The Stacks project

99.2 Conventions

We have intentionally placed this chapter, as well as the chapters “Examples of Stacks”, “Sheaves on Algebraic Stacks”, “Criteria for Representability”, and “Artin's Axioms” before the general development of the theory of algebraic stacks. The reason for this is that starting with the next chapter (see Properties of Stacks, Section 100.2) we will no longer distinguish between a scheme and the algebraic stack it gives rise to. Thus our language will become more flexible and easier for a human to parse, but also less precise. These first few chapters, including the initial chapter “Algebraic Stacks”, lay the groundwork that later allow us to ignore some of the very technical distinctions between different ways of thinking about algebraic stacks. But especially in the chapters “Artin's Axioms” and “Criteria of Representability” we need to be very precise about what objects exactly we are working with, as we are trying to show that certain constructions produce algebraic stacks or algebraic spaces.

Unfortunately, this means that some of the notation, conventions and terminology is awkward and may seem backwards to the more experienced reader. We hope the reader will forgive us!

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.

Let $S$ be a scheme and let $X$ be an algebraic space over $S$. In this chapter and the following we will write $X \times _ S X$ for the product of $X$ with itself (in the category of algebraic spaces over $S$), instead of $X \times X$.