## 37.22 Cohen-Macaulay morphisms

Compare with Section 37.20. Note that, as pointed out in Algebra, Section 10.167 and Varieties, Section 33.13 “geometrically Cohen-Macaulay” is the same as plain Cohen-Macaulay.

Definition 37.22.1. Let $f : X \to Y$ be a morphism of schemes. Assume that all the fibres $X_ y$ are locally Noetherian schemes.

1. Let $x \in X$, and $y = f(x)$. We say that $f$ is Cohen-Macaulay at $x$ if $f$ is flat at $x$, and the local ring of the scheme $X_ y$ at $x$ is Cohen-Macaulay.

2. We say $f$ is a Cohen-Macaulay morphism if $f$ is Cohen-Macaulay at every point of $X$.

Here is a translation.

Lemma 37.22.2. Let $f : X \to Y$ be a morphism of schemes. Assume all fibres of $f$ are locally Noetherian. The following are equivalent

1. $f$ is Cohen-Macaulay, and

2. $f$ is flat and its fibres are Cohen-Macaulay schemes.

Proof. This follows directly from the definitions. $\square$

Lemma 37.22.3. Let $f : X \to Y$ be a morphism of locally Noetherian schemes which is locally of finite type and Cohen-Macaulay. For every point $x$ in $X$ with image $y$ in $Y$,

$\dim _ x(X) = \dim _ y(Y) + \dim _ x(X_ y),$

where $X_ y$ denotes the fiber over $y$.

Proof. After replacing $X$ by an open neighborhood of $x$, there is a natural number $d$ such that all fibers of $X \to Y$ have dimension $d$ at every point, see Morphisms, Lemma 29.29.4. Then $f$ is flat, locally of finite type and of relative dimension $d$. Hence the result follows from Morphisms, Lemma 29.29.6. $\square$

Lemma 37.22.4. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of schemes. Assume that the fibres of $f$, $g$, and $g \circ f$ are locally Noetherian. Let $x \in X$ with images $y \in Y$ and $z \in Z$.

1. If $f$ is Cohen-Macaulay at $x$ and $g$ is Cohen-Macaulay at $f(x)$, then $g \circ f$ is Cohen-Macaulay at $x$.

2. If $f$ and $g$ are Cohen-Macaulay, then $g \circ f$ is Cohen-Macaulay.

3. If $g \circ f$ is Cohen-Macaulay at $x$ and $f$ is flat at $x$, then $f$ is Cohen-Macaulay at $x$ and $g$ is Cohen-Macaulay at $f(x)$.

4. If $g \circ f$ is Cohen-Macaulay and $f$ is flat, then $f$ is Cohen-Macaulay and $g$ is Cohen-Macaulay at every point in the image of $f$.

Proof. Consider the map of Noetherian local rings

$\mathcal{O}_{Y_ z, y} \to \mathcal{O}_{X_ z, x}$

and observe that its fibre is

$\mathcal{O}_{X_ z, x}/\mathfrak m_{Y_ z, y}\mathcal{O}_{X_ z, x} = \mathcal{O}_{X_ y, x}$

Thus the lemma this follows from Algebra, Lemma 10.163.3. $\square$

Lemma 37.22.5. Let $f : X \to Y$ be a flat morphism of locally Noetherian schemes. If $X$ is Cohen-Macaulay, then $f$ is Cohen-Macaulay and $\mathcal{O}_{Y, f(x)}$ is Cohen-Macaulay for all $x \in X$.

Proof. After translating into algebra this follows from Algebra, Lemma 10.163.3. $\square$

Lemma 37.22.6. Let $f : X \to Y$ be a morphism of schemes. Assume that all the fibres $X_ y$ are locally Noetherian schemes. Let $Y' \to Y$ be locally of finite type. Let $f' : X' \to Y'$ be the base change of $f$. Let $x' \in X'$ be a point with image $x \in X$.

1. If $f$ is Cohen-Macaulay at $x$, then $f' : X' \to Y'$ is Cohen-Macaulay at $x'$.

2. If $f$ is flat at $x$ and $f'$ is Cohen-Macaulay at $x'$, then $f$ is Cohen-Macaulay at $x$.

3. If $Y' \to Y$ is flat at $f'(x')$ and $f'$ is Cohen-Macaulay at $x'$, then $f$ is Cohen-Macaulay at $x$.

Proof. Note that the assumption on $Y' \to Y$ implies that for $y' \in Y'$ mapping to $y \in Y$ the field extension $\kappa (y')/\kappa (y)$ is finitely generated. Hence also all the fibres $X'_{y'} = (X_ y)_{\kappa (y')}$ are locally Noetherian, see Varieties, Lemma 33.11.1. Thus the lemma makes sense. Set $y' = f'(x')$ and $y = f(x)$. Hence we get the following commutative diagram of local rings

$\xymatrix{ \mathcal{O}_{X', x'} & \mathcal{O}_{X, x} \ar[l] \\ \mathcal{O}_{Y', y'} \ar[u] & \mathcal{O}_{Y, y} \ar[l] \ar[u] }$

where the upper left corner is a localization of the tensor product of the upper right and lower left corners over the lower right corner.

Assume $f$ is Cohen-Macaulay at $x$. The flatness of $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$ implies the flatness of $\mathcal{O}_{Y', y'} \to \mathcal{O}_{X', x'}$, see Algebra, Lemma 10.100.1. The fact that $\mathcal{O}_{X, x}/\mathfrak m_ y\mathcal{O}_{X, x}$ is Cohen-Macaulay implies that $\mathcal{O}_{X', x'}/\mathfrak m_{y'}\mathcal{O}_{X', x'}$ is Cohen-Macaulay, see Varieties, Lemma 33.13.1. Hence we see that $f'$ is Cohen-Macaulay at $x'$.

Assume $f$ is flat at $x$ and $f'$ is Cohen-Macaulay at $x'$. The fact that $\mathcal{O}_{X', x'}/\mathfrak m_{y'}\mathcal{O}_{X', x'}$ is Cohen-Macaulay implies that $\mathcal{O}_{X, x}/\mathfrak m_ y\mathcal{O}_{X, x}$ is Cohen-Macaulay, see Varieties, Lemma 33.13.1. Hence we see that $f$ is Cohen-Macaulay at $x$.

Assume $Y' \to Y$ is flat at $y'$ and $f'$ is Cohen-Macaulay at $x'$. The flatness of $\mathcal{O}_{Y', y'} \to \mathcal{O}_{X', x'}$ and $\mathcal{O}_{Y, y} \to \mathcal{O}_{Y', y'}$ implies the flatness of $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$, see Algebra, Lemma 10.100.1. The fact that $\mathcal{O}_{X', x'}/\mathfrak m_{y'}\mathcal{O}_{X', x'}$ is Cohen-Macaulay implies that $\mathcal{O}_{X, x}/\mathfrak m_ y\mathcal{O}_{X, x}$ is Cohen-Macaulay, see Varieties, Lemma 33.13.1. Hence we see that $f$ is Cohen-Macaulay at $x$. $\square$

Lemma 37.22.7. Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Let

$W = \{ x \in X \mid f\text{ is Cohen-Macaulay at }x\}$

Then

1. $W = \{ x \in X \mid \mathcal{O}_{X_{f(x)}, x}\text{ is Cohen-Macaulay}\}$,

2. $W$ is open in $X$,

3. $W$ dense in every fibre of $X \to S$,

4. the formation of $W$ commutes with arbitrary base change of $f$: For any morphism $g : S' \to S$, consider the base change $f' : X' \to S'$ of $f$ and the projection $g' : X' \to X$. Then the corresponding set $W'$ for the morphism $f'$ is equal to $W' = (g')^{-1}(W)$.

Proof. As $f$ is flat with locally Noetherian fibres the equality in (1) holds by definition. Parts (2) and (3) follow from Algebra, Lemma 10.130.5. Part (4) follows either from Algebra, Lemma 10.130.7 or Varieties, Lemma 33.13.1. $\square$

Lemma 37.22.8. Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Let $x \in X$ with image $s \in S$. Set $d = \dim _ x(X_ s)$. The following are equivalent

1. $f$ is Cohen-Macaulay at $x$,

2. there exists an open neighbourhood $U \subset X$ of $x$ and a locally quasi-finite morphism $U \to \mathbf{A}^ d_ S$ over $S$ which is flat at $x$,

3. there exists an open neighbourhood $U \subset X$ of $x$ and a locally quasi-finite flat morphism $U \to \mathbf{A}^ d_ S$ over $S$,

4. for any $S$-morphism $g : U \to \mathbf{A}^ d_ S$ of an open neighbourhood $U \subset X$ of $x$ we have: $g$ is quasi-finite at $x$ $\Rightarrow$ $g$ is flat at $x$.

Proof. Openness of flatness shows (2) and (3) are equivalent, see Theorem 37.15.1.

Choose affine open $U = \mathop{\mathrm{Spec}}(A) \subset X$ with $x \in U$ and $V = \mathop{\mathrm{Spec}}(R) \subset S$ with $f(U) \subset V$. Then $R \to A$ is a flat ring map of finite presentation. Let $\mathfrak p \subset A$ be the prime ideal corresponding to $x$. After replacing $A$ by a principal localization we may assume there exists a quasi-finite map $R[x_1, \ldots , x_ d] \to A$, see Algebra, Lemma 10.125.2. Thus there exists at least one pair $(U, g)$ consisting of an open neighbourhood $U \subset X$ of $x$ and a locally1 quasi-finite morphism $g : U \to \mathbf{A}^ d_ S$.

Claim: Given $R \to A$ flat and of finite presentation, a prime $\mathfrak p \subset A$ and $\varphi : R[x_1, \ldots , x_ d] \to A$ quasi-finite at $\mathfrak p$ we have: $\mathop{\mathrm{Spec}}(\varphi )$ is flat at $\mathfrak p$ if and only if $\mathop{\mathrm{Spec}}(A) \to \mathop{\mathrm{Spec}}(R)$ is Cohen-Macaulay at $\mathfrak p$. Namely, by Theorem 37.16.2 flatness may be checked on fibres. The same is true for being Cohen-Macaulay (as $A$ is already assumed flat over $R$). Thus the claim follows from Algebra, Lemma 10.130.1.

The claim shows that (1) is equivalent to (4) and combined with the fact that we have constructed a suitable $(U, g)$ in the second paragraph, the claim also shows that (1) is equivalent to (2). $\square$

Lemma 37.22.9. Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. For $d \geq 0$ there exist opens $U_ d \subset X$ with the following properties

1. $W = \bigcup _{d \geq 0} U_ d$ is dense in every fibre of $f$, and

2. $U_ d \to S$ is of relative dimension $d$ (see Morphisms, Definition 29.29.1).

Proof. This follows by combining Lemma 37.22.7 with Morphisms, Lemma 29.29.4. $\square$

Lemma 37.22.10. Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Suppose $x' \leadsto x$ is a specialization of points of $X$ with image $s' \leadsto s$ in $S$. If $x$ is a generic point of an irreducible component of $X_ s$ then $\dim _{x'}(X_{s'}) = \dim _ x(X_ s)$.

Proof. The point $x$ is contained in $U_ d$ for some $d$, where $U_ d$ as in Lemma 37.22.9. $\square$

Lemma 37.22.11. The property $\mathcal{P}(f)=$“the fibres of $f$ are locally Noetherian and $f$ is Cohen-Macaulay” is local in the fppf topology on the target and local in the syntomic topology on the source.

Proof. We have $\mathcal{P}(f) = \mathcal{P}_1(f) \wedge \mathcal{P}_2(f)$ where $\mathcal{P}_1(f)=$“$f$ is flat”, and $\mathcal{P}_2(f)=$“the fibres of $f$ are locally Noetherian and Cohen-Macaulay”. We know that $\mathcal{P}_1$ is local in the fppf topology on the source and the target, see Descent, Lemmas 35.23.15 and 35.27.1. Thus we have to deal with $\mathcal{P}_2$.

Let $f : X \to Y$ be a morphism of schemes. Let $\{ \varphi _ i : Y_ i \to Y\} _{i \in I}$ be an fppf covering of $Y$. Denote $f_ i : X_ i \to Y_ i$ the base change of $f$ by $\varphi _ i$. Let $i \in I$ and let $y_ i \in Y_ i$ be a point. Set $y = \varphi _ i(y_ i)$. Note that

$X_{i, y_ i} = \mathop{\mathrm{Spec}}(\kappa (y_ i)) \times _{\mathop{\mathrm{Spec}}(\kappa (y))} X_ y.$

and that $\kappa (y_ i)/\kappa (y)$ is a finitely generated field extension. Hence if $X_ y$ is locally Noetherian, then $X_{i, y_ i}$ is locally Noetherian, see Varieties, Lemma 33.11.1. And if in addition $X_ y$ is Cohen-Macaulay, then $X_{i, y_ i}$ is Cohen-Macaulay, see Varieties, Lemma 33.13.1. Thus $\mathcal{P}_2$ is fppf local on the target.

Let $\{ X_ i \to X\}$ be a syntomic covering of $X$. Let $y \in Y$. In this case $\{ X_{i, y} \to X_ y\}$ is a syntomic covering of the fibre. Hence the locality of $\mathcal{P}_2$ for the syntomic topology on the source follows from Descent, Lemma 35.17.2. Combining the above the lemma follows. $\square$

[1] If $S$ is quasi-separated, then $g$ will be quasi-finite.

Comment #5015 by Yujie Luo on

Lemma \cite{0C0W} (4), a small typo $f\circ g$ should be $g\circ f$.

Comment #5876 by Sasha on

In the statement of Lemma 054T it is not clear what is $W$.

Comment #6087 by on

@#5877: The lemma defines $W$ in the statement! You can say more but the lemma does not claim more.

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