General schemes

Scheme{BaseRingType<:Ring}

A scheme over a ring $𝕜$ of type BaseRingType.

SchemeMor{DomainType, CodomainType, MorphismType, BaseMorType}

A morphism of schemes $f : X → Y$ of type MorphismType with $X$ of type DomainType and $Y$ of type CodomainType.

When $X$ and $Y$ are defined over schemes $BX$ and $BY$ other than $Spec(𝕜)$, BaseMorType is the type of the underlying morphism $BX → BY$; otherwise, it can be set to Nothing.

irreducible_components(X::Scheme)

Return the irreducible components of $X$ with their reduced structure.

Note that the irreducible components are defined over the base ring of $X$. An irreducible component may split into several irreducible components after a base change, e.g. a field extension.

base_change(phi::Any, X::Scheme)

For a Scheme $X$ over a base_ring $𝕜$ and a map $φ : 𝕜 → R$ we compute $X' = X ×ₖ Spec(R)$ and return a pair (X', f) where $f : X' → X$ is the canonical morphism.

base_change(phi::Any, f::SchemeMor;
    domain_map::SchemeMor, codomain_map::SchemeMor
  )

For a morphism $f : X → Y$ with both $X$ and $Y$ defined over a base_ring $𝕜$ and a map $φ : 𝕜 → R$ return a triple (a, F, b) where $a : X' → X$ is the morphism from base_change(phi, X), $b : Y' → Y$ the one for $Y$, and $F : X' → Y'$ the induced morphism on those fiber products.