Fppf coverings and the fppf site

Fix a base scheme $S$ (Schemes, Schemes and morphisms over a base). An fppf covering of an $S$-scheme $T$ is a family $\{T_i\to T{\}}_{i\in I}$ of morphisms of $S$-schemes (Morphisms of schemes) such that each $T_i\to T$ is flat (Flat morphism of schemes) and locally of finite presentation (Locally finite presentation morphisms), and the images of the underlying maps $\mathrm{\mid }{T}_i\mathrm{\mid }\to \mathrm{\mid }T\mathrm{\mid }$ cover $T$. The fppf site $(Sch\mathrm{/}S{)}_{fppf}$ is the category of $S$-schemes with the pretopology whose coverings of $T$ are the fppf coverings of $T$, with the identity refinements and the usual composition of coverings of an fppf topology.

The covering condition is a topological surjectivity condition on the index family together with the two morphism properties; the index set $I$ need not be finite. The images may overlap. The maps need not be open immersions. We work inside one fixed big fppf site of $S$-schemes; nothing below uses size questions beyond those conventions.

Three standard properties are used constantly and are recorded here with their proofs. First, a Zariski open cover is an fppf covering (Open immersions of schemes): an open immersion is flat and locally of finite presentation, and its underlying map is an open topological embedding, so the images of the members of an open cover of $T$ cover $T$. Second, fppf coverings are stable under base change: for a morphism $T^{\mathrm{\prime }}\to T$ the base-changed family $\{T_i{\times }_T{T}^{\mathrm{\prime }}\to T^{\mathrm{\prime }}\}$ is fppf, because flatness and local finite presentation are stable under base change and images of the base-changed maps still cover $T^{\mathrm{\prime }}$ (Fibre product of schemes). Third, fppf coverings are stable under composition: if $\{T_i\to T{\}}_{i\in I}$ is fppf and $\{T_{ij}\to T_i{\}}_{j\in J_i}$ is fppf for every $i$, then the composites $\{T_{ij}\to T\}$ form an fppf covering of $T$, because a composite of flat morphisms is flat, a composite of locally finitely presented morphisms is locally of finite presentation, and the images of the composites cover $T$ by the covering property of the two families together.

No choice principle is used to state the definition or its consequences: a covering is a single family of morphisms, and the stability assertions are element-wise. A covering may be indexed by an empty set only when the target $T$ is empty, in which case the covering condition is vacuous.