Lemma 31.9.1. Let f : T \to S be a morphism of schemes. Let \mathcal{F} be a finite type quasi-coherent \mathcal{O}_ S-module. Then f^{-1}\text{Fit}_ i(\mathcal{F}) \cdot \mathcal{O}_ T = \text{Fit}_ i(f^*\mathcal{F}).
31.9 Fitting ideals
This section is the continuation of the discussion in More on Algebra, Section 15.8. Let S be a scheme. Let \mathcal{F} be a finite type quasi-coherent \mathcal{O}_ S-module. In this situation we can construct the Fitting ideals
0 = \text{Fit}_{-1}(\mathcal{F}) \subset \text{Fit}_0(\mathcal{F}) \subset \text{Fit}_1(\mathcal{F}) \subset \ldots \subset \mathcal{O}_ S
as the sequence of quasi-coherent ideals characterized by the following property: for every affine open U = \mathop{\mathrm{Spec}}(A) of S if \mathcal{F}|_ U corresponds to the A-module M, then \text{Fit}_ i(\mathcal{F})|_ U corresponds to the ideal \text{Fit}_ i(M) \subset A. This is well defined and a quasi-coherent sheaf of ideals because if f \in A, then the ith Fitting ideal of M_ f over A_ f is equal to \text{Fit}_ i(M) A_ f by More on Algebra, Lemma 15.8.4.
Alternatively, we can construct the Fitting ideals in terms of local presentations of \mathcal{F}. Namely, if U \subset X is open, and
\bigoplus \nolimits _{i \in I} \mathcal{O}_ U \to \mathcal{O}_ U^{\oplus n} \to \mathcal{F}|_ U \to 0
is a presentation of \mathcal{F} over U, then \text{Fit}_ r(\mathcal{F})|_ U is generated by the (n - r) \times (n - r)-minors of the matrix defining the first arrow of the presentation. This is compatible with the construction above because this is how the Fitting ideal of a module over a ring is actually defined. Some details omitted.
Proof. Follows immediately from More on Algebra, Lemma 15.8.4 part (3). \square
Lemma 31.9.2. Let S be a scheme. Let \mathcal{F} be a finitely presented \mathcal{O}_ S-module. Then \text{Fit}_ r(\mathcal{F}) is a quasi-coherent ideal of finite type.
Proof. Follows immediately from More on Algebra, Lemma 15.8.4 part (4). \square
Lemma 31.9.3. Let S be a scheme. Let \mathcal{F} be a finite type, quasi-coherent \mathcal{O}_ S-module. Let Z_0 \subset S be the closed subscheme cut out by \text{Fit}_0(\mathcal{F}). Let Z \subset S be the scheme theoretic support of \mathcal{F}. Then
Z \subset Z_0 \subset S as closed subschemes,
Z = Z_0 = \text{Supp}(\mathcal{F}) as closed subsets,
there exists a finite type, quasi-coherent \mathcal{O}_{Z_0}-module \mathcal{G}_0 with
(Z_0 \to X)_*\mathcal{G}_0 = \mathcal{F}.
Proof. Recall that Z is locally cut out by the annihilator of \mathcal{F}, see Morphisms, Definition 29.5.5 (which uses Morphisms, Lemma 29.5.4 to define Z). Hence we see that Z \subset Z_0 scheme theoretically by More on Algebra, Lemma 15.8.4 part (6). On the other hand we have Z = \text{Supp}(\mathcal{F}) set theoretically by Morphisms, Lemma 29.5.4 and we have Z_0 = Z set theoretically by More on Algebra, Lemma 15.8.4 part (7). Finally, to get \mathcal{G}_0 as in part (3) we can either use that we have \mathcal{G} on Z as in Morphisms, Lemma 29.5.4 and set \mathcal{G}_0 = (Z \to Z_0)_*\mathcal{G} or we can use Morphisms, Lemma 29.4.1 and the fact that \text{Fit}_0(\mathcal{F}) annihilates \mathcal{F} by More on Algebra, Lemma 15.8.4 part (6). \square
Lemma 31.9.4. Let S be a scheme. Let \mathcal{F} be a finite type, quasi-coherent \mathcal{O}_ S-module. Let s \in S. Then \mathcal{F} can be generated by r elements in a neighbourhood of s if and only if \text{Fit}_ r(\mathcal{F})_ s = \mathcal{O}_{S, s}.
Proof. Follows immediately from More on Algebra, Lemma 15.8.7. \square
Lemma 31.9.5. Let S be a scheme. Let \mathcal{F} be a finite type, quasi-coherent \mathcal{O}_ S-module. Let r \geq 0. The following are equivalent
\mathcal{F} is finite locally free of rank r
\text{Fit}_{r - 1}(\mathcal{F}) = 0 and \text{Fit}_ r(\mathcal{F}) = \mathcal{O}_ S, and
\text{Fit}_ k(\mathcal{F}) = 0 for k < r and \text{Fit}_ k(\mathcal{F}) = \mathcal{O}_ S for k \geq r.
Proof. Follows immediately from More on Algebra, Lemma 15.8.8. \square
Lemma 31.9.6. Let S be a scheme. Let \mathcal{F} be a finite type, quasi-coherent \mathcal{O}_ S-module. The closed subschemes
S = Z_{-1} \supset Z_0 \supset Z_1 \supset Z_2 \ldots
defined by the Fitting ideals of \mathcal{F} have the following properties
The intersection \bigcap Z_ r is empty.
The functor (\mathit{Sch}/S)^{opp} \to \textit{Sets} defined by the rule
T \longmapsto \left\{ \begin{matrix} \{ *\} & \text{if }\mathcal{F}_ T\text{ is locally generated by } \leq r\text{ sections} \\ \emptyset & \text{otherwise} \end{matrix} \right.is representable by the open subscheme S \setminus Z_ r.
The functor F_ r : (\mathit{Sch}/S)^{opp} \to \textit{Sets} defined by the rule
T \longmapsto \left\{ \begin{matrix} \{ *\} & \text{if }\mathcal{F}_ T\text{ locally free rank }r \\ \emptyset & \text{otherwise} \end{matrix} \right.is representable by the locally closed subscheme Z_{r - 1} \setminus Z_ r of S.
If \mathcal{F} is of finite presentation, then Z_ r \to S, S \setminus Z_ r \to S, and Z_{r - 1} \setminus Z_ r \to S are of finite presentation.
Proof. Part (1) is true because over every affine open U there is an integer n such that \text{Fit}_ n(\mathcal{F})|_ U = \mathcal{O}_ U. Namely, we can take n to be the number of generators of \mathcal{F} over U, see More on Algebra, Section 15.8.
For any morphism g : T \to S we see from Lemmas 31.9.1 and 31.9.4 that \mathcal{F}_ T is locally generated by \leq r sections if and only if \text{Fit}_ r(\mathcal{F}) \cdot \mathcal{O}_ T = \mathcal{O}_ T. This proves (2).
For any morphism g : T \to S we see from Lemmas 31.9.1 and 31.9.5 that \mathcal{F}_ T is free of rank r if and only if \text{Fit}_ r(\mathcal{F}) \cdot \mathcal{O}_ T = \mathcal{O}_ T and \text{Fit}_{r - 1}(\mathcal{F}) \cdot \mathcal{O}_ T = 0. This proves (3).
Assume \mathcal{F} is of finite presentation. Then each of the morphisms Z_ r \to S is of finite presentation as \text{Fit}_ r(\mathcal{F}) is of finite type (Lemma 31.9.2 and Morphisms, Lemma 29.21.7). This implies that Z_{r - 1} \setminus Z_ r is a retrocompact open in Z_ r (Properties, Lemma 28.24.1) and hence the morphism Z_{r - 1} \setminus Z_ r \to Z_ r is of finite presentation as well. \square
Lemma 31.9.6 notwithstanding the following lemma does not hold if \mathcal{F} is a finite type quasi-coherent module. Namely, the stratification still exists but it isn't true that it represents the functor F_{flat} in general.
Lemma 31.9.7. Let S be a scheme. Let \mathcal{F} be an \mathcal{O}_ S-module of finite presentation. Let S = Z_{-1} \supset Z_0 \supset Z_1 \supset \ldots be as in Lemma 31.9.6. Set S_ r = Z_{r - 1} \setminus Z_ r. Then S' = \coprod _{r \geq 0} S_ r represents the functor
F_{flat} : \mathit{Sch}/S \longrightarrow \textit{Sets},\quad \quad T \longmapsto \left\{ \begin{matrix} \{ *\}
& \text{if }\mathcal{F}_ T\text{ flat over }T
\\ \emptyset
& \text{otherwise}
\end{matrix} \right.
Moreover, \mathcal{F}|_{S_ r} is locally free of rank r and the morphisms S_ r \to S and S' \to S are of finite presentation.
Proof. Suppose that g : T \to S is a morphism of schemes such that the pullback \mathcal{F}_ T = g^*\mathcal{F} is flat. Then \mathcal{F}_ T is a flat \mathcal{O}_ T-module of finite presentation. Hence \mathcal{F}_ T is finite locally free, see Properties, Lemma 28.20.2. Thus T = \coprod _{r \geq 0} T_ r, where \mathcal{F}_ T|_{T_ r} is locally free of rank r. This implies that
F_{flat} = \coprod \nolimits _{r \geq 0} F_ r
in the category of Zariski sheaves on \mathit{Sch}/S where F_ r is as in Lemma 31.9.6. It follows that F_{flat} is represented by \coprod _{r \geq 0} (Z_{r - 1} \setminus Z_ r) where Z_ r is as in Lemma 31.9.6. The other statements also follow from the lemma. \square
Example 31.9.8. Let R = \prod _{n \in \mathbf{N}} \mathbf{F}_2. Let I \subset R be the ideal of elements a = (a_ n)_{n \in \mathbf{N}} almost all of whose components are zero. Let \mathfrak m be a maximal ideal containing I. Then M = R/\mathfrak m is a finite flat R-module, because R is absolutely flat (More on Algebra, Lemma 15.104.6). Set S = \mathop{\mathrm{Spec}}(R) and \mathcal{F} = \widetilde{M}. The closed subschemes of Lemma 31.9.6 are S = Z_{-1}, Z_0 = \mathop{\mathrm{Spec}}(R/\mathfrak m), and Z_ i = \emptyset for i > 0. But \text{id} : S \to S does not factor through (S \setminus Z_0) \amalg Z_0 because \mathfrak m is a nonisolated point of S. Thus Lemma 31.9.7 does not hold for finite type modules.
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