Basic properties of schemes #
We provide some basic properties of schemes
Main definition #
To show that a statement
P holds for all open subsets of all schemes, it suffices to show that
- In any scheme
Pholds for an open cover of
- For an open immerison
f : X ⟶ Y, if
Pholds for the entire space of
Pholds for the image of
Pholds for the entire space of an affine scheme.
- nonempty : Nonempty ↑↑X.toPresheafedSpace
X is integral if its carrier is nonempty,
𝒪ₓ(U) is an integral domain for each
U ≠ ∅.