Counter-examples to non-noetherian Elkik's approximation theorem

TL;DR

This paper demonstrates that Elkik's approximation theorem, valid for noetherian henselian rings, fails for certain non-noetherian henselian rings lacking weak proregularity, revealing new limitations of the theorem.

Contribution

It identifies specific non-noetherian henselian rings where Elkik's theorem does not hold, highlighting the importance of noetherianness and weak proregularity conditions.

Findings

Elkik's theorem fails in some non-noetherian henselian rings.

Weak proregularity is crucial for the theorem's validity.

Pathologies arise when conditions weaker than noetherianness are not met.

Abstract

Elkik established a remarkable theorem that can be applied for any noetherian henselian ring. For algebraic equations with a formal solution (restricted by some smoothness assumption), this theorem provides a solution adically close to the formal one in the base ring. In this paper, we show that the theorem would fail for some non-noetherian henselian rings. These rings do not satisfy several conditions weaker than noetherianness, such as weak proregularity (due to Grothendieck et al.) of the defining ideal. We describe the resulting pathologies.