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Gerko's conjecture on chains of semidualizing modules

Let (R,m,k)(R,\mathfrak{m},k) be an Artinian local ring, with all modules finite. Let [Cn]⊲⋯⊲[C1]⊲[C0][C_n]\vartriangleleft\cdots\vartriangleleft[C_1]\vartriangleleft[C_0] be a chain in G0(R)\mathfrak{G}_0(R) of length nn. Gerko's conjecture. The chain satisfies

mn≠0.\mathfrak{m}^n\neq 0.

Moreover, if mn+1=0\mathfrak{m}^{n+1}=0, then the Poincaré series of kk has the form

1∏i=1n(1−dit)\frac{1}{\prod_{i=1}^n(1-d_i t)}

for some positive integers did_i. The preceding theorem establishes these conclusions under the stronger hypothesis that the associated modules are non-free and strongly Tor⁡\operatorname{Tor}-independent; the conjecture asks for them for every chain in G0(R)\mathfrak{G}_0(R) and remains unresolved in the supplied source.

Sources & referencesView supporting material

Primary source

Ensiyeh Amanzadeh, “Chains of semidualizing modules”, arXiv:1611.05790 (2016).

commutative-algebrapoincare-seriessemidualizing-modules

Progress summary

Refreshed
Open

The conjecture remains open: only a stronger version for special chains is proved, and no newer proof or counterexample was found.

A 2016 paper formulates Gerko’s conjecture for chains in G0(R)\mathfrak{G}_0(R): a chain of length nn should force mn≠0\mathfrak{m}^n\neq 0, with the stated product formula for the Poincaré series when mn+1=0\mathfrak{m}^{n+1}=0. The full assertion for arbitrary chains was explicitly left unresolved.

Known results

  • For suitable chains, Theorem 3.5 proves mn≠0\mathfrak{m}^n\neq 0.
  • Under the same suitability hypothesis and mn+1=0\mathfrak{m}^{n+1}=0, it proves PkR(t)=1/∏i=1n(1−dit)P_k^R(t)=1/\prod_{i=1}^n(1-d_i t) for positive integers did_i.
  • Strongly Tor⁡\operatorname{Tor}-independent modules yield related conclusions, but this does not cover every chain in G0(R)\mathfrak{G}_0(R).

Current status (as of August 2026): The conjecture is proved for suitable, in particular stronger strongly Tor⁡\operatorname{Tor}-independent, chains; its validity for arbitrary chains in G0(R)\mathfrak{G}_0(R) remains open.

Sources

Solutions 0

No solutions have been posted yet.