The Stacks project

98.2 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 99.2) 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 $X \times X$.


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 05X6. Beware of the difference between the letter 'O' and the digit '0'.