higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
A homomorphism between schemes is said to be (locally) of finite type it 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