English
There are finitely many associated primes of a finitely generated module over a Noetherian ring.
Русский
Т finitely порожденный модуль над Noetherovским кольцом имеет конечное число ассоциированных прайм-идеалов.
LaTeX
$$$\\operatorname{Ass}_A(M) \\\\text{ is finite.}$$$
Lean4
/-- There are only finitely many associated primes of a finitely generated module
over a Noetherian ring. -/
@[stacks 00LC]
theorem finite : (associatedPrimes A M).Finite := by
induction ‹Module.Finite A M› using IsNoetherianRing.induction_on_isQuotientEquivQuotientPrime A with
| subsingleton N => simp [associatedPrimes.eq_empty_of_subsingleton]
| quotient N p f =>
have := associatedPrimes.eq_singleton_of_isPrimary p.2.isPrimary
simp [LinearEquiv.AssociatedPrimes.eq f, this]
| exact N₁ N₂ N₃ f g hf _ hfg h₁ h₃ => exact (h₁.union h₃).subset (associatedPrimes.subset_union_of_exact hf hfg)
