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# Contents

## Idea

A homomorphism between schemes is said to be (locally) of finite type if it behaves like a finite covering space.

## 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, $U_i \simeq Spec B_i$;

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

$\array{ U_{i j_i} &\to& U_i \\ \downarrow && \downarrow \\ X &\stackrel{f}{\to}& Y }$

for all $i,j$,

we have that the morphism of algebras $B_i \to A_{i j}$ formally dual to $U_{i j} \to U_i$ exhibits $A_{i j}$ 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.

Introductory disucssoon over the complex numbers (with an eye towards GAGA) is in

• Amnon Neeman, section 3.10 Algebraic and analytic geometry, London Math. Soc. Lec. Note Series 345, 2007 (publisher)