Formal Gaga on Artin Stacks
نویسنده
چکیده
Suppose X is a locally noetherian Deligne–Mumford stack. Definition 1.2 has an obvious variant X̂ét using the underlying smaller étale site Xét and the restriction Oc Xét of Oc X to this site. By [3, 12.7.4], the category of cartesian Oc X -modules on Xlis-ét is equivalent to the category of Oc Xét-modules on Xét: (1.1) ModXlis-ét,cart(Oc X ) ' ModXét(Oc Xét) Definition 1.3. Let X be a locally noetherian stack and X0 ⊆ |X | a closed subset, and let X̂ be the completion of X along X0. The category Coh(X̂ ) of coherent sheaves is the full subcategory of cartesian Oc X -modules on Xlis-ét that are locally of finite presentation. If X is a locally noetherian Deligne–Mumford stack then the category Coh(X̂ét) of coherent sheaves is the full subcategory of Oc Xét-modules on Xét that are locally of finite presentation.
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