Reconstructing complete local rings from quotients

DefinitionComplete local ring reconstruction conjecture#✓fully proved

The statement that any two commutative Noetherian local rings \(A\) and \(B\), with maximal ideals \(\mathfrak m_A\) and \(\mathfrak m_B\), that are complete and Hausdorff for the \(\mathfrak m_A\)-adic and \(\mathfrak m_B\)-adic topologies are isomorphic as rings whenever \(A/\mathfrak m_A^r\) and \(B/\mathfrak m_B^r\) are isomorphic as rings for every natural number \(r\).

def conjecture : Prop :=
  ∀ (A B : Type*) [CommRing A] [CommRing B] [IsLocalRing A] [IsLocalRing B]
    [IsNoetherianRing A] [IsNoetherianRing B]
    [IsAdicComplete (IsLocalRing.maximalIdeal A) A]
    [IsAdicComplete (IsLocalRing.maximalIdeal B) B],
    (∀ r : ℕ, Nonempty (A ⧸ (IsLocalRing.maximalIdeal A) ^ r ≃+*
      B ⧸ (IsLocalRing.maximalIdeal B) ^ r)) →
    Nonempty (A ≃+* B)

Sources

• OWR-1453-004.

Used by 1

Used byThe complete local ring reconstruction conjecture is false

TheoremThe complete local ring reconstruction conjecture is false#●ready to prove

There exist commutative Noetherian local rings \(A\) and \(B\), with maximal ideals \(\mathfrak m_A\) and \(\mathfrak m_B\), that are complete and Hausdorff for the \(\mathfrak m_A\)-adic and \(\mathfrak m_B\)-adic topologies, such that \(A/\mathfrak m_A^r\) and \(B/\mathfrak m_B^r\) are isomorphic as rings for every natural number \(r\), but \(A\) and \(B\) are not isomorphic as rings.

Van den Dries proves that such rings are isomorphic when the residue field is algebraic over its prime field, and reports examples by Gabber showing that the answer is no in general.

public theorem_wanted conjecture_refuted : ¬ conjecture

Sources

• Lou van den Dries, Isomorphism of complete local noetherian rings and strong approximation, Proc. Amer. Math. Soc. 136 (2008), 3435-3448, https://doi.org/10.1090/S0002-9939-08-09401-X.

Uses 1

Statement usesComplete local ring reconstruction conjecture

1 fully proved1 ready to prove

The dependency explorer needs JavaScript and its graph data.

Download the complete graph data.