GENERIC FIBER RINGS OF MIXED POWER SERIES/POLYNOMIAL RINGS

William Heinzer, Christel Rotthaus and Sylvia Wiegand

Abstract. Let K be a field, m and n positive integers, and X = {x_{1}, …, x_{n}}, and Y = {y_{1}, …, y_{m}} sets of independent variables over K. Let A be the localized polynomial ring K[X]_{(X)}. We prove that every prime ideal P in Â = K[[X]] that is maximal with respect to P ∩ A = (0) has height n − 1. We consider the mixed power series/polynomial rings B := K[[X]] [Y]_{(X,Y )} and C := K[Y]_{(Y )}[[X]]. For each prime ideal P of B̂ = Ĉ that is maximal with respect to either P ∩ B = (0) or P ∩ C = (0), we prove that P has height n + m − 2. We also prove each prime ideal P of K[[X, Y]] that is maximal with respect to P ∩ K[[X]] = (0) is of height either m or n + m − 2.

1. Introduction and Background.

Let (R, m) be a Noetherian local integral domain and let R̂ denote the m-adic completion of R. The generic formal fiber ring of R is the localization (R \ (0))^{−1}R̂ of R̂. The formal fibers of R are the fibers of the morphism Spec R̂ → Spec R; for a prime ideal P of R, the formal fiber over P is Spec( (R_{P} /P R_{P}) ⊗_{R} R̂). The formal fibers encode important information about the structure of R. For example, the local ring R is excellent provided it is universally catenary and has geometrically regular formal fibers [2, (7.8.3), page 214].

Let R ↪ S be an injective homomorphism of commutative rings. If R is an integral domain, the generic fiber ring of the map R ↪ S is the localization (R\(0))^{−1}S of S. In this article we study generic fiber rings for “mixed” polynomial and power series rings over a field. More precisely, for K a field, m and n positive integers, and X = {x_{1}, …, x_{n}} and Y = {y_{1}, …, y_{m}} sets of variables over K, we consider the local rings A := K[X]_{(X)}, B := K[[X]] [Y]_{(X,Y)} and C := K[Y]_{(Y)}[[X]], as well as their completions Â = K[[X]] and B̂ = Ĉ = K[[X, Y]]. Notice that there is a canonical inclusion map B ↪ C.

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The authors are grateful for the hospitality and cooperation of Michigan State, Nebraska and Purdue, where several work sessions on this research were conducted. Wiegand thanks the National Security Agency for its support.

We have the following local embeddings.

A := K[X]_{(X)} ↪ Â:= K[[X]], Â ↪ B̂ = Ĉ = K[[X, Y]] and
B := K[[X]] [Y]_{(X,Y)} ↪ C := K[Y]_{(Y)}[[X]] ↪ B̂ = Ĉ = K[[X]] [[Y]].

Matsumura proves in [7] that the generic formal fiber ring of A has dimension n − 1 = dim A − 1, and the generic formal fiber rings of B and C have dimension n + m − 2 = dim B − 2 = dim C − 2. However he does not address the question of whether all maximal ideals of the generic formal fiber rings for A, B and C have the same height. If the field K is countable, it follows from [3, Prop. 4.10, page 36] that all maximal ideals of the generic formal fiber ring of A have the same height.

In answer to a question raised by Matsumura in [7], Rotthaus in [10] establishes the following result. Let n be a positive integer. Then there exist excellent regular local rings R such that dim R = n and such that the generic formal fiber ring of R has dimension t, where the value of t may be taken to be any integer between 0 and dim R − 1. It is also shown in [10, Corollary 3.2] that there exists an excellent regular local domain having the property that its generic formal fiber ring contains maximal ideals of different heights.

Let T̂ be a complete Noetherian local ring and let C be a finite set of incomparable prime ideals of T̂. Charters and Loepp in [1] (see also [6, Theorem 17]) determine necessary and sufficient conditions for T̂ to be the completion of a Noetherian local domain T such that the generic formal fiber of T has as maximal elements precisely the prime ideals in C. If T̂ is of characteristic zero, Charters and Loepp give necessary and sufficient conditions to obtain such a domain T that is excellent. The finite set C may be chosen to contain prime ideals of different heights. This provides many examples where the generic formal fiber ring contains maximal ideals of different heights.

Our main results may be summarized as follows.

1.1 Theorem. With the above notation, we prove that all maximal ideals of the generic formal fiber rings of A, B and C have the same height. In particular, we prove:

(1) If P is a prime ideal of Â maximal with respect to P ∩ A = (0), then ht(P) = n − 1.

GENERIC FIBER RINGS 3

(2) If P is a prime ideal of B̂ maximal with respect to P ∩ B = (0), then ht(P) = n + m − 2.

(3) If P is a prime ideal of Ĉ maximal with respect to P ∩ C = (0), then ht(P) = n + m − 2.

(4) In addition, there are at most two possible values for the height of a maximal

ideal of the generic fiber ring (Â \ (0))^{−1}Ĉ of the inclusion map Â ↪ Ĉ. (a) If n ≥ 2 and P is a prime ideal of Ĉ maximal with respect to P ∩ Â = (0), then either ht P = n + m − 2 or ht P = m.

(b) If n = 1, then all maximal ideals of the generic fiber ring (Â \ (0))^{−1}Ĉ have height m.

We were motivated to consider generic fiber rings for the embeddings displayed above because of questions related to [4] and [5] and ultimately because of the following question posed by Melvin Hochster.

1.2 Question. Let R be a complete local domain. Can one describe or somehow classify the local maps of R to a complete local domain S such that U^{−1}S is a field, where U = R \ (0), i.e., such that the generic fiber of R ↪ S is trivial?

Hochster remarks that if, for example, R is equal characteristic zero, one obtains such extensions by starting with

(1.2.1)  R = K[[x_{1}, ..., x_{n}]] ↪ T = L[[x_{1}, ..., x_{n}, y_{1}, ..., y_{m}]] → T/P = S,

where K is a subfield of L, the x_{i}, y_{j} are formal indeterminates, and P is a prime ideal of T maximal with respect to being disjoint from the image of R \ {0}. Of course, such prime ideals P correspond to the maximal ideals of the generic fiber (R \ (0))^{−1}T.

In Theorem 7.2, we answer Question 1.2 in the special case where the extension arises from the embedding in (1.2.1) with the field L = K. We prove in this case that the dimension of the extension ring S must be either 2 or n.

In [5] we study extensions of integral domains R ↪^{φ} S such that, for every nonzero Q ∈ Spec S, we have Q ∩ R ≠ (0). Such extensions are called trivial generic fiber extensions or TGF extensions in [5]. One obtains such an extension by considering a composition R ↪ T → T/P = S, where T is an extension ring of R and P ∈ Spec T is maximal with respect to P ∩ R = (0). Thus the generic fiber ring and so also Theorem 1.1 give information regarding TGF extensions in the case where the smaller ring is a mixed polynomial/power series ring.

4 HEINZER, ROTTHAUS AND WIEGAND

In addition, Theorem 1.1 is useful in the study of (1.2.1), because the map in (1.2.1) factors through:

R = K[[x_{1}, …, x_{n}]] ↪ K[[x_{1}, …, x_{n}]] [y_{1}, …, y_{m}] ↪ T = L[[x_{1}, …, x_{n}, y_{1}, …, y_{n}]].

Section 2 contains implications of Weierstrass’ Preparation Theorem to the prime ideals of power series rings. We first prove a technical proposition regarding a change of variables that provides a “nice” generating set for a given prime ideal P of a power series ring; then in Theorem 2.3 we prove that, in certain circumstances, a larger prime ideal can be found with the same contraction as P to a certain subring. In Sections 3 and 4, we prove parts 2 and 3 of Theorem 1.1 stated above. In Section 5 we use a result of Valabrega for the two-dimensional case. We then apply this result in Section 6 to prove part 1 of Theorem 1.1, and in Section 7 we prove part 4.

2. Variations on a theme of Weierstrass.

In this section, we apply the Weierstrass Preparation Theorem [12, Theorem 5, page 139, and Corollary 1, page 145] to examine the structure of a given prime ideal P in the power series ring Â = K[[X]], where X = {x_{1}, …, x_{n}} is a set of n variables over the field K. Here A = K[X]_{(X)} is the localized polynomial ring in these variables. Our procedure is to make a change of variables that yields a regular sequence in P of a nice form.

2.1 Notation. By a change of variables, we mean a finite sequence of ‘polynomial’ change of variables of the type described below, where X = {x_{1}, …, x_{n}} is a set of n variables over the field K. For example, with e_{i}, f_{i} ∈ ℕ, consider

x_{1} ↦ x_{1} + x_{n}^{e_{1}} = z_{1},  x_{2} ↦ x_{2} + x_{n}^{e_{2}} = z_{2},  …,
x_{n−1} ↦ x_{n−1} + x_{n}^{e_{n−1}} = z_{n−1}, x_{n} ↦ x_{n} = z_{n},

followed by:

z_{1} ↦ z_{1} = t_{1},  z_{2} ↦ z_{2} + z_{1}^{f_{2}} = t_{2},  …,
z_{n−1} ↦ z_{n−1} + z_{1}^{f_{n−1}} = t_{n−1}, x_{n} ↦ z_{n} + z_{1}^{f_{n}} = t_{n}.

Thus a change of variables defines an automorphism of Â that restricts to an automorphism of A.

We also consider a change of variables for subrings of A and Â. For example, if A_{1} = K[x_{2}, …, x_{n}] ⊆ A and S = K[[x_{2}, …, x_{n}]] ⊆ Â, then by a change of variables

GENERIC FIBER RINGS 5

inside A_{1} and S, we mean a finite sequence of automorphisms of A and Â of the type described above on x_{2}, …, x_{n} that leave the variable x_{1} fixed. In this case we obtain an automorphism of Â that restricts to an automorphism on each of S, A and A_{1}.

2.2 Proposition. Let Â := K[[X]] = K[[x_{1}, …, x_{n}]] and let P ∈ Spec Â with x_{1} ∉ P and ht P = r, where 1 ≤ r ≤ n−1. There exists a change of variables x_{1} ↦ z_{1} := x_{1} (x_{1} is fixed), x_{2} ↦ z_{2}, …, x_{n} ↦ z_{n} and a regular sequence f_{1}, …, f_{r} ∈ P so that, upon setting Z_{1} = {z_{1}, …, z_{n−r}}, Z_{2} = {z_{n−r+1}, …, z_{n}} and Z = Z_{1} ∪Z_{2}, we have

f_{1} ∈ K[[Z_{1}]] [z_{n−r+1}, …, z_{n−1}] [z_{n}]  is monic as a polynomial in z_{n}
f_{2} ∈ K[[Z_{1}]] [z_{n−r+1}, …, z_{n−2}] [z_{n−1}] is monic as a polynomial in z_{n−1}, etc
⋮
f_{r} ∈ K[[Z_{1}]] [z_{n−r+1}]  is monic as a polynomial in z_{n−r+1}.

In addition:

(1) P is a minimal prime of the ideal (f_{1}, …, f_{r})Â.

(2) The (Z_{2})-adic completion of K[[Z_{1}]] [Z_{2}]_{(Z)} is identical to the (f_{1}, …, f_{r})- adic completion and both equal Â = K[[X]] = K[[Z]].

(3) If P_{1} := P ∩ K[[Z_{1}]] [Z_{2}]_{(Z)}, then P_{1}Â = P, that is, P is extended from K[[Z_{1}]] [Z_{2}]_{(Z)}.

(4) The ring extension:

K[[Z_{1}]] ↪ K[[Z_{1}]] [Z_{2}]_{(Z)}/P_{1} ≅ K[[Z]]/P

is finite (and integral).

Proof. Since Â is a unique factorization domain, there exists a nonzero prime element f in P. The power series f is therefore not a multiple of x_{1}, and so f must contain a monomial term x_{2}^{i_{2}} … x_{n}^{i_{n}} with a nonzero coefficient in K. This nonzero coefficient in K may be assumed to be 1. There exists an automorphism σ : Â → Â defined by the change of variables:

x_{1} ↦ x_{1}  x_{2} ↦ t_{2} := x_{2}+x_{n}^{e_{2}} … x_{n−1} ↦ t_{n−1} := x_{n−1}+x_{n}^{e_{n−1}}  x_{n} ↦ x_{n},

with e_{2}, …, e_{n−1} ∈ ℕ chosen suitably so that f written as a power series in the variables x_{1}, t_{2}, …, t_{n−1}, x_{n} contains a term a_{n}x_{n}^{s_{n}}, where s_{n} is a positive integer,

6 HEINZER, ROTTHAUS AND WIEGAND

and a_{n} ∈ K is nonzero. We assume that the integer s_{n} is minimal among all integers i such that a term ax_{n}^{i} occurs in f with a nonzero coefficient a ∈ K; we further assume that the coefficient a_{n} = 1. By Weierstrass we have that:

f = mϵ,

where m ∈ K[[x_{1}, t_{2}, …, t_{n−1}]] [x_{n}] is a monic polynomial in x_{n} of degree s_{n} and ϵ is a unit in Â. Since f ∈ P is a prime element, m ∈ P is also a prime element. Using Weierstrass again, every element g ∈ P can be written as:

g = mh + q,

where h ∈ K[[x_{1}, t_{2}, …, t_{n−1}, x_{n}]] = Â and q ∈ K[[x_{1}, t_{2}, …, t_{n−1}]] [x_{n}] is a polynomial in x_{n} of degree less than s_{n}. Note that

K[[x_{1}, t_{2}, …, t_{n−1}]] ↪ K[[x_{1}, t_{2}, …, t_{n−1}]] [x_{n}]/(m)

is an integral (finite) extension. Thus the ring K[[x_{1}, t_{2}, …, t_{n−1}]] [x_{n}]/(m) is complete. Moreover, the two ideals (x_{1}, t_{2}, …, t_{n−1}, m) = (x_{1}, t_{2}, …, t_{n−1}, x_{n}^{s_{n}}) and (x_{1}, t_{2}, …, t_{n−1}, x_{n}) of B_{0} := K[[x_{1}, t_{2}, …, t_{n−1}]] [x_{n}] have the same radical. Therefore Â is the (m)-adic and the (x_{n})-adic completion of B_{0} and P is extended from B_{0}.

This implies the statement for r = 1, with f_{1} = m, z_{n} = x_{n}, z_{1} = x_{1}, z_{2} = t_{2}, …, z_{n−1} = t_{n−1}, Z_{1} = {x_{1}, t_{2}, …, t_{n−1}} and Z_{2} = {z_{n}} = {x_{n}}. In particular, when r = 1, P is minimal over mÂ, so P = mÂ.

For r > 1 we continue by induction on r. Let P_{0} := P ∩K[[x_{1}, t_{2}, …, t_{n−1}]]. Since m ∉ K[[x_{1}, t_{2}, …, t_{n−1}]] and P is extended from B_{0} := K[[x_{1}, t_{2}, …, t_{n−1}]] [x_{n}], then P ∩ B_{0} has height r and ht P_{0} = r − 1. Since x_{1} ∉ P, we have x_{1} ∉ P_{0}, and by the induction hypothesis there is a change of variables t_{2} ↦ z_{2}, …, t_{n−1} ↦ z_{n−1} of K[[x_{1}, t_{2}, …, t_{n−1}]] and elements f_{2}, …, f_{r} ∈ P_{0} so that:

f_{2} ∈ K[[x_{1}, z_{2} …, z_{n−r}]] [z_{n−r+1}, …, z_{n−2}] [z_{n−1}] is monic in z_{n−1}
f_{3} ∈ K[[x_{1}, z_{2} …, z_{n−r}]] [z_{n−r+1}, …, z_{n−3}] [z_{n−2}] is monic in z_{n−2}, etc
⋮
f_{r} ∈ K[[x_{1}, z_{2}, …, z_{n−r}]] [z_{n−r+1}]  is monic in z_{n−r+1},

GENERIC FIBER RINGS 7

and f_{2}, …, f_{r} satisfy the assertions of Proposition 2.2 for P_{0}.

It follows that m, f_{2}, …, f_{r} is a regular sequence of length r and that P is a minimal prime of the ideal (m, f_{2}, …, f_{r})Â. Set z_{n} = x_{n}. We now prove that m may be replaced by a polynomial f_{1} ∈ K[[x_{1}, z_{2}, …, z_{n−r}]] [z_{n−r+1}, …, z_{n}]. Write

m = ∑_{i=0}^{s_{n}} a_{i}z_{n},

where the a_{i} ∈ K[[x_{1}, z_{2}, …, z_{n−1}]]. For each i < s_{n}, apply Weierstrass to a_{i} and f_{2} in order to obtain:

a_{i} = f_{2}h_{i} + q_{i},

where h_{i} is a power series in K[[x_{1}, z_{2}, …, z_{n−1}]] and q_{i} ∈ K[[x_{1}, z_{2}, …, z_{n−2}]] [z_{n−1}] is a polynomial in z_{n−1}. With q_{s_{n}} = 1 = a_{s_{n}}, we define

m_{1} = ∑_{i=0}^{s_{n}} q_{i}z_{n}^{i}.

Now (m_{1}, f_{2}, …, f_{r})Â = (m, f_{2}, …, f_{r})Â and we may replace m by m_{1} which is a polynomial in z_{n−1} and z_{n}. To continue, for each i < s_{n}, write:

q_{i} = ∑_{j,k} b_{ij}z_{n−1}^{j} with b_{ij} ∈ K[[x_{1}, z_{2}, …, z_{n−2}]].

For each b_{ij}, we apply Weierstrass to b_{ij} and f_{3} to obtain:

b_{ij} = f_{3}h_{ij} + q_{ij},

where q_{ij} ∈ K[[x_{1}, z_{2}, …, z_{n−3}]] [z_{n−2}]. Set

m_{2} = ∑_{i,j} q_{ij}z_{n−1}^{j}z_{n}^{i} ∈ K[[x_{1}, z_{2}, …, z_{n−3}]] [z_{n−2}, z_{n−1}, z_{n}]

with q_{s_{n}0} = 1. It follows that (m_{2}, f_{2}, …, f_{r})Â = (m, f_{2}, …, f_{r})Â. Continuing this process by applying Weierstrass to the coefficients of z_{n−2}^{k}z_{n−1}^{j}z_{n}^{i} and f_{4}, we establish the existence of a polynomial f_{1} ∈ K[[Z_{1}]] [z_{n−r+1}, …, z_{n}] that is monic in z_{n} so that (f_{1}, f_{2}, …, f_{r})Â = (m, f_{2}, …, f_{r})Â. Therefore P is a minimal prime of (f_{1}, …, f_{r})Â.

The extension

K[[Z_{1}]] ⟶ K[[Z_{1}]] [Z_{2}]/(f_{1}, …, f_{r})

8 HEINZER, ROTTHAUS AND WIEGAND

is integral and finite. Thus the ring K[[Z_{1}]] [Z_{2}]/(f_{1}, …, f_{r}) is complete. This implies Â = K[[x_{1}, z_{2}, …, z_{n}]] is the (f_{1}, …, f_{r})-adic (and the (Z_{2})-adic) completion of K[[Z_{1}]] [Z_{2}]_{(Z)} and that P is extended from K[[Z_{1}]] [Z_{2}]_{(Z)}. This completes the proof of Proposition 2.2. □

The following theorem is the technical heart of the paper.

2.3 Theorem. Let K be a field and let y and X = {x_{1}, …, x_{n}} be variables over K. Assume that V is a discrete valuation domain with completion V̂ = K[[y]] and that K[y] ⊆ V ⊆ K[[y]]. Also assume that the field K((y)) = K[[y]] [1/y] has uncountable transcendence degree over the quotient field Q(V) of V. Set R_{0} := V [[X]] and R = R̂_{0} = K[[y, X]]. Let P ∈ Spec R be such that:

(i) P ⊆ (X)R (so y ∉ P), and (ii) dim(R/P) > 2.

Then there is a prime ideal Q ∈ Spec R such that

(1) P ⊂ Q ⊂ XR,

(2) dim(R/Q) = 2, and

(3) P ∩ R_{0} = Q ∩ R_{0}.

In particular, P ∩ K[[X]] = Q ∩ K[[X]].

Proof. Assume that P has height r. Since dim(R/P) > 2, we have 0 ≤ r < n − 1. If r > 0, then there exist a transformation x_{1} ↦ z_{1}, …, x_{n} ↦ z_{n} and elements f_{1}, …, f_{r} ∈ P, by Proposition 2.2, so that the variable y is fixed, and

f_{1} ∈ K[[y, z_{1}, …, z_{n−r}]] [z_{n−r+1}, …, z_{n}] is monic in z_{n}, f_{2} ∈ K[[y, z_{1}, …, z_{n−r}]] [z_{n−r+1}, …, z_{n−1}] is monic in z_{n−1} etc,

⋮ f_{r} ∈ K[[y, z_{1}, …, z_{n−r}]] [z_{n−r+1}] is monic in z_{n−r+1},

and the assertions of Proposition 2.2 are satisfied. In particular, P is a minimal prime of (f_{1}, …, f_{r})R. Let Z_{1} = {z_{1}, …, z_{n−r}} and Z_{2} = {z_{n−r+1}, …, z_{n−1}, z_{n}}. By Proposition 2.2, if D := K[[y, Z_{1}]] [Z_{2}]_{(Z)} and P_{1} := P ∩ D, then P_{1}R = P.

The following diagram shows these rings and ideals.

R = K[[y, X]] = K[[y, Z_{1}, Z_{2}]] (X)R  D = K[[y, Z_{1}]] [Z_{2}]_{(Z)} P = P_{1}R  P_{1} = P ∩ D

Note that f_{1}, …, f_{r} ∈ P_{1}. Let g_{1}, …, g_{s} ∈ P_{1} be other generators such that

P_{1} = (f_{1}, …, f_{r}, g_{1}, …, g_{s})D. Then P = P_{1}R = (f_{1}, …, f_{r}, g_{1}, …, g_{s})R. For each (i) := (i_{1}, …, i_{n}) ∈ ℕ^{n} and j, k with 1 ≤ j ≤ r, 1 ≤ k ≤ s, let a_{j,(i)}, b_{k,(i)} denote the coefficients in K[[y]] of the f_{j}, g_{k}, so that

f_{j} = ∑_{(i)∈ℕ^{n}} a_{j,(i)}z_{1}^{i_{1}} … z_{n}^{i_{n}} , g_{k} = ∑_{(i)∈ℕ^{n}} b_{k,(i)}z_{1}^{i_{1}} … z_{n}^{i_{n}} ∈ K[[y]] [[Z]].

Define

Δ := \{ \begin{matrix}\{a_{j,(i)}, b_{k,(i)}\} ⊆ K[[y]], & for r > 0 \\ ∅, & for r = 0. \end{matrix}

A key observation here is that in either case the set Δ is countable.

To continue the proof, we consider S := Q(V (Δ)) ∩ K[[y]], a discrete valuation domain, and its field of quotients L := Q(V (Δ)). Since Δ is a countable set, the field K((y)) is (still) of uncountable transcendence degree over L. Let γ_{2}, …, γ_{n−r} be elements of K[[y]] that are algebraically independent over L. We define

T := L(γ_{2}, …, γ_{n−r}) ∩ K[[y]] and E := Q(T) = L(γ_{2}, …, γ_{n−r}).

The diagram below shows the prime ideals P and P_{1} and the containments between the relevant rings.

R = K[[y, Z]] P = ({f_{j}, g_{k}})R D := K[[y, Z_{1}]] [Z_{2}]_{(Z)} P_{1} = ({f_{j}, g_{k}})D  Q(K[[y]]) = K[[y]] [1/y] = K((y)) K[[y]]  E := Q(T) = L(γ_{2}, …, γ_{n−r}) T := L(γ_{2}, …, γ_{n−r}) ∩ K[[y]] S := Q(V (Δ)) ∩ K[[y]] L := Q(S) = Q(V (Δ)) V  Q(V) K[y]

Let P_{2} := P ∩ S[[Z_{1}]] [Z_{2}]_{(Z)}. Since f_{1}, …, f_{r}, g_{1}, …, g_{s} ∈ S[[Z_{1}]] [Z_{2}]_{(Z)}, we have P_{2}R = P. Since P ⊆ (x_{1}, …, x_{n})R = (Z)R, there is a prime ideal P̃ in L[[Z]] that is minimal over P_{2}L[[Z]]. Since L[[Z]] is flat over S[[Z]], P̃ ∩S[[Z]] = P_{2}S[[Z]]. Note that L[[X]] = L[[Z]] is the (f_{1}, …, f_{r})-adic (and the (Z_{2})-adic) completion of L[[Z_{1}]][Z_{2}]_{(Z)}. In particular,

L[[Z_{1}]] [Z_{2}]/(f_{1}, …, f_{r}) = L[[Z_{1}]] [[Z_{2}]]/(f_{1}, …, f_{r})

and this also holds with the field L replaced by its extension field E.

Since L[[Z]]/P̃ is a homomorphic image of L[[Z]]/(f_{1}, …, f_{r}), it follows that L[[Z]]/P̃ is integral (and finite) over L[[Z_{1}]]. This yields the commutative diagram:

(2.3.0)  \begin{matrix}E[[Z_{1}]] & ⟶ & E[[Z_{1}]] [[Z_{2}]]/P̃E[[Z]] \\ ↑ & ↑ \\ L[[Z_{1}]] & ⟶ & L[[Z_{1}]] [[Z_{2}]]/P̃ \end{matrix}

GENERIC FIBER RINGS 11

with injective integral (finite) horizontal maps. Recall that E is the subfield of K((y)) obtained by adjoining γ_{2}, …, γ_{n−r} to the field L. Thus the vertical maps of (2.3.0) are faithfully flat.

Let q := (z_{2} − γ_{2}z_{1}, …, z_{n−r} − γ_{n−r}z_{1})E[[Z_{1}]] ∈ Spec(E[[Z_{1}]]) and let W̃ be a minimal prime of the ideal (P̃, q)E[[Z]]. Since

f_{1}, …, f_{r}, z_{2} − γ_{2}z_{1}, …, z_{n−r} − γ_{n−r}z_{1}

is a regular sequence in T [[Z]] the prime ideal W := W̃∩T [[Z]] has height n−1. Let Q̃ be a minimal prime of W̃K((y))[[Z]] and let Q := Q̃ ∩ R. Then W = Q ∩ T [[Z]], P ⊂ Q ⊂ ZR = XR, and pictorially we have:

(W̃) ⊆ Q̃ ⊂ K((y))[[Z]] R := K[[y, Z]] P = ({f_{j}, g_{k}})R ⊆ Q ⊂ R D := K[[y, Z_{1}]] [Z_{2}]_{(Z)} P_{1} = ({f_{j}, g_{k}})D ⊂ D  (P̃, q) ⊆ W̃ ⊂ E[[Z]] (P_{2}) ⊆ P̃ ⊂ L[[Z]] P_{2} = ({f_{j}, g_{k}}) ⊂ S[[Z_{1}]] [Z_{2}]_{(Z)} W ⊂ T [[Z]] S[[Z]]  L[[Z_{1}]] [Z_{2}]_{(Z)} q ⊂ E[[Z_{1}]] L[[Z_{1}]]

Notice that q is a prime ideal of height n − r − 1. Also, since K((y))[[Z]] is flat over K[[y, Z]] = R, we have ht Q = n − 1 and dim(R/Q) = 2. We clearly have P_{2} ⊆ W ∩ S[[Z_{1}]] [Z_{2}]_{(Z)}.

2.3.1 Claim. q ∩ L[[Z_{1}]] = (0).

To show this we argue as in [7]: Suppose that

h = ∑_{m∈ℕ} H_{m} ∈ q ∩ L[[z_{1}, …, z_{n−r}]],

where H_{m} ∈ L[z_{1}, …, z_{n−r}] is a homogeneous polynomial of degree m:

H_{m} = ∑_{|(i)|=m} c_{(i)}z_{1}^{i_{1}} … z_{n−r}^{i_{n−r}},

12 HEINZER, ROTTHAUS AND WIEGAND

where (i) := (i_{1}, …, i_{n−r}) ∈ ℕ^{n−r}, |(i)| := i_{1} + ⋯ + i_{n−r} and c_{(i)} ∈ L. Consider the E-algebra homomorphism π : E[[Z_{1}]] → E[[z_{1}]] defined by π(z_{1}) = z_{1} and π(z_{i}) = γ_{i}z_{1} for 2 ≤ i ≤ n − r. Then ker π = q, and for each m ∈ ℕ:

π(H_{m}) = π( ∑_{|(i)|=m} c_{(i)}z_{1}^{i_{1}} … z_{n−r}^{i_{n−r}}) = ∑_{|(i)|=m} c_{(i)}γ_{2}^{i_{2}} … γ_{n−r}^{i_{n−r}}z_{1}^{m}

and

π(h) = ∑_{m∈ℕ} π(H_{m}) = ∑_{m∈ℕ} ∑_{|(i)|=m} c_{(i)}γ_{2}^{i_{2}} … γ_{n−r}^{i_{n−r}}z_{1}^{m}.

Since h ∈ q, π(h) = 0. Since π(h) is a power series in E[[z_{1}]], each of its coefficients is zero, that is, for each m ∈ ℕ,

∑_{|(i)|=m} c_{(i)}γ_{2}^{i_{2}} … γ_{n−r}^{i_{n−r}} = 0.

Since the γ_{i} are algebraically independent over L, each c_{(i)} = 0. Therefore h = 0, and so q ∩ L[[Z_{1}]] = (0). This proves Claim 2.3.1.

Using the commutativity of the displayed diagram (2.3.0) and that the horizonal maps of this diagram are integral extensions, we deduce that (W̃∩E[[Z_{1}]]) = q, and q∩L[[Z_{1}]] = (0) implies W̃∩L[[Z_{1}]] = (0). We conclude that Q∩S[[Z]] = P ∩S[[Z]] and therefore Q ∩ R_{0} = P ∩ R_{0}. □

We record the following corollary. 2.4 Corollary. Let K be a field and let R = K[[y, X]], where X = {x_{1}, …, x_{n}} and y are independent variables over K. Assume P ∈ Spec R is such that:

(i) P ⊆ (x_{1}, …, x_{n})R and (ii) dim(R/P) > 2.

Then there is a prime ideal Q ∈ Spec R so that (1) P ⊂ Q ⊂ (x_{1}, …, x_{n})R,

(2) dim(R/Q) = 2, and

(3) P ∩ K[y]_{(y)}[[X]] = Q ∩ K[y]_{(y)}[[X]]. In particular, P ∩ K[[x_{1}, …, x_{n}]] = Q ∩ K[[x_{1}, …, x_{n}]]. Proof. With notation as in Theorem 2.3, let V = K[y]_{(y)}.

3. Weierstrass implications for the ring B = K[[X]] [Y]_{(X,Y)}.

As before K denotes a field, n and m are positive integers, and X = {x_{1}, …, x_{n}} and Y = {y_{1}, …, y_{m}} denote sets of variables over K. Let B := K[[X]] [Y]_{(X,Y)} = K[[x_{1}, …, x_{n}]] [y_{1}, …, y_{m}]_{(x_{1},...,x_{n},y_{1},...,y_{m})}. The completion of B is B̂ = K[[X, Y]].

3.1 Theorem. With the notation as above, every ideal Q of B̂ = K[[X, Y]] maximal with the property that Q ∩ B = (0) is a prime ideal of height n + m − 2.

Proof. Suppose first that Q is such an ideal. Then clearly Q is prime. Matsumura shows in [7, Theorem 3] that the dimension of the generic formal fiber of B is at most n + m − 2. Therefore ht Q ≤ n + m − 2.

Now suppose P ∈ Spec B̂ is an arbitrary prime ideal of height r < n + m − 2 with P ∩ B = (0). We construct a prime Q ∈ Spec B̂ with P ⊂ Q, Q ∩ B = (0), and ht Q = n + m − 2. This will show that all prime ideals maximal in the generic fiber have height n + m − 2.

For the construction of Q we consider first the case where P ⊈ XB̂. Then there exists a prime element f ∈ P that contains a term θ := y_{1}^{i_{1}} ⋯ y_{m}^{i_{m}}, where the i_{j}’s are nonnegative integers and at least one of the i_{j} is positive. Notice that m ≥ 2 for otherwise with y = y_{1} we have f ∈ P contains a term y^{i}. By Weierstrass it follows that f = gϵ, where g ∈ K[[X]] [y] is a nonzero monic polynomial in y and ϵ is a unit of B̂. But g ∈ P and g ∈ B implies P ∩ B ≠ (0), a contradiction to our assumption that P ∩ B = (0).

For convenience we now assume that the last exponent i_{m} appearing in θ above is positive. We apply a change of variables: y_{m} → t_{m} := y_{m} and, for 1 ≤ ℓ < m, let y_{ℓ} → t_{ℓ} := y_{ℓ} + t_{m}^{e_{ℓ}}, where the e_{ℓ} are chosen so that f, expressed in the variables t_{1}, …, t_{m}, contains a term t_{m}^{q}, for some positive integer q. This change of variables induces an automorphism of B. By Weierstrass f = g_{1}h, where h is a unit in B̂ and g_{1} ∈ K[[X, t_{1}, …, t_{m−1}]] [t_{m}] is monic in t_{m}. Set P_{1} = P ∩K[[X, t_{1}, …, t_{m−1}]]. If P_{1} ⊆ XK[[X, t_{1}, …, t_{m−1}]], we stop the procedure and take s = m − 1 in what follows. If P_{1} ⊈ XK[[X, t_{1}, …, t_{m−1}]], then there exists a prime element f̃ ∈ P_{1} that contains a term t_{1}^{j_{1}} ⋯ t_{m−1}^{j_{m−1}}, where the j_{k}’s are nonnegative integers and at least one of the j_{k} is positive. We then repeat the procedure using the prime ideal P_{1}. That is, we replace t_{1}, … t_{m−1} with a change of variables so that a prime element of P_{1} contains a term monic in some one of the new variables. After a suitable finite iteration of changes of variables, we obtain an automorphism of B̂ that restricts to an automorphism of B and maps y_{1}, …, y_{m} ↦ z_{1}, …, z_{m}. Moreover, there exist a positive integer s ≤ m − 1 and elements g_{1}, … g_{m−s} ∈ P such that

g_{1} ∈ K[[X, z_{1}, …, z_{m−1}]] [z_{m}]  is monic in z_{m}
g_{2} ∈ K[[X, z_{1}, …, z_{m−2}]] [z_{m−1}] is monic in z_{m−1}, etc
⋮
g_{m−s} ∈ K[[X, z_{1}, …, z_{s}]] [z_{s+1}]  is monic in z_{s+1},

and such that, for R_{s} := K[[X, z_{1}, …, z_{s}]] and P_{s} := P ∩ R_{s}, we have P_{s} ⊆ XR_{s}.

As in the proof of Proposition 2.2 we replace the regular sequence g_{1}, ….g_{m−s} by a regular sequence f_{1}, …, f_{m−s} so that:

f_{1} ∈ R_{s}[z_{s+1}, …, z_{m}]  is monic in z_{m}
f_{2} ∈ R_{s}[z_{s+1}, …, z_{m−1}] is monic in z_{m−1}, etc
⋮
f_{m−s} ∈ R_{s}[z_{s+1}]  is monic in z_{s+1}.

and (g_{1}, ….g_{m−s})B̂ = (f_{1}, …, f_{m−s})B̂.

Let G := K[[X, z_{1}, …, z_{s}]] [z_{s+1}, …, z_{m}] = R_{s}[z_{s+1}, …, z_{m}]. By Proposition 2.2, P is extended from G. Let q := P ∩ G and extend f_{1}, …, f_{m−s} to a generating system of q, say, q = (f_{1}, …, f_{m−s}, h_{1}, …, h_{t})G. For integers k, ℓ with 1 ≤ k ≤ m − s and 1 ≤ ℓ ≤ t, express the f_{k} and h_{ℓ} in G as power series in B̂ = K[[z_{1}]][[z_{2}, …, z_{m}]] [[X]] with coefficients in K[[z_{1}]]:

f_{k} = ∑ a_{k(i)(j)}z_{2}^{i_{2}} … z_{m}^{i_{m}}x_{1}^{j_{1}} … x_{n}^{j_{n}} and h_{ℓ} = ∑ b_{ℓ(i)(j)}z_{2}^{i_{2}} … z_{m}^{i_{m}}x_{1}^{j_{1}} … x_{n}^{j_{n}} ,

where a_{k(i)(j)}, b_{ℓ(i)(j)} ∈ K[[z_{1}]], (i) = (i_{2}, …, i_{m}) and (j) = (j_{1}, …, j_{n}). The set Δ = {a_{k(i)(j)}, b_{ℓ(i)(j)}} is countable. We define V := K(z_{1}, Δ) ∩ K[[z_{1}]]. Then V is a discrete valuation domain with completion K[[z_{1}]] and K((z_{1})) has uncountable transcendence degree over Q(V). Let V_{s} := V [[X, z_{2}, …, z_{s}]] ⊆ R_{s}. Notice that R_{s} = V̂_{s}, the completion of V_{s}. Also f_{1}, …, f_{m−s} ∈ V_{s}[z_{s+1}, …, z_{m}] ⊆ G and (f_{1}, …, f_{m−s})G ∩ R_{s} = (0). Furthermore the extension

V_{s} := V [[X, z_{2}, …, z_{s}]] ↪ V_{s}[z_{s+1}, …, z_{m}]/(f_{1}, …, f_{m−s})

is finite. Set P_{0} := P ∩ V_{s}. Then P_{0} ⊆ XR_{s} ∩ V_{s} = XV_{s}.

Consider the commutative diagram:

(3.1.1)  \begin{matrix}R_{s} := K[[X, z_{1}, …, z_{s}]] & ⟶ & R_{s}[[z_{s+1}, …, z_{m}]]/(f_{1}, …, f_{m−s}) \\ ↑ & ↑ \\ V_{s} := V [[X, z_{2}, …, z_{s}]] & ⟶ & V_{s}[z_{s+1}, …, z_{m}]/(f_{1}, …, f_{m−s}) . \end{matrix}

The horizontal maps are injective and finite and the vertical maps are completions.

The prime ideal q̄ := P R_{s}[[z_{s+1}, …, z_{m}]]/(f_{1}, …, f_{m−s}) lies over P_{s} in R_{s}. By assumption P_{s} ⊆ (X)R_{s} and by Theorem 2.3 there is a prime ideal Q_{s} of R_{s} such that P_{s} ⊆ Q_{s} ⊆ (X)R_{s}, Q_{s} ∩ V_{s} = P_{s} ∩ V_{s} = P_{0}, and dim(R_{s}/Q_{s}) = 2. There is a prime ideal Q̄ in R_{s}[[z_{s+1}, …, z_{m}]]/(f_{1}, …, f_{m−s}) lying over Q_{s} with q̄ ⊆ Q̄ by the “going-up theorem” [8, Theorem 9.4]. Let Q be the preimage in B̂ = K[[X, z_{1}, …, z_{m}]] of Q̄. We show the rings and ideals of Theorem 3.1 below.

B̂ = K[[X, Y]] = K[[X, z_{1}, …, z_{m}]] = R_{s}[[z_{s+1}, …, z_{m}]] (q, Q_{s})B̂ ⊆ Q P ⊈ XB̂ G := R_{s}[z_{s+1}, …, z_{m}] q := P ∩ G q = ({f_{i}, h_{j}})G f_{i} ∉ R_{s} := K[[X, z_{1}, …, z_{s}]] P_{s} ⊆ Q_{s} ⊂ R_{s} P_{s} := P ∩ R_{s} ⊆ XR_{s}  V̂ = K[[z_{1}]] V_{s} := V [[X, z_{2}, …, z_{s}]] P_{0} := P ∩ V_{s}  V := K(z_{1}, Δ) ∩ K[[z_{1}]]

Then Q has height n + s − 2 + m − s = n + m − 2. Moreover, from diagram (3.1.1), it follows that Q and P have the same contraction to V_{s}[z_{s+1}, …, z_{m}]. This implies that Q ∩ B = (0) and completes the proof in the case where P ⊈ XB̂.

In the case where P ⊆ XB̂, let h_{1}, …, h_{t} ∈ B̂ be a finite set of generators of P, and as above, let b_{ℓ(i)(j)} ∈ K[[z_{1}]] be the coefficients of the h_{ℓ}’s. Consider the countable set Δ = {b_{ℓ(i)(j)}} and the valuation domain V := K(z_{1}, Δ) ∩ K[[z_{1}]]. Set P_{0} := P ∩ V [[X, z_{2}, …, z_{m}]]. By Theorem 2.3, there exists a prime ideal Q of B̂ =

16 HEINZER, ROTTHAUS AND WIEGAND

K[[X, z_{1}, …, z_{m}]] of height n+m−2 such that P ⊂ Q and Q∩V [[X, z_{2}, …, z_{m}]] = P ∩ V [[X, z_{2}, …, z_{m}]] = P_{0}. Therefore Q ∩ B = (0). This completes the proof of Theorem 3.1. □

4. Weierstrass implications for the ring C = K[Y]_{(Y)}[[X]].

As before K denotes a field, n and m are positive integers, and X = {x_{1}, …, x_{n}} and Y = {y_{1}, …, y_{m}} denote sets of variables over K. Consider the ring C = K[y_{1}, …, y_{m}]_{(y_{1},...,y_{m})}[[x_{1}, …, x_{n}]] = K[Y]_{(Y)}[[X]]. Then the completion of C is Ĉ = K[[Y, X]]. 4.1 Theorem. With notation as above, let Q ∈ Spec Ĉ be maximal with the property that Q ∩ C = (0). Then ht Q = n + m − 2. Proof. Let B = K[[X]] [Y]_{(X,Y)} ⊂ C. If P ∈ Spec Ĉ = Spec B̂ and P ∩ C = (0), then P ∩ B = (0), so ht P ≤ n + m − 2 by Theorem 3.1. Consider a nonzero prime P ∈ Spec Ĉ with P ∩ C = (0) and ht P = r < n + m − 2. If P ⊆ XĈ then Theorem 2.3 implies the existence of Q ∈ Spec Ĉ with ht Q = n + m − 2 such that P ⊂ Q and Q ∩ C = (0).

Assume that P is not contained in XĈ and consider the ideal J := (P, X)Ĉ. Since C is complete in the XC-adic topology, [9, Lemma 2] implies that if J is primary for the maximal ideal of Ĉ, then P is extended from C. Since we are assuming P ∩ C = (0), J is not primary for the maximal ideal of Ĉ and we have ht J = n + s < n + m, where 0 < s < m. Let W ∈ Spec Ĉ be a minimal prime of J such that ht W = n + s. Let W_{0} = W ∩ K[[Y]]. Then W = (W_{0}, X)Ĉ and W_{0} is a prime ideal of K[[Y]] with ht W_{0} = s. By Proposition 2.2 applied to K[[Y]] and the prime ideal W_{0} ∈ Spec K[[Y]], there exists a change of variables Y ↦ Z with y_{1} ↦ z_{1}, …, y_{m} ↦ z_{m} and elements f_{1}, …, f_{s} ∈ W_{0} so that with Z_{1} = {z_{1}, …, z_{m−s}}, we have

f_{1} ∈ K[[Z_{1}]] [z_{m−s+1}, …, z_{m}]  is monic in z_{m}
f_{2} ∈ K[[Z_{1}]] [z_{m−s+1}, …, z_{m−1}] is monic in z_{m−1}, etc
⋮
f_{s} ∈ K[[Z_{1}]] [z_{m−s+1}]  is monic in z_{m−s+1}.

Now z_{1}, …, z_{m−s}, f_{1}, …, f_{s} is a regular sequence in K[[Z]] = K[[Y]]. Let T = {t_{m−s+1}, …, t_{m}} be a set of additional variables and consider the map:

φ : K[[Z_{1}, T]] ⟶ K[[z_{1}, …, z_{m}]]

defined by z_{i} ↦ z_{i} for all 1 ≤ i ≤ m − s and t_{m−i+1} ↦ f_{i} for all 1 ≤ i ≤ s. The embedding φ is finite (and free) and so is the extension to power series rings in X:

ρ : K[[Z_{1}, T]] [[X]] ⟶ K[[z_{1}, …, z_{m}]] [[X]] = Ĉ.

Since W ∈ Spec Ĉ is of height n+s, so is its contraction ρ^{−1}(W) ∈ Spec K[[Z_{1}, T, X]]. Moreover ρ^{−1}(W) contains (T, X)K[[Z_{1}, T, X]], a prime ideal of height n+s. Therefore ρ^{−1}(W) = (T, X)K[[Z_{1}, T, X]]. By construction, P ⊆ W which yields that ρ^{−1}(P) ⊆ (T, X)K[[Z_{1}, T, X]].

To complete the proof we construct a suitable base ring related to C. Consider the expressions for the f_{i}’s as power series in z_{2}, …, z_{m} with coefficients in K[[z_{1}]]:

f_{j} = ∑ a_{j(i)}z_{2}^{i_{2}} … z_{m}^{i_{m}},

where (i) := (i_{2}, …, i_{m}), 1 ≤ j ≤ s, a_{j(i)} ∈ K[[z_{1}]]. Also consider a finite generating system g_{1}, …, g_{q} for P and expressions for the g_{k}, where 1 ≤ k ≤ q, as power series in z_{2}, …, z_{m}, x_{1}, …, x_{n} with coefficients in K[[z_{1}]]:

g_{k} = ∑ b_{k(i)(ℓ)}z_{2}^{i_{2}} … z_{m}^{i_{m}}x_{1}^{ℓ_{1}} … x_{n}^{ℓ_{n}},

where (i) := (i_{2}, …, i_{m}), (ℓ) := (ℓ_{1}, …, ℓ_{n}), and b_{k(i)(ℓ)} ∈ K[[z_{1}]]. We take the subset Δ = {a_{j(i)}, b_{k(i)(ℓ)}} of K[[z_{1}]] and consider the discrete valuation domain:

V := K(z_{1}, Δ) ∩ K[[z_{1}]].

Since V is countably generated over K(z_{1}), the field K((z_{1})) has uncountable transcendence degree over Q(V) = K(z_{1}, Δ). Moreover, by construction the ideal P is extended from V [[z_{2}, …, z_{m}]] [[X]]. Consider the embedding:

ψ : V [[z_{2}, …, z_{m−s}, T]] ⟶ V [[z_{2}, …, z_{m}]],

which is the restriction of φ above, so that z_{i} ↦ z_{i} for all 2 ≤ i ≤ m − s and t_{m−i+1} ↦ f_{i} for all i with 1 ≤ i ≤ s.

Let σ be the extension of ψ to the power series rings:

σ : V [[z_{2}, …, z_{m−s}, T]] [[X]] ⟶ V [[z_{2}, …, z_{m}]] [[X]]

with σ(x_{i}) = x_{i} for all i with 1 ≤ i ≤ n.

Notice that ρ defined above is the completion σ̂ of the map σ, that is, the extension of σ to the completions. Consider the commutative diagram:

(4.1.0)  \begin{matrix}K[[Z_{1}, T]] [[X]] & ⟶^{σ̂=ρ} & K[[Z]] [[X]] = Ĉ \\ ↑ & & ↑ \\ V [[z_{2}, …, z_{m−s}, T]] [[X]] & ⟶^{σ} & V [[z_{2}, …, z_{m}]] [[X]] \end{matrix}

where σ̂ = ρ is a finite map.

Recall that ρ^{−1}(W) = (T, X)K[[Z_{1}, T, X]], and so ρ^{−1}(P) ⊆ (T, X)K[[Z_{1}, T, X]] by Diagram 4.1.0. By Theorem 2.3, there exists a prime ideal Q_{0} of the ring K[[Z_{1}, T, X]] such that ρ^{−1}(P) ⊆ Q_{0}, ht Q_{0} = n + m − 2, and

Q_{0} ∩ V [[z_{2}, …, z_{m−s}, T]] [[X]] = ρ^{−1}(P) ∩ V [[z_{2}, …, z_{m−s}, T]] [[X]].

By the “going-up theorem” [8, Theorem 9.4], there is a prime ideal Q ∈ Spec Ĉ that lies over Q_{0} and contains P. Moreover, Q also has height n + m − 2. The commutativity of diagram (4.1.0) implies that

P_{1} := P ∩ V [[z_{2}, …, z_{m−s}, T]] [[X]] ⊆ Q_{1} := Q ∩ V [[z_{2}, …, z_{m−s}, T]] [[X]].

Consider the finite homomorphism:

λ : V [[z_{2}, …, z_{m−s}]] [T]_{(Z_{1},T)}[[X]] ⟶ V [[z_{2}, …, z_{m−s}]] [z_{m−s+1}, …, z_{m}]_{(Z)}[[X]]

(determined by t_{i} ↦ f_{i} for 1 ≤ i ≤ m) and the commutative diagram:

V [[z_{2}, …, z_{m−s}]] [[T]] [[X]] ⟶^{σ}  V [[z_{2}, …, z_{m}]] [[X]]
↑ ↑
V [[z_{2}, …, z_{m−s}]] [T]_{(Z_{1},T)}[[X]] ⟶^{λ} V [[z_{2}, …, z_{m−s}]] [z_{m−s+1}, …, z_{m}]_{(Z)}[[X]].

Since Q ∩ V [[z_{2}, …, z_{m−s}, T]] [[X]] = P ∩ V [[z_{2}, …, z_{m−s}, T]] [[X]] and since λ is a finite map we conclude that

Q_{1} ∩ V [[z_{2}, …, z_{m−s}]] [z_{m−s+1}, …, z_{m}]_{(Z)}[[X]]
= P_{1} ∩ V [[z_{2}, …, z_{m}]] [z_{m−s+1}, …, z_{m}]_{(Z)}[[X]].

Since C ⊆ V [[z_{2}, …, z_{m−s}]] [z_{m−s+1}, …, z_{m}]_{(Z)}[[X]], we obtain that Q ∩ C = P ∩ C = (0). This completes the proof of Theorem 4.1. □

4.2 Remark. With B and C as in Sections 3 and 4, we have

B = K[[X]] [Y]_{(X,Y)} ↪ K[Y]_{(Y)}[[X]] = C and B̂ = K[[X, Y]] = Ĉ.

Thus for P ∈ Spec K[[X, Y]], if P ∩ C = (0), then P ∩ B = (0). By Theorems 3.1 and 4.1, each prime of K[[X, Y]] maximal in the generic formal fiber of B or C has height n + m − 2. Therefore each P ∈ Spec K[[X, Y]] maximal with respect to P ∩C = (0) is also maximal with respect to P ∩B = (0). However, if n+m ≥ 3, the generic fiber of B ↪ C is nonzero [4], so there exist primes of K[[X, Y]] maximal in the generic formal fiber of B that are not in the generic formal fiber of C.

5. Subrings of the power series ring K[[z, t]].

In this section we establish properties of certain subrings of the power series ring K[[z, t]] that will be useful in considering the generic formal fiber of localized polynomial rings over the field K.

5.1. Notation. Let K be a field and let z and t be independent variables over K. Consider countably many power series:

α_{i}(z) = ∑_{j=0}^{∞} a_{ij}z^{j} ∈ K[[z]]

with coefficients a_{ik} ∈ K. Let s be a positive integer and let ω_{1}, …, ω_{s} ∈ K[[z, t]] be power series in z and t, say:

ω_{i} = ∑_{j=0}^{∞} β_{ij}t^{j}, where β_{ij}(z) = ∑_{k=0}^{∞} b_{ijk}z^{k} ∈ K[[z]] and b_{ijk} ∈ K,

for each i with 1 ≤ i ≤ s. Consider the subfield K(z, {α_{i}}, {β_{ij}}) of K((z)) and the discrete rank-one valuation domain

V := K(z, {α_{i}}, {β_{ij}}) ∩ K[[z]].

The completion of V is V̂ = K[[z]]. Assume that ω_{1}, …, ω_{r} are algebraically independent over Q(V)(t) and that the elements ω_{r+1}, …, ω_{s} are algebraic over the field Q(V)(t, {ω_{i}}_{i=1}^{r}). Notice that the set {α_{i}} ∪ {β_{ij}} is countable, and that also the set of coefficients of the α_{i} and β_{ij}

Δ := {a_{ij}} ∪ {b_{ijk}}

20 HEINZER, ROTTHAUS AND WIEGAND

is a countable subset of the field K. Let K_{0} denote the prime subfield of K and let F denote the algebraic closure in K of the field K_{0}(Δ). The field F is countable and the power series α_{i}(z) and β_{ij}(z) are in F [[z]]. Consider the subfield F (z, {α_{i}}, {β_{ij}}) of F ((z)) and the discrete rank-one valuation domain

V_{0} := F (z, {α_{i}}, {β_{ij}}) ∩ F [[z]].

The completion of V_{0} is V̂_{0} = F [[z]]. Since Q(V_{0})(t) ⊆ Q(V)(t), the elements ω_{1}, …, ω_{r} are algebraically independent over the field Q(V_{0})(t).

Consider the subfield E_{0} := Q(V_{0})(t, ω_{1}, …, ω_{r}) of Q(V_{0}[[t]]) and the subfield E := Q(V)(t, ω_{1}, …, ω_{r}) of Q(V [[t]]). A result of Valabrega [11] implies that the integral domains:

(5.1.1)  D_{0} := E_{0} ∩ V_{0}[[t]]  and  D := E ∩ V [[t]]

are two-dimensional regular local rings with completions D̂_{0} = F [[z, t]] and D̂ = K[[z, t]], respectively. Moreover, Q(D_{0}) = E_{0} is a countable field.

5.2 Proposition. Let D_{0} be as defined in (5.1). Then there exists a power series γ ∈ zF [[z]] such that the prime ideal (t − γ)F [[z, t]] ∩ D_{0} = (0), i.e., (t − γ)F [[z, t]] is in the generic formal fiber of D_{0}.

Proof. Since D_{0} is countable there are only countably many prime ideals in D_{0} and since D_{0} is Noetherian there are only countably many prime ideals in D̂_{0} = F [[z, t]] that lie over a nonzero prime of D_{0}. There are uncountably many primes in F [[z, t]], which are generated by elements of the form t − σ for some σ ∈ zF [[z]]. Thus there must exist an element γ ∈ zF [[z]] with (t − γ)F [[z, t]] ∩ D_{0} = (0). □

For ω_{i} = ω_{i}(t) = ∑_{j=0}^{∞} β_{ij}t^{j} as in (5.1) and γ an element of zK[[z]], let ω_{i}(γ) denote the following power series in K[[z]]:

ω_{i}(γ) := ∑_{j=0}^{∞} β_{ij}γ^{j} ∈ K[[z]].

5.3 Proposition. Let D be as defined in (5.1.1). For an element γ ∈ zK[[z]] the following conditions are equivalent:

(i) (t − γ)K[[z, t]] ∩ D = (0).

(ii) The elements γ, ω_{1}(γ), …, ω_{r}(γ) are algebraically independent over Q(V).

GENERIC FIBER RINGS 21

Proof. (i) ⇒ (ii): Assume by way of contradiction that the set {γ, ω_{1}(γ), …, ω_{r}(γ)} is algebraically dependent over Q(V) and let d_{(k)} ∈ V be finitely many elements such that

∑_{(k)} d_{(k)}ω_{1}(γ)^{k_{1}} … ω_{r}(γ)^{k_{r}}γ^{k_{r+1}} = 0

is a nontrivial equation of algebraic dependence for γ, ω_{1}(γ), …, ω_{r}(γ), where each (k) = (k_{1} …, k_{r}, k_{r+1}) is an (r + 1)-tuple of nonnegative integers. It follows that

∑_{(k)} d_{(k)}ω_{1}^{k_{1}} … ω_{r}^{k_{r}}t^{k_{r+1}} ∈ (t − γ)K[[z, t]] ∩ D = (0).

Since ω_{1}, …, ω_{r} are algebraically independent over Q(V)(t), we have d_{(k)} = 0 for all (k), a contradiction. This completes the proof that (i) ⇒ (ii).

(ii) ⇒ (i): If (t − γ)K[[z, t]] ∩ D ≠ (0), then there exists a nonzero element

τ = ∑_{(k)} d_{(k)}ω_{1}^{k_{1}} … ω_{r}^{k_{r}}t^{k_{r+1}} ∈ (t − γ)K[[z, t]] ∩ V [t, ω_{1}, …, ω_{r}].

But this implies that

τ(γ) = ∑_{(k)} d_{(k)}ω_{1}(γ)^{k_{1}} … ω_{r}(γ)^{k_{r}}γ^{k_{r+1}} = 0.

Since γ, ω_{1}(γ), …, ω_{r}(γ) are algebraically independent over Q(V), it follows that all the coefficients d_{(k)} = 0, a contradiction to the assumption that τ is nonzero. □

Let γ ∈ zF [[z]] be as in Proposition 5.2 with (t − γ)F [[z, t]] ∩ D_{0} = (0). Then:

5.4 Proposition. With notation as above, we also have (t − γ)K[[z, t]] ∩ D = (0), that is, (t − γ)K[[z, t]] is in the generic formal fiber of D.

Proof. Let {t_{i}}_{i∈I} be a transcendence basis of K over F and let L := F ({t_{i}}_{i∈I}). Then K is algebraic over L. Let {α_{i}}, {β_{ij}} ⊂ F [[z]] be as in (5.1) and define

V_{1} = L(z, {α_{i}}, {β_{ij}}) ∩ L[[z]] and D_{1} = Q(V_{1})(t, ω_{1}, …, ω_{r}) ∩ L[[z, t]].

Then V_{1} is a discrete rank-one valuation domain with completion L[[z]] and D_{1} is a two-dimensional regular local domain with completion D̂_{1} = L[[z, t]]. Note that Q(V) and Q(D) are algebraic over Q(V_{1}) and Q(D_{1}), respectively. Since (t−γ)K[[z, t]]∩L[[z, t]] = (t−γ)L[[z, t]], it suffices to prove that (t−γ)L[[z, t]]∩D_{1} = (0). By Proposition 5.3, it suffices to show that γ, ω_{1}(γ), …, ω_{r}(γ) are algebraically independent over Q(V_{1}). The commutative diagram

F [[z]]

{t_{i}}algebraically ind.

⟶ L[[z]]

↑  ↑
Q(V_{0})

transcendence basis {t_{i}}

⟶ Q(V_{1})

implies that the set {γ, ω_{1}(γ), …, ω_{r}(γ)} ∪ {t_{i}} is algebraically independent over Q(V_{0}). Therefore {γ, ω_{1}(γ), …, ω_{r}(γ)} is algebraically independent over Q(V_{1}), which completes the proof of Proposition 5.4. □ 5.5 Remark. We remark that with ω_{r+1}, …, ω_{s} algebraic over Q(V)(ω_{1}, …, ω_{r}) as in (5.1), if we define

D̃ := Q(V)(t, ω_{1}, …, ω_{s}) ∩ V [[t]],

then again by Valabrega [11], D̃ is a two-dimensional regular local domain with completion K[[z, t]]. Moreover, Q(D̃) is algebraic over Q(D) and (t − γ)K[[z, t]] ∩ D = (0) implies that (t − γ)K[[z, t]] ∩ D̃ = (0).

6. Weierstrass implications for the localized polynomial ring A = K[X]_{(X)}. Let n be a positive integer, let X = {x_{1}, …, x_{n}} be a set of n variables over a field

K, and let A := K[x_{1}, …, x_{n}]_{(x_{1},...,x_{n})} = K[X]_{(X)} denote the localized polynomial ring in these n variables over K. Then the completion of A is Â = K[[X]]. 6.1 Theorem. For the localized polynomial ring A = K[X]_{(X)} defined above, if Q is an ideal of Â maximal with respect to Q ∩ A = (0), then Q is a prime ideal of height n − 1. Proof. Again it is clear that Q as described in the statement is a prime ideal. Also the assertion holds for n = 1. Thus we assume n ≥ 2. By Proposition 5.4, there exists a nonzero prime p in K[[x_{1}, x_{2}]] such that p ∩ K[x_{1}, x_{2}]_{(x_{1},x_{2})} = (0). It follows that pÂ ∩ A = (0). Thus the generic formal fiber of A is nonzero.

Let P ∈ Spec Â be a nonzero prime ideal with P ∩A = (0) and ht P = r < n−1. We construct Q ∈ Spec Â of height n − 1 with P ⊆ Q and Q ∩ A = (0). By Proposition 2.2, there exists a change of variables x_{1} ↦ z_{1}, …, x_{n} ↦ z_{n} and polynomials

f_{1} ∈ K[[z_{1}, …, z_{n−r}]] [z_{n−r+1}, …, z_{n}]  monic in z_{n}
f_{2} ∈ K[[z_{1}, …, z_{n−r}]] [z_{n−r+1}, …, z_{n−1}] monic in z_{n−1}, etc
⋮
f_{r} ∈ K[[z_{1}, …, z_{n−r}]] [z_{n−r+1}]  monic in z_{n−r+1},

GENERIC FIBER RINGS 23

so that P is a minimal prime of (f_{1}, …, f_{r})Â and P is extended from

R := K[[z_{1}, …, z_{n−r}]] [z_{n−r+1}, …, z_{n}].

Let P_{0} := P ∩ R and extend f_{1}, …, f_{r} to a system of generators of P_{0}, say:

P_{0} = (f_{1}, …, f_{r}, g_{1}, …, g_{s})R.

Using an argument similar to that in the proof of Theorem 2.3, write

f_{j} = ∑_{(i)∈ℕ^{n−1}} a_{j,(i)}z_{2}^{i_{2}} … z_{n}^{i_{n}} and g_{k} = ∑_{(i)∈ℕ^{n−1}} b_{k,(i)}z_{2}^{i_{2}} … z_{n}^{i_{n}},

where a_{j,(i)}, b_{k,(i)} ∈ K[[z_{1}]]. Let

V_{0} := K(z_{1}, a_{j,(i)}, b_{k,(i)}) ∩ K[[z_{1}]].

Then V_{0} is a discrete rank-one valuation domain with completion K[[z_{1}]], and K((z_{1})) has uncountable transcendence degree over the field of fractions Q(V_{0}) of V_{0}. Let γ_{3}, …, γ_{n−r} ∈ K[[z_{1}]] be algebraically independent over Q(V_{0}) and define

q := (z_{3} − γ_{3}z_{2}, z_{4} − γ_{4}z_{2}, …, z_{n−r} − γ_{n−r}z_{2})K[[z_{1}, …, z_{n−r}]].

We see that q ∩ V_{0}[[z_{2}, …, z_{n−r}]] = (0) by an argument similar to that in [7] and in Claim 2.3.1. Let R_{1} := V_{0}[[z_{2}, …, z_{n−r}]] [z_{n−r+1}, …, z_{n}], let P_{1} := P ∩ R_{1} and consider the commutative diagram:

K[[z_{1}, …, z_{n−r}]]⟶ R/P_{0}
↑  ↑
V_{0}[[z_{2}, …, z_{n−r}]]⟶R_{1}/P_{1}

The horizontal maps are injective finite integral extensions. Let W be a minimal prime of (q, P)Â. Then ht W = n − 2 and q ∩ V_{0}[[z_{2}, …, z_{n−r}]] = (0) implies that W ∩ R_{1} = P_{1}. We have found a prime ideal W ∈ Spec Â such that ht W = n − 2, W ∩ A = (0) and P ⊆ W. Since f_{1}, …, f_{r} ∈ W and since Â = K[[z_{1}, …, z_{n}]] is the (f_{1}, …, f_{r})-adic completion of K[[z_{1}, …, z_{n−r}]] [z_{n−r+1}, …, z_{n}], the prime ideal W is extended from K[[z_{1}, …, z_{n−r}]] [z_{n−r+1}, …, z_{n}].

We claim that W is actually extended from K[[z_{1}, z_{2}]] [z_{3}, …, z_{n}]. To see this let g ∈ W ∩ K[[z_{1}, …, z_{n−r}]] [z_{n−r+1}, …, z_{n}] and write:

g = ∑_{(i)} a_{(i)}z_{n−r+1}^{i_{n−r+1}} … z_{n}^{i_{n}} ∈ K[[z_{1}, …, z_{n−r}]] [z_{n−r+1}, …, z_{n}],

where the sum is over all (i) = (i_{n−r}, …, i_{n}) and a_{(i)} ∈ K[[z_{1}, …, z_{n−r}]]. For all a_{(i)} by Weierstrass we can write

a_{(i)} = (z_{n−r} − γ_{n−r}z_{2})h_{(i)} + q_{(i)},

where h_{(i)} ∈ K[[z_{1}, … z_{n−r}]] and q_{(i)} ∈ K[[z_{1}, …, z_{n−r−1}]]. If n − r > 3, we write

q_{(i)} = (z_{n−r−1} − γ_{n−r−1}z_{2})h′_{(i)} + q′_{(i)},

where h′_{(i)} ∈ K[[z_{1}, … z_{n−r−1}]] and q′_{(i)} ∈ K[[z_{1}, …, z_{n−r−2}]]. In this way we replace a generating set for W in K[[z_{1}, …, z_{n−r}]] [z_{n−r+1}, …, z_{n}] by a generating set for W in K[[z_{1}, z_{2}]] [z_{3}, …, z_{n}].

In particular, we can replace the elements f_{1}, …, f_{r} by elements:

h_{1} ∈ K[[z_{1}, z_{2}]] [z_{3}, …, z_{n}]  monic in z_{n}
h_{2} ∈ K[[z_{1}, z_{2}]] [z_{3}, …, z_{n−1}] monic in z_{n−1}, etc
⋮
h_{r} ∈ K[[z_{1}, z_{2}]] [z_{3} …, z_{n−r+1}] monic in z_{n−r+1}

and set h_{r+1} = z_{3}−γ_{3}z_{2}, …, h_{n−2} = z_{n−r}−γ_{n−r}z_{2}, and then extend to a generating set h_{1}, …, h_{n+s−2} for

W_{0} = W ∩ K[[z_{1}, z_{2}]] [z_{3}, …, z_{n}]

such that W_{0}Â = W. Consider the coefficients in K[[z_{1}]] of the h_{j}:

h_{j} = ∑_{(i)} c_{j(i)}z_{2}^{i_{2}} … z_{n}^{i_{n}}

with c_{j(i)} ∈ K[[z_{1}]]. The set {c_{j(i)}} is countable. Define

V := Q(V_{0})({c_{j(i)}}) ∩ K[[z_{1}]]

Then V is a rank-one discrete valuation domain that is countably generated over K[z_{1}]_{(z_{1})} and W is extended from V [[z_{2}]] [z_{3}, …, z_{n}].

We may also write each h_{i} as a polynomial in z_{3}, …, z_{n} with coefficients in V [[z_{2}]]:

h = ∑ ω_{(i)}z_{3}^{i_{3}} … z_{n}^{i_{n}}

GENERIC FIBER RINGS 25

with ω_{(i)} ∈ V [[z_{2}]] ⊆ K[[z_{1}, z_{2}]]. By the result of Valabraga [11], the integral domain

D := Q(V)(z_{2}, {ω_{(i)}}) ∩ K[[z_{1}, z_{2}]]

is a two-dimensional regular local domain with completion D̂ = K[[z_{1}, z_{2}]]. Let W_{1} := W ∩ D[z_{3}, …, z_{n}]. Then W_{1}Â = W. We have shown in Section 5 that there exists a prime element q ∈ K[[z_{1}, z_{2}]] with qK[[z_{1}, z_{2}]] ∩ D = (0). Consider the finite extension

D ⟶ D[z_{3}, …, z_{n}]/W_{1}.

Let Q ∈ Spec Â be a minimal prime of (q, W)Â. Since ht W = n − 2 and q ∉ W, ht Q = n − 1. Moreover, P ⊆ W implies P ⊆ Q. We claim that

Q ∩ D[z_{3}, …, z_{n}] = W_{1} and therefore Q ∩ A = (0).

To see this consider the commutative diagram:

K[[z_{1}, z_{2}]]⟶ K[[z_{1}, …, z_{n}]]/W
↑  ↑
D ⟶ D[z_{3}, …, z_{n}]/W_{1} ,

which has injective finite horizontal maps. Since qK[[z_{1}, z_{2}]] ∩ D = (0), it follows that Q ∩ D[z_{3}, …, z_{n}] = W_{1}. This completes the proof of Theorem 6.1. □

7. Generic fibers of power series ring extensions.

In this section we apply the Weierstrass machinery from Section 2 to the generic fiber rings of power series extensions.

7.1 Theorem. Let n ≥ 2 be an integer and let y, x_{1}, …, x_{n} be variables over the field K. Let X = {x_{1}, …, x_{n}}. Consider the formal power series ring R_{1} = K[[X]] and the extension R_{1} ↪ R_{1}[[y]] = R. Let U = R_{1} \ (0). For P ∈ Spec R such that P ∩ U = ∅ we have:

(1) If P ⊈ XR, then dim R/P = n and P is maximal in the generic fiber U^{−1}R.

(2) If P ⊆ XR, then there exists Q ∈ Spec R such that P ⊆ Q, dim R/Q = 2

and Q is maximal in the generic fiber U^{−1}R. If n > 2 for each prime ideal Q maximal in the generic fiber U^{−1}R, we have

dim R/Q = \{ \begin{matrix}n & and R_{1} ↪ R/Q is finite, or \\ 2 & and Q ⊂ XR. \end{matrix}

26 HEINZER, ROTTHAUS AND WIEGAND

Proof. Let P ∈ Spec R be such that P ∩ U = ∅ or equivalently P ∩ R_{1} = (0). Then R_{1} embeds in R/P. If dim(R/P) ≤ 1, then the maximal ideal of R_{1} generates an ideal primary for the maximal ideal of R/P. By [8, Theorem 8.4] R/P is finite over R_{1}, and so dim R_{1} = dim(R/P), a contradiction. Thus dim(R/P) ≥ 2.

If P ⊈ XR, then there exists a prime element f ∈ P that contains a term y^{s} for some positive integer s. By Weierstrass, it follows that f = gϵ, where g ∈ K[[X]] [y] is a nonzero monic polynomial in y and ϵ is a unit of R. We have fR = gR ⊆ P is a prime ideal and R_{1} ↪ R/gR is a finite integral extension. Since P ∩ R_{1} = (0), we must have gR = P.

If P ⊆ XR and dim(R/P) > 2, then Theorem 2.3 implies there exists Q ∈ Spec R such that dim(R/Q) = 2, P ⊂ Q ⊂ XR and P ∩ R_{1} = (0) = Q ∩ R_{1}, and so P is not maximal in the generic fiber. Thus Q ∈ Spec R maximal in the generic fiber of R_{1} ↪ R implies that the dimension of dim(R/Q) is 2, or equivalently that ht Q = n − 1. □

7.2 Theorem. Let n and m be positive integers, and let X = {x_{1}, …, x_{n}} and Y = {y_{1}, …, y_{m}} be sets of independent variables over the field K. Consider the formal power series rings R_{1} = K[[X]] and R = K[[X, Y]] and the extension R_{1} ↪ R_{1}[[Y]] = R. Let U = R_{1} \ (0). Let Q ∈ Spec R be maximal with respect to Q ∩ U = ∅. If n = 1, then dim R/Q = 1 and R_{1} ↪ R/Q is finite.

If n ≥ 2, there are two possibilities

(1) R_{1} ↪ R/Q is finite, in which case dim R/Q = dim R_{1} = n, or

(2) dim R/Q = 2.

Proof. First assume n = 1, and let x = x_{1}. Since Q is maximal with respect to Q ∩ U = ∅, for each P ∈ Spec R with Q ⊊ P we have P ∩ U is nonempty and therefore x ∈ P. It follows that dim R/Q = 1, for otherwise,

Q = ⋂{P | P ∈ Spec R and Q ⊊ P },

which implies x ∈ Q. By [8, Theorem 8.4], R_{1} ↪ R/Q is finite.

It remains to consider the case where n ≥ 2. We proceed by induction on m. Theorem 7.1 yields the assertion for m = 1. Suppose Q ∈ Spec R is maximal with respect to Q∩U = ∅. As in the proof of Theorem 7.1, we have dim R/Q ≥ 2. If Q ⊆ (X, y_{1}, …, y_{m−1})R, then by Theorem 2.3 with R_{0} = K[y_{m}]_{(y_{m})}[[X, y_{1}, …, y_{m−1}]], there exists Q′ ∈ Spec R with Q ⊆ Q′, dim R/Q′ = 2, and Q ∩ R_{0} = Q′ ∩ R_{0}. Since R_{1} ⊆ R_{0}, we have Q′ ∩ U = ∅. Since Q is maximal with respect to Q ∩ U = ∅, we

have Q = Q′, so dim R/Q = 2.

Otherwise, if Q ⊈ (X, y_{1}, …, y_{m−1})R, then there exists a prime element f ∈ Q

that contains a term y_{m}^{s} for some positive integer s. Let R_{2} = K[[X, y_{1}, …, y_{m−1}]].

By Weierstrass, it follows that f = gϵ, where g ∈ R_{2}[y_{m}] is a nonzero monic

polynomial in y_{m} and ϵ is a unit of R. We have fR = gR ⊆ Q is a prime ideal and

R_{2} ↪ R/gR is a finite integral extension. Thus R_{2}/(Q ∩ R_{2}) ↪ R/Q is an integral

extension. It follows that Q ∩ R_{2} is maximal in R_{2} with respect to being disjoint

from U. By induction dim R_{2}/(Q ∩ R_{2}) is either n or 2. Since R/Q is integral over

R_{2}/(Q ∩ R_{2}), dim R/Q is either n or 2. □

7.3 Remark. In the notation of Theorem 1.1, Theorem 7.2 proves the second part

of the theorem, since dim R = n + m. Thus if n = 1, ht Q = m. If n ≥ 2, the two

cases are (i) ht Q = m and (ii) ht Q = n + m − 2, as in (a) and (b) of Theorem 1.1.

Using the TGF terminology discussed in the introduction, we have the following

corollary to Theorem 7.2.

7.4 Corollary. With the notation of Theorem 7.2, assume P ∈ Spec R is such that

R_{1} ↪ R/P =: S is a TGF extension. Then dim S = dim R_{1} = n or dim S = 2.

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

Department of Mathematics, Michigan State University, East Lansing, MI 48824-28 HEINZER, ROTTHAUS AND WIEGAND 1027

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