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Math 711: Lecture of November 1, 2006
Before attacking the problem of comparing symbolic powers of primes, we want to discuss some techniques that will be needed. One is connected with enlarging the residue class field of a local ring.
Proposition. Let (R, m, K) be a local ring, and let θ be an element of the algebraic closure of K with minimal monic irreducible polynomial f (x) ∈ K[x]. Let F (x) be a monic polynomial of the same degree d as f that lifts F to R[x]. Let S = R[x]/(F ). Then S is module-finite, free of rank d, and local over R. Hence, S is R-flat. The residue field of S is isomorphic with L = K[θ], and S has maximal ideal mS.
Proof. S is module-finite and free of rank d over R by the division algorithm. Hence, every maximal ideal of S must lie over m, and the maximal ideals of S correspond bijectively to those of S/mS = R[x]/(mR[x] + F R[x]) ∼= K[x]/f K[x] ∼= K[θ], which shows that mS is maximal and that it is the only maximal ideal of S. This also shows that S/mS ∼= K[θ].
Discussion: getting reductions such that the number of generators is the analytic spread. Let (R, m, K) be local and I an ideal with analytic spread h. One way of enlarging the residue field so as to guarantee the existence of a reduction of I with h generators is to replace R by R(t), so that the residue class field becomes infinite. For this purpose, it is not necessary to enlarge R so that K becomes infinite. One only needs that K have sufficiently large cardinality. When K is finite, one can choose a primitive element θ for a larger finite field extension L: the cardinality of the finite field L may be taken a large as one likes, and a primitive element exists because the extension is separable. Recall that the issue is to give one-forms of B = K ⊗R grI (R) that are a homogeneous system of parameters. After making the type of extension in the Proposition, one has, because mS is the maximal ideal of S, that
L ⊗S grIS (S) ∼= L ⊗S
S ⊗R grI (R)
∼= L ⊗R grI (R) ∼= L ⊗K
K ⊗R grI (R)
If one makes a base change to K ⊗K B, where K is the algebraic closure of K, one certainly has a linear homogeneous system of parameters. The coefficients will lie in L for any sufficiently large choice of finite field L.
Proposition. Let (R, m, K) be any complete local ring. Then R has a faithfully flat extension (S, n, L) such that n = mS and L is the algebraic closure of K. If R is regular, then S is regular.
Proof. We may take R to be a homomorphic image of T = K[[x 1 ,... , xd]], where K is a field, or of T = V [[x 1 ,... , xd]], where (V, πV, K) is a complete DVR such that the induced map of residue class fields is an isomorphism. In the first case, let L be the algebraic closure of K. Then T 1 = L[[x 1 ,... , xn]] is faithfully flat over T , and the expansion of 1
2
(x 1 ,... , xd)T to T 1 is the maximal ideal of T 1. Here , faithful flatness follows using the Lemma on p. 2 of the Lecture Notes of October 18, becasuse every system of parameters for T is a system of parameters for T 1 , and so a regular sequence on T 1 , since T 1 is Cohen- Macaulay. Then S = T 1 ⊗T R is faithfully flat over R, has residue class field L, and m expands to the maximal ideal.
We can solve the problem in the same way in mixed characteristic provided that we can solve the problem for V : if (W, πW, L) is a complete DVR that is a local extension of V with residue class field L, then T 1 = W [[x 1 ,... , xd]] will solve the problem for T , and T 1 ⊗T R will solve the problem for R, just as above.
We have therefore reduced to studying the case where the ring is a complete DVR V. Furthermore, if (W, πW, L) solves the problem but is not necessarily complete, we may
use Ŵ to give a solution that is a complete DVR.
Next note that if (Vλ, πVλ, Kλ) is a direct limit system of DVRs, all with the same generator π for their maximal ideals, such that the maps are local and injective, then lim −→ λ^ Vλ is DVR with maximal ideal generated by π. It is then clear that the residue class
field is lim −→ λ^
Kλ. The reason is that every nonzero element of the direct limit may be
viewed as arising from some Vλ, and in that ring it may be written as a unit times a power of π. Thus, every nonzero element of the direct limit is a unit times a power of π.
We now construct the required DVR as a direct limit of DVRs, where the index set is given by a well-ordering of the field L, the algebraic closure of K, in which 0 is the least element. We shall construct the family {(Vλ, π, , Kλ)}λ∈L in such a way that for every λ ∈ L,
{μ ∈ L : μ ≤ λ} ⊆ Kλ ⊆ L.
This will complete the proof, since the direct limit of the family will be the required DVR with residue class field L.
Take V 0 = V. If λ ∈ L and Vμ has been constructed for μ < λ such that for all μ < λ,
{ν ∈ L : ν ≤ μ} ⊆ Kμ ⊆ L,
then we proceed as follows to construct Vλ. There are two cases.
(1) If λ has an immediate predecessor μ and λ ∈ Kμ we simply let Vλ = Vμ, while if λ /∈ Kμ, we take θ = λ in the first Proposition to construct Vλ.
(2) If λ is a limit ordinal, we first let (V ′, πV ′, K′) = lim −→ μ<λ^ Vμ. If λ is in the residue
class field of V ′, we let Vλ = V ′. If not, we use the first Proposition to extend V ′^ so that its residue class field is K′[λ].
To prove the theorem on comparison of symbolic powers in regular rings, we shall also need some results on valuation domains that are not necessarily Noetherian. In particular, we need the following method of constructing such valuation domains.
