38.24 Flattening in the local case

In this section we start applying the earlier material to obtain a shadow of the flattening stratification.

Theorem 38.24.1. In Situation 38.20.3 assume $A$ is henselian, $B$ is essentially of finite type over $A$, and $M$ is a finite $B$-module. Then there exists an ideal $I \subset A$ such that $A/I$ corepresents the functor $F_{lf}$ on the category $\mathcal{C}$. In other words given a local homomorphism of local rings $\varphi : A \to A'$ with $B' = B \otimes _ A A'$ and $M' = M \otimes _ A A'$ the following are equivalent:

1. $\forall \mathfrak q \in V(\mathfrak m_{A'}B' + \mathfrak m_ B B') \subset \mathop{\mathrm{Spec}}(B') : M'_{\mathfrak q}\text{ is flat over }A'$, and

2. $\varphi (I) = 0$.

If $B$ is essentially of finite presentation over $A$ and $M$ of finite presentation over $B$, then $I$ is a finitely generated ideal.

Proof. Choose a finite type ring map $A \to C$ and a finite $C$-module $N$ and a prime $\mathfrak q$ of $C$ such that $B = C_{\mathfrak q}$ and $M = N_{\mathfrak q}$. In the following, when we say “the theorem holds for $(N/C/A, \mathfrak q)$ we mean that it holds for $(A \to B, M)$ where $B = C_{\mathfrak q}$ and $M = N_{\mathfrak q}$. By Lemma 38.20.6 the functor $F_{lf}$ is unchanged if we replace $B$ by a local ring flat over $B$. Hence, since $A$ is henselian, we may apply Lemma 38.6.6 and assume that there exists a complete dévissage of $N/C/A$ at $\mathfrak q$.

Let $(A_ i, B_ i, M_ i, \alpha _ i, \mathfrak q_ i)_{i = 1, \ldots , n}$ be such a complete dévissage of $N/C/A$ at $\mathfrak q$. Let $\mathfrak q'_ i \subset A_ i$ be the unique prime lying over $\mathfrak q_ i \subset B_ i$ as in Definition 38.6.4. Since $C \to A_1$ is surjective and $N \cong M_1$ as $C$-modules, we see by Lemma 38.20.5 it suffices to prove the theorem holds for $(M_1/A_1/A, \mathfrak q'_1)$. Since $B_1 \to A_1$ is finite and $\mathfrak q_1$ is the only prime of $B_1$ over $\mathfrak q'_1$ we see that $(A_1)_{\mathfrak q'_1} \to (B_1)_{\mathfrak q_1}$ is finite (see Algebra, Lemma 10.41.11 or More on Morphisms, Lemma 37.43.4). Hence by Lemma 38.20.5 it suffices to prove the theorem holds for $(M_1/B_1/A, \mathfrak q_1)$.

At this point we may assume, by induction on the length $n$ of the dévissage, that the theorem holds for $(M_2/B_2/A, \mathfrak q_2)$. (If $n = 1$, then $M_2 = 0$ which is flat over $A$.) Reversing the last couple of steps of the previous paragraph, using that $M_2 \cong \mathop{\mathrm{Coker}}(\alpha _2)$ as $B_1$-modules, we see that the theorem holds for $(\mathop{\mathrm{Coker}}(\alpha _1)/B_1/A, \mathfrak q_1)$.

Let $A'$ be an object of $\mathcal{C}$. At this point we use Lemma 38.10.1 to see that if $(M_1 \otimes _ A A')_{\mathfrak q'}$ is flat over $A'$ for a prime $\mathfrak q'$ of $B_1 \otimes _ A A'$ lying over $\mathfrak m_{A'}$, then $(\mathop{\mathrm{Coker}}(\alpha _1) \otimes _ A A')_{\mathfrak q'}$ is flat over $A'$. Hence we conclude that $F_{lf}$ is a subfunctor of the functor $F'_{lf}$ associated to the module $\mathop{\mathrm{Coker}}(\alpha _1)_{\mathfrak q_1}$ over $(B_1)_{\mathfrak q_1}$. By the previous paragraph we know $F'_{lf}$ is corepresented by $A/J$ for some ideal $J \subset A$. Hence we may replace $A$ by $A/J$ and assume that $\mathop{\mathrm{Coker}}(\alpha _1)_{\mathfrak q_1}$ is flat over $A$.

Since $\mathop{\mathrm{Coker}}(\alpha _1)$ is a $B_1$-module for which there exist a complete dévissage of $N_1/B_1/A$ at $\mathfrak q_1$ and since $\mathop{\mathrm{Coker}}(\alpha _1)_{\mathfrak q_1}$ is flat over $A$ by Lemma 38.10.2 we see that $\mathop{\mathrm{Coker}}(\alpha _1)$ is free as an $A$-module, in particular flat as an $A$-module. Hence Lemma 38.10.1 implies $F_{lf}(A')$ is nonempty if and only if $\alpha \otimes 1_{A'}$ is injective. Let $N_1 = \mathop{\mathrm{Im}}(\alpha _1) \subset M_1$ so that we have exact sequences

$0 \to N_1 \to M_1 \to \mathop{\mathrm{Coker}}(\alpha _1) \to 0 \quad \text{and}\quad B_1^{\oplus r_1} \to N_1 \to 0$

The flatness of $\mathop{\mathrm{Coker}}(\alpha _1)$ implies the first sequence is universally exact (see Algebra, Lemma 10.82.5). Hence $\alpha \otimes 1_{A'}$ is injective if and only if $B_1^{\oplus r_1} \otimes _ A A' \to N_1 \otimes _ A A'$ is an isomorphism. Finally, Theorem 38.23.3 applies to show this functor is corepresentable by $A/I$ for some ideal $I$ and we conclude $F_{lf}$ is corepresentable by $A/I$ also.

To prove the final statement, suppose that $A \to B$ is essentially of finite presentation and $M$ of finite presentation over $B$. Let $I \subset A$ be the ideal such that $F_{lf}$ is corepresented by $A/I$. Write $I = \bigcup I_\lambda$ where $I_\lambda$ ranges over the finitely generated ideals contained in $I$. Then, since $F_{lf}(A/I) = \{ *\}$ we see that $F_{lf}(A/I_\lambda ) = \{ *\}$ for some $\lambda$, see Lemma 38.20.4 part (2). Clearly this implies that $I = I_\lambda$. $\square$

Remark 38.24.2. Here is a scheme theoretic reformulation of Theorem 38.24.1. Let $(X, x) \to (S, s)$ be a morphism of pointed schemes which is locally of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Assume $S$ henselian local with closed point $s$. There exists a closed subscheme $Z \subset S$ with the following property: for any morphism of pointed schemes $(T, t) \to (S, s)$ the following are equivalent

1. $\mathcal{F}_ T$ is flat over $T$ at all points of the fibre $X_ t$ which map to $x \in X_ s$, and

2. $\mathop{\mathrm{Spec}}(\mathcal{O}_{T, t}) \to S$ factors through $Z$.

Moreover, if $X \to S$ is of finite presentation at $x$ and $\mathcal{F}_ x$ of finite presentation over $\mathcal{O}_{X, x}$, then $Z \to S$ is of finite presentation.

At this point we can obtain some very general results completely for free from the result above. Note that perhaps the most interesting case is when $E = X_ s$!

Lemma 38.24.3. Let $S$ be the spectrum of a henselian local ring with closed point $s$. Let $X \to S$ be a morphism of schemes which is locally of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Let $E \subset X_ s$ be a subset. There exists a closed subscheme $Z \subset S$ with the following property: for any morphism of pointed schemes $(T, t) \to (S, s)$ the following are equivalent

1. $\mathcal{F}_ T$ is flat over $T$ at all points of the fibre $X_ t$ which map to a point of $E \subset X_ s$, and

2. $\mathop{\mathrm{Spec}}(\mathcal{O}_{T, t}) \to S$ factors through $Z$.

Moreover, if $X \to S$ is locally of finite presentation, $\mathcal{F}$ is of finite presentation, and $E \subset X_ s$ is closed and quasi-compact, then $Z \to S$ is of finite presentation.

Proof. For $x \in X_ s$ denote $Z_ x \subset S$ the closed subscheme we found in Remark 38.24.2. Then it is clear that $Z = \bigcap _{x \in E} Z_ x$ works!

To prove the final statement assume $X$ locally of finite presentation, $\mathcal{F}$ of finite presentation and $Z$ closed and quasi-compact. First, choose finitely many affine opens $W_ j \subset X$ such that $E \subset \bigcup W_ j$. It clearly suffices to prove the result for each morphism $W_ j \to S$ with sheaf $\mathcal{F}|_{X_ j}$ and closed subset $E \cap W_ j$. Hence we may assume $X$ is affine. In this case, More on Algebra, Lemma 15.19.4 shows that the functor defined by (1) is “limit preserving”. Hence we can show that $Z \to S$ is of finite presentation exactly as in the last part of the proof of Theorem 38.24.1. $\square$

Remark 38.24.4. Tracing the proof of Lemma 38.24.3 to its origins we find a long and winding road. But if we assume that

1. $f$ is of finite type,

2. $\mathcal{F}$ is a finite type $\mathcal{O}_ X$-module,

3. $E = X_ s$, and

4. $S$ is the spectrum of a Noetherian complete local ring.

then there is a proof relying completely on more elementary algebra as follows: first we reduce to the case where $X$ is affine by taking a finite affine open cover. In this case $Z$ exists by More on Algebra, Lemma 15.20.3. The key step in this proof is constructing the closed subscheme $Z$ step by step inside the truncations $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}/\mathfrak m_ s^ n)$. This relies on the fact that flattening stratifications always exist when the base is Artinian, and the fact that $\mathcal{O}_{S, s} = \mathop{\mathrm{lim}}\nolimits \mathcal{O}_{S, s}/\mathfrak m_ s^ n$.

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