English
A property codescends along a given morphism when, under a suitable setup, it holds after a specific pushout-like transfer.
Русский
Свойство.codescendsAlong сохраняется в рамках заданной конструкций перехода через гомоморфизм, аналог pushout.
LaTeX
$$$\text{CodescendsAlong}(P,Q) := \forall R,S,R',S',\text{условия} \Rightarrow Q(\text{algebraMap } R S) \Rightarrow P(\text{algebraMap } S (S \otimes_R T)).$$$
Lean4
/-- A property of ring homomorphisms `Q` codescends along `Q'` if whenever
`R' →+* R' ⊗[R] S` satisfies `Q` and `R →+* R'` satisfies `Q'`, then `R →+* S` satisfies `Q`. -/
def CodescendsAlong : Prop :=
∀ (R S R' S' : Type u) [CommRing R] [CommRing S] [CommRing R'] [CommRing S'],
∀ [Algebra R S] [Algebra R R'] [Algebra R S'] [Algebra S S'] [Algebra R' S'],
∀ [IsScalarTower R S S'] [IsScalarTower R R' S'],
∀ [Algebra.IsPushout R S R' S'], Q (algebraMap R R') → P (algebraMap R' S') → P (algebraMap R S)
