The Stacks project

38.3 The local structure of a finite type module

The key technical lemma that makes a lot of the arguments in this chapter work is the geometric Lemma 38.3.2.

Lemma 38.3.1. Let $f : X \to S$ be a finite type morphism of affine schemes. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Let $x \in X$ with image $s = f(x)$ in $S$. Set $\mathcal{F}_ s = \mathcal{F}|_{X_ s}$. Then there exist a closed immersion $i : Z \to X$ of finite presentation, and a quasi-coherent finite type $\mathcal{O}_ Z$-module $\mathcal{G}$ such that $i_*\mathcal{G} = \mathcal{F}$ and $Z_ s = \text{Supp}(\mathcal{F}_ s)$.

Proof. Say the morphism $f : X \to S$ is given by the ring map $A \to B$ and that $\mathcal{F}$ is the quasi-coherent sheaf associated to the $B$-module $M$. By Morphisms, Lemma 29.15.2 we know that $A \to B$ is a finite type ring map, and by Properties, Lemma 28.16.1 we know that $M$ is a finite $B$-module. In particular the support of $\mathcal{F}$ is the closed subscheme of $\mathop{\mathrm{Spec}}(B)$ cut out by the annihilator $I = \{ x \in B \mid xm = 0\ \forall m \in M\} $ of $M$, see Algebra, Lemma 10.40.5. Let $\mathfrak q \subset B$ be the prime ideal corresponding to $x$ and let $\mathfrak p \subset A$ be the prime ideal corresponding to $s$. Note that $X_ s = \mathop{\mathrm{Spec}}(B \otimes _ A \kappa (\mathfrak p))$ and that $\mathcal{F}_ s$ is the quasi-coherent sheaf associated to the $B \otimes _ A \kappa (\mathfrak p)$ module $M \otimes _ A \kappa (\mathfrak p)$. By Morphisms, Lemma 29.5.3 the support of $\mathcal{F}_ s$ is equal to $V(I(B \otimes _ A \kappa (\mathfrak p)))$. Since $B \otimes _ A \kappa (\mathfrak p)$ is of finite type over $\kappa (\mathfrak p)$ there exist finitely many elements $f_1, \ldots , f_ m \in I$ such that

\[ I(B \otimes _ A \kappa (\mathfrak p)) = (f_1, \ldots , f_ n)(B \otimes _ A \kappa (\mathfrak p)). \]

Denote $i : Z \to X$ the closed subscheme cut out by $(f_1, \ldots , f_ m)$, in a formula $Z = \mathop{\mathrm{Spec}}(B/(f_1, \ldots , f_ m))$. Since $M$ is annihilated by $I$ we can think of $M$ as an $B/(f_1, \ldots , f_ m)$-module. In other words, $\mathcal{F}$ is the pushforward of a finite type module on $Z$. As $Z_ s = \text{Supp}(\mathcal{F}_ s)$ by construction, this proves the lemma. $\square$

Lemma 38.3.2. Let $f : X \to S$ be morphism of schemes which is locally of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Let $x \in X$ with image $s = f(x)$ in $S$. Set $\mathcal{F}_ s = \mathcal{F}|_{X_ s}$ and $n = \dim _ x(\text{Supp}(\mathcal{F}_ s))$. Then we can construct

  1. elementary étale neighbourhoods $g : (X', x') \to (X, x)$, $e : (S', s') \to (S, s)$,

  2. a commutative diagram

    \[ \xymatrix{ X \ar[dd]_ f & X' \ar[dd] \ar[l]^ g & Z' \ar[l]^ i \ar[d]^\pi \\ & & Y' \ar[d]^ h \\ S & S' \ar[l]_ e & S' \ar@{=}[l] } \]
  3. a point $z' \in Z'$ with $i(z') = x'$, $y' = \pi (z')$, $h(y') = s'$,

  4. a finite type quasi-coherent $\mathcal{O}_{Z'}$-module $\mathcal{G}$,

such that the following properties hold

  1. $X'$, $Z'$, $Y'$, $S'$ are affine schemes,

  2. $i$ is a closed immersion of finite presentation,

  3. $i_*(\mathcal{G}) \cong g^*\mathcal{F}$,

  4. $\pi $ is finite and $\pi ^{-1}(\{ y'\} ) = \{ z'\} $,

  5. the extension $\kappa (s') \subset \kappa (y')$ is purely transcendental,

  6. $h$ is smooth of relative dimension $n$ with geometrically integral fibres.

Proof. Let $V \subset S$ be an affine neighbourhood of $s$. Let $U \subset f^{-1}(V)$ be an affine neighbourhood of $x$. Then it suffices to prove the lemma for $f|_ U : U \to V$ and $\mathcal{F}|_ U$. Hence in the rest of the proof we assume that $X$ and $S$ are affine.

First, suppose that $X_ s = \text{Supp}(\mathcal{F}_ s)$, in particular $n = \dim _ x(X_ s)$. Apply More on Morphisms, Lemmas 37.43.2 and 37.43.3. This gives us a commutative diagram

\[ \xymatrix{ X \ar[dd] & X' \ar[l]^ g \ar[d]^\pi \\ & Y' \ar[d]^ h \\ S & S' \ar[l]_ e } \]

and point $x' \in X'$. We set $Z' = X'$, $i = \text{id}$, and $\mathcal{G} = g^*\mathcal{F}$ to obtain a solution in this case.

In general choose a closed immersion $Z \to X$ and a sheaf $\mathcal{G}$ on $Z$ as in Lemma 38.3.1. Applying the result of the previous paragraph to $Z \to S$ and $\mathcal{G}$ we obtain a diagram

\[ \xymatrix{ X \ar[dd]_ f & Z \ar[l] \ar[dd]_{f|_ Z} & Z' \ar[l]^ g \ar[d]^\pi \\ & & Y' \ar[d]^ h \\ S & S \ar@{=}[l] & S' \ar[l]_ e } \]

and point $z' \in Z'$ satisfying all the required properties. We will use Lemma 38.2.1 to embed $Z'$ into a scheme étale over $X$. We cannot apply the lemma directly as we want $X'$ to be a scheme over $S'$. Instead we consider the morphisms

\[ \xymatrix{ Z' \ar[r] & Z \times _ S S' \ar[r] & X \times _ S S' } \]

The first morphism is étale by Morphisms, Lemma 29.36.18. The second is a closed immersion as a base change of a closed immersion. Finally, as $X$, $S$, $S'$, $Z$, $Z'$ are all affine we may apply Lemma 38.2.1 to get an étale morphism of affine schemes $X' \to X \times _ S S'$ such that

\[ Z' = (Z \times _ S S') \times _{(X \times _ S S')} X' = Z \times _ X X'. \]

As $Z \to X$ is a closed immersion of finite presentation, so is $Z' \to X'$. Let $x' \in X'$ be the point corresponding to $z' \in Z'$. Then the completed diagram

\[ \xymatrix{ X \ar[dd] & X' \ar[dd] \ar[l] & Z' \ar[l]^ i \ar[d]^\pi \\ & & Y' \ar[d]^ h \\ S & S' \ar[l]_ e & S' \ar@{=}[l] } \]

is a solution of the original problem. $\square$

Lemma 38.3.3. Assumptions and notation as in Lemma 38.3.2. If $f$ is locally of finite presentation then $\pi $ is of finite presentation. In this case the following are equivalent

  1. $\mathcal{F}$ is an $\mathcal{O}_ X$-module of finite presentation in a neighbourhood of $x$,

  2. $\mathcal{G}$ is an $\mathcal{O}_{Z'}$-module of finite presentation in a neighbourhood of $z'$, and

  3. $\pi _*\mathcal{G}$ is an $\mathcal{O}_{Y'}$-module of finite presentation in a neighbourhood of $y'$.

Still assuming $f$ locally of finite presentation the following are equivalent to each other

  1. $\mathcal{F}_ x$ is an $\mathcal{O}_{X, x}$-module of finite presentation,

  2. $\mathcal{G}_{z'}$ is an $\mathcal{O}_{Z', z'}$-module of finite presentation, and

  3. $(\pi _*\mathcal{G})_{y'}$ is an $\mathcal{O}_{Y', y'}$-module of finite presentation.

Proof. Assume $f$ locally of finite presentation. Then $Z' \to S$ is locally of finite presentation as a composition of such, see Morphisms, Lemma 29.21.3. Note that $Y' \to S$ is also locally of finite presentation as a composition of a smooth and an étale morphism. Hence Morphisms, Lemma 29.21.11 implies $\pi $ is locally of finite presentation. Since $\pi $ is finite we conclude that it is also separated and quasi-compact, hence $\pi $ is actually of finite presentation.

To prove the equivalence of (1), (2), and (3) we also consider: (4) $g^*\mathcal{F}$ is a $\mathcal{O}_{X'}$-module of finite presentation in a neighbourhood of $x'$. The pullback of a module of finite presentation is of finite presentation, see Modules, Lemma 17.11.4. Hence (1) $\Rightarrow $ (4). The étale morphism $g$ is open, see Morphisms, Lemma 29.36.13. Hence for any open neighbourhood $U' \subset X'$ of $x'$, the image $g(U')$ is an open neighbourhood of $x$ and the map $\{ U' \to g(U')\} $ is an étale covering. Thus (4) $\Rightarrow $ (1) by Descent, Lemma 35.7.3. Using Descent, Lemma 35.7.10 and some easy topological arguments (see More on Morphisms, Lemma 37.43.4) we see that (4) $\Leftrightarrow $ (2) $\Leftrightarrow $ (3).

To prove the equivalence of (a), (b), (c) consider the ring maps

\[ \mathcal{O}_{X, x} \to \mathcal{O}_{X', x'} \to \mathcal{O}_{Z', z'} \leftarrow \mathcal{O}_{Y', y'} \]

The first ring map is faithfully flat. Hence $\mathcal{F}_ x$ is of finite presentation over $\mathcal{O}_{X, x}$ if and only if $g^*\mathcal{F}_{x'}$ is of finite presentation over $\mathcal{O}_{X', x'}$, see Algebra, Lemma 10.83.2. The second ring map is surjective (hence finite) and finitely presented by assumption, hence $g^*\mathcal{F}_{x'}$ is of finite presentation over $\mathcal{O}_{X', x'}$ if and only if $\mathcal{G}_{z'}$ is of finite presentation over $\mathcal{O}_{Z', z'}$, see Algebra, Lemma 10.36.23. Because $\pi $ is finite, of finite presentation, and $\pi ^{-1}(\{ y'\} ) = \{ x'\} $ the ring homomorphism $\mathcal{O}_{Y', y'} \leftarrow \mathcal{O}_{Z', z'}$ is finite and of finite presentation, see More on Morphisms, Lemma 37.43.4. Hence $\mathcal{G}_{z'}$ is of finite presentation over $\mathcal{O}_{Z', z'}$ if and only if $\pi _*\mathcal{G}_{y'}$ is of finite presentation over $\mathcal{O}_{Y', y'}$, see Algebra, Lemma 10.36.23. $\square$

Lemma 38.3.4. Assumptions and notation as in Lemma 38.3.2. The following are equivalent

  1. $\mathcal{F}$ is flat over $S$ in a neighbourhood of $x$,

  2. $\mathcal{G}$ is flat over $S'$ in a neighbourhood of $z'$, and

  3. $\pi _*\mathcal{G}$ is flat over $S'$ in a neighbourhood of $y'$.

The following are equivalent also

  1. $\mathcal{F}_ x$ is flat over $\mathcal{O}_{S, s}$,

  2. $\mathcal{G}_{z'}$ is flat over $\mathcal{O}_{S', s'}$, and

  3. $(\pi _*\mathcal{G})_{y'}$ is flat over $\mathcal{O}_{S', s'}$.

Proof. To prove the equivalence of (1), (2), and (3) we also consider: (4) $g^*\mathcal{F}$ is flat over $S$ in a neighbourhood of $x'$. We will use Lemma 38.2.3 to equate flatness over $S$ and $S'$ without further mention. The étale morphism $g$ is flat and open, see Morphisms, Lemma 29.36.13. Hence for any open neighbourhood $U' \subset X'$ of $x'$, the image $g(U')$ is an open neighbourhood of $x$ and the map $U' \to g(U')$ is surjective and flat. Thus (4) $\Leftrightarrow $ (1) by Morphisms, Lemma 29.25.13. Note that

\[ \Gamma (X', g^*\mathcal{F}) = \Gamma (Z', \mathcal{G}) = \Gamma (Y', \pi _*\mathcal{G}) \]

Hence the flatness of $g^*\mathcal{F}$, $\mathcal{G}$ and $\pi _*\mathcal{G}$ over $S'$ are all equivalent (this uses that $X'$, $Z'$, $Y'$, and $S'$ are all affine). Some omitted topological arguments (compare More on Morphisms, Lemma 37.43.4) regarding affine neighbourhoods now show that (4) $\Leftrightarrow $ (2) $\Leftrightarrow $ (3).

To prove the equivalence of (a), (b), (c) consider the commutative diagram of local ring maps

\[ \xymatrix{ \mathcal{O}_{X', x'} \ar[r]_\iota & \mathcal{O}_{Z', z'} & \mathcal{O}_{Y', y'} \ar[l]^\alpha & \mathcal{O}_{S', s'} \ar[l]^\beta \\ \mathcal{O}_{X, x} \ar[u]^\gamma & & & \mathcal{O}_{S, s} \ar[lll]_\varphi \ar[u]_\epsilon } \]

We will use Lemma 38.2.4 to equate flatness over $\mathcal{O}_{S, s}$ and $\mathcal{O}_{S', s'}$ without further mention. The map $\gamma $ is faithfully flat. Hence $\mathcal{F}_ x$ is flat over $\mathcal{O}_{S, s}$ if and only if $g^*\mathcal{F}_{x'}$ is flat over $\mathcal{O}_{S', s'}$, see Algebra, Lemma 10.39.9. As $\mathcal{O}_{S', s'}$-modules the modules $g^*\mathcal{F}_{x'}$, $\mathcal{G}_{z'}$, and $\pi _*\mathcal{G}_{y'}$ are all isomorphic, see More on Morphisms, Lemma 37.43.4. This finishes the proof. $\square$


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