Lechs conjecture in dimension three Linquan Ma Midwest Commutative - - PowerPoint PPT Presentation

Lech’s conjecture in dimension three

Linquan Ma

Midwest Commutative algebra Conference at Purdue University lquanma1019@gmail.com

August 4th

Linquan Ma (Purdue University) August 4th 1 / 23

Overview

1

Lech’s Conjecture

2

Main Results

3

Sketch of Proof of Theorem A

4

Further Questions

Linquan Ma (Purdue University) August 4th 2 / 23

Lech’s Conjecture

Lech’s Conjecture

Around 1960, C. Lech made the following remarkable conjecture on the Hilbert-Samuel multiplicities: Conjecture (Lech’s Conjecture) Let (R, m) → (S, n) be a flat local extension of local rings. Then eR ≤ eS. Here, eR denotes the classical Hilbert-Samuel multiplicity of R (with respect to the maximal ideal): eR = limt→∞ d! · lR(R/mt)

td

Linquan Ma (Purdue University) August 4th 3 / 23

Lech’s Conjecture

Remarks and Reductions

We start by observing the following:

1 Trivial if R is regular; True if S is regular. 2 We may assume R, S both complete, with infinite residue field.

(in particular, may assume minimal reductions of m and n exist.)

3 We may assume dim R = dim S and R is a domain.

(localization theorem: If P is a prime ideal of an excellent local ring R such that dim R/P + htP = dim R, then eRP ≤ eR.)

Linquan Ma (Purdue University) August 4th 4 / 23

Lech’s Conjecture

Related Questions

We could ask some related questions/generalizations:

1 Hilbert functions: Is H0

R(t) := l(mt/mt+1) ≤ l(nt/nt+1) for all t for

flat local extensions with dim R = dim S? Or more generally, we could define Hi

R(t) = t j=0 Hi−1 R

(j) and ask whether there exists i such that Hi

R(t) ≤ Hi S(t) for all t? This implies Lech’s conjecture because

the multiplicity is the leading coefficient of Hi

R(t) up to a constant.

These stronger questions are all open in general. True for t ≤ 1 (Lech 60’), i.e., we always have edimR ≤ edimS.

2 Localization formula in general: Does the localization theorem on

multiplicities hold for non-excellent R? It turns out that this is equivalent to Lech’s conjecture! (Larfeldt-Lech 80’)

3 Other measures of singularities: Is eHK(R) ≤ eHK(S) for every flat

local extensions? Yes! (Hanes 00’)

Linquan Ma (Purdue University) August 4th 5 / 23

Lech’s Conjecture

Past results on Lech’s Conjecture

Lech’s Conjecture was known in the following cases:

1 (Lech 60’) dim R ≤ 2. 2 (Lech 60’) S/mS is a complete intersection. 3 (follows from Backelin-Herzog-Ulrich 90’) R is a strict complete

intersection: grmR is a complete intersection (e.g., hypersurfaces).

4 (Hanes 00’) R is a three-dimensional standard graded k-algebra and k

is perfect of characteristic p > 0.

5 (Hanes 00’) R, S both standard graded and the map sends a minimal

reduction of m to homogeneous elements of S.

Linquan Ma (Purdue University) August 4th 6 / 23

Lech’s Conjecture

• B. Herzog’s work
• B. Herzog made a very deep study on Lech’s conjecture: more precisely,
• n Lech’s inequality Hi

R(t) ≤ Hi S(t). Most of his results are obtained by

putting some (technical) conditions on the closed fibre S/mS, which can generalize and recover Lech’s result when S/mS is a complete intersection. He also made an extensive study of examples/classifications. Here is one experiment of B. Herzog: Let R → S be a flat local extension with R/m → S/n separable and dim R = dim S. Suppose S/mS is isomorphic to k[x, y, z]/I, where I is generated by power products of x, y, z of degrees two and three, and k is the finite field with 31991 elements. In this very special case, B. Herzog classified S/mS into 115 classes, and he could show that, among 83 out of the 115 class of singularities, there exists i with Hi

R(t) ≤ Hi S(t) for all t.

Linquan Ma (Purdue University) August 4th 7 / 23

Lech’s Conjecture

Ulrich modules and Lech’s Conjecture

Theorem/Definition: For any maximal Cohen-Macaulay module M over (R, m), we have e(M) ≥ ν(M). If equality holds, then M is called a Ulrich module over R. Observation (Hochster): If R admits a sequence of maximal Cohen-Macaulay modules {Mi} such that lim e(Mi)

ν(Mi) = 1, then Lech’s

Conjecture holds for R: eR = lim e(Mi) rank(Mi) = lim ν(Mi) rank(Mi) · lim e(Mi) ν(Mi) = lim ν(Mi ⊗ S) rank(Mi ⊗ S) · lim e(Mi) ν(Mi) ≤ e(Mi ⊗ S) rank(Mi ⊗ S) · lim e(Mi) ν(Mi) = eS.

Linquan Ma (Purdue University) August 4th 8 / 23

Lech’s Conjecture

Existence of Ulrich modules

In general, the existence of Ulrich modules seems very difficult to prove, even for Cohen-Macaulay rings. The following related results are known:

1 (Backelin-Herzog-Ulrich) If R is a strict complete intersection, then R

admits Ulrich modules.

2 (Hanes) If R is a three-dimensional standard graded k-algebra and k

is perfect of characteristic p > 0, then R admits a a sequence of maximal Cohen-Macaulay modules {Mi} such that lim e(Mi)

ν(Mi) = 1.

In general, we don’t even know whether R admits a finitely generated maximal Cohen-Macaulay module! This is open as long as dim R ≥ 3. In characteristic p > 0, we have a “natural” sequence {R1/pe}. Unfortunately these are neither maximal Cohen-Macaulay modules in general nor does e(R1/pe )

ν(R1/pe ) converge to 1 (it tends to e(R) eHK (R) ≥ 1).

Linquan Ma (Purdue University) August 4th 9 / 23

Main Results

Main Result

Our main theorem on Lech’s conjecture is the following: Theorem (-) Let (R, m) → (S, n) be a flat local extension of local rings of equal characteristic p > 0. Suppose dim R = 3 and [k : kp] < ∞ where k = R/m. Then eR ≤ eS. The above result obviously generalizes Hanes’s result in the standard graded case. Moreover, it seems promising that we can use standard reduction to characteristic p > 0 technique to prove the corresponding theorem in equal characteristic 0.

Linquan Ma (Purdue University) August 4th 10 / 23

Main Results

Two general estimates

Theorem A (-) Let (R, m) → (S, n) be a flat local extension between complete local rings

• f dimension d and characteristic p > 0. Suppose [k : kp] = pα < ∞

where k = R/m. If edimS − edimR ≥ max{d, d! + d − 2d}, then eR ≤ eS. Theorem B (-) Let (R, m) → (S, n) be a flat local extension between complete local rings

• f dimension d and characteristic p > 0. Then we have the following:

(i) If edimS − edimR ≤ 1, then eR ≤ eS (this is known to experts). (ii) If edimS − edimR = 2 and R is equidimensional with depthR ≥ d − 2, then eR ≤ eS. (iii) If edimS − edimR = 3 and R is Cohen-Macaulay, then eR ≤ eS.

Linquan Ma (Purdue University) August 4th 11 / 23

Sketch of Proof of Theorem A

A weaker result of Hanes

We first prove a result which is weaker than Theorem A, however this result (and its proof) motivates the idea behind Theorem A. To avoid technicality we assume that R is a complete local domain with k = R/m perfect throughout. Theorem (Hanes) Let (R, m) → (S, n) be a flat local extension between complete local Cohen-Macaulay rings of dimension d and characteristic p > 0. If edimS − edimR ≥ d! + d, then eR ≤ eS.

Linquan Ma (Purdue University) August 4th 12 / 23

Sketch of Proof of Theorem A

Proof of Hanes’s theorem

Pick a minimal reduction x = x1, . . . , xd of n. We have: ped · eS = ped · e(x, S) = e(x, R1/pe ⊗ S) = l( R1/pe ⊗ S (x) · (R1/pe ⊗ S)) ≥ l( R1/pe ⊗ S (m + x) · (R1/pe ⊗ S)) = l

• (

S (m + x)S )dim(R1/pe /mR1/pe )

• Now divide by ped and let e → ∞, this computation gives:

eS ≥ eHK(R) · l( S (m + x)S ) ≥ eHK(R) · d! ≥ eR.

Linquan Ma (Purdue University) August 4th 13 / 23

Sketch of Proof of Theorem A

Three crucial Lemmas

Lemma Let (S, n) be a local ring of dimension d and N be a finitely generated S-module. Then for every system of parameters x = x1, . . . , xd of S, we have e(x, N) ≥ νS(nN) + (1 − d)νS(N) − χ1(x, N). Proof Sketch: e(x, N) =

d

• i=0

(−1)ilS(Hi(x, N)) = lS( N (x)N ) − χ1(x, N) = lS( N n(x)N ) − lS( (x)N n(x)N ) − χ1(x, N) ≥ lS(N/n2N) − d · νS(N) − χ1(x, N) = νS(nN) + (1 − d)νS(N) − χ1(x, N)

Linquan Ma (Purdue University) August 4th 14 / 23

Sketch of Proof of Theorem A

Three crucial Lemmas–continued

Eventually we would apply the previous lemma to N = R1/pe ⊗ S, just as in the proof of Hanes’s theorem. To get rid of χ1, we need: Lemma Let (R, m) → (S, n) be a flat local extension between complete local rings

• f dimension d. Then for every system of parameters x = x1, . . . , xd of S

and every i > 0, we have lim

e→∞

lS(Hi(x, R1/pe ⊗ S)) ped = 0. This lemma essentially follows from earlier results of Dutta, Roberts, Hochster-Huneke...... (in various forms)

Linquan Ma (Purdue University) August 4th 15 / 23

Sketch of Proof of Theorem A

Three crucial Lemmas–continued

Theorem (Roberts) Let (R, m) be a complete local ring of characteristic p > 0 and dimension

• d. Let G• be a bounded complex of finite free R-modules of length d with

homology of finite length. Then for every i ≥ 1 we have lim

e→∞

lR(Hi(G• ⊗ R1/pe)) ped = 0. This theorem easily implies our lemma if x is a system of parameters of R: apply theorem to G• = K(x, R) and then tensor with S. More generally we can show that if it holds for one system of parameters, then it holds for all system of parameters (details omitted).

Linquan Ma (Purdue University) August 4th 16 / 23

Sketch of Proof of Theorem A

Three crucial Lemmas–continued

Lemma (Hanes) Let (R, m) → (S, n) be a local map and let M be a finitely generated module over R. Then: νS(nM′) ≥ νR(mM) + (edimS − edimR) · νR(M) where M′ = S ⊗R M. Idea of Proof: This is trivial if M = Rn is free. The general case follows by an easy induction, that is, we can prove the statement for N = M/Ry assuming it is true for M.

Linquan Ma (Purdue University) August 4th 17 / 23

Sketch of Proof of Theorem A

Sketch of Proof of Theorem A

Apply the lemmas to N = R1/pe ⊗ S. We have e(x, R1/pe ⊗ S) ≥ νS(n(R1/pe ⊗ S)) + (1 − d) · νS(R1/pe ⊗ S) −χ1(x, R1/pe ⊗ S) ≥ νR(mR1/pe) + (edimS − edimR + 1 − d) · νR(R1/pe) −χ1(x, R1/pe ⊗ S) Now we observe that when e → ∞, we have: e(x, S ⊗R R1/pe) = e(x, S) · ped, ν(R1/pe) = lR( R1/pe mR1/pe ) → eHK(R) · ped, χ1(x, R1/pe ⊗ S) =

d

• i=1

(−1)i−1lS(Hi(x, R1/pe ⊗ S)) = o(ped),

Linquan Ma (Purdue University) August 4th 18 / 23

Sketch of Proof of Theorem A

Sketch of Proof of Theorem A–continued

ν(mR1/pe) = lR( mR1/pe m2R1/pe ) = lR( R1/pe m2R1/pe ) − lR( R1/pe mR1/pe ) → (eHK(m2, R) − eHK(R)) · ped. Eventually we get (details omitted), taking x to be a minimal reduction of n and using edimS − edimR ≥ max{d, d! + d − 2d}, that: eS = e(x, S) ≥ eHK(m2, R) + (edimS − edimR − d) · eHK(R) ≥ 1 d!e(m2, R) + 1 d!(edimS − edimR − d) · eR = edimS − edimR + 2d − d d! · eR ≥ eR

Linquan Ma (Purdue University) August 4th 19 / 23

Further Questions

On the constraints of residue field

Question 1: Can we reduce Lech’s conjecture, at least in equal characteristic, to the case that k = R/m is perfect or algebraically closed? One could hope to apply − ⊗kk to both R and S. However the subtle point is that the image of k might not be contained in a coefficient field of S, and that S ⊗kk might not be Noetherian.

Linquan Ma (Purdue University) August 4th 20 / 23

Further Questions

Cohen-Factorization

Cohen Factorization (Avramov-Foxby-Herzog 90’) A local homomorphism (R, m) → (S, n) with S complete can be factored as (R, m) → (T, nT) → (S, n) such that (R, m) → (T, nT) is flat local with T/mT regular, (T, nT) is complete, and (T, nT) → (S, n) is

• surjective. Moreover, if S has finite flat dimension over R (e.g., S is flat
• ver R), then S has finite projective dimension over T.

Question 2: Let (R, m) be a local ring and let J ⊂ R be an ideal of finite projective dimension. Is eR ≤ eR/J? The above result of Avramov-Foxby-Herzog easily implies that Question 2 implies Lech’s conjecture. On the other hand, we don’t know how to prove Question 2 even in the case that R/J has finite length i.e., eR ≤ l(R/J)? (we could show that even this case would imply Lech’s conjecture for Cohen-Macaulay rings)

Linquan Ma (Purdue University) August 4th 21 / 23

Further Questions

Cohen-Factorization–continued

Nonetheless, the idea of using Cohen-Factorization can yield many partial cases of Lech’s conjecture. For example, when S/mS is a complete intersection, it can be shown that once we do the Cohen-Factorization, the map T → S is simply killing a regular sequence. Since it is well known that multiplicity does not drop when we kill a regular sequence, this gives a very quick proof of Lech’s result that eR ≤ eS when S/mS is a complete intersection. Cohen-Factorization also plays a crucial rule in our proof of Theorem B.

Linquan Ma (Purdue University) August 4th 22 / 23

Further Questions

Thank you!

Linquan Ma (Purdue University) August 4th 23 / 23