The Stacks project

Remark 46.3.4. Consider the category $\textit{Alg}_{fp, A}$ whose objects are $A$-algebras $B$ of the form $B = A[x_1, \ldots , x_ n]/(f_1, \ldots , f_ m)$ and whose morphisms are $A$-algebra maps. Every $A$-algebra $B$ is a filtered colimit of finitely presented $A$-algebra, i.e., a filtered colimit of objects of $\textit{Alg}_{fp, A}$. By Lemma 46.3.3 we conclude every adequate functor $F$ is determined by its restriction to $\textit{Alg}_{fp, A}$. For some questions we can therefore restrict to functors on $\textit{Alg}_{fp, A}$. For example, the category of adequate functors does not depend on the choice of the big $\tau $-site chosen in Section 46.2.