Open questions for moderately ramified $\mathbb{Z}/p\mathbb{Z}$-actions in positive characteristic

Lorenzini and Schröer (Math. Z. 295, 2020) introduced moderately ramified actions of $G=\mathbb{Z}/p\mathbb{Z}$ on formal power series rings $A=k[[u_1,\dots,u_n]]$ over a field of characteristic $p>0$ — a class of wild actions ramified precisely at the origin whose invariant rings are hypersurfaces for $n=2$ and quasi-Gorenstein in general — and posed a precise list of questions completing the theory. The surviving open core of that list is: (1) whether every $k$-automorphism of $A^G$ extends to $A$ without any hypothesis on the field of representatives $k$; (2) whether $\operatorname{Frac}(A^G)/\operatorname{Frac}(R)$ over the norm subring $R$ can fail to be separable for actions ramified precisely at the origin; (3) whether the number of vertices of valency at least three in minimal resolution graphs of $\mathbb{Z}/p\mathbb{Z}$-quotient surface singularities is unbounded; (4) whether every exponent $s\equiv 1\pmod p$, $s>1$, occurs with $|\det N|=p^s$ for the intersection matrix of a resolution when the action is required to be ramified precisely at the origin (the unrestricted form was resolved affirmatively in 2023); (5) whether the integral closure of a regular local ring in a cyclic wild Galois extension of prime order $p\geq 5$ must be Cohen–Macaulay; and (6) whether Artin's normal form for wild $\mathbb{Z}/2\mathbb{Z}$-actions in dimension two holds over an arbitrary ground field.

Background

Let $p$ be a prime and let $k$ be a field of characteristic $p$. The ring of formal power series $A=k[[u_1,\dots,u_n]]$ is the prototype of a complete regular local ring with residue field $k$; the embedded copy of $k$ is called a field of representatives. If a finite group $G$ acts on $A$ by continuous $k$-algebra automorphisms, the action is tame when $p\nmid|G|$ and wild otherwise, and the central object is the invariant ring $A^G$, whose spectrum is the quotient singularity. The fixed-point ideal $I=(\sigma(a)-a: a\in A,\sigma\in G)$ defines the fixed-point scheme, and following Lorenzini–Schröer the action is called ramified precisely at the origin when the radical of $I$ is the maximal ideal, i.e. the quotient map is étale away from the closed point. Wild actions of $G=\mathbb{Z}/p\mathbb{Z}$ ramified precisely at the origin are never given by matrices, since every order-$p$ matrix in characteristic $p$ has an eigenvector with eigenvalue $1$. Artin (1975) classified them for $p=2$ in dimension two: the invariant ring is a hypersurface $k[[x,y,z]]/(z^2-abz-ya^2+xb^2)$ for a suitable system of parameters $a,b\in k[[x,y]]$ (two elements forming a parameter ideal). Peskin (1983) described an explicit class of examples for $p=3$; for a general wild action the invariant ring need not even be Gorenstein.

For an action of $G=\mathbb{Z}/p\mathbb{Z}$, a regular system of parameters $u_1,\dots,u_n$ can be chosen so that the norm elements $x_i=\prod_{\sigma\in G}\sigma(u_i)$ generate a norm subring $R=k[[x_1,\dots,x_n]]$ over which $A$ is finite of rank $p^n$; following Lorenzini–Schröer (Definition 2.2), a system of parameters with this property is called admissible, and the term is used in exactly this sense in the problem statement below. Lorenzini and Schröer (2020) defined a moderately ramified action by five axioms on this situation; the key consequences are that $\operatorname{Frac}(A)/\operatorname{Frac}(R)$ is Galois with elementary abelian group $(\mathbb{Z}/p\mathbb{Z})^n$, that in dimension two $A^G\simeq k[[x,y,z]]/(z^p-a^{p-1}b^{p-1}z-a^py+b^px)$ and hence is a hypersurface (a complete intersection, therefore Gorenstein), and that in arbitrary dimension $A^G$ is quasi-Gorenstein (its canonical module, the dualizing object measuring distance from Gorensteinness, is free of rank one) while for $n\geq 3$ it fails to be Cohen–Macaulay (a local ring is Cohen–Macaulay when depth equals dimension, equivalently when some system of parameters is a regular sequence). Their paper poses a series of precise questions left open by these results, all concerning the same objects.

Two of the questions live on the geometry of resolutions. Every normal surface singularity has a minimal resolution dominating all others; its exceptional curves are encoded in a dual graph whose vertices are the irreducible exceptional curves, with an edge for each intersection point. The valency of a vertex is its number of edges, and a vertex of valency at least three is called a node. The intersection matrix $N=(C_i\cdot C_j)$ of the exceptional curves is negative definite, its determinant is independent of the chosen resolution, and the order of the finite discriminant group $\mathbb{Z}^r/N\mathbb{Z}^r$ equals $|\det N|$; for wild $\mathbb{Z}/p\mathbb{Z}$-quotient surface singularities this order is a power $p^s$ of $p$. Ito and Schröer (2015) proved that the dual graph of a wild quotient surface singularity ramified precisely at the origin always contains a node, and constructed examples with two nodes for non-cyclic wild groups (diagonal quotients of Hermitian curves by Heisenberg-type special $p$-groups).

One further question is homological. By a theorem of Roberts (1980), if $R$ is regular local and $L/\operatorname{Frac}(R)$ is Galois with abelian group of order prime to the residue characteristic, the integral closure $B$ of $R$ in $L$ is Cohen–Macaulay. In the wild case this fails: moderately ramified actions with $n\geq 3$ give counterexamples over a regular base for elementary abelian groups, and Koh (1986) gave a mixed-characteristic counterexample with $H$ cyclic of order $3$. An automorphism $\sigma:B\to B$ of finite order is a generalized reflection when $(\sigma-\mathrm{id})(B)\subseteq xB$ for some regular non-unit $x\in B$, i.e. its fixed-point scheme contains an effective Cartier divisor; for order prime to the residue characteristic this agrees with the classical notion of a pseudo-reflection, and in the tame setting finite groups generated by pseudo-reflections are exactly those with well-behaved invariant rings — an analogue that fails in the wild setting.

Problem Statement

Throughout, $p$ is a prime, $k$ a field of characteristic $p$, $A=k[[u_1,\dots,u_n]]$ with field of representatives $k$, and $G=\mathbb{Z}/p\mathbb{Z}$ acts on $A$ by continuous $k$-algebra automorphisms, with invariant ring $A^G$ and norm subring $R=k[[x_1,\dots,x_n]]$ for an admissible system of parameters; items (1)–(4) use this setting, while items (5) and (6) concern more general or different objects and specify their settings and notation locally. The problem is to resolve, definitively and without any hypothesis on the ground field $k$ beyond characteristic $p$, the following six questions left open by the theory of moderately ramified actions.

(1) Automorphism extension (source: Remark 3.5(i) versus Theorem 3.4(ii)). For every action of $G$ on $A$ ramified precisely at the origin: does every $k$-automorphism $\varphi:A^G\to A^G$ extend to a $k$-automorphism $\varphi':A\to A$; and is it true that whenever a subring $R\subset A^G$ makes $R\subset A^G$ finite with $\operatorname{Frac}(A^G)/\operatorname{Frac}(R)$ Galois, then $\operatorname{Frac}(A)/\operatorname{Frac}(R)$ is Galois as well? Both statements are known under the hypothesis that $k$ has no Galois extension of degree $p$; the question is whether that hypothesis can be removed.

(2) Separability of the norm extension (source: Remark 3.5(ii)). Does there exist an action of $G$ on $A$ ramified precisely at the origin, together with an admissible system of parameters defining $R$, such that the finite extension $\operatorname{Frac}(A^G)/\operatorname{Frac}(R)$ is not separable? (Theorem 3.4(i) shows it is never purely inseparable; when the action is not ramified precisely at the origin, inseparable examples are easy.) A negative answer — a proof that the extension is always separable — equally resolves the question.

(3) Unbounded nodes (source: Introduction). In the class of minimal resolutions of $\mathbb{Z}/p\mathbb{Z}$-quotient singularities of surfaces, is there no upper bound on the number of vertices of valency at least three in the minimal resolution graph? At least one such vertex always occurs when the action is ramified precisely at the origin (Ito–Schröer); examples with at least two nodes are known only for non-cyclic wild groups (diagonal quotients of Hermitian curves by Heisenberg-type special $p$-groups) and, in mixed characteristic, for the wild $\mathbb{Z}/2\mathbb{Z}$-quotient singularities over the $2$-adic numbers constructed by Lorenzini — in positive characteristic no $\mathbb{Z}/p\mathbb{Z}$-quotient example with two or more nodes is known; the question is whether the number is unbounded (equivalently, whether for every $m$ some such quotient singularity has a minimal resolution graph with at least $m$ nodes).

(4) Determinants with ramification precisely at the origin (source: Introduction, refined form of the question resolved by Lorenzini–Schröer 2023 in general). For $p>2$: does every exponent $s\equiv 1\pmod p$ with $s>1$ occur as the exponent in $|\det N|=p^s$ for the intersection matrix $N$ of a resolution of some wild $\mathbb{Z}/p\mathbb{Z}$-quotient surface singularity whose action is ramified precisely at the origin? (All exponents $s\not\equiv 1\pmod p$ are known with this ramification restriction, and all exponents are known without it.)

(5) Cohen–Macaulayness over a regular base (source: Remark 6.6). Let $R$ be a regular local ring of residue characteristic $p$, and let $L/\operatorname{Frac}(R)$ be a Galois extension whose Galois group $H$ is cyclic of order $p\geq 5$. Must the integral closure $B$ of $R$ in $L$ be Cohen–Macaulay? (Known: yes when $|H|$ is prime to the residue characteristic, by Roberts; counterexamples exist for elementary abelian $H$ over a regular base, for $H$ cyclic of order $3$ in mixed characteristic, by Koh, and — when $R$ is merely Cohen–Macaulay and $H$ is generated by a generalized reflection — within the source paper itself.)

(6) Artin's normal form over an arbitrary field (source: Remark 7.8). Does every wild $\mathbb{Z}/2\mathbb{Z}$-action on $k[[u,v]]$ ramified precisely at the origin, over an arbitrary field $k$ of characteristic $2$, have invariant ring of the form $k[[x,y,z]]/(z^2-abz-ya^2+xb^2)$ for some system of parameters $a,b\in k[[x,y]]$, as stated by Artin without any hypothesis on $k$? The known derivation requires that $k$ have no separable quadratic extension.

Each of (1)–(6) is an independent open question, and the two statements bundled in (1) resolve separately. The six questions arise from the common theory of moderately ramified actions and their quotient singularities, and a result on any one of them stands on its own.

References

1. D. Lorenzini, S. Schröer, Moderately ramified actions in positive characteristic, Mathematische Zeitschrift 295 (2020), 1095–1142. DOI: 10.1007/s00209-019-02408-4. https://doi.org/10.1007/s00209-019-02408-4 (arXiv:1904.08371, https://arxiv.org/abs/1904.08371)
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