply Lemma 37.21.4 and set $S' = S$. $\square$

The following lemma shows that sheaves for the fppf topology are the same thing as sheaves for the “quasi-finite, flat, finite presentation” topology.

Lemma 37.21.6. Let $S$ be a scheme. Let $\mathcal{U} = \{ S_ i \to S\} _{i \in I}$ be an fppf covering of $S$, see Topologies, Definition 34.7.1. Then there exists an fppf covering $\mathcal{V} = \{ T_ j \to S\} _{j \in J}$ which refines (see Sites, Definition 7.8.1) $\mathcal{U}$ such that each $T_ j \to S$ is locally quasi-finite.

Proof. For every $s \in S$ there exists an $i \in I$ such that $s$ is in the image of $S_ i \to S$. By Lemma 37.21.5 we can find a morphism $g_ s : T_ s \to S$ such that $s \in g_ s(T_ s)$ which is flat, locally of finite presentation and locally quasi-finite and such that $g_ s$ factors through $S_ i \to S$. Hence $\{ T_ s \to S\} $ is the desired covering of $S$ that refines $\mathcal{U}$. $\square$


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