[Paper Review] Regularity of quotients by an automorphism of order $p$

Stefan Wewers|arXiv (Cornell University)|Jan 5, 2010

Finite Group Theory Research8 references3 citations

TL;DR

This paper establishes a sufficient condition for the regularity of the invariant ring $ A = B^G $, where $ B $ is a regular local ring and $ G $ is a cyclic group of order $ p $ acting on $ B $, with $ p $ equal to the residue characteristic of $ B $. The key result shows that if there exists a regular parameter $ \pi \in \mathfrak{m} \setminus \mathfrak{m}^2 $ such that the $ G $-invariant ideal $ \mathcal{I}_\sigma = (\pi^\delta) $ and either $ \mathcal{I}_\sigma = (\sigma(\pi) - \pi) $ or $ \sigma(\pi) - \pi \in (\pi^{\delta+1}) $, then $ A $ is regular, providing a criterion for wild quotient singularities to be regular.

ABSTRACT

Let $B$ be a regular local ring and $G\subset\Aut(B)$ a finite group of local automorphisms. Assume that $G$ is cyclic of prime order $p$, where $p$ is equal to the residue characteristic of $B$. We give conditions under which the ring of invariants $A=B^G$ is again regular.

Motivation & Objective

• To address the problem of when the ring of invariants $ A = B^G $ is regular when $ G $ is a cyclic group of order $ p $, and $ p $ equals the residue characteristic of $ B $, a case not covered by classical tame quotient singularity theory.
• To provide a criterion for regularity of the quotient scheme $ X/G $ in the wild case, where $ G $ acts on a regular local ring $ B $, extending results known in the tame case.
• To complement existing combinatorial studies of wild quotient singularities by offering a cohomological and ideal-theoretic criterion applicable to dimension two and higher.
• To establish a sufficient condition under which the quotient of a regular local ring by a wild $ p $-group action remains regular, using properties of the action on the maximal ideal and its powers.

Proposed method

• The method centers on analyzing the $ G $-invariant ideal $ \mathcal{I}_\sigma = \langle \sigma(x) - x \mid x \in B \rangle_B $, which measures the nontriviality of the group action on $ B $.
• It uses the existence of a regular parameter $ \pi \in \mathfrak{m} \setminus \mathfrak{m}^2 $ such that $ \mathcal{I}_\sigma = (\pi^\delta) $, linking the ideal structure to the group action's order and fixed points.
• The proof constructs a $ G $-invariant element $ \lambda = u\pi $ with $ u \in B^\times $, using an inductive lifting process in the power series ring to satisfy valuation conditions on $ \sigma(\lambda) - \lambda $.
• It applies a key lemma involving the map $ \theta_\pi $, defined via the action of $ \sigma $, to reduce the problem to solving a linear equation modulo $ \pi $, relying on the surjectivity of $ \theta_\pi $ on the residue field.
• The quotient ring $ \bar{A} = A/A\lambda $ is shown to equal $ \bar{B} \cap \bar{K} $, where $ \bar{B} = B/B\pi $, and this is used to prove that $ \bar{A} $ is regular, implying $ A $ is regular.
• The argument uses the fact that $ \theta_\pi $ is surjective on $ \bar{B} $, which follows from the existence of certain lifts in the ring of integers of a $ p $-adic field, and applies Corollary 2.2 to solve the required equations.

Experimental results

Research questions

• RQ1 Under what conditions is the ring of invariants $ A = B^G $ regular when $ G $ is a cyclic group of order $ p $, and $ p $ is equal to the residue characteristic of $ B $?
• RQ2 Can a sufficient condition be given for the regularity of the quotient scheme $ X/G $ in the wild case, where the group order is not invertible in the residue field?
• RQ3 How does the structure of the $ G $-invariant ideal $ \mathcal{I}_\sigma $ relate to the regularity of $ A = B^G $, particularly when $ \mathcal{I}_\sigma $ is a power of a regular parameter?
• RQ4 What role does the action of $ \sigma $ on a regular parameter $ \pi $ play in determining whether $ A $ is regular, especially when $ \sigma(\pi) - \pi \in (\pi^{\delta+1}) $?
• RQ5 Is there a cohomological or ideal-theoretic criterion that ensures regularity of $ A $ in the wild case, even when the action induces the trivial automorphism on the cotangent space?

Key findings

• If there exists a regular parameter $ \pi \in \mathfrak{m} \setminus \mathfrak{m}^2 $ such that $ \mathcal{I}_\sigma = (\pi^\delta) $, then the $ G $-invariant ideal is generated by a power of $ \pi $, linking the group action to the local structure of $ B $.
• When $ \mathcal{I}_\sigma = (\sigma(\pi) - \pi) $ or $ \sigma(\pi) - \pi \in (\pi^{\delta+1}) $, the action is sufficiently controlled to ensure regularity of the invariant ring $ A $.
• The existence of a $ G $-invariant element $ \lambda = u\pi $ with $ u \in B^\times $ allows reduction to the residue field, enabling the use of cohomological techniques on $ \bar{B} = B/B\pi $.
• The quotient ring $ \bar{A} = A/A\lambda $ is shown to equal $ \bar{B} \cap \bar{K} $, which is a key step in proving regularity of $ \bar{A} $ and hence of $ A $.
• The regularity of $ \bar{A} $ follows from the surjectivity of the map $ \theta_\pi $ on the residue field, which is established via lifting arguments in the ring of integers of a $ p $-adic field.
• The final conclusion is that under the stated conditions, $ A = B^G $ is regular, providing a new criterion for wild quotient singularities to be regular in positive characteristic.