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
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
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