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 Lech’s Conjecture 1 Main Results 2 Sketch of Proof of Theorem A 3 Further Questions 4 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 e R ≤ e S . Here, e R denotes the classical Hilbert-Samuel multiplicity of R (with respect to the maximal ideal): e R = lim t →∞ d ! · l R ( R / m t ) t d 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 e R P ≤ e R .) Linquan Ma (Purdue University) August 4th 4 / 23
Lech’s Conjecture Related Questions We could ask some related questions/generalizations: 1 Hilbert functions: Is H 0 R ( t ) := l ( m t / m t +1 ) ≤ l ( n t / n t +1 ) for all t for flat local extensions with dim R = dim S ? Or more generally, we could R ( t ) = � t j =0 H i − 1 define H i ( j ) and ask whether there exists i such R that H i R ( t ) ≤ H i S ( t ) for all t ? This implies Lech’s conjecture because the multiplicity is the leading coefficient of H i 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 e HK ( R ) ≤ e HK ( 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 / m S is a complete intersection. 3 (follows from Backelin-Herzog-Ulrich 90’) R is a strict complete intersection: gr m R 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, on Lech’s inequality H i R ( t ) ≤ H i S ( t ). Most of his results are obtained by putting some (technical) conditions on the closed fibre S / m S , which can generalize and recover Lech’s result when S / m S 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 / m S 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 / m S into 115 classes, and he could show that, among 83 out of the 115 class of singularities, there exists i with H i R ( t ) ≤ H i 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 { M i } such that lim e ( M i ) ν ( M i ) = 1, then Lech’s Conjecture holds for R : e ( M i ) rank ( M i ) · lim e ( M i ) ν ( M i ) e R = lim rank ( M i ) = lim ν ( M i ) rank ( M i ⊗ S ) · lim e ( M i ) ν ( M i ⊗ S ) rank ( M i ⊗ S ) · lim e ( M i ) e ( M i ⊗ S ) = lim ν ( M i ) ≤ ν ( M i ) = e S . 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 { M i } such that lim e ( M i ) ν ( M i ) = 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 { R 1 / p e } . Unfortunately these are neither maximal Cohen-Macaulay modules in general nor does e ( R 1 / pe ) e ( R ) ν ( R 1 / pe ) converge to 1 (it tends to e HK ( 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 : k p ] < ∞ where k = R / m . Then e R ≤ e S . 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 of dimension d and characteristic p > 0 . Suppose [ k : k p ] = p α < ∞ where k = R / m . If edimS − edimR ≥ max { d , d ! + d − 2 d } , then e R ≤ e S . Theorem B (-) Let ( R , m ) → ( S , n ) be a flat local extension between complete local rings of dimension d and characteristic p > 0 . Then we have the following: (i) If edimS − edimR ≤ 1 , then e R ≤ e S (this is known to experts). (ii) If edimS − edimR = 2 and R is equidimensional with depthR ≥ d − 2 , then e R ≤ e S . (iii) If edimS − edimR = 3 and R is Cohen-Macaulay, then e R ≤ e S . 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 e R ≤ e S . Linquan Ma (Purdue University) August 4th 12 / 23
Sketch of Proof of Theorem A Proof of Hanes’s theorem Pick a minimal reduction x = x 1 , . . . , x d of n . We have: p ed · e ( x , S ) = e ( x , R 1 / p e ⊗ S ) p ed · e S = R 1 / p e ⊗ S = l ( ( x ) · ( R 1 / p e ⊗ S )) R 1 / p e ⊗ S ≥ l ( ( m + x ) · ( R 1 / p e ⊗ S )) � � ( m + x ) S ) dim( R 1 / pe / m R 1 / pe ) S = l ( Now divide by p ed and let e → ∞ , this computation gives: S e S ≥ e HK ( R ) · l ( ( m + x ) S ) ≥ e HK ( R ) · d ! ≥ e R . 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 = x 1 , . . . , x d of S, we have e ( x , N ) ≥ ν S ( n N ) + (1 − d ) ν S ( N ) − χ 1 ( x , N ) . Proof Sketch : d � ( − 1) i l S ( H i ( x , N )) = l S ( N e ( x , N ) = ( x ) N ) − χ 1 ( x , N ) i =0 n ( x ) N ) − l S ( ( x ) N N = l S ( n ( x ) N ) − χ 1 ( x , N ) l S ( N / n 2 N ) − d · ν S ( N ) − χ 1 ( x , N ) ≥ = ν S ( n N ) + (1 − d ) ν S ( N ) − χ 1 ( 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 = R 1 / p e ⊗ S , just as in the proof of Hanes’s theorem. To get rid of χ 1 , we need: Lemma Let ( R , m ) → ( S , n ) be a flat local extension between complete local rings of dimension d. Then for every system of parameters x = x 1 , . . . , x d of S and every i > 0 , we have l S ( H i ( x , R 1 / p e ⊗ S )) lim = 0 . p ed e →∞ This lemma essentially follows from earlier results of Dutta, Roberts, Hochster-Huneke...... (in various forms) Linquan Ma (Purdue University) August 4th 15 / 23
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