ON A CLASS OF COHERENT REGULAR RINGS

CHRISTEL ROTTHAUS AND LIANA M. ŞEGA

Abstract. The paper investigates a special class of quasi-local rings. It is shown that these rings are coherent and regular in the sense that every finitely generated submodule of a free module has a finite free resolution.

1. Introduction

In a series of papers, W. Heinzer, S. Wiegand and C. Rotthaus investigated a class of quasi-local rings which can be described as a peculiar class of birational extensions of polynomial rings over fields. The construction method used for these rings has been introduced by Akizuki and Nagata in order to produce specific examples of Noetherian local rings. Roughly these rings can be described as follows:

Let K be a field and x, y_{1}, …, y_{n} be variables over K and let τ_{1}, …, τ_{m} ∈ xK[[x]] be power series in x which are algebraically independent over K(x). Fix an element

f ∈ K[x, y_{1}, …, y_{n}, τ_{1}, …, τ_{m}]

which is algebraically independent over K(x, y_{1}, …, y_{n}). The polynomial f can be considered as an element of the (x)-adic completion K[y_{1}, …, y_{n}]_{(y_{1},...,y_{n})}[[x]] of the localized polynomial ring R = K[x, y_{1}, …, y_{n}]_{(x,y_{1},...,y_{n})} and thus can be approximated in a natural way by polynomials r_{n} ∈ R so that f −r_{n} ∈ x^{n}K[y_{1}, …, y_{n}][[x]] for all n ∈ ℕ. The elements

f_{n} = x^{−n}(f − r_{n}) ∈ K[y_{1}, …, y_{n}][[x]]

are the so-called endpieces of f. The rings B under investigation in this paper are obtained by adjoining f and all its endpieces f_{n} to R and by localizing at the maximal ideal which is the contraction of the maximal ideal of R̂ = K[[x, y_{1}, …, y_{n}]]. In a way the ring B can be understood as a birational extension of the polynomial ring in n + 2 variables x, y_{1}, …, y_{n}, f over K. Moreover, B can be written as a direct limit of localized polynomial rings in the n + 2 variables x, y_{1}, …, y_{n} and f_{n} over K.

In [3], Heinzer, Wiegand, and Rotthaus showed that a simple flatness criterion on rings essentially of finite type over the field K decides if the constructed ring B is Noetherian or not. In previous papers an emphasis has been put on the construction of Noetherian rings and in a way these examples are well understood. In the Noetherian case B is a regular local ring of dimension n + 1 with completion R̂ = K[[x, y_{1}, …, y_{n}]]. An appropriate choice of f allows the construction of rings with strange formal fibers. For example, if f is not reduced in the completion K[[x, y_{1}, …, y_{n}]] the quotient ring B/(f) is an analytically ramified domain.

2000 Mathematics Subject Classification: 13E15, 13C10, 13D02; keywords: coherent rings, regular rings, modules of finite presentation.

Not very much is known in the non-Noetherian case. If B is not Noetherian, the maximal ideal 𝔪 of B is still n + 1 generated by x, y_{1}, …, y_{n} and the 𝔪-adic completion of B equals the formal power series ring R̂ = K[[x, y_{1}, …, y_{n}]]. However, if B is not Noetherian then B has Krull dimension n+2. The purpose of this paper is to show that B is a coherent regular quasilocal ring in the sense that every finitely generated submodule of a free B-module has a finite free resolution. Our proof makes use of the fact that B is the nested union of localized polynomial rings B_{n} in n + 2 variables over K.

The paper is organized as follows. In section 1 we discuss the construction method and the main properties of B. In section 2 we prove a main result which shows that every finitely generated submodule of a free B-module is of finite presentation. Since B is the nested union of localized polynomial rings of dimension n + 2, and thus of regular local rings, we show in the last section that every finitely generated submodule of a free B-module has projective dimension at most n + 1.

2. The rings under consideration

2.1. Construction. Let K be a field, x, y_{1}, …, y_{n} variables in degree 1 over K and let R_{0} = K[x, y_{1}, …, y_{n}] be the polynomial ring in n + 1 variables over K. Set R = (R_{0})_{(x,y_{1},...,y_{n})}, the localized polynomial ring with maximal ideal 𝔪_{R} = (x, y_{1}, …, y_{n}). Let τ_{1}, …, τ_{m} ∈ xK[[x]] be power series over K and assume that the set {τ_{1}, …, τ_{m}} is algebraically independent over K(x). Assume also that the τ′_{i}s can be represented as:

τ_{i} = ∑_{i=1}^{∞} a_{ij}x^{j} ∈ xK[[x]]

where a_{ij} ∈ K. Let

f(x, y_{1}, …, y_{n}, τ_{1}, …, τ_{m}) ∈ K[x, y_{1}, …, y_{n}, τ_{1}, …, τ_{m}]

be a polynomial in these variables. It can be considered as a power series in K[y_{1}, …, y_{n}][[x]]. Throughout the paper we assume that f is transcendental over L = Q(R), the fraction field of R. Since only the variables τ_{1}, …, τ_{m} are of interest for our investigation we write:

f(τ_{1}, …, τ_{m}) = f(x, y_{1}, …, y_{n}, τ_{1}, …, τ_{m}).

The end pieces f_{t} of f are defined as follows: For all t ∈ ℕ set

f_{t} = x^{−t}(f − f(∑_{i=1}^{t} a_{1j}x^{j}, …, ∑_{i=1}^{t} a_{mj}x^{j}))

where f(∑_{i=1}^{t} a_{1j}x^{j}, …, ∑_{i=1}^{t} a_{mj}x^{j}) is obtained from f(τ_{1}, …, τ_{m}) by replacing each τ_{i} by its front piece ∑_{j=1}^{t} a_{ij}x^{j}. By the Taylor formula f_{t} is again an element of K[y_{1}, …, y_{n}][[x]]. Moreover, for all t, k ∈ ℕ the end pieces f_{t} and f_{t+k} are related as follows:

(2.1.1)  f_{t} = x^{k}f_{t+k} + r_{t,k}

where r_{t,k} ∈ K[x, y_{1}, …, y_{n}]. In this paper we have fixed the definition of the remainder r_{t,k}. It has been shown in [6, 4.3 Proposition] that for the subsequent construction of B we could have chosen any element r_{t,k} ∈ R so that f_{t} −r_{t,k} ∈ x^{k}R̂ and defined the end pieces f_{t+k} accordingly. For all t ∈ ℕ set

B_{t} = R[f_{t}]_{(𝔪_{R},f_{t})}.

This ring can be considered as a polynomial ring in n+2 variables over K, localized at the irrelevant maximal ideal. It is naturally a subring of the power series ring R̂ = K[[x, y_{1}, …, y_{n}]]. For all t ∈ ℕ the inclusion:

ι_{t,1} : B_{t} ↪ B_{t+1}

is induced by the identity on R and by f_{t} ↦ xf_{t+1} + r_{t,1}. The ring of interest is given by the union:

B = ⋃_{t=1}^{∞} B_{t} = \underrightarrow{lim}_{t∈ℕ} B_{t}.

This ring B has been studied in [3]. It is relatively easy to decide if B is Noetherian or not. Consider the homomorphism of algebras essentially of finite type over K:

φ : R[f] ⟶ (R[τ_{1}, …, τ_{n}]_{(𝔪_{R},τ_{1},...,τ_{m})})_{x}

and let

ψ : (R[τ_{1}, …, τ_{n}]_{(𝔪_{R},τ_{1},...,τ_{m})})_{x} ⟶ R̂_{x} = K[[x, y_{1}, …, y_{n}]]_{x}

denote the naturally embedding. The homomorphism ψ is always faithfully flat [3]. According to [3, Theorem 3.2], B is a Noetherian ring if and only if φ is flat at all prime ideals of (R[τ_{1}, …, τ_{n}]_{(𝔪_{R},τ)})_{x} which lie under prime ideals of R̂_{x} via ψ. Equivalently B is Noetherian if and only if the composition:

R[f] ⟶^{φ} (R[τ_{1}, …, τ_{n}]_{(𝔪_{R},τ)})_{x} ⟶^{ψ} R̂_{x}

is flat. In all cases the maximal ideal 𝔪_{B} of B is generated by x, y_{1}, …, y_{n}, every ideal of B which contains the element x is finitely generated, and the completion of B with respect to the maximal ideal 𝔪_{B} of B equals R̂. The equation (2.1.1) shows that for all t, k ∈ ℕ we have (B_{t})_{x} = (B_{t+k})_{x}. In particular, B_{x} = (B_{t})_{x} for all t ∈ ℕ, and the rings B and B_{t} have the same field of quotients. Another interesting property of the ring B has been studied in [7]. As a consequence of [7, Theorem 2.1] we obtain that B is Noetherian if and only if the Krull dimension of B equals n + 1. In the non-Noetherian case B has dimension n + 2.

Our aim is to show that the ring B defined above is a coherent regular quasi-local domain. In the Noetherian case this is a well known fact about regular local rings. Our proof does not distinguish between the Noetherian and the non-Noetherian case, that is, we will not make use of the nature of the map φ.

2.2. Example. The following example has been studied extensively in [4] and [5].

As above, let K be a field and consider the localized polynomial ring in 2 variables R = K[x, y]_{(x,y)}. Let τ ∈ xK[[x]] be an algebraically independent element over K(x):

τ = ∑_{i=1}^{∞} a_{i}x^{i} ; a_{i} ∈ K.

For all t ∈ ℕ let

τ_{t} = ∑_{i=t+1}^{∞} a_{i}x^{i−t}

be the end pieces of τ in K[[x, y]] and let q = p_{1} … p_{s} ∈ (x, y)R be a product of finitely many mutually not associated prime elements p_{i} ∈ R \ (x). Set f = q · τ. Then for all t ∈ ℕ the end pieces of f are given by:

f_{t} = q · τ_{t} = q( ∑_{i=t+1}^{∞} a_{i}x^{i−t}).

For all t, k ∈ ℕ, notice that there is the following formula relating f_{t} and f_{t+k}:

(2.2.1)  f_{t} = x^{k}f_{t+k} + q( ∑_{j=t+1}^{t+k} a_{j}x^{j−t}).

In this case the remainders are given by: r_{t,k} = q( ∑_{j=t+1}^{t+k} a_{j}x^{j−t}). Set B_{t} =

R[f_{t}]_{(x,y,f_{t})} and B = ⋃_{t∈ℕ}B_{t} ⊆ Q(R)(τ).

Next we show that the map

φ : R[f] ⟶ (R[τ]_{(x,y,τ)})_{x}

is not flat. For every prime ideal P_{i} = (p_{i})(R[τ]_{(x,y,τ)})_{x} (which is the contraction of Q_{i} = p_{i}K[[x, y]]_{x}) we have that φ^{−1}(P_{i}) = (p_{i}, f) is a prime ideal of height 2 in R[f]. In [4] and [5] it has been shown that B is a 3- dimensional quasi-local domain with maximal ideal 𝔪_{B} = (x, y). Moreover, B has exactly s prime ideals of height 2. These prime ideals are exactly the non-finitely generated primes of B.

2.3. Let now B and B_{t} be as defined in 2.1, with B ⊆ K[[x, y_{1}, …, y_{n}]] and with f(x, y_{1}, …, y_{n}, τ_{1}, …, τ_{m}) ∈ K[x, y_{1}, …, y_{n}, τ_{1}, …, τ_{m}]. Recall that the inclusion ι_{t,k}: B_{t} ↪ B_{t+k} is the identity on R and maps f_{t} into x^{k}f_{t+k} + r_{t,k}. It induces a surjective map

ψ_{t,k}: B_{t}[Z_{t,k}] → B_{t+k}

given by

ψ_{t,k}|_{B_{t}} = ι_{t,k} and ψ_{t,k}(Z_{t,k}) = f_{t+k}.

Since B_{t} is a regular local ring of dimension n + 2, the kernel of ψ_{t,k} is the principal prime ideal generated by the polynomial:

h_{t,k} = x^{k}Z_{t+k} + r_{t,k} − f_{t}

and the map ψ_{t,k} induces an isomorphism:

\overline{ψ}_{t,k} : \frac{B_{t}[Z_{t,k}]}{(h_{t,k})} ⟶^{≅} B_{t+k}

For each triple (t, k, m) ∈ ℕ^{3} we have the following commutative diagram:

[figure: 0; (h_{t,k+m}); φ̃; B_{t}[Z_{t,k+m}]; ψ_{t,k+m}; φ; B_{t+k+m}; 0; 0; (h_{t+k,m}); B_{t+k}[Z_{t+k,m}]; ψ_{t+k,m}; B_{t+k+m}; 0]

where φ = φ_{t,k,m}: B_{t}[Z_{t,k+m}] → B_{t+k}[Z_{t+k,m}] is defined by

φ|_{B_{t}} = ι_{t,k} and φ(Z_{t,k+m}) = Z_{t+k,m}

and the map φ̃ is the restriction of φ. Notice that we can write the ring B_{t+k+m} as a homomorphic image of the ring B_{t}[Z_{t,m+k}] and as a homomorphic image of B_{t+k}[Z_{t+k,m}] and that φ is an extension of the natural map: ι_{t,k} : B_{t} ⟶ B_{t+k}. φ maps the kernel of ψ_{t,k+m} into the kernel of ψ_{t+k,m}.

It is crucial for the proof in Section 3 to understand how h_{t,k+m} relates to h_{t+k,m} under the map φ.

2.4. Lemma. The following relation holds for all t, k, m:

φ̃(h_{t,m+k}) = x^{k}h_{t+k,m}

Proof. First note that

r_{t,m+k} = f_{t} − x^{k+m}f_{t+m+k}
= f_{t} − x^{k}f_{t+k} + x^{k}f_{t+k} − x^{k+m}f_{t+m+k}
= r_{t,k} + x^{k}r_{t+k,m}.

Thus

φ̃(h_{t,m+k}) = φ(h_{t,m+k})
= φ(x^{m+k}Z_{t,m+k} + r_{t,m+k} − f_{t})
= x^{m+k}Z_{t+k,m} + r_{t,m+k} − f_{t}
= x^{m+k}Z_{t+k,m} + r_{t,k} + x^{k}r_{t+k,m} − f_{t}.

Using the equation (2.1.1), we further have:

φ̃(h_{t,m+k}) = x^{m+k}Z_{t+k,m} + r_{t,k} + x^{k}r_{t+k,m} − f_{t}
= x^{m+k}Z_{t+k,m} + x^{k}r_{t+k,m} − x^{k}f_{t+k}
= x^{k}h_{t+k,m}.
□

3. Coherence

A ring B is said to be coherent if every finitely generated ideal of B is of finite presentation. We refer to [2] for basic results concerning this notion.

In this section we show that the ring B defined in 2.1 is coherent. Since the proof will also be used in the next section, we show a more general statement, namely that every finitely generated submodule of a finitely generated free B-module is of finite presentation.

3.1. Theorem. Let B ⊆ K[[x, y_{1}, …, y_{n}]] be a ring as in 2.1. If r ∈ ℕ and M = ∑_{i=1}^{s} Bm_{i} ⊆ B^{r} is finitely generated submodule of B^{r}, then M is of finite presentation.

The lemma used in the proof of the Theorem is independent of the more complex structure of the ring in 2.1, hence we prove it in greater generality, for a ring B as described below.

3.2. Let B be a commutative ring, {B_{n}}_{n≥0} a sequence of Noetherian subrings of B with B_{n} ⊆ B_{n+1} for all n ∈ ℕ and such that B = ∪_{n≥0}B_{n}.

Let M = ∑_{i=1}^{s} Bm_{i} ⊆ B^{r} be a finitely generated submodule of the finite free B-module B^{r} and let q ∈ ℕ be an integer so that

m_{i} ∈ B_{n}^{r} for all n ≥ q .

For all n ≥ q set

M_{n} = ∑_{i=1}^{s} B_{n}m_{i} ⊆ B_{n}^{r} .

Let χ: B^{s} → M denote the B-module homomorphism which sends the standard basis of B^{s} to m_{1}, …, m_{s}. For each n ≥ q let χ_{n}: B_{n}^{s} → M_{n} denote the similarly defined B_{n}-homomorphism which is the restriction of χ. Set L_{n} = Ker(χ_{n}) and L = Ker(χ). Note that L = ∪_{n≥q}L_{n}. For each pair i, t with i ≥ t ≥ q the following diagram is commutative:

\xymatrix{0 \ar[r] & L_{t} \ar[r] \ar@{^{(}->}[d] & B_{t}^{s} \ar[r]^{χ_{t}} \ar@{^{(}->}[d] & M_{t} \ar[r] \ar@{^{(}->}[d] & 0 \\ 0 \ar[r] & L_{i} \ar[r] \ar@{^{(}->}[d] & B_{i}^{s} \ar[r]^{χ_{i}} \ar@{^{(}->}[d] & M_{i} \ar[r] \ar@{^{(}->}[d] & 0 \\ 0 \ar[r] & L \ar[r] & B^{s} \ar[r]^{χ} & M \ar[r] & 0}

For i, t as above we let

β_{t,i}: L_{t} ⊗_{B_{t}} B_{i} → L_{i}

denote the map induced by the inclusion L_{t} ↪ L_{i} and we let

β_{t}: L_{t} ⊗_{B_{t}} B → L

denote the map induced by the inclusion L_{t} ↪ L.

3.3. Lemma. With the notation in 3.2, the following hold:

(1) The map β_{t,i} is surjective if and only if Tor_{1}^{B_{t}}(B_{i}, B_{t}^{r}/M_{t}) = 0.

(2) If there exists an increasing sequence of integers {t_{n}}_{n≥q} such that β_{t_{n},t_{n+1}} is surjective for all n ≥ q, then β_{t} is surjective for all t ≥ t_{q} and the B-module M is of finite presentation.

Proof. (1) For i, t with i ≥ t ≥ q we have a commutative diagram, with vertical maps induced by the inclusions M_{t} ↪ M_{i}, B_{t}^{s} ↪ B_{i}^{r}, respectively L_{t} ↪ L_{i}.

\xymatrix{ & L_{t} ⊗_{B_{t}} B_{i} \ar[r] \ar[d]^{β_{t,i}} & B_{t}^{s} ⊗_{B_{t}} B_{i} \ar[r] \ar[d]^{≅} & M_{t} ⊗_{B_{t}} B_{i} \ar[r] \ar[d]^{α_{t,i}} & 0 \\ 0 \ar[r] & L_{i} \ar[r] & B_{i}^{s} \ar[r]^{χ_{i}} & M_{i} \ar[r] & 0}.

The map β_{t,i} is surjective if and only if α_{t,i} is injective. In order to understand the latter map, consider the short exact sequence:

0 → M_{t} → B_{t}^{r} → B_{t}^{r}/M_{t} → 0

and the induced sequence:

0 → Tor_{1}^{B_{t}}(B_{t}^{r}/M_{t}, B_{i}) → M_{t} ⊗_{B_{t}} B_{i} → B_{i}^{r}.

Notice that the image of the rightmost map is precisely M_{i} and we have a short exact sequence:

0 → Tor_{1}^{B_{t}}(B_{t}^{r}/M_{t}, B_{i}) → M_{t} ⊗_{B_{t}} B_{i} ⟶^{α_{t,i}} M_{i} → 0.

In consequence, the map β_{t,i} is surjective whenever Tor_{1}^{B_{t}}(B_{i}, B_{t}^{r}/M_{t}) = 0.

(2) Under the given hypothesis, if {l_{1}, …, l_{d}} is a generating set of the B_{t_{n}}- module L_{t_{n}} for some n ≥ q, then l_{1}, …, l_{d} also generate the B_{t_{k}}-module L_{t_{k}} for all k ≥ n. Since L = ∪_{k≥n}L_{t_{k}}, it follows that L is generated by l_{1}, …, l_{d}. In particular, this shows that β_{t_{n}} is surjective for all n ≥ q. Note that for t ≥ t_{q} we have β_{t_{q}} = β_{t} ∘ (β_{t_{q},t} ⊗_{B_{t}} B). Since β_{t_{q}} is surjective, it follows that β_{t} is surjective. □

In order to prove that a given finitely generated module M is of finite presentation, our approach will be to show that there exists an increasing sequence of integers {t_{n}}_{n≥q} such that Tor_{1}^{B_{tn}} (B_{t_{n+1}}, B_{t_{n}}^{r}/M_{t_{n}}) = 0 for all n ≥ q. This will be achieved by using certain maps between homology modules. The purpose of the next paragraph is to define and set up notation for these maps.

3.4. Let π: A → A′ be a ring homomorphism, M, N be A-modules, M′, N′ be A′-modules and let f: M → M′ and g: N → N′ be A-module homomorphisms. (We make the convention that an A′-module inherits a structure of A-module via π.) Let F, respectively G, be a free A-resolution of M, respectively N, and let F′, respectively G′, be a free A′-resolution of M′, respectively N′.

Note that the map f induces a map M ⊗_{A} A′ → M′, which lifts (uniquely, up to homotopy) to a homomorphism of complexes of A′-modules F ⊗_{A} A′ → F′. By composing with the natural homomorphism F → F ⊗_{A} A′, we obtain thus a homomorphism of complexes of A-modules f̃: F → F′. Similarly, we can define g̃: G → G′.

With Tor^{A}(M, N) = H(F ⊗_{A} G) and Tor^{A′}(M′, N′) = H(F′ ⊗_{A′} G′), we define

Tor^{π}(f, g): Tor^{A}(M, N) → Tor^{A′}(M′, N′)

to be the map induced in homology by the map f̃⊗_{A} g̃: F ⊗_{A} G → F′ ⊗_{A′} G′.

When M = M′ and f is the identity map M → M, we will write Tor^{π}(M, g) instead of Tor^{π}(f, g).

Proof of Theorem 3.1. Let B ⊆ K[[x, y_{1}, …, y_{n}]] be defined as in 2.1 and let M = ∑_{i=1}^{s} Bm_{i} ⊆ B^{r} be a finitely generated submodule of the finite free module B^{r}. We use the notation introduced in 3.2, with this choice of B and M.

Fix an integer t with t ≥ q and choose an integer m = m_{t} such that

(3.1.1)  (M_{t}: x^{m})_{B_{t}^{r}} = (M_{t}: x^{n})_{B_{t}^{r}} for all n ≥ m .

We will show that Tor_{1}^{B_{t+k}}(B_{t+k+m}, B_{t+k}^{r}/M_{t+k}) = 0 for all k ≥ m. Then Lemma 3.3 implies that M has a finite presentation.

Let t, k, m be as above and set p = t + k + m. Recall that ι_{t,k}: B_{t} → B_{t+k} denotes the natural inclusion, as in Section 1. Since we fixed the indices t and k, for the remaining of the proof we will simply denote this map by ι. Consider the commutative diagram

\xymatrix{0 \ar[r] & M_{t} \ar[r] \ar@{^{(}->}[d]^{ε} & B_{t}^{r} \ar[r] \ar@{^{(}->}[d]^{ι^{r}} & B_{t}^{r}/M_{t} \ar[r] \ar[d]^{π} & 0 \\ 0 \ar[r] & M_{t+k} \ar[r] & B_{t+k}^{r} \ar[r] & B_{t+k}^{r}/M_{t+k} \ar[r] & 0}

Now apply Tor^{B_{t}}(B_{p}, −), respectively Tor^{B_{t+k}}(B_{p}, −), to these short exact sequences and write the long exact sequences in homology. Basic homological algebra arguments, as in [1, V 4.3, VI 4.4], show that the following diagram is commutative. (The horizontal maps on the left are the connecting homomorphisms.)

[figure: 0; Tor_{1}^{B_{t}}(B_{p}, B_{t}^{r}/M_{t}); Tor_{1}^{ι}(B_{p},π); B_{p} ⊗_{B_{t}} M_{t}; Tor_{0}^{ι}(B_{p},ε); M_{p}; 0; 0; Tor_{1}^{B_{t+k}}(B_{p}, B_{t+k}^{r}/M_{t+k}); B_{p} ⊗_{B_{t+k}} M_{t+k}; M_{p}; 0].

The map Tor_{0}^{ι}(B_{p}, ε) sends 1 ⊗_{B_{t}} m_{i} to 1 ⊗_{B_{t+k}} m_{i}, for all i with 0 ≤ i ≤ s. Recall that for n ≥ q the B_{n}-module M_{n} is generated by m_{1}, … m_{s}. In consequence, Tor_{0}^{ι}(B_{p}, ε) is surjective and the Snake Lemma gives that the map Tor_{1}^{ι}(B_{p}, π) is surjective.

In order to prove Tor_{1}^{B_{t+k}}(B_{p}, B_{t+k}^{r}/M_{t+k}) = 0, we will show that the surjective map Tor_{1}^{ι}(B_{p}, π) is equal to zero.

To compute the map Tor_{1}^{ι}(B_{p}, π), we use the commutative diagram in 2.3. We apply Tor^{B_{t}}(−, B_{t}^{r}/M_{t}), respectively Tor^{B_{t+k}}(−, B_{t}^{r}/M_{t}), to the exact sequences in this diagram and we write the induced long exact sequences. As above, we obtain the following commutative diagram:

\xymatrix{0 \ar[r] & Tor_{1}^{B_{t}} (B_{p}, \frac{B_{t}^{r}}{M_{t}} ) \ar[r] \ar[d]_{Tor_{1}^{ι}(B_{p},π)} & (h_{t,k+m}) ⊗_{B_{t}} \frac{B_{t}^{r}}{M_{t}} \ar[r] \ar[d]_{Tor_{0}^{ι}(φ̃,π)} & \frac{B_{t}[Z_{t,k+m}]^{r}}{M_{t}^{∗}} \ar[d]_{Tor_{0}^{ι}(φ,π)} \\ 0 \ar[r] & Tor_{1}^{B_{t+k}} (B_{p}, \frac{B_{t+k}^{r}}{M_{t+k}} ) \ar[r] & (h_{t+k,m}) ⊗_{B_{t+k}} \frac{B_{t+k}^{r}}{M_{t+k}} \ar[r] & \frac{B_{t+k}[Z_{t+k,m}]^{r}}{M_{t+k}^{∗}}}

Here M_{t}^{∗} and M_{t+k}^{∗} are the submodules of B_{t}[Z_{t,k+m}]^{r}, respectively B_{t+k}[Z_{t+k,m}]^{r}, which are generated by m_{1}, …, m_{s}. The vertical map in the middle sends the element h_{t,k+m} ⊗_{B_{t}} ē_{j} to the element φ̃(h_{t,m+k}) ⊗_{B_{t+k}} ē_{j} where ē_{j} denotes the image of the canonical basis element e_{j} in B_{t}^{r}/M_{t}, respectively B_{t+k}^{r}/M_{t+k}.

An element in Tor_{1}^{B_{t}}(B_{p}, B_{t}^{r}/M_{t}) has the form

ω = ∑_{i=0}^{n} (b_{i}Z_{t,m+k}^{i}h_{t,k+m} ⊗ u_{i}), with b_{i} ∈ B_{t} and u_{i} ∈ B_{t}^{r}/M_{t}.

By Lemma 2.4 we have φ̃(h_{t,m+k}) = x^{k}h_{t+k,m} and we obtain

Tor_{1}^{ι}(B_{p}, π)(ω) = ∑_{i=0}^{n} (b_{i}Z_{t+m,k}^{i}x^{k}h_{t+k,m} ⊗ u_{i}).

We claim that x^{k}u_{i} = 0 for each i. In order to see this, first note that the ideal (h_{t,k+m}) is a free B_{t}-module with basis {h_{t,k+m}Z^{i}}_{i≥0}. Thus

(h_{t,k+m}) ⊗_{B_{t}} B_{t}^{r}/M_{t} ≅ ⨁_{i=0}^{∞} (h_{t,k+m}Z^{i} ⊗ B_{t}^{r}/M_{t}) ≅ ⨁_{i=0}^{∞} B_{t}^{r}/M_{t}.

Since (B_{t+k})_{x} = (B_{t})_{x}, we have

Tor_{1}^{B_{t}}(B_{t+k}, B_{t}^{r}/M_{t})_{x} = 0

and ω is equal to zero when regarded as an element of ((h_{t,k+m}) ⊗_{B_{t}} B_{t}^{r}/M_{t})_{x}.

This implies that for each i we have u_{i}x^{s_{i}} ∈ M_{t} for some s_{i} ∈ ℕ, hence:

u_{i} ∈ (M_{t} : x^{m})_{B_{t}^{r}} for all i

Since we have chosen k such that k ≥ m, it follows from (3.1.1) that x^{k}u_{i} = 0 for all i, and hence Tor_{1}^{ι}(B_{r}, π)(ω) = 0. We have thus Tor_{1}^{ι}(B_{r}, π) = 0 and this finishes the proof, as discussed above. □

4. Regularity

A coherent ring is said to be regular if every finitely generated submodule of a finitely generated free B-module has a finite free resolution. In this section we show that the rings B in 2.1 are regular.

4.1. Theorem. Let B ⊆ K[[x, y_{1}, …, y_{n}]] be a ring as in 2.1. If M is a finitely generated submodule of a finitely generated free B-module, then pd_{B}M ≤ n + 1. In particular, B is regular.

Proof. Let M = ∑_{i=1}^{s} Bm_{i} ⊆ B^{r} be a finitely generated submodule of the finite free module B^{r}. Let B_{n} = R[f_{n}]_{(𝔪_{R},f_{n})} be the localized polynomial rings in n + 2 variables as in 2.1. Then B = ∪_{n=1}^{∞}B_{n} and we will further use the notation in 3.2. We set L^{1} = L, L_{n}^{1} = L_{n} and s_{0} = s, where L, L_{n} are as in 3.2.

In the proof of Theorem 3.1 we showed that the ring B in 2.1 satisfies the hypothesis of Lemma 3.3(2) for a certain strictly increasing sequence {t_{n}}_{n≥q}. In particular, by renaming B_{t_{n}} as B_{n}, we may assume that the maps β_{n,n+1} : L_{n} ⊗_{B_{n}} B_{n+1} → L_{n+1} are surjective for all n ≥ q, while the equalities B = ∪_{n=q}^{∞}B_{n} and L^{1} = ∪_{n=q}^{∞}L_{n}^{1} still hold. Note that Lemma 3.3 also implies that the map β_{q}: L_{q} ⊗_{B_{q}} B → L is surjective.

Since L_{q}^{1} is a finitely generated B_{q}-module, we have

L_{q}^{1} = ∑_{i=1}^{s_{1}} B_{q}ℓ_{1i} with ℓ_{1i} ∈ L_{q}^{1} .

The surjectivity of the maps β_{q,n} and β_{q} implies that we also have:

L_{t}^{1} = ∑_{i=1}^{s_{1}} B_{t}ℓ_{1i} and L^{1} = ∑_{i=1}^{s_{1}} Bℓ_{1i}.

We will use next the constructions in 3.2, only with L^{1} = ∑_{i=1}^{s_{1}} Bℓ_{1i} ⊆ B^{s_{0}} instead of M = ∑_{i=1}^{s_{0}} Bm_{i} ⊆ B^{r}. For all t ≥ q we obtain thus exact sequences:

0 → L_{t}^{2} → B_{t}^{s_{1}} → L_{t}^{1} → 0

and

0 → L^{2} → B^{s_{1}} → L^{1} → 0

with L^{2} = ∪_{n=q}^{∞}L_{n}^{2}. As above, we may assume L^{2} = ∑_{i=1}^{s_{2}} Bℓ_{2i} is a finitely generated B-module, and L_{t}^{2} = ∑_{i=1}^{s_{2}} B_{t}ℓ_{2i} for each t ≥ q. Repeating this process we obtain for t ≥ q resolutions of M_{t}:

0 → L_{t}^{n+1} → B_{t}^{s_{n}} → B_{t}^{s_{n−1}} → ⋯ → B_{t}^{s_{1}} → B_{t}^{s_{0}} → M_{t} → 0

and for the B-module M:

0 → L^{n+1} → B^{s_{n}} → B^{s_{n−1}} → ⋯ → B^{s_{1}} → B^{s_{0}} → M → 0.

Again L^{n+1} = ∪_{t≥q}L_{t}^{n+1} and L^{n+1} = ∑_{i=1}^{s_{n+1}} Bℓ_{n+1,i} is a finitely generated Bmodule and L_{t}^{n+1} = ∑_{i=1}^{s_{n+1}} B_{t}ℓ_{n+1,i} for each t ≥ q. Each B_{t} is a Noetherian regular local ring of dimension n + 2. For each t, the B_{t}-module M_{t} has depth at least 1 (because it is contained in a free module), hence the projective dimension of M_{t} is at most n + 1. Thus all n + 1th syzygies L_{t}^{n+1} are free B_{t}-modules. Since for all t ∈ ℕ the quotient fields Q(B_{t}) and Q(B) are identical and since for sufficiently large t ∈ ℕ the generators of L_{t}^{n+1} generate the B-module L^{n+1}, we conclude that L^{n+1} is a free B-module. The B-module M has thus a finite free resolution:

0 → B^{s_{n+1}} → B^{s_{n}} → ⋯ → B^{s_{0}} → M → 0 .  □

4.2. A projective resolution for the residue field. Let B ⊆ K[[x, y_{1}, …, y_{n}]] be again as in 2.1. Recall that B is a local ring with maximal ideal 𝔪_{B} = (x, y_{1}, …, y_{n})B. If B is Noetherian, then a minimal projective resolution of the residue field K is the Koszul complex K^{B} on the sequence x, y_{1}, …, y_{n}. We claim that this remains the case when B is assumed non-Noetherian (and hence the Krull dimension of B is n + 2).

Proposition. The Koszul complex K^{B} on the sequence x, y_{1}, …, y_{n} is a free resolution of K over B. In particular, pd_{B}K = n + 1.

Proof. The construction in 3.2 applied to the module M := 𝔪_{B} = (x, y_{1}, …, y_{n})B gives that M_{t} is the B_{t}-ideal generated by the regular sequence x, y_{1}, …, y_{n}. The Koszul complex K^{B_{t}} on this sequence represents a free resolution of B_{t}/M_{t} over B_{t}, and hence K_{⩾1}^{B_{t}} is a free resolution of M_{t}. The argument in the proof of Theorem 4.1 then shows that K_{⩾1}^{B} is a free resolution of M(= 𝔪_{B}) over B, hence K^{B} is a free resolution of K = B/𝔪_{B}. □

References

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

   E-mail address: rotthaus@math.msu.edu

Department of Mathematics and Statistics, University of Missouri, Kansas City, MO 64110-2499

   E-mail address: segal@umkc.edu