Open Immersion Base Change

Follows immediately from Lemma 18.7 and the commutativity of (17.6).  

The square a is a fiber square, since the canonical map / — U, xx XI is an image dense closed open immersion, and is an isomorphism. 

To prove the theorem, it suffices to show that the map in question is an isomorphism after applying ( ) for any i G /. By Proposition 18.14, Lemma 19.2, and [26, (3.7.2), (iii)], the problem is reduced to the flat base change theorem (in fact open immersion base change theorem is enough) for schemes [41, Theorem 2, and we are done.  

Lemma 20.2. Let the notation be as above. Conditions 1 and 8—14 in Definition 16.1 are satisfied. Moreover, any pi-sqvMre is a fiber square. 

1 i (r) is an open immersion whose scheme theoretic image is Z, r) (i.e., for any j, j r)j is an open immersion whose scheme theoretic image is Z, r)j). In particular, j, r) is an image dense open immersion. 

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Proof. As pointed out in [35, section 6], this is an immediate consequence of the open immersion base change [41, Theorem 2], ... 

The proof of (4.8.1) presented here is based on a formal method of Deligne for pasting pseudofunctors (see Proposition (4.8.4)), and on the compactifi-cation theorem of Nagata, that any finite-type separable map of noetherian schemes factors as an open immersion followed by a proper map (see [Lt], [C j, [Vj]). The proof of (4.8.3) is based on a formal pasting procedure for base-change setups (see (4.8.2), (4.8.5)). 

The diagonal of a separated etale map is an open-and-closed immersion [EGA IV, (17.4.2) (b)] and maps which are etale (resp. separated, resp. proper) remain so after arbitrary base change [EGA IV, (17.3.3)(iii)]. Therefore the category E of separated etale maps (resp. proper etale maps) satisfies the hypotheses of (4.8.10.5) with respect to (H, E) (resp. (B, E)). Keeping in mind the uniqueness part of (4.8.10), one see that the resulting... 

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