To appear in Crelles Journal

FACTORIAL EXTENSIONS OF REGULAR LOCAL RINGS

AND INVARIANTS OF FINITE GROUPS

Luchezar L. Avramov and Adam Borek

Dedicated to the memory of Maurice Auslander

Introduction

It is often useful to approach the singularity of a noetherian commutative local domain S from a regular local subring P over which S is finite as a module and residually trivial (that is, the induced extension of residue fields is trivial). Such subrings exist in a variety of situations: in commutative algebra they are given by Cohen’s structure theory for complete local rings; in analytic geometry they are in place by definition; in algebraic geometry they are sometimes produced by Noether normalization.

If S is normal (here meaning noetherian and integrally closed) and divisorially unramified over P, then S = P by the classical purity property of the branch locus, cf. [2]. Thus, when studying finite normal local extensions of a regular local ring P a natural next step is to allow a controlled amount of ramification in codimension one.

An early example appears in [15], where Serre uses purity to show that if P ↪ S is ramified in codimension one only at principal primes of S, and is generically Galois in the sense that the induced extension of fraction fields P_{0} ↪ S_{0} is Galois, then Gal(S_{0}|P_{0}) is generated by generalized reflections (an automorphism h of S is a generalized reflection if (h − id)(S) ⊆ (x) for some non-unit x ∈ S).

The preceding result is applicable, in particular, when S is factorial. Recently Griffith has obtained in [9] structure theorems for factorial domains, with the condition on the fraction field extension weakened to one of ‘uniform’ divisorial ramification. His results assume that the regular ring P is complete with algebraically closed residue field of characteristic zero, and his arguments strongly depend on these additional conditions.

This raises the question of describing the structure of local factorial extensions of more general regular local rings. Our hypotheses appear in a diagram of inclusions of local rings

\frac{1}{n} ∈ P_{regular}\overbrace{⟶^{generically Galois} D_{normal} ⟶^{divisorially unramified} S_{factorial} ⟶}^{residually trivial and generically Galois} E_{normal} ∋ \sqrt[n]{1}

where n = rank_{P} E and \sqrt[n]{1} denotes a primitive n’th root of unity.

Denoting by Q (respectively, R) the integral closure of P in the largest abelian (respectively, largest Galois) extension of P_{0} contained in S_{0}, and by T its integral closure in the smallest Galois extension of P_{0} containing S_{0}, we extend and sharpen Griffith’s results by

Typeset by AMS-TEX

1980 Mathematics Subject Classification (1985 Revision). 13B15, 13B05, 13F15.

L. L. A. was partially supported by NSF Grant No. DMS-9102951 .

exhibiting a canonical splitting of the divisorial ramification of S over P into a sequence of ‘nice’ intermediate extensions. It is described by a second picture

P \overbrace{⟶^{Kummer} Q_{normal complete intersection}⟶^{divisorially unramified}}^{Galois group generated by generalized reflections} R_{factorial complete intersection}⟶^{divisorially unramified} S ⟶^{divisorially unramified} T_{normal}

where a Kummer extension (or Kummer covering, cf. Grothendieck and Murre [10]) is a tamely ramified abelian extension, obtained by adjunction of suitable roots.

The extensions P ↪ R and R ↪ T are handled in very different ways.

In view of Serre’s result, the first inclusion is studied in the framework of a problem of invariant theory: the action on a local domain (R, 𝔪, k) of a finite group G generated by generalized reflections, whose order is invertible in R. In Section 1 we determine the invariants of the commutator subgroup G′ for the induced action on the symmetric algebra of the k–vector space 𝔪/𝔪^{2}; on it G acts as a linear group generated by pseudo-reflections, so we have at our disposal an attractive chapter of classical invariant theory. In Section 2 we use this explicit computation together with the comprehensive local results of Singh [16] and Avramov [4] to prove that Q is a Kummer extension of P. When, in addition, R is integrally closed, we establish in Section 3 that Q is divisorially unramified over P.

The factorial ring S appears in Section 4, where we show that R is factorial and R ↪ T is divisorially unramified by combining arguments from [9] with work in preceding sections.

Finally, a note on terminology. We say that a ring extension is normal if both rings are integrally closed noetherian domains. To avoid confusion with other uses of ‘normal extension’, we adopt Bourbaki’s term quasi-Galois extension for the inclusion of a field into the splitting field of a polynomial; of course, the prefix ‘quasi’ is dropped when the field extension is separable. For background in ramification theory we refer to Abhyankar [1], or to Auslander and Buchsbaum [3].

1. Polynomial invariants

In this section we focus on a finite group G⩽ GL_{k}(V) acting on a finite k–vector space V, and assume that the order n of G is invertible in k. An element h ∈ G is a pseudoreflection if rank_{k}(h − id) = 1. We denote by P the set of pseudo-reflections of G, and with each h ∈ P we associate the following objects:

• L[h] := Im(h − id) ⊆ V, the reflected line of h.

• H[h] := Ker(h − id) ⊆ V, the reflecting hyperplane of h.

• G[h] := {g ∈ G | g(v) = v for all v ∈ H[h]}⩽G, the stabilizer of H[h]. Clearly, G acts on P by conjugation. This induces actions on the sets L = {L[h]}_{h∈P} and H = {H[h]}_{h∈P} by translation: g(L[h]) = L[ghg^{−1}] and g(H[h]) = H[ghg^{−1}], and on G = {G[h]}_{h∈P} by conjugation: g(G[h])g^{−1} = G[ghg^{−1}].

It follows from [4], (13), that the natural maps L ⟵ P ⟶ H ⟶ G induce G-equivariant bijections L ↔ H ↔ G. In particular, there is a bijective correspondence L_{i} ↔ G_{i} between the orbits L_{1}, …, L_{s} and G_{1}, …, G_{s} of the actions of G on the sets L and G. It follows from Maschke’s theorem that each G[h] is cyclic and its non-identity elements are pseudoreflections. For i = 1, …, s we choose in G_{i} a cyclic group G_{i} = ⟨h_{i}⟩ and set n_{i} = |G_{i}|.

Proposition 1. Let V be a finite vector space over a field k, let G⩽ GL_{k}(V) be a finite group with order invertible in k, and consider the natural actions of G and of its commutator subgroup G′ on the symmetric algebra C = Sym_{k}(V).

Let Y_{1}, …, Y_{s} be variables over A = C^{G} of degree |L_{1}|, …, |L_{s}|, respectively, and con-

sider the action of the group ∏_{i=1}^{s} G_{i} on A[Y_{1}, …, Y_{s}] by A–algebra automorphisms defined

by (g_{1}, …, g_{s})(Y_{i}) = det(g_{i})Y_{i}. For each pseudo-reflection h ∈ P choose a non-zero vector

e[h] ∈ L[h], and for i = 1, …, s set e_{i} = ∏_{L[h]∈L_{i}} e[h] ∈ C.

If G is generated by pseudo-reflections, then the elements e_{i} are in B = C^{G′}, the elements f_{i} = e_{i}^{n_{i}} are in A, and the degree zero homomorphism of graded A–algebras A[Y_{1}, …, Y_{s}] ⟶ C which maps Y_{i} to e_{i} induces a ρ–equivariant isomorphism

φ: \frac{A[Y_{1}, …, Y_{s}]}{(Y_{1}^{n_{1}} − f_{1}, …, Y_{s}^{n_{s}} − f_{s})} ≅ B

where ρ: ∏_{i=1}^{s} G_{i} ⟶ G_{ab} is a group isomorphism given by ρ(g_{1}, …, g_{s}) = (g_{1} ⋯ g_{s})G′.

Remark. The k–algebra A is generated by rank_{k} V algebraically independent forms by the Chevalley–Shephard–Todd Theorem, cf. [6], Theorem 4 or [5], Theorem 7.2.1.

Proof. Dualizing [17], Theorem 4.3.4, or [18], Theorem 3.1, for each h ∈ P we get

h(e_{i}) = \{ \begin{matrix}det(h)e_{i} & if L[h] ∈ L_{i} ; \\ e_{i} & if L[h] ∉ L_{i} . \end{matrix}

As det(h) is a primitive root of unity of order |h|, direct computations yield g(f_{i}) = f_{i} and ghg^{−1}h^{−1}(e_{i}) = e_{i} for all g, h ∈ G, so f_{i} is G–invariant and e_{i} is G′–invariant for i = 1, …, s. The formulas also show that the homomorphism φ is ρ–equivariant.

For each h ∈ P, there exist g ∈ G and i ∈ {1, …, s} such that gh^{−1}g^{−1} ∈ G_{i}, hence h is congruent modulo G′ to a power of h_{i}. Thus, h_{1}G′, …, h_{s}G′ generate G_{ab}, so ρ is surjective. This observation produces the inequality below

∏_{i=1}^{s} n_{i} = | ∏_{i=1}^{s} G_{i}| ≥ |G_{ab}|

and shows that if equality holds, then ρ is bijective.

In view of the transitivity formulas A = C^{G} = (C^{G′})^{G/G′} = B^{G_{ab}}, Galois theory yields

|G_{ab}| = rank_{A} B .

The finite extension B of the graded polynomial ring A is a Cohen–Macaulay ring by [11], Proposition 15, and thus a free A–module. It follows that

rank_{A} B = rank_{k} \overline{B}

where an overbar denotes the result of tensoring over A with k = A/A_{+}.

For the composition of φ with the injection of A–algebras ι: B ⟶ C we have

(ιφ)(∏_{i=1}^{s} Y_{i}^{n_{i}−1}) = ∏_{i=1}^{s} e_{i}^{n_{i}−1} = ∏_{i=1}^{s} ∏_{L[h]∈L_{i}} e[h]^{n_{i}−1} = ∏_{i=1}^{s} ∏_{G[h]∈G_{i}} ∏_{\begin{matrix}g∈G[h] \\ g≠id \end{matrix}} e[g] = ∏_{h∈P} e[h] .

It is shown in [4], §5, Step 4, that the image of the last element in \overline{C} is not zero, hence

y = ∏_{i=1}^{s} \overline{Y} _{i}^{n_{i}−1} is not in the kernel of \overline{ιφ}. As y generates the socle of the artinian com-

plete intersection k[Y_{1}, …, Y_{s}]/(Y_{1}^{n_{1}}, …, Y_{s}^{n_{s}}), it follows that \overline{ιφ} is injective. Thus, \overline{φ} is injective, and as a consequence

rank_{k} \overline{B} ≥ rank_{k} \frac{k[Y_{1}, …, Y_{s}]}{(Y_{1}^{n_{1}}, …, Y_{s}^{n_{s}})} = ∏_{i=1}^{s} n_{i} .

Putting together the (in)equalities obtained above, we get equalities

∏_{i=1}^{s} n_{i} = |G_{ab}|  and  rank_{k} \frac{k[Y_{1}, …, Y_{s}]}{(Y_{1}^{n_{1}}, …, Y_{s}^{n_{s}})} = rank_{k} \overline{B} .

As already noted, the first one shows that ρ is bijective. In view of the injectivity of \overline{φ}, the second one implies that \overline{φ} is bijective. Since \overline{φ} = k ⊗_{A} φ, and φ is an A–linear homomorphism of graded free A–modules, it is an isomorphism by Nakayama. □

2. Local invariants

In this section G is a finite subgroup of the group of automorphisms of a local domain (R, 𝔪, k) which induce the trivial action on k, and we assume that the order of G is invertible in R. An element h ∈ G is a generalized reflection if h ≠ id and there exists a principal ideal 𝔞 ≠ R such that h(a) − a ∈ 𝔞 for all a ∈ R. We know by [12], Corollary 3.4, that R is finite over R^{G}, by [16], Lemma 3, that the natural homomorphism of groups ε: G ⟶ GL_{k}(𝔪/𝔪^{2}) is injective, and by [4], (12), that h is a generalized reflection if and only if ε(h) is a pseudo-reflection. The ideal 𝔞 in the definition of a generalized reflection is uniquely defined as the inertia ideal 𝔦[h], generated by {h(a) − a | a ∈ R}, cf. [4], (12).

We identify through ε the set of generalized reflections in G and the set of pseudoreflections in ε(G), and write P for either set. For each h ∈ P the inertia subgroup {g ∈ G | g(a) − a ∈ 𝔦[h] for all a ∈ R} corresponds through ε to the stabilizer of the reflecting hyperplane of h, so we denote it by G[h]. The group G acts by conjugation on the set P of generalized reflections. There are induced actions by conjugation on the family G = {G[h]}_{h∈P} and by translation g(𝔦[h]) = 𝔦[ghg^{−1}] on the family I = {𝔦[h]}_{h∈P}.

By [4], (13), there is a G–equivariant bijection between these families, and thus a bijection of orbits {G_{1}, …, G_{s}} ↔ {I_{1}, …, I_{s}}. As before, we choose G_{i} ∈ G_{i} and set n_{i} = |G_{i}|.

Theorem 2. Let G be a group of automorphisms of a local domain (R, 𝔪, k) which induce the identity on k, and let G′⩽G be the commutator subgroup; thus, Q = R^{G′} is the integral closure of P = R^{G} in the largest abelian extension of P_{0} in R_{0}.

If G is generated by generalized reflections, |G| is invertible in P, and R has a primitive root of unity of order equal to the least common multiple of {|h|}_{h∈P}, then for i = 1, …, s there exists a generator x_{i} ∈ Q of the ideal ∏_{𝔦[h]∈I_{i}} 𝔦[h] ⊆ R and there exists a group

homomorphism ω_{i}: G ⟶ P^{×} such that g(x_{i}) = ω_{i}(g)x_{i}.

Let Y_{1}, …, Y_{s} be indeterminates over P, let ∏_{i=1}^{s} G_{i} act on P [Y_{1}, …, Y_{s}] by P –algebra

automorphisms such that (g_{1}, …, g_{s})(Y_{i}) = ω_{i}(g_{i})Y_{i}.

The elements y_{i} = x_{i}^{n_{i}} are in P for i = 1, …, s, and the homomorphism of P –algebras P [Y_{1}, …, Y_{s}] ⟶ R which maps Y_{i} to x_{i} induces a ρ–equivariant isomorphism of local rings

ψ: \frac{P [Y_{1}, …, Y_{s}]}{(Y_{1}^{n_{1}} − y_{1}, …, Y_{s}^{n_{s}} − y_{s})} ≅ Q

where ρ: ∏_{i=1}^{s} G_{i} ⟶ G_{ab} is a group isomorphism given by ρ(g_{1}, …, g_{s}) = (g_{1} ⋯ g_{s})G′.

The proof uses a modest amount of Galois cohomology, for which we set the stage. Let (R, 𝔪, k) be a local ring, let R^{×} denote its group of units and let H be a finite

group acting on R by ring automorphisms which induce the identity on k. The 1-cocycles Z^{1}(H, R^{×}) are the maps τ: H ⟶ R^{×} such that τ(gh) = τ(g)g(τ(h)), the 1-coboundaries B^{1}(H, R^{×}) are the cocycles of the form g ↦ b^{−1}g(b) for some b ∈ R^{×}, and H^{1}(H, R^{×}) = Z^{1}(H, R^{×})/ B^{1}(H, R^{×}) is the first cohomology group of H with coefficients in R^{×}.

Lemma 3. For R and H as above, the exact sequence of multiplicative groups

1 ⟶ 1 + 𝔪 ⟶ R^{×} ⟶^{κ} k^{×} ⟶ 1

induces a homomorphism κ_{∗}: H^{1}(H, R^{×}) ⟶ Hom(H, k^{×}) with κ_{∗}(cls (τ)) = κτ. If |H| is invertible in R, then κ_{∗} is injective. If furthermore R has a primitive root of unity of order equal to the exponent of the

group H_{ab}, then κ_{∗} is an isomorphism, and for each τ ∈ Z^{1}(H, R^{×}) there exists b ∈ R^{×} such that ω(g) = b^{−1}g(b)τ(g) defines a homomorphism of groups H ⟶ (R^{H})^{×}.

Proof. The injectivity of κ_{∗} is well known, cf. [16], Lemma 2.

Next we assume that R has the appropriate primitive root of unity, and show that each homomorphism χ : H ⟶ k^{×} lifts to a homomorphism ω: H ⟶ (R^{H})^{×}. Note that Im χ ∈ k^{×} is cyclic, generated by a primitive root of unity of order m = | Im χ|. As χ factors through H_{ab}, R has a primitive m’th root of unity ζ. By [16], Lemma 4, this root is actually in R^{H}, and κ(ζ) ∈ k is a primitive m’th root of unity. Thus, κ(ζ) generates Im χ, so one can find a homomorphism ω: H ⟶ (R^{H})^{×} with χ = κω.

The composition of ω with the inclusion (R^{H})^{×} ⊆ R^{×} is clearly in Z^{1}(H, R^{×}). It follows that that κ_{∗} is surjective, and thus an isomorphism.

Finally, consider a τ ∈ Z^{1}(H, R^{×}). By the surjectivity of κ_{∗} there is a σ ∈ Z^{1}(H, R^{×}) with Im σ ⊆ (R^{H})^{×} and κτ = κσ. The injectivity of κ_{∗} implies that τ and σ differ by a coboundary, that is, σ(g) = b^{−1}g(b)τ(g) for some b ∈ R^{×} and all g ∈ H. □

Proof of Theorem 2. The bijectivity of ρ is obtained from that of the similarly named map in Proposition 1, by means of the isomorphism ε.

The principal ideal ∏_{𝔦[h]∈I_{i}} 𝔦[h] ⊆ R is stable for the action of G. Thus, if a_{i} is a generator,

then g(a_{i}) ∈ R^{×}a_{i}. It is well known and easily verified that g ↦ a_{i}^{−1}g(a_{i}) defines a 1cocycle in Z^{1}(G, R^{×}). By Lemma 3 there exist a unit b_{i} in R and a homomorphism ω_{i}: G ⟶ P^{×}, such that x_{i} = a_{i}b_{i} satisfies g(x_{i}) = ω_{i}(g)x_{i} for all g ∈ G. It is clear that x_{i} generates ∏_{𝔦[h]∈I_{i}} 𝔦[h], and it is immediately checked that x_{i} is in Q = R^{G′}, that x_{i}^{n_{i}} is in

P = R^{G}, and that the homomorphism of algebras ψ is ρ–equivariant.

Along with the action of G on R we consider its induced actions on the graded k– algebras C = Sym_{k}(V) and gr(R), where V = 𝔪/𝔪^{2} and gr denotes an associated graded ring for the 𝔪-adic filtration. The natural surjection C ⟶ gr(R) of graded k–algebras is G–linear, and hence sends C_{+}^{G} to (gr_{+}(R))^{G}. It is easily seen that (gr_{+}(R))^{G} is contained in the kernel of the induced homomorphism gr(R) ⟶ gr(R/𝔪^{G}R), cf. [4], (11), for details, and thus there are natural surjective G–linear homomorphisms of graded k–algebras

\overline{C} = \frac{C}{(C_{+}^{G})C} ↠ \frac{gr(R)}{(gr_{+}(R))^{G} gr(R)} ↠ gr(\frac{R}{𝔪^{G}R}) .

By [4], (16), these maps are in fact bijective, and so induce an isomorphism \overline{C}^{G′} ≅ gr(R/𝔪^{G}R)^{G′}. As |G′| is invertible in R, the formation of G′–invariants is an exact functor on the category of modules over the group ring R[G′]. Thus, denoting by gr′ an associated graded ring for the filtration induced from an 𝔪-adic filtration of an ambient ring, we get

(gr(R/𝔪^{G}R))^{G′} ≅ gr′((R/𝔪^{G}R)^{G′}) ≅ gr′(Q/𝔪^{G}Q) .

Let \overline{x}_{i} stand for the image of x_{i} in Q/𝔪^{G}Q, write x_{i}^{∗} for the initial form of \overline{x}_{i} in gr′(Q/𝔪^{G}Q), and note that x_{i}^{∗} is a non-zero multiple of the form e_{i} introduced in Proposition 1. That result yields an isomorphisms of graded k–algebras

\frac{k[Y_{1}, …, Y_{s}]}{(Y_{1}^{n_{1}}, …, Y_{s}^{n_{s}})} ⟶^{≅} gr′(\frac{Q}{𝔪^{G}Q})

whose image is equal to k[x_{1}^{∗}, …, x_{s}^{∗}]. We conclude first that k[\overline{x}_{1}, …, \overline{x}_{s}] = Q/𝔪^{G}Q, and then by Nakayama that P [x_{1}, …, x_{s}] = Q.

Thus, the homomorphism ψ is surjective. Its source is a free P –module of rank ∏_{i=1}^{s} n_{i},

and its target is a finite torsion-free P –module of rank |G_{ab}|. We see that ψ ⊗_{P} P_{0} is a surjective homomorphism of P_{0}–vector spaces of the same rank, and so is injective. It follows that ψ is injective, and hence an isomorphism. □

3. Normal extensions

Given a finite extension P ↪ R of integrally closed domains, there is a bijective correspondence between the subfields of R_{0} which contain P_{0}, and the integrally closed subrings of R which contain P. In particular, if P is noetherian and R_{0} is Galois over P_{0} with Galois group G, then the usual Galois correspondence restricts to a bijection from the subgroups of G to the normal domains Q intermediate between P and R. Note that the group Gal(R_{0}|Q_{0}) operates on R by ring automorphisms, and its fixed ring is Q.

Recall that a finite ring extension P ↪ R is unramified at a prime 𝔮 ⊂ R if 𝔮R_{𝔮} = 𝔭R_{𝔮} for 𝔭 = 𝔮 ∩ P and the extension of residue fields P_{𝔭}/𝔭P_{𝔭} ↪ R_{𝔮}/𝔮R_{𝔮} is separable. In codimension one the ramification of R over P is described by the (Dedekind) different 𝔇_{R|P}, a divisorial ideal such that a height one prime 𝔮 ⊆ R is ramified over P if and only if it contains 𝔇_{R|P}, cf. [5], Theorem 3.10.2. Thus, Q ↪ R is divisorially unramified (that is, unramified at all height one primes of R) if and only if 𝔇_{R|P} = R, cf. [3], Proposition 3.6.

Theorem 4. If in Theorem 2 the ring R is normal, then Q ↪ R is divisorially unramified.

Proof. We use the notation of Theorem 2 and its proof. In particular, we recall that

R x_{i} = ∏_{𝔦[h]∈I_{i}} 𝔦[h] for i = 1, …, s, and set z = ∏_{i=1}^{s} x_{i}^{n_{i}−1}.

As G is generated by generalized reflections, [4], Theorem 4, yields the first equality in

𝔇_{R|P} = ∏_{h∈P} 𝔦[h] = ∏_{i=1}^{s} ∏_{G[h]∈G_{i}} ∏_{\begin{matrix}g∈G[h] \\ g≠id \end{matrix}} 𝔦[g] = ∏_{i=1}^{s} ∏_{𝔦[h]∈I_{i}} 𝔦[h]^{n_{i}−1} = ∏_{i=1}^{s}(R x_{i})^{n_{i}−1} = R z .

On the other hand, Theorem 2 shows that P is the fixed subring of Q under the action of the group G_{ab} = G/G′, whose generators ρ(h_{1}), …, ρ(h_{s}) are generalized reflections with inertia ideals in Q generated by x_{1}, …, x_{s}, respectively. The set P′ of generalized

reflections in G_{ab} is the disjoint union ⨆_{i=1}^{s}(ρ(G_{i}) ∖ 1). Thus, denoting by 𝔦[h′] the inertia

ideal in Q of h′ ∈ P′, by another application of [4], Theorem 4, we obtain

𝔇_{Q|P} = ∏_{h′∈P′} 𝔦[h′] = ∏_{i=1}^{s} ∏_{\begin{matrix}h′∈ρ(G_{i}) \\ h′≠id \end{matrix}} 𝔦[h′] = ∏_{i=1}^{s} 𝔦[ρ(h_{i})]^{n_{i}−1} = ∏_{i=1}^{s}(Q x_{i})^{n_{i}−1} = Q z .

Given a fractional ideal 𝔞 of R, we denote by \overline{𝔞} its divisorial hull (R : (R : 𝔞)), where colons are computed in the fraction field R_{0}. We then have equalities

R z = 𝔇_{R|P} = \overline{𝔇_{R|Q} 𝔇_{Q|P}} = \overline{z 𝔇_{R|Q}} = z \overline{𝔇_{R|Q}} = z 𝔇_{R|Q}

where the first and third come from the computations above, the second is the transitivity of the different, cf. [5], Lemma 3.10.1, the fourth follows from the definition, and the last is due to the divisoriality of the different. The result is 𝔇_{R|Q} = R, as desired. □

4. Factorial extensions

We start with generically Galois factorial extensions of regular local rings. The first two assertions of the next result are already contained in [8], Theorem 2.1. The remaining ones give a rather more general and somewhat more precise version of [8], Theorem 2.4.

Theorem 5. Let (R, 𝔪) be a local factorial domain which is the integral closure of a regular local ring P′ in a Galois extension of the fraction field P′_{0}. The following then hold.

The group G = {g ∈ Gal(R_{0}|P′_{0}) | g(a) − a ∈ 𝔪 for all a ∈ R} of automorphisms of R is generated by generalized reflections, the ring P = R^{G} is regular, and P is étale over P′.

If |G| is invertible in P, then R is a complete intersection.

If furthermore R has a primitive root of unity of order equal to the exponent of G_{ab}, and Q is the integral closure of P in the largest abelian extension of P_{0} in R_{0}, then there exist y_{1}, …, y_{s} ∈ P such that Q is a complete intersection P [Y_{1}, …, Y_{s}]/(Y_{1}^{n_{1}} −y_{1}, …, Y_{s}^{n_{s}} −y_{s}) and the extension Q ↪ R is divisorially unramified.

Proof. It is classical that P is étale over P′, cf. [13], Théorème X.1. This implies that P is regular. It is shown in the proof of [15], Théorème 2′, that a finite group G which acts on a factorial local ring with a regular ring of invariants is generated by generalized reflections. If |G| is invertible, then R is a complete intersection by [4], Corollary 3.(i). The remaining assertions have been gleaned from Theorems 2 and 4 above. □

To relax the condition on the fraction field extension in Theorem 5, we focus on two field extensions associated with a finite field extension K ↪ K′. The (quasi-)Galois closure of K in K′ is the unique largest (quasi-)Galois extension of K contained in K′; a (quasi-)Galois shell of K′ over K is the unique up to isomorphism smallest quasi-Galois extension of K which contains K′.

We say that an extension S of P is normally divisorially ramified over P if there exists a normal ring D ⊆ S, such that P is the fixed ring of a finite group of automorphisms of D, and S is divisorially unramified over D.

Theorem 6. Consider a finite residually trivial extension of a regular local ring P by a normal local domain E, such that the fraction field extension P_{0} ↪ E_{0} is Galois, n = rank_{P} E is invertible in P, and E_{0} has a primitive n’th root of unity.

Let S be a ring between P and E, let R be the integral closure of P in the Galois closure of P_{0} in S_{0}, and let T be the integral closure of P in a Galois shell of S_{0} over P_{0}.

If S is factorial and normally divisorially ramified over P, then the extensions R ↪ S ↪ T are divisorially unramified and the ring R is factorial (so Theorem 5 applies to R).

The integral closure of a henselian domain P in a quasi-Galois shell of a finite field extension of P_{0} is local, finite, and residually trivial over P, so we have

Corollary 7. If P is a henselian (e.g., complete) regular local ring with algebraically closed residue field of characteristic zero, P ↪ S is a finite normally divisorially ramified extension, and S is factorial, then the conclusions of the theorem hold. □

Remark. The corollary is closely related to [9], Theorem 4.2, which works from the hypothesis that the divisorial ramification is uniform in the sense that all height one primes of S lying over the same prime of P have equal ramification indices (an additional condition that S has an isolated singularity is never used in the proof). The assumption of ‘uniformity’ is applied through [9], Theorem 1.5, but there is a gap in the proof of that result: the equality starting on its second line holds only after multiplication by some u ∈ S^{×}, so the balance of the argument requires the existence in S of an appropriate root of u.

Such a root exists when P is complete with algebraically closed residue field of characteristic zero. By [9], Theorem 1.5, if S is factorial, P is regular, and the extension is uniformly ramified in codimension one, then it is normally divisorially ramified.

We precede the proof of the theorem by a variation on [9], Proposition 1.2.

Lemma 8. Let P ↪ D ↪ S be finite extensions of normal domains such that D is generically Galois over P, and let P_{0} ↪ L be a quasi-Galois shell of S_{0} over P_{0}. If S is divisorially unramified over D, then so is the integral closure T of P in L.

Proof. As S is divisorially unramified over D, its field of fractions S_{0} is separable over D_{0}, so S_{0} is separable over P_{0}, hence L is Galois over P_{0}. Note that T is the integral closure in L of the composite of the rings {g(S) | g ∈ Gal(L|P_{0})}. Each g(S) is normal, finite, and divisorially unramified over g(D). As g(D) = D for all g (because P_{0} ↪ D_{0} is Galois), it suffices to show that if S_{1} and S_{2} are normal domains which are finite and divisorially unramified extensions of D, then so is the integral closure (S_{1}S_{2})′ of their composite S_{1}S_{2}.

Let 𝔭 ⊂ D be a prime of height one. The semi–local Dedekind domains (S_{1})_{𝔭} and (S_{2})_{𝔭} are unramified over D_{𝔭}, hence by [3], Proposition A.1, the extension D_{𝔭} ↪ (S_{1})_{𝔭}(S_{2})_{𝔭} is unramified. As D_{𝔭} is regular, so is (S_{1})_{𝔭}(S_{2})_{𝔭} = (S_{1}S_{2})_{𝔭}. In particular, (S_{1}S_{2})_{𝔭} is integrally closed. Since integral closure commutes with localization, we have (S_{1}S_{2})_{𝔭} = (S_{1}S_{2})′_{𝔭}. It follows that (S_{1}S_{2})′ is divisorially unramified over D. □

The proof of Theorem 6 draws upon that of [9], Theorem 4.2; the preceding results allow for an extension of the scope of that argument, and for significant simplifications.

Proof of Theorem 6. By assumption, there is a normal domain D such that P ↪ D ↪ S, the fraction field extension P_{0} ↪ D_{0} is Galois, and S is divisorially unramified over D. Lemma 8 shows T is divisorially unramified over D. We set G = Gal(T_{0}|P_{0}), and note that S = T^{H}, where H = Gal(T_{0}|S_{0})⩽G. If N is the the smallest normal subgroup of G which contains H, then T_{0}^{N} is the largest Galois extension of P_{0} contained in S_{0}. The integral closure R of P in T_{0}^{N} is equal to T^{N}, and R necessarily contains D. As D ↪ T is divisorially unramified, by [3], Proposition A.2, so are the intermediate extensions R ↪ S ↪ T.

Next we analyze the homomorphisms of abelian groups in the diagram

Cl(S) ⟶_{≅}^{α} H^{1}(H, T^{×}) ⟶_{≅}^{β} Hom(H, k^{×})
↑^{γ}
Cl(R) ⟶_{α′}^{≅} H^{1}(N, T^{×}) ⟶_{β′} Hom(N, k^{×})

where Cl( ) denotes divisor class groups and k stands for the common residue field of all the local rings under consideration. The maps α and α′ are given by Galois descent; they are bijective because T is divisorially unramified over S and over R, cf. [14], Theorem III.1.1, or [7], Theorem 16.1. The map β′ is injective by Lemma 3. The roots of unity in E_{0} live in its integrally closed subring E, so the same lemma proves β is bijective.

The isomorphisms in the top row of the diagram and the factoriality of S yield Hom(H, k^{×}) = 0. Assuming Hom(N, k^{×}) ≠ 0, let ξ: N ⟶ k^{×} be a nontrivial group homomorphism. As N is the subgroup of G generated by all the conjugates of H, there exists g ∈ G such that ξ(gHg^{−1}) ≠ 1. But then the composition of ξ with the isomorphism H ⟶ gHg^{−1} given by conjugation provides a nontrivial homomorphism H ⟶ k^{×}, which contradicts the already established triviality of Hom(H, k^{×}). Thus, Hom(N, k^{×}) = 0. The injectivity of the maps in the bottom row implies Cl(R) = 0, hence R is factorial. □

Acknowledgements

We thank Phil Griffith for the motivating work in [8], [9], and for suggestions incorporated in the proof of Lemma 8, and Dale Cutkosky for a reference to [10].

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Department of Mathematics, Purdue University, West Lafayette, Indiana 47907, U. S. A. E-mail: avramov@math.purdue.edu aborek@math.purdue.edu