APPROXIMATING DISCRETE VALUATION

RINGS BY REGULAR LOCAL RINGS

A REMARK ON LOCAL UNIFORMIZATION

William Heinzer, Christel Rotthaus and Sylvia Wiegand, June 30

June 30, 1998

Abstract. Let k be a field of characteristic zero and let (V, n) be a discrete rankone valuation domain containing k with V/n = k. Assume that the fraction field L of V has finite transcendence degree s over k. For every positive integer d ≤ s, we prove that V can be realized as a directed union of regular local k-subalgebras of V of dimension d.

1. Introduction. Suppose (R, m) is a local Noetherian domain with fraction

field F and (V, n) is a valuation domain that dominates R in the sense that R is a

subring of V and n ∩ R = m. Classically, in the case where R is essentially finitely

generated over a field, local uniformization of R along V, in algebraic terms, means

the existence of a regular local domain S between R and F, dominated by V and

such that S is essentially finitely generated over R. If R is a regular local ring and

P is a prime ideal of R, embedded local uniformization translates to the existence

of S as above having a prime ideal Q with Q ∩ R = P and S/Q a regular local

domain.

The consideration of embedded local uniformization sometimes yields a repre-

sentation of the valuation domain V as a directed union of regular local rings of

dimension equal to that of R [A, Lemma 12]. In this article we prove that certain

rank-one discrete valuation rings (DVRs) can be represented as a directed union of

Typeset by AMS-TEX 2 WILLIAM HEINZER, CHRISTEL ROTTHAUS AND SYLVIA WIEGAND, JUNE 30

1991 Mathematics Subject Classification. Primary: 13F30, 13H05; Secondary: 13E05, 13G05, 13J05, 13J15.

Key words and phrases. discrete rank-one valuation domain, étale extension, excellent ring, Henselization, local uniformization, regular local domain,.

The authors would like to thank the National Science Foundation and the National Security Agency for support for this research. In addition they are grateful for the hospitality and cooperation of Michigan State, Nebraska and Purdue, where several work sessions on this research were conducted.

regular local domains of dimension d for every positive integer d less than or equal to the dimension of R. We use a technique inspired by Nagata [N1] and developed in our earlier work for the construction of new Noetherian domains [HRW1], [HRW2], [HRW3].

The classical approach for obtaining embedded local uniformization, introduced by Zariski in the 1940’s [Z1], uses local quadratic transforms of R along V. The first local quadratic transform of R along V is defined to be R_{1} = R[m/a]_{m_{1}}, where a ∈ m is such that mV = aV and m_{1} := n ∩ R[m/a]. R_{1} is also called the dilatation of R by the ideal m along V [N2, page 141]. More generally, if I ⊆ m is a nonzero ideal of R, the dilatation of R by I along V is the ring R[I/a]_{m_{1}}, where a ∈ m is such that IV = aV and m_{1} = n ∩ R[I/a]. It is straightforward to check that R_{1} is uniquely determined by R, V and the ideal I [N2, page 141]. For each positive integer i, we define the (i + 1)^{th} quadratic transform R_{i+1} of R along V inductively: R_{i+1} is the first local quadratic transform of R_{i} along V. Then R_{i+j} = (R_{i})_{j} for all i, j ≥ 0.

Using monoidal transforms (a special type of dilatation), S. Cutkosky in [C1] and [C2] has recently done interesting work on birational factorization and local uniformization. 1.1 Background remarks.

(1) If R is a regular local ring, it is well known that a quadratic transform R_{1} of R is again a regular local ring [N2, (38.1)]

(2) If R is a pseudogeometric regular local domain of dimension 2 and V is a valuation domain that birationally dominates^{1} R, a classical result of Zariski and Abhyankar asserts that V = ∪_{n=1}^{∞}R_{n} [A, Lemma 12].

(3) Suppose R is a regular local ring of dimension d ≥ 3. For certain valuation rings V that birationally dominate R the union ∪_{n=1}^{∞}R_{n} of the local quadratic transforms of R along V is strictly smaller than V [S, (4.13)]. In many cases Shannon in [S, (4.5), page 308] proves that V is a directed union of iterated monoidal transforms of R, where a monoidal transform of R is a dilatation of R by a prime ideal P for which R/P is again regular.

(4) If (V, n) is a DVR that birationally dominates a regular local ring R, then

1We say that V birationally dominates R if V dominates R and has the same fraction field as R.

APPROXIMATING DISCRETE VALUATION RINGS, JUNE 30 3

V = ∪_{n=1}^{∞}R_{n},^{2} where R_{n} is the n^{th} quadratic transform of R along V [A, page 336], [Z2, page 27-28].

A proof of item (4) goes as follows: a nonzero element η of V has the form η = b/c, where b, c ∈ R. If (b, c)V = V, then b/c ∈ V implies cV = V. Since V dominates R, it follows that cR = R, so b/c ∈ R in this case. If η = b/c, with b, c ∈ R and (b, c)V = n^{k}, we prove by induction that η ∈ R_{k}. We have already done the case where k = 0. Assume for every regular local domain (S, p) birationally dominated by V, and every nonzero element β/γ ∈ V with β, γ ∈ S, (β, γ)V = n^{j} and 0 ≤ j < k we have β/γ ∈ S_{j}, where S_{j} is the j-th iterated local quadratic transform of S along V. Suppose β, γ ∈ S, β/γ ∈ V with (β, γ)V = n^{k}. Let S_{1} = S[p/a]_{p_{1}}, where a ∈ p is such that pV = aV and p_{1} := n ∩ S[p/a]; that is, S_{1} is the first local quadratic transform of S along V. Then β_{1} := β/a and γ_{1} := γ/a are in S_{1}. Thus a ∈ n implies (β_{1}, γ_{1})V = n^{j} where 0 ≤ j < k, so by induction

β/γ = β_{1}/γ_{1} ∈ (S_{1})_{j} = S_{j+1} ⊆ S_{k},

which completes the proof.

In this article we prove the following theorem:

1.2 Main Theorem. Let k be a field of characteristic zero and let (V, n) be a DVR containing k with V/n = k. Assume that the fraction field L of V has finite transcendence degree s over k. Then for every integer d with 1 ≤ d ≤ s, there exists a nested family {C_{n}^{(α)} : n ∈ ℕ, α ∈ Γ} of d-dimensional regular local k-subalgebras of V such that V is the directed union of the C_{n}^{(α)} and V dominates each C_{n}^{(α)}.

If the field L is finitely generated over k, then V is a countable union ∪_{n=1}^{∞}C_{n}, where, for each n ∈ ℕ,

(1) C_{n} is a d-dimensional regular local k-subalgebra of V,

(2) C_{n} has fraction field L,

(3) C_{n+1} dominates C_{n} and

(4) V dominates C_{n}.

We have the following corollary to the main theorem.

2It may happen that R_{n} = V for some n, in which case R_{n+i} = R_{n} for all i ≥ 0

4 WILLIAM HEINZER, CHRISTEL ROTTHAUS AND SYLVIA WIEGAND, JUNE 30

1.3 Corollary. Let k be a field of characteristic zero and let (R, m) be a local domain essentially of finite type over k with coefficient field k = R/m and field of fractions L. Let (V, n) be a DVR birationally dominating R with V/n = k. For every integer d with 1 ≤ d ≤ s, s = trdeg_{k}(L), there exists a sequence of ddimensional regular local k-subalgebras C_{n} of V such that V = ∪_{n=1}^{∞}C_{n}, and for each n ∈ ℕ, C_{n+1} dominates C_{n} and V dominates C_{n}. Moreover C_{n} dominates R for all sufficiently large n.

Proof. By (1.2), V = ∪_{n=1}^{∞}C_{n}, where the C_{n} are regular local rings satisfying all the conditions of (1.3) except possibly that of dominating R. Since R is essentially of finite type over k and is dominated by V, it follows that R is dominated by C_{n} for all sufficiently large n. □

1.4 Discussion. (1) If L/k is finitely generated of transcendence degree s, then the fact that V is a directed union of s-dimensional regular local rings follows from classical theorems of Zariski. The local uniformization theorem of Zariski [Z1] implies the existence of a regular local domain (R, m) containing the field k such that V birationally dominates R. Since k is a coefficient field for V, we have

(a) k ↪ V ↠ V/n ≅ k; thus k is relatively algebraically closed in L.

(b) R/m = k (because V dominates R).

(c) Every iterated local quadratic transform of R along V has dimension s. Now by (1.1.4), V is a directed union of s-dimensional RLRs.

(2) If d = 1, the main theorem is trivially true by taking each C_{n} = V. Thus if L/k is finitely generated of transcendence degree s = 2, then the theorem is saying nothing new.

(3) If s > 2, then the classical local uniformization theorem says nothing about expressing V as a directed union of d-dimensional RLRs, where 2 ≤ d ≤ s − 1. If (S, p) is a local Noetherian domain containing k and birationally dominated by V with dim(S) = d < s, then S does not satisfy the dimension formula. It follows that S is not essentially finitely generated over k [M1, page 119].

We use the following notation in the proof of the main theorem.

1.5 Endpieces and related localized polynomial rings.

APPROXIMATING DISCRETE VALUATION RINGS, JUNE 30 5

Let (R, m) be a local Noetherian domain with fraction field K. Let y ∈ m be a nonzero element, let R^{∗} = { \atop (R, (y))} be the (y)-adic completion of R and let R_{y}^{∗} = R^{∗}[1/y]. Suppose τ_{1}, …, τ_{s} ∈ yR^{∗} are regular elements^{3} of R^{∗} that are algebraically independent over K. We consider the localized polynomial ring

B_{0} := R[τ_{1}, …, τ_{s}]_{(m,τ_{1},...,τ_{s})}.

It is clear that B_{0} is a local Noetherian domain with dim(B_{0}) = dim(R) + s such that R^{∗} dominates B_{0} and B_{0} dominates R.

For every γ ∈ R^{∗} and every n > 0, we define the n^{th}-endpiece γ_{n} with respect to y of γ to be

(1.5.1)  γ_{n} := ∑_{j=n+1}^{∞} c_{j}y^{j−n}, where γ := ∑_{j=1}^{∞} c_{j}y^{j} and each c_{j} ∈ R.

In particular, we represent each of the τ_{i} by a power series expansion in y; we use these representations to obtain for each positive integer n the n^{th}-endpieces τ_{in} and corresponding n^{th}-localized polynomial ring B_{n}. For 1 ≤ i ≤ s, and τ_{i} := ∑_{j=1}^{∞} a_{ij}y^{j}, where the a_{ij} ∈ R, we have

τ_{in} := ∑_{j=n+1}^{∞} a_{ij}y^{j−n}, B_{n} := R[τ_{1n}, …, τ_{sn}]_{(m,τ_{1n},...,τ_{sn})}, for each n ∈ ℕ.

The definition of B_{n} is independent of the representation of the τ_{i} as power series in y [HRW1, (2.3)]. The localized polynomial ring B_{n+1} birationally dominates B_{n}. We define

(1.5.2)  B := ∪_{n=0}^{∞}B_{n} = \underrightarrow{lim} B_{n}  and  A := K(τ_{1}, …, τ_{s}) ∩ R^{∗}.

It is readily seen that A birationally dominates B.

In the proof of our main theorem, we make use of the following result from previous work that we summarize as Theorem 1.6. 1.6 Theorem. [HRW1] or [HRW2] Suppose (R, m) is a local Noetherian domain, y ∈ m is a nonzero element, and τ_{1}, …, τ_{s} are elements of the (y)-adic completion R^{∗} of R that are algebraically independent over R. Then the following conditions are equivalent:

(1) The ring R^{∗}[1/y] is flat over B_{0} = R[τ_{1}, …, τ_{s}]_{(m,τ_{1},...,τ_{s})}.

3We say an element of a ring is a regular element if it is not a zero divisor.

6 WILLIAM HEINZER, CHRISTEL ROTTHAUS AND SYLVIA WIEGAND, JUNE 30

(2) The ring R^{∗}[1/y] is flat over R[τ_{1}, …, τ_{s}].

(3) The directed union B := ∪_{n=0}^{∞}B_{n} is Noetherian. Moreover, if these equivalent conditions hold, then the domain A := K(τ_{1}, …, τ_{s})∩ R^{∗} is equal to B, and thus both are Noetherian.

2. Proof of the main theorem.

Let y ∈ n be such that yV = n. Then the n-adic completion V̂ of V is k[[y]]. Hence we have

k[y]_{(y)} ⊆ V ⊆ k[[y]].

Then V = L ∩ k[[y]], for example since V ↪ k[[y]] is flat. Since trdeg_{k(y)} L = s − 1, there are s − 1 elements σ_{1}, …, σ_{s−d}, τ_{1}, …, τ_{d−1} ∈ yV such that L is algebraic over K := k(y, σ_{1}, …, σ_{s−d}, τ_{1}, …, τ_{d−1}).

Set F := k(y, σ_{1}, …, σ_{s−d}) and R := V ∩ F. Then R is a DVR and the (y)-adic completion of R is R^{∗} = k[[y]], so R_{y}^{∗} is flat as an R[τ_{1}, …, τ_{d−1}]-module. Hence Theorem 1.6 applies. In the notation of (1.5), B_{0} := R[τ_{1}, …, τ_{d−1}]_{(y,τ_{1},...,τ_{d−1})} is a d-dimensional regular local ring and V_{0} := V ∩ K is a DVR that birationally dominates B_{0} and has y-adic completion V̂_{0} = k[[y]]. The following diagram displays these domains:

k ⟶^{⊆}  F  ⟶^{⊆} K  K  ⟶^{⊆} L := Q(V)
∥↑  ^{∪|}↑  ^{∪|}↑  ^{∪|}↑  ^{∪|}↑
k ⟶^{⊆} R := V ∩ F ⟶^{⊆} B_{0} ⟶^{⊆} V_{0} := V ∩ K ⟶^{⊆}  V

Let τ_{in} be the n^{th} endpiece of τ_{i} for each i; since V_{0} = K ∩ k[[y]], the τ_{in} ∈ V_{0} and B_{n} = R[τ_{1n}, …, τ_{d−1,n}]_{(y,τ_{1n},...,τ_{d−1,n})} is, for each n ∈ ℕ, the first quadratic transform of B_{n−1} along V_{0}.

By Theorem 1.6, V_{0} = ∪_{n=1}^{∞}B_{n}.

We have

V_{0} ⟶_{alg} V ⟶ k[[y]],

where V_{0} and V are DVRs of characteristic zero having completion k[[y]], Since V_{0} and V are excellent, their Henselizations V_{0}^{h} and V^{h} are the set of elements of k[[y]] algebraic over V_{0} or V [N2, (44.3)]. Thus V_{0}^{h} = V^{h} and V is a directed union of étale extensions of V_{0} [R, (1.3)].

APPROXIMATING DISCRETE VALUATION RINGS, JUNE 30 7

2.1 Proposition. Let B_{n}^{h} denote the Henselization of B_{n}. Then ∪_{n=1}^{∞}B_{n}^{h} = V_{0}^{h} = V^{h}.

Proof. The ring C := ∪B_{n}^{h} is Henselian and contains V_{0}, so V_{0}^{h} = V^{h} ⊆ C. Moreover, the inclusion map V → C = ∪B_{n}^{h} extends to a map V^{h} →^{σ} C = ∪B_{n}^{h}.

On the other hand, the maps B_{n} → V extend to maps: B_{n}^{h} → V^{h} yielding a map ρ : C → V^{h} with σρ = 1_{C} , and ρσ = 1_{V^{h}}. Thus ∪_{n=1}^{∞}B_{n}^{h} = V^{h}. □

Continuation with proof of main theorem. First assume L is finitely generated over k. It follows that L is finite algebraic over K. Since ∪_{n=1}^{∞}B_{n}^{h} = V^{h}, we have ∪_{n=1}^{∞}Q(B_{n}^{h}) = Q(V^{h}) and L ⊆ Q(V^{h}). Since L/K is finite algebraic, L ⊆ Q(B_{n}^{h}) for all sufficiently large n. By relabeling, we may assume L ⊆ Q(B_{n}^{h}) for all n. Let C_{n} := B_{n}^{h} ∩ L. Since B_{n} is a regular local ring, C_{n} is a regular local ring with C_{n}^{h} = B_{n}^{h}. [R, (1.3)]

Claim. For every n, C_{n+1} dominates C_{n} and V dominates C_{n}. Also ∪_{n=1}^{∞}C_{n} = V.

Proof of claim. Since B_{n+1} dominates B_{n}, we see that B_{n+1}^{h} dominates B_{n}^{h} and hence C_{n+1} = B_{n+1}^{h} ∩L dominates C_{n} = B_{n}^{h} ∩L. Since C_{n} = B_{n}^{h} ∩L ⊆ V^{h} ∩L = V, it follows that V dominates C_{n} and V_{0} ⊆ ∪_{n=1}^{∞}C_{n} ⊆ V. Since V birationally dominates ∪_{n=1}^{∞}C_{n}, it suffices to show that ∪_{n=1}^{∞}C_{n} is a DVR.

But by the same argument as before, ∪_{n=1}^{∞}C_{n}^{h} = (∪_{n=1}^{∞}C_{n})^{h} = V^{h}. This shows that ∪_{n=1}^{∞}C_{n} is a DVR, and therefore ∪_{n=1}^{∞}C_{n} = V. Thus in the case where L/k is finitely generated we have completed the proof of the main theorem including the “Moreover” statement.

2.2 Remark. An alternate approach to the definition of C_{n} is as follows. Since V is a directed union of étale extensions of V_{0} and ℚ(V) = L is finite algebraic over ℚ(V_{0}) = K, V is étale over V_{0} and therefore V = V_{0}[θ] = V_{0}[X]/(f(X)), where f(X) is a monic polynomial such that f(θ) = 0 and f′(θ) is a unit of V. Let B′_{n} denote the integral closure of B_{n} in L and let C_{n} = (B′_{n})_{(n∩B′_{n})}. Since ∪_{n=1}^{∞}B_{n} = V_{0}, it follows that ∪_{n=1}^{∞}C_{n} = V. Moreover, for all sufficiently large n, f(X) ∈ B_{n}[X] and f′(θ) is a unit of C_{n}. Therefore C_{n} is a regular local ring for all sufficiently large n [N2, (38.6)]. As we note in (2.3) below, this allows us to deduce a version of the main theorem also in the case where k has characteristic p > 0 provided the field K can be chosen so that L/K is separable.

8 WILLIAM HEINZER, CHRISTEL ROTTHAUS AND SYLVIA WIEGAND, JUNE 30

Completion of proof of main theorem. If L is not finitely generated over k, we choose a nested family of fields L_{α}, with α ∈ Γ, such that

(1) K ⊆ L_{α}, for all α.

(2) L_{α} is finite algebraic over K.

(3) ∪_{n=1}^{∞}L_{α} = L.

The rings V_{α} = L_{α}∩V are DVRs with ∪_{n=1}^{∞}V_{α} = V and V_{α}^{h} = V^{h}, since V_{0} ⊆ V_{α}, for each α ∈ Γ.

As above, ∪_{n=1}^{∞}B_{n}^{h} = V^{h}, ∪_{n=1}^{∞}Q(B_{n}^{h}) = Q(V^{h}) and L ⊆ Q(V^{h}). Thus we see that for each α ∈ Γ, there is an n_{α} ∈ ℕ such that L_{α} ⊆ Q(B_{n}^{h}) for all n ≥ n_{α}.

Put C_{n}^{(α)} = L_{α} ∩B_{n}^{h} for each n ≥ n_{α}. Then V_{α} = ∪_{n=n_{α}}^{∞}C_{n}^{(α)} and V_{α} birationally dominates C_{n}^{(α)}. Hence

V = ⋃_{α∈Γ,n≥n_{α}}C_{n}^{(α)}.

This completes the proof of the main theorem. □

2.3 Remark. If the characteristic of k is p > 0 then the Henselization V_{0}^{h} of V_{0} = K ∩ k[[y]] may not equal the Henselization V^{h} of V = L ∩ k[[y]], because the algebraic field extension L/K may not be separable. But in the case where L/K is separable algebraic, the fact that the DVRs V and V_{0} have the same completion implies that V is a directed union of étale extensions of V_{0} (see, for example, [AH, Theorem 2.7]). Therefore in the case where L/K is separable algebraic, V is a directed union of regular local rings of dimension d.

Thus for a local domain (R, m) essentially of finite type over a field k of characteristic p > 0, a result analogous to Corollary 1.3 is true provided there exists a subfield K of L such that K is purely transcendental over k, L/K is separable algebraic, and K contains a generator for the maximal ideal of V.

In characteristic p > 0, with V excellent and the extension separable, the ring V_{0} need not be excellent (see for example [HRS, (3.3) and (3.4)]).

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Department of Mathematics, Purdue University, West Lafayette, IN 47907-1395

Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027

Department of Mathematics and Statistics, University of Nebraska, Lincoln, NE 68588-0323