The localic reflection of the petit étale topos

Recall that the petit étale topos of a scheme X is the topos of sheaves on the category of schemes étale over X (with respect to the étale topology). Since étale morphisms are open, any closed sieve on X must contain the union of the (open) images of all its members. Thus, the closed sieves on X are classified by the open subspaces of X, so the localic reflection of the petit étale topos of X is the petit Zariski topos of X.

Note also that the same argument works for the petit fppf topos of X – the point being that flat morphisms locally of finite presentation are open.

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