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Math 711: Lecture of October 23, 2006
One of our goals is to discuss what is known about the following conjecture of C. Lech, which has been an open question for over forty years.
Conjecture. If R → S is a flat local map of local rings, then e(R) ≤ e(S).
This is open even in dimension three when S is module-finite and free over R. Note that one can immediately reduce to the case where both rings are complete.
Under mild conditions, a local ring R of multiplicity 1 is regular: it suffices if the completion R̂ has no associated prime P such that dim (R/P̂ ) < dim ( R̂ ). Therefore, the following result is related to Lech’s conjecture:
Theorem. If S is faithfully flat over R and S is regular then R is regular. In particular, if (R, m, K) → (S, n, L) is a flat local map of local rings and S is regular, then R is regular.
Proof. The second statement implies the first, for if P is any prime of R then some prime Q of S lies over P , and we can apply the second statement to RP → SQ to conclude that RP is regular.
To prove the second statement, let
(∗) · · · → Gn → · · · → G 1 → G 0 → R → R/m → 0
be a minimal resolution of R/m over R. Then the matrix αi of the map Gi → Gi− 1 has entries in m for all i ≥ 1. Since S is R-flat, the complex obtained by applying S ⊗R , namely
(∗∗) · · · → S ⊗R Gn → · · · → S ⊗R G 1 → S ⊗R G 0 → S/mS → 0
gives an S-free resolution of S/mS over R. Moreover, the entries of the matrix of the map S ⊗ Gi → S ⊗R Gi− 1 are simply the images of the entries of the matrix αi in S: these are in n, and so the complex given in (∗∗) is a minimal free resolution of S/mS over S. Thus, all of its terms are eventually 0, and this implies that all of the terms of (∗) are eventually
- Hence, K has finite projective dimension over R, which implies that R is regular.
Before treating Lech’s conjecture itself, we want to give several other characterizations of eI (M ) when I is generated by a system of parameters. There is a particularly simple characterization in the Cohen-Macaulay case. We first recall some facts about regular sequences. The results we state in the Proposition below are true for an arbitrary regular sequence on an arbitrary module. However, we only indicate proofs for the situation where R is local, M is a finitely generated R-module, and x 1 ,... , xd are elements of the maximal 1
ideal of R. The proofs are valid whenever we are in a situation where regular sequences are permutable, which makes the arguments much easier. (There is a treatment of the case where the regular sequence is not assumed to be permutable in the Extra Credit problems in Problem Sets #2 and #3 from Math 615, Winter, 2004. It is assumed that M = R there, but the proofs are completely unchanged in the module case.) Recall that, in all cases, by virtue of the definition, the fact that x 1 ,... , xd is a regular sequence on M implies that (x 1 ,... , xd)M 6 = M.
Proposition. Let x 1 ,... , xd ∈ R, let I = (x 1 ,... , xd)R, and let M be an R-module.
(a) Let t 1 ,... , td be nonnegative integers. Then x 1 ,... , xd is a regular sequence if and only if xt 11 ,... , xt dd is a regular sequence on M.
(b) If x 1 ,... , xd is a regular sequence on M , and a 1 ,... , ad are nonnegative integers, then xa 11 · · · xa dd w ∈ (xa 11 +1,... , xa dd +1)M implies that w ∈ (x 1 ,... , xd)M.
(c) If x 1 ,... , xd is a regular sequence on M , μ 1 ,... , μN are the monomials of degree n in x 1 ,... , xd, and w 1 ,... , wN are elements of M such that
∑N
j=1 μj^ wj^ ∈^ I n+1M , then every wj ∈ IM. (d) If x 1 ,... , xd is a regular sequence on M , then grI (M ) may be identified with
(M/IM ) ⊗R/I (R/I)[X 1 ,... , Xd],
where the Xj are indeterminates and for nonnegative integers a 1 ,... , ad such that ∑d j=1 aj^ =^ n, the image of^ x
a 1 1 · · ·^ x
ad d M^ in^ I
nM/In+1M corresponds to
(M/IM )X 1 a 1 · · · X da d.
(e) If x 1 ,... , xd is a regular sequence on M , then M/In+1M has a filtration in which the factors are
(n+d d
copies of M/IM.
Proof. (a) It suffices to prove the statement in the case where just one of ti is different from 1: we can adjust the exponents on one element at a time. Since R-sequences are permutable, it suffices to do the case where only td is different from 1, and for this purpose we may work with M/(x 1 ,... , xd− 1 )M. Thus, we may assume that d = 1, and the assertion we need is that xt^ is a nonzerodivisor if and only if x is. Clearly, if xw = 0 then xtw = 0, while if xtw = 0 for t chosen as small as possible and w 6 = 0 then x(xt−^1 w) = 0.
(b) If all the ai are zero then we are already done. If not, we use induction on the number of ai > 0. Since we are assuming a situation in which R-sequences on a module are permutable we may assume that ad > 0. Then
xa 1 1 · · · xa dd w =
d∑− 1
j=
xa j j^ +1wj + xa dd +1wd
This gives the result, for we then have
`(M/In+1M ) =
n + d d
`(M/IM ),
and the leading term of
n + d d
is
nd d!
Theorem. Let (R, m, K) be module-finite over a regular local ring A such that x 1 ,... , xd is a regular system of parameters of A, and let M be an R-module of dimension d. Let I = (x 1 ,... , xd)R. Then eI (M ) is the torsion-free rank of M over A.
Proof. From the definition, it does not matter whether we think of M as an R-module, or whether we think of it as an A-module with maximal ideal n = (x 1 ,... , xd)A. In the latter case, if ρ is the torsion-free rank of M as an A-module, we have an exact sequence of A-modules 0 → Aρ^ → M → C → 0
where C is a torsion A-module, so that dim (C) < d. It follows that
eI (M ) = en(M ) = ren(A) + 0 = r`(A/(x 1 ,... , xd)A
= r · 1 = r.
Discussion. If R is equicharacteristic, we can always reach the situation of the Theorem above. The mulltiplicity does not change if we replace R by R̂. But then we can choose a coefficient field K, and the structure theorems for complete local rings guarantee that R
is module-finite over A = K[[x 1 ,... , xd]] ⊆ R̂.
More generally:
Theorem. Let R be module-finite over a Cohen-Macaulay local ring B such that x 1 ,... , xd is a system of parameters for B. Let M be an R-module of dimension d. Let I = (x 1 ,... , xd)R. If B is a domain, eI (M ) = `(B/IB)ρ, where ρ is the torsion-free rank of M over B. When B is not a domain, if there is a short exact sequence
0 → Bρ^ → M → C → 0
with dim (C) < d, then eI (M ) = `(B/IB)ρ.
Proof. (M/In+1)M is independent of whether one thinks of x 1 ,... , xd as in B or in R. Thus, we can replace R by B. The result is then immediate from our results on additivity of multiplicity and the fact that when B is Cohen-Macaulay, eI (B) =(B/I).
We want to give a different characterization of multiplicities due to C. Lech. If n = n 1 ,... , nd is a d-tuple of nonnegative integers and f is a real-valued function of n, we
write lim n→∞ f (n) = r, where r ∈ R, to mean that for all > 0 there exists N such that
for all n = n 1 ,... , nd satisfying ni ≥ N , 1 ≤ i ≤ d, we have that |f (n) − r| < . One might also write lim min n→∞
f (n) = r with the same meaning. If x = x 1 ,... , xd is a system
of parameters for R, we temporarily define the Lech multiplicity eL x (M ) to be
lim n→∞
`
M/(xn 1 1 ,... , xn d d)M
n 1 · · · nd
We shall show that the limit always exists, is 0 if dim (M ) < d, and, with I = (x 1 ,... , xd)R, is eI (M ) when dim (M ) = d.
We first prove:
Lemma. Let x = x 1 ,... , xd, d ≥ 1 , be a system of parameters for a local ring (R, m, K) and let M ′, M , and M ′′^ be finitely generated R-modules. Given n = n 1 ,... , nd, let In = (xn 1 1 ,... , xn d d)R, and let Ln(M ) = `(M/InM )/n 1 · · · nd.
(a) If 0 → M ′^ → M → M ′′^ → 0
is exact, then for any m-primary ideal J,
(M ′′/JM ′′) ≤(M/JM ) ≤ (M ′/JM ′) +(M ′′/JM ′′),
i.e., 0 ≤ (M/JM ) −(M ′′/JM ′′) ≤ `(M ′/JM ′).
Hence, for all n, Ln(M ′′) ≤ Ln(M ) ≤ Ln(M ′) + Ln(M ′′),
i.e., 0 ≤ Ln(M ) − Ln(M ′′) ≤ Ln(M ′).
Therefore, if the three limits exist,
eL x (M ′′) ≤ eL x (M ) ≤ eL x (M ′) + eL x (M ′′),
If eL x (M ′) = 0 and eL x (M ′′) = 0, then eL x (M ) = 0.
(b) If M has a finite filtration with factors Nj we have that for any m-primary ideal J, `(M/JM ) ≤
j `(Nj^ /JNj^ ). Hence, for all^ n,^ Ln(M^ )^ ≤^
j Ln(Nj^ ), and^ e
L x (M^ ) = 0 whenever eL x (Nj ) = 0 for all j.
(c) If dim (M ) < d then eL x (M ) = 0.
(d) If 0 → M ′^ → M → M ′′^ → 0 is exact and dim (M ′) < d, then eL x (M ) and eL x (M ′′) exist or not alike, and if they exist they are equal.
