Fppf topology¶

A Grothendieck topology on schemes whose covering families are jointly surjective flat morphisms locally of finite presentation.

Version: v1 · 2026-10-31 · History
Domain-specific #: 14499
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Algebraic Geometry, Sites and Topoi → Mathematics

Core Idea¶

An fppf cover is a family of scheme maps whose members are each flat and locally of finite presentation and whose images together cover the target. Coverage belongs to the family: one member, such as an affine chart inclusion, need not cover the whole target. The fppf topology makes such families the covers of a Grothendieck site. A particular big site also requires a size convention; the cover test and that convention are separate.^[1]

How would you explain it like I'm…

No faithful explanation at this level. All three generators judged a five-year-old picture of covering collapses into patches cut from one shape, the open-subset notion fppf goes beyond, losing that covers are families of maps tested one by one for flatness and finite presentation and only jointly for surjectivity.

Rules for What Counts as Covering

In a very advanced part of math, mathematicians study spaces called schemes and need rules for when a bunch of maps into a space 'covers' it. The fppf topology is one such rule. Each map in the bunch has to pass two technical tests, called flat and locally of finite presentation. Then, all together, the maps have to reach every point of the space. One map alone doesn't have to reach everywhere, as long as the whole group does.

Flat Finitely Presented Covers

In algebraic geometry, a 'topology' in Grothendieck's sense is a rule saying which families of maps count as covers of an object. The fppf topology is one such rule for schemes. A family of morphisms {f_i : T_i → T} is an fppf covering if each f_i is flat and locally of finite presentation (two technical conditions checked on each map separately), and the images of all the T_i together cover T (a condition checked on the whole family). So one member, such as an open immersion, can fail to be surjective by itself and still belong to a cover. These covers obey the rules for a site: isomorphisms are covers, covers of covers give covers, and pulling back a cover gives a cover. Results like descent and fppf cohomology use this rule but don't define it.

The fppf topology (fidèlement plate de présentation finie) specifies which families of scheme morphisms are coverings. A family {f_i : T_i → T} is an fppf covering when each f_i is flat and locally of finite presentation and the images of the T_i jointly cover T. Flatness and finite presentation are conditions on each individual morphism, whereas surjectivity is a condition on the family as a whole; consequently an open immersion that is not surjective can still be a member of a covering. These families satisfy the axioms of a Grothendieck topology: isomorphisms are coverings, fppf coverings compose, and the base change of a covering along any morphism of schemes is again an fppf covering. When a site is meant, one works with a size-controlled big site (Sch/S)_fppf over a specified base scheme S, and the covering predicate should be stated separately from the site's size convention. The topology is the rule for admissible covers; descent for quasi-coherent sheaves and fppf cohomology are applications requiring further theorems and choices of coefficients, and they do not define what an fppf cover is.

Scope of Application¶

This covering rule is used for sheaves and local arguments in algebraic geometry. Every Zariski or étale cover satisfies the fppf test, and some fppf covers are not étale. Descent and cohomology can use the site, but each conclusion requires a theorem about the relevant objects or coefficients. The cover definition alone does not say that arbitrary objects glue.^[1]^[2]^[3]

Clarity¶

Check three things: flatness of every map, local finite presentation of every map, and joint coverage of the target. A standard affine fppf cover over an affine target is a convenient finite refinement, not the definition of every fppf family. Fpqc has a different affine-local cover condition; every fppf cover is fpqc, but an arbitrary fpqc cover need not satisfy the fppf presentation condition.^[1]^[4]

Manages Complexity¶

The rule turns a large class of possible local presentations into one test that is stable under composition and base change. For affine targets, a finite affine refinement helps calculations without changing what the original covering family was. Any later descent or cohomology calculation must state its own hypotheses.^[1]^[2]

Abstract Reasoning¶

First identify the schemes and the base. Prove both map properties for every arrow, then prove their images jointly cover the target. A known fppf cover may be pulled back or composed using the covering stability lemma. Only after the cover is established should one ask what data can be glued or what cohomology group is being computed.^[1]^[2]

The strict parent is Grothendieck topology, whose general covering axioms do not require flat locally finitely presented maps of schemes. The paired parent repair removes its unsupported link to the open-set Topology Prime. That Prime remains a comparison neighbor rather than an inherited parent.^[1]

Knowledge Transfer¶

The test works for unlike scheme families: a pair of Zariski chart inclusions and a single non-étale characteristic-\(p\) torsor map both qualify when the three cover tests hold. This transfer is within scheme theory. An ordinary open cover or a vague local-to-global analogy does not supply the scheme-morphism, flatness and presentation obligations.^[5]^[6]^[1]

Example¶

The two usual affine opens of \(\mathbf P^1_k\) jointly cover the projective line. Each inclusion is flat and locally of finite presentation, though neither inclusion is surjective alone. Mapped back: the projective line is the carrier; the two maps meet the per-map tests; their combined images meet the family test. This gives an fppf cover within the stated big site.^[5]^[1]

Relationships to Other Abstractions¶

Current abstraction Fppf topology Domain-specific

Parents (1) — outgoing typed relations

Fppf topology is a kind of Grothendieck topology Domain-specific

The fppf covering rule is a Grothendieck topology on schemes, with flat locally finitely presented jointly covering families as its differentia.

Hierarchy path (1) — routes to 1 parentless root

Fppf topology → is a kind of Grothendieck topology

Neighborhood in Abstraction Space¶

Fppf topology sits in a crowded region of the domain-specific corpus (37^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Graph Structures & Combinatorial Objects (48 abstractions)

Nearest neighbors

Sheaf — 0.88
A-paracompact Space — 0.88
Closed Immersion — 0.88
Algebraic stack — 0.87
Descent (Mathematics) — 0.87

Computed from structural-signature embeddings · 2026-10-31

Not to Be Confused With¶

Fpqc topology has a different cover test. Étale topology permits a smaller class of local morphisms. Faithfully flat morphism describes one map, whereas an fppf covering family may contain maps that are not individually surjective. Flat cohomology is an application using a site and coefficients, and descent is a separate recovery question about local objects.^[1]^[4]^[3]

References¶

[1] The Stacks Project. “The Stacks Project”. Section 34.7, “The fppf topology,” tag 021L. Primary collaborative mathematical reference, full web text inspected. Definition 34.7.1 gives the family test; Lemmas 34.7.2–34.7.4 give inclusions, stability and affine refinement; Definitions 34.7.5, 34.7.6 and 34.7.8 give standard affine covers and the size-controlled big site. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i

[2] The Stacks Project. “The Stacks Project”. Section 35.5, “Fpqc descent of quasi-coherent sheaves,” tag 023R. Primary collaborative mathematical reference, full web text inspected. Proposition 35.5.2 states the effective-descent theorem for the specified object class. registry ↩a ↩b ↩c

[3] The Stacks Project. “The Stacks Project”. Lemma 59.28.5, tag 040Q. Primary collaborative mathematical reference, full web text inspected. It identifies \(H^1_{fppf}(S,\mu_n)\) with specified line-bundle/root pairs and states an étale identification when \(n\) is invertible on \(S\). registry ↩a ↩b

[4] The Stacks Project. “The Stacks Project”. Section 34.9, “The fpqc topology,” tag 022A. Primary collaborative mathematical reference, full web text inspected. Definition 34.9.1 states the affine-local fpqc test; Lemma 34.9.7 proves every fppf cover is fpqc. registry ↩a ↩b

[5] The Stacks Project. “The Stacks Project”. Example 26.21.8, tag 01KQ. Primary collaborative mathematical reference, full web text inspected. The projective line over a field has two affine-open charts. registry ↩a ↩b

[6] The Stacks Project. “The Stacks Project”. Section 110.55, “A non-algebraic classifying stack,” tag 077C. Primary collaborative mathematical reference, full web text inspected. The paragraph beginning “The first objection” supplies the characteristic-\(p\) \(\mu_p\) torsor and its fppf versus étale local-triviality contrast. registry ↩