Grothendieck's SGA 1, Exposé VIII proves that faithfully flat quasi-compact (fpqc) morphisms are of descent for the category of schemes (pullback is fully faithful) and, in §7, gives sufficient conditions for a descent datum to be effective: affine or quasi-affine structure, a compatible relatively ample line bundle, radicial (universally injective) morphisms, and finite locally free morphisms with an affine-orbit condition. Remarques 7.10 states that these exhaust the known — and probably the useful — effectivity criteria, and records two counterexamples showing the hypotheses are genuinely needed. The intrinsic characterization problem is still open. Later work does not settle this general problem but shows that fppf descent data on schemes are always effective in Artin's category of algebraic spaces; the surviving open core is therefore precisely when the descended algebraic space is a scheme, together with the classification of effective descent morphisms for the category of schemes, where Grothendieck's remark after Corollary 7.5 records the radicial case as the only known one and conjectures it to be essentially the only one (an expectation whose intended scope requires interpretation, since open covers are effective for trivial reasons; see part (2) of the Problem Statement). Adjacent cores (modules over rings; étale morphisms) are fully resolved, which sharpens the contrast.

Background

Grothendieck's descent theory answers the question of which objects defined only after a cover glue to a global object. Fix a morphism of schemes g:S′→Sg:S'\to S that is faithfully flat and quasi-compact (fpqc) — that is, surjective, flat (pulling back modules along gg preserves exact sequences), and quasi-compact (preimages of affine opens are covered by finitely many affine opens); typical examples are a finite Galois field extension, or the cover of a scheme by a coproduct of its affine open subschemes. A descent datum on a scheme X′X' over S′S' is an isomorphism ϕ:p1∗X′→p2∗X′\phi:p_1^*X'\to p_2^*X' between the two pullbacks of X′X' along the projections of the fiber product S′×SS′S'\times_S S', satisfying a cocycle condition on the triple product — the algebraic analogue of gluing data for a cover by two open sets. The datum is effective if (X′,ϕ)(X',\phi) is isomorphic to the pullback of a scheme XX over SS; in that case XX is unique up to unique isomorphism. The morphism gg is an effective descent morphism for the category of schemes if every descent datum relative to gg is effective. SGA 1 (Séminaire de Géométrie Algébrique du Bois Marie 1960–61; the word 'prescheme' used there means what is now called a scheme) proves in Exposé VIII that every fpqc morphism is at least of descent, i.e. pullback from schemes over SS to schemes over S′S' with descent data is fully faithful. Effectivity is subtler.

Exposé VIII, §7 first reduces the problem (Proposition 7.1): a descent datum is effective if and only if the flat equivalence relation RR on X′X' that it determines is effective — meaning the quotient X′/RX'/R exists as a scheme — and the natural map X′→X′/RX'\to X'/R is fpqc. The sufficient criteria proved there are: (a) the affine case (Corollary 7.4, stated for affine SS and S′S': necessary and sufficient that X′X' be covered by affine opens stable under the datum); (b) radicial fpqc morphisms (Corollary 7.5), radicial meaning universally injective, informally a bijection on points after any base change; (c) finite locally free morphisms (finite and flat, equivalently finite projective over affine opens) whose equivalence classes of points of X′X' over a fixed point of SS lie in affine opens (Corollary 7.6), a condition automatic when X′X' is quasi-projective over S′S' (quasi-projective meaning isomorphic to a locally closed subscheme of a projective space over the base; Corollary 7.7 — the hypothesis used in most applications, such as Galois descent of quasi-projective varieties); (d) existence on X′X' of a relatively ample line bundle (ampleness is the standard measure of projective-like behavior of a morphism) compatible with the datum (Proposition 7.8); and (e) quasi-affine morphisms (a morphism that factors as an open immersion into an affine morphism; Corollary 7.9).

Remarques 7.10 records two counterexamples showing the hypotheses cannot simply be dropped. One, attributed to Serre: a smooth proper threefold over a quadratic Galois field extension carrying a non-effective descent datum, built from Hironaka's example of a non-projective threefold with a free involution admitting two congruent points contained in no common affine open. The other: over a torsor under the constant two-element group covering the spectrum of a regular local ring of dimension 3, with the points over the closed point removed, a projective regular scheme with a non-effective descent datum. The same Remarques assert that the results of the section exhaust the currently known — and probably the useful — effectivity criteria; a footnote added in the current edition records that this opinion was partially wrong, pointing to later results of Murre, Néron, and Raynaud, which concern group schemes and related special classes.

The modern framework relocates rather than solves the question. An algebraic space, in the sense of Artin and Knutson, is a sheaf for the étale topology (the Grothendieck topology on schemes whose covering families are jointly surjective étale morphisms, étale morphisms being the algebraic-geometric analogue of local isomorphisms) that is étale-locally an affine scheme — a mild but strictly larger generalization of the notion of scheme. As recorded in Vistoli's survey, by a result of Artin the category of algebraic spaces is a stack for the fppf topology (flat morphisms locally of finite presentation; Vistoli states he does not know whether the fpqc analogue holds), and Rydh has shown that geometric quotients of finite flat groupoids — the equivalence relations arising from finite flat descent data — exist as algebraic spaces in full generality. Consequently fppf descent data on schemes are always effective in the category of algebraic spaces: when gluing fails to produce a scheme, the descended object exists but is a non-scheme algebraic space. The Stacks Project formulates the dictionary precisely: a descent datum is effective if and only if the sheaf it defines is representable by a scheme (with the fpqc topology deliberately excluded from that equivalence for technical reasons). The open question is therefore exactly: when is the descended algebraic space a scheme?

Problem Statement

Let g:S′→Sg:S'\to S be a faithfully flat quasi-compact morphism of schemes, let X′X' be a scheme over S′S', and let ξ\xi be a descent datum on X′X' relative to gg (an isomorphism between the two pullbacks of X′X' along the projections of the fiber product S′×SS′S'\times_S S', satisfying the cocycle condition on the triple product, as in the background). SGA 1, Exposé VIII, §7 proves the sufficient effectivity criteria listed in the background and states in Remarques 7.10 that these exhaust the currently known — and probably the useful — effectivity criteria (a footnote added in the current edition records that this opinion was partially wrong: additional criteria were later obtained for special classes, notably group schemes, by Murre, Néron, and Raynaud, as noted in the background), so the general problem of characterizing effectivity in the category of schemes is left open. The open problems are the following. The two parts are independent — neither presupposes the other:

(1) Give an intrinsic characterization of the effectivity of ξ\xi in the category of schemes: necessary and sufficient conditions — geometric, cohomological, or stack-theoretic — checkable on (g,X′,ξ)(g,X',\xi) itself, going beyond the known sufficient hypotheses (affine or quasi-affine structure, a compatible relatively ample line bundle, radiciality of gg, finite local freeness of gg with the affine-orbit condition). By Proposition 7.1 of the same section this is equivalent to characterizing when the flat equivalence relation RR associated with ξ\xi admits a quotient scheme X′/RX'/R with X′→X′/RX'\to X'/R faithfully flat and quasi-compact. For covers in the Zariski, fppf, étale, smooth or syntomic topologies, the modern reformulation (Stacks Project, Lemma 35.39.1, tag 02W4) shows this is further equivalent to characterizing when the sheaf associated with the descent datum is representable by a scheme; that lemma deliberately excludes the fpqc topology ("the fact that fpqc is missing is not a typo"), and for general fpqc covers the existence of a descended algebraic space is itself not established — Vistoli (Remark 4.48) records that he does not know whether algebraic spaces form a stack in the fpqc topology.

(2) At the level of the cover: determine which morphisms gg are effective descent morphisms for the category of all schemes. The radicial fpqc case is effective by Corollary 7.5, and Grothendieck's accompanying remark states that this is the only known case of an effective descent morphism in the category of preschemes, and probably the only one, even restricting to noetherian schemes or to schemes of finite type over a field. Taken literally, the remark is contradicted by open covers (finite coproducts of open immersions), which are effective descent morphisms by Zariski gluing; the intended scope of the remark is therefore itself an open interpretive question subsidiary to the classification demand and not a substitute for it, and the expectation as recorded remains neither proved nor refuted.

The known counterexamples delimit the problem: non-effective descent data exist already for a smooth proper threefold over a quadratic Galois field extension (Serre, via Hironaka's non-projective threefold), for a projective regular family over a torsor under the constant two-element group covering the spectrum of a regular local ring of dimension 3 with the points over the closed point removed (both in SGA 1, Remarques 7.10), for a projective morphism over a degree-2 étale cover (Stacks Project, tag 08KE, where the descended object is a smooth separated (separated being the scheme-theoretic analogue of the Hausdorff property) algebraic space of dimension 3), and in dimension 1 over a degree-2 Galois field extension (Stacks Project, Example 65.14.2, tag 02Z0).

References

  1. Alexander Grothendieck (with Michele Raynaud, co-credited in the revised edition), Exposé VIII: Descente fidèlement plate, in Revêtements étales et groupe fondamental (SGA 1), Lecture Notes in Mathematics 224, Springer, Berlin, 1971, chapter DOI 10.1007/BFb0058663, https://doi.org/10.1007/BFb0058663; revised edition, arXiv:math/0206203, https://arxiv.org/abs/math/0206203
  2. Angelo Vistoli, Notes on Grothendieck topologies, fibered categories and descent theory, arXiv:math/0412512 (version 4, revised 24 May 2007), https://arxiv.org/abs/math/0412512
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  7. Yves André and Luisa Fiorot, On the canonical, fpqc, and finite topologies on affine schemes. The state of the art, arXiv:1912.04957 (2019), https://arxiv.org/abs/1912.04957; published in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, DOI 10.2422/2036-2145.201912_006, https://doi.org/10.2422/2036-2145.201912_006
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  10. The Stacks Project Authors, The Stacks Project, electronic reference, https://stacks.math.columbia.edu; cited by stable tag: Descent, Lemma 35.38.1 (tag 0247, https://stacks.math.columbia.edu/tag/0247); Descent, Section 35.4, Descent for universally injective morphisms (tag 08WE, https://stacks.math.columbia.edu/tag/08WE); Descent, Lemma 35.39.1 (tag 02W4, https://stacks.math.columbia.edu/tag/02W4); Examples, Section 110.68, Non-effective descent data for projective schemes (tag 08KE, https://stacks.math.columbia.edu/tag/08KE); Algebraic Spaces, Section 65.14, Examples of algebraic spaces, Example 65.14.2 (tag 02Z0, https://stacks.math.columbia.edu/tag/02Z0)
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