Separated and Proper Morphisms
نویسنده
چکیده
Last quarter, we introduced the closed diagonal condition for a prevariety to be a prevariety, and the “universally closed” condition for a variety to be complete. Earlier this quarter, we verified that in the case of complex prevarieties, these conditions are equivalent to the analytic topology being Hausdorff and compact, respectively. As we’ve discussed, in the scheme context a prevariety corresponds not to an individual scheme, but to a scheme over Spec k, and therefore the above two conditions are rephrased by Grothendieck as conditions on morphisms of schemes. These are called separatedness and properness.
منابع مشابه
Formal GAGA for good moduli spaces
We prove formal GAGA for good moduli space morphisms under an assumption of “enough vector bundles” (which holds for instance for quotient stacks). This supports the philosophy that though they are non-separated, good moduli space morphisms largely behave like proper morphisms.
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