The Stacks project

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

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. $\square$

Proof. Consider the map of Noetherian local rings

and observe that its fibre is

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

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

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

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 $x$. $\square$

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. $\square$

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). $\square$

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

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

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

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. $\square$