The Stacks project

65.2 General remarks

We work in a suitable big fppf site $\mathit{Sch}_{fppf}$ as in Topologies, Definition 34.7.6. So, if not explicitly stated otherwise all schemes will be objects of $\mathit{Sch}_{fppf}$. In Section 65.15 we discuss what changes if you change the big fppf site.

We will always work relative to a base $S$ contained in $\mathit{Sch}_{fppf}$. And we will then work with the big fppf site $(\mathit{Sch}/S)_{fppf}$, see Topologies, Definition 34.7.8. The absolute case can be recovered by taking $S = \mathop{\mathrm{Spec}}(\mathbf{Z})$.

If $U, T$ are schemes over $S$, then we denote $U(T)$ for the set of $T$-valued points over $S$. In a formula: $U(T) = \mathop{\mathrm{Mor}}\nolimits _ S(T, U)$.

Note that any fpqc covering is a universal effective epimorphism, see Descent, Lemma 35.13.7. Hence the topology on $\mathit{Sch}_{fppf}$ is weaker than the canonical topology and all representable presheaves are sheaves.