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fpqc site

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topos theory

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Definition

Fix some scheme SS.

Definition

The fpqc-site (over SS) is the site

  • whose underlying category is the category Aff/SAff/S of affine schemes over SS;

  • whose coverage has as covering families {f:U iX}\{f : U_i \to X\} those families of morphisms that are such that

    • each f if_i is a flat morphism;

    • for every affine open WXW \hookrightarrow X there exists n0n \geq 0, a function a:{1,,n}Ia : \{1, \cdots, n\} \to I and affine opens V jT a(j)V_j \hookrightarrow T_{a(j)} with

      j=1 nf a(j)(V j)=W. \cup_{j = 1}^{n} f_{a(j)}(V_j) = W \,.

This appears as (de Jong, def. 27.8.1).

Remark

The last condition does imply that if i(U i)=X\cup_i f_i(U_i) = X.

Remark

The abbreviation “fpqc” is for fidèlement plat quasi-compacte : faithfully flat and quasi-compact.

Remark

Because the collection of fpqc covers of a scheme does not have a small collection of refinements, working with the fpqc topology can be set-theoretically tricky. Indeed, in 1975, Waterhouse gave an example of a functor on schemes that admits no fpqc sheafification. This contradicts many claims in the literature that fpqc sheafification and stackification is functorial (and such claims continue to be made).

fpqc-site \to fppf-site \to syntomic site \to étale site \to Nisnevich site \to Zariski site

References

Chaper 27.8 in

  • W. C. Waterhouse, Basically bounded functors and flat sheaves, Pacific Journal of Mathematics 57 (1975), no. 2, 597–610 MR396578, euclid

Revised on September 5, 2011 09:41:46 by Urs Schreiber (89.204.153.80)