This article is about homomorphisms between schemes which are locally of finite type. For the notion of locally finite type in dependent type theory, see locally finite type.
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A homomorphism between schemes is said to be (locally) of finite type if it behaves like a finite covering space.
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)
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
Last revised on July 25, 2023 at 16:09:33. See the history of this page for a list of all contributions to it.