The Stacks project

Definition 29.5.5. Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite type. The scheme theoretic support of $\mathcal{F}$ is the closed subscheme $Z \subset X$ constructed in Lemma 29.5.4.

Comments (2)

Comment #7212 by DatPham on

I am wondering if there is any reason why the support defined in this way deserves the name "scheme theoretic support", and not the one defined using the Fitting ideal sheaf of ? I think the latter bahaves nicer in famillies as its formation commutes with arbitrary base change, which is not the case for the former.

Comment #7327 by on

This is the definition used most often in the literature.

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