Processing math: 100%

The Stacks project

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


Comments (0)


Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.