nLab
morphism of finite type

For schemes

A morphism $f:X\to Y$ of schemes is locally of finite type if

• for every open cover $\{U_i\to Y\}$ by affine schemes, $U_i\simeq \mathrm{Spec}{B}_i$;

• and every cover $\{U_{i{j}_i}\to X\}$ by affine schemes $U_{i{j}_i}=A_{i{j}_i}$, fitting into a commuting diagram (this always exists, see coverage)

  $$\begin{array}{ccc}{U}_{i{j}_i}& \to & U_i\\ \downarrow & & \downarrow \\ X& \stackrel{f}{\to }& Y\end{array}$$\array{ U_{i j_i} &\to& U_i \\ \downarrow && \downarrow \\ X &\stackrel{f}{\to}& Y }

  for all $i,j$,

we have that the morphism of algebras $B_i\to A_{ij}$ formally dual to $U_{ij}\to U_i$ exhibits $A_{ij}$ as a finitely generated algebra over $B_i$.

If for fixed $i$ the $j_i$ range only over a finite set, then the morphism is said to be of finite type.

Related concepts

• morphism of finite presentation