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finitely presented
Ring morphisms
A ring map \(R\to A\) is finitely presented iff $A\cong R[x_1,\cdots, x_n]/\left\langle{f_1,\cdots, f_m}\right\rangle\in {}_{R} \mathsf{Alg} $. 
Scheme morphisms
- For affines: a morphism \(f\in {\mathsf{Sch}}(\operatorname{Spec}A, \operatorname{Spec}B)\) is of finite presentation iff the induced ring morphism \(B\to A\) should be of finite presentation.
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For arbitrary schemes: \(f\in {\mathsf{Sch}}(X, Y)\) is of finite presentation iff
- \(f\) is locally of finite presentation, so there are affine open covers \({\mathcal{U}}\rightrightarrows X, {\mathcal{V}}\rightrightarrows Y\) with \(f(U) \subseteq V\) and the induced ring morphism \(U = \operatorname{Spec}B \to V=\operatorname{Spec}B\) is of finite presentation, and
- \(f\) is quasicompact and quasiseparated.
- In terms of compact objects:
