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@doc raw""" | ||
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. | ||
!!! note | ||
We do not restrict `phi` to be a `Hecke.Map` so that one can also use | ||
coercion, anonymous functions, etc. | ||
""" | ||
function base_change(phi::Any, X::Scheme) | ||
error("base_change not implemented for input of type $(typeof(X))") | ||
end | ||
|
||
@doc raw""" | ||
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. | ||
!!! note | ||
We do not restrict `phi` to be a `Hecke.Map` so that one can also use | ||
coercion, anonymous functions, etc. | ||
!!! note | ||
The morphisms ``a`` and ``b`` can be passed as the optional arguments | ||
`domain_map` and `codomain_map`, respectively. | ||
""" | ||
function base_change(phi::Any, f::SchemeMor; | ||
domain_map::SchemeMor, codomain_map::SchemeMor | ||
) | ||
error("base_change not implemented for input of type $(typeof(f))") | ||
end |