The Stacks project

87.3 Conventions and notation

The conventions from now on will be similar to the conventions in Properties of Spaces, Section 66.2. Thus from now on the standing assumption is that all schemes are contained in a big fppf site $\mathit{Sch}_{fppf}$. And all rings $A$ considered have the property that $\mathop{\mathrm{Spec}}(A)$ is (isomorphic) to an object of this big site. For topological rings $A$ we assume only that all discrete quotients have this property (but usually we assume more, compare with Remark 87.11.5).

Let $S$ be a scheme and let $X$ be a “space” over $S$, i.e., a sheaf on $(\mathit{Sch}/S)_{fppf}$. In this chapter we will write $X \times _ S X$ for the product of $X$ with itself in the category of sheaves on $(\mathit{Sch}/S)_{fppf}$ instead of $X \times X$. Moreover, if $X$ and $Y$ are “spaces” then we say "let $f : X \to Y$ be a morphism" to indicate that $f$ is a natural transformation of functors, i.e., a map of sheaves on $(\mathit{Sch}/S)_{fppf}$. Similarly, if $U$ is a scheme over $S$ and $X$ is a “space” over $S$, then we say "let $f : U \to X$ be a morphism" or "let $g : X \to U$ be a morphism" to indicate that $f$ or $g$ is a map of sheaves $h_ U \to X$ or $X \to h_ U$ where $h_ U$ is as in Categories, Example 4.3.4.