Lim Ulrich sequence: proof of Lech's conjecture for graded base rings
aa r X i v : . [ m a t h . A C ] M a y LIM ULRICH SEQUENCE: PROOF OF LECH’S CONJECTURE FORGRADED BASE RINGS
LINQUAN MA
Abstract.
The long standing Lech’s conjecture in commutative algebra states that for aflat local extension ( R, m ) −→ ( S, n ) of Noetherian local rings, we have an inequality on theHilbert–Samuel multiplicities: e ( R ) ≤ e ( S ). In general the conjecture is wide open as longas dim R >
3, even in equal characteristic. In this paper, we prove Lech’s conjecture in alldimensions, provided ( R, m ) is a standard graded ring over a perfect field (localized at thehomogeneous maximal ideal).We introduce the notions of lim Ulrich and weakly lim Ulrich sequence. Roughly speakingthese are sequences of finitely generated modules that are not necessarily Cohen–Macaulay,but asymptotically behave like Ulrich modules. We prove that the existence of these se-quences imply Lech’s conjecture. Though the existence of Ulrich modules is known in verylimited cases, we construct weakly lim Ulrich sequences for all standard graded domainsover perfect field of positive characteristic. Introduction and preliminaries
Around 1960, Lech made the following remarkable conjecture on the Hilbert–Samuel mul-tiplicities [Lec60]:
Conjecture (Lech’s conjecture) . Let ( R, m ) −→ ( S, n ) be a flat local extension of Noetherianlocal rings. Then e ( R ) ≤ e ( S ) . This conjecture has now stood for sixty years and remains open in most cases (despiteit is very simple to state). In [Lec60, Lec64] the conjecture is proven when dim R ≤ S/ m S is a complete intersection. In [Ma17] the conjecture is provenwhen dim R = 3 and R has equal characteristic. For some other partial progress and relatedresults on Lech’s conjecture, see [Her94, Han99, Han05, Ma14, Ma17]. The main result ofthis paper settles Lech’s conjecture for a large class of rings, in arbitrary dimension. Theorem A (=Theorem 3.8) . Let ( R, m ) −→ ( S, n ) be a flat local extension of Noetherianlocal rings. Suppose ( R, m ) is a standard graded ring over a perfect field (localized at thehomogeneous maximal ideal). Then e ( R ) ≤ e ( S ) . Our main new ingredient in the proof of Theorem A (and which we hope to attack Lech’sconjecture in general) is a notion called (weakly) lim Ulrich sequence , which is a special typeof (weakly) lim Cohen–Macaulay sequence developed by Bhatt, Hochster and the author in[BHM] (see also [Hoc17]).Roughly speaking, a sequence of finitely generated modules of maximal dimension is limCohen–Macaulay (resp. weakly lim Cohen–Macaulay) if the lengths of their higher Koszulhomology modules (resp. their first higher Euler characteristics) with respect to a system of That is, N -graded and generated by degree one forms. arameters grow relative slowly compared to their minimal number of generators. Further-more, a (weakly) lim Cohen–Macaulay sequence is called (weakly) lim Ulrich if their minimalnumber of generators are approaching to their Hilbert–Samuel multiplicities. Clearly, a small(i.e., finitely generated) maximal Cohen–Macaulay module induces a constant lim Cohen–Macaulay sequence, and a Ulrich module induces a constant lim Ulrich sequence.One of the main results in [BHM] is that the existence of lim Cohen–Macaulay sequencesimplies Serre’s conjecture on positivity of intersection multiplicities, which greatly extendsthe earlier observation that the existence of small Cohen–Macaulay modules implies Serre’sconjecture. Similarly, it was an earlier observation of Hochster–Huneke and Hanes that theexistence of Ulrich modules implies Lech’s conjecture, see [Han99]. We largely generalizethis idea and prove the following Theorem B (=Theorem 2.8) . Let ( R, m ) −→ ( S, n ) be a flat local extension of Noetherianlocal rings such that R is a domain and dim R = dim S . Suppose R admits a weakly limUlrich sequence. Then e ( R ) ≤ e ( S ) . Since Ulrich modules were introduced in [Ulr84], a fundamental open question is theirexistence. One difficulty in the general case is that we do not know the existence of smallCohen–Macaulay modules. However, even if we restrict ourselves to Cohen–Macaulay rings,the existence of Ulrich modules is not known (even in dimension two or graded dimensionthree!). In fact, only in some very limited cases (e.g., rings with strong combinatorial proper-ties) could we establish the existence of Ulrich modules, and the method is usually difficult:for example see [HUB91] or [ESW03].The main contribution of this paper shows that, on the other hand, weakly lim Ulrichsequences always exist for standard graded rings of positive characteristic. This vastly gen-eralizes, in certain sense, our understanding of Ulrich-like modules, and it leads to theaforementioned result on Lech’s conjecture in characteristic p >
0. The characteristic 0 caseof Theorem A then follows from a reduction to characteristic p >
Theorem C (=Theorem 3.5) . Let ( R, m ) be a Noetherian standard graded domain over aninfinite F -finite field of characteristic p > (localized at the homogeneous maximal ideal).Then R admits a weakly lim Ulrich sequence. It should be pointed out that, even when ( R, m ) is Cohen–Macaulay, the constructedmodules in the weakly lim Ulrich sequence in Theorem C are not maximal Cohen–Macaulay.Thus it is very important that we allow the weakly lim Cohen–Macaulay property, that is,the variations on the asymptotic behavior of the higher Koszul homology modules (ratherthan requiring them to be zero).Throughout the rest of this paper, all rings are commutative, Noetherian, with multiplica-tive identity. We use ν ( M ) or ν R ( M ) to denote the minimal number of generators of an R -module M , e ( I, M ) to denote the Hilbert–Samuel multiplcity of M with respect to an m -primary ideal I ⊆ R , H i ( x, M ) to denote the Koszul homology module of M with respectto a system of parameters x , and χ ( x, M ) := P di =1 ( − i − ℓ ( H i ( x, M )) to denote the firsthigher Euler characteristic. For basic properties on Hilbert–Samuel multiplicities and higherEuler characteristics, we refer to [Ser65, Eis95, HS06]. That is, a small maximal Cohen–Macaulay module whose minimal number of generators equals to itsHilbert–Samuel multiplicity [Ulr84]. . Weakly lim Cohen–Macaulay and weakly lim Ulrich sequence
In this section we introduce lim Ulrich and weakly lim Ulrich sequence. These definitionsdepend on the notion of lim Cohen–Macaulay sequence developed in [BHM] as well as itsvariations. The notion of weakly lim Cohen–Macaulay sequence we define below appeared in[Hoc17, Section 9] (in relation with the monomial property of system of parameters). Herewe formally introduce this concept and investigate its properties.
Definition 2.1.
Let ( R, m ) be a local ring of dimension d . A sequence of finitely generated R -modules { M n } of dimension d is called lim Cohen–Macaulay , if there exists a system ofparameters x = x , . . . , x d of R such that for all i ≥ ℓ ( H i ( x, M n )) = o ( ν ( M n )). { M n } iscalled weakly lim Cohen–Macaulay , if there exists a system of parameters x = x , . . . , x d of R such that χ ( x, M n ) = o ( ν ( M n )). Remark . It is worth to point out that, under the above definitions, there do exist weaklylim Cohen–Macaulay sequences that are not lim Cohen–Macaulay, see [Hoc17, paragraphbefore Conjecture 10.1].We begin by collecting some simple facts about weakly lim Cohen–Macaulay sequence.
Lemma 2.3.
Let ( R, m ) −→ ( S, n ) be a flat local extension of local rings such that dim R =dim S . If { M n } is a (weakly) lim Cohen–Macaulay sequence for R , then { M n ⊗ R S } is a(weakly) lim Cohen–Macaulay sequence for S .Proof. This is immediate by noting that, since ( R, m ) −→ ( S, n ) is flat local with dim R =dim S , we have ℓ S ( H i ( x, M n ⊗ R S )) = ℓ R ( H i ( x, M n )) · ℓ S ( S/ m S )and ν S ( M n ⊗ R S ) = ν R ( M n ). (cid:3) Lemma 2.4.
Let ( R, m ) be a local ring of dimension d . Then a sequence of finitely generatedmodules { M n } of dimension d is weakly lim Cohen–Macaulay if and only if there exists asystem of parameters x = x , . . . , x d of R such that lim n −→ ∞ e ( x, M n ) ℓ ( M n / ( x ) M n ) = 1 ( or equivalently, lim n −→ ∞ χ ( x, M n ) ℓ ( M n / ( x ) M n ) = 0) . Moreover, if R is a domain and { M n } is weakly lim Cohen–Macaulay, then there exists aconstant C such that for all n , rank R ( M n ) ≤ ν ( M n ) ≤ C · rank R ( M n ) . In particular, if R is a domain then we can use rank R ( M n ) in place of ν ( M n ) in the definitionof lim Cohen–Macaulay and weakly lim Cohen–Macaulay sequence.Proof. The first conclusion is clear since ν ( M n ) ≤ ℓ ( M n / ( x ) M n ) ≤ ν ( M n ) · ℓ ( R/ ( x )) (thusasymptotically it doesn’t matter whether we use ℓ ( M n / ( x ) M n ) or ν ( M n ) in the denominator).To see the second conclusion, we note thatrank R ( M n ) · e ( x, R ) = e ( x, M n ) = ℓ ( M n / ( x ) M n ) − χ ( x, M n ) ≥ ν ( M n ) − χ ( x, M n ) . Dividing by ν ( M n ) we obtain thatrank R ( M n ) · e ( x, R ) ν ( M n ) ≥ − χ ( x, M n ) ν ( M n ) . ince { M n } is weakly lim Cohen–Macaulay, the right hand side tends to 1 when n −→ ∞ .Thus there exists ǫ > n sufficiently large, ν ( M n ) ≤ (1 + ǫ ) e ( x, R ) rank R ( M n ) . We now simply pick C ≫ (1 + ǫ ) e ( x, R ) that also works for all small values of n . (cid:3) In [BHM], it is proved that the definition of lim Cohen–Macaulay sequence is independentof the choice of the system of parameters x . Here we prove the analogous statement forweakly lim Cohen–Macaulay sequence. The proof is quite non-obvious (however, we pointout that this is needed, because eventually we can only show that our construction leads toweakly lim Cohen–Macaulay sequence: see Theorem 3.5). Proposition 2.5.
Let ( R, m ) be a local ring of dimension d . If { M n } is a weakly lim Cohen-Macaulay sequence, then ( † ) lim n −→ ∞ e ( x, M n ) ℓ ( M n / ( x ) M n ) = 1 ( or equivalently, lim n −→ ∞ χ ( x, M n ) ℓ ( M n / ( x ) M n ) = 0) for every system of parameters x = x , . . . , x d of R . As a consequence, if { M n } is a weaklylim Cohen-Macaulay sequence, then χ ( x, M n ) = o ( ν ( M n )) for every system of parameters x = x , . . . , x d of R .Proof. We first note that if ( † ) holds for x = x , . . . , x d , then it holds for x t = x t , . . . , x t d d .This is because e ( x t , M ) = t · · · t d · e ( x, M ) while ℓ ( M n / ( x t ) M n ) ≤ t · · · t d · ℓ ( M n / ( x ) M n ),and the limit in ( † ) is always ≤ x = x , . . . , x d and y = y , . . . , y d of R ,we can always connect x, y by a chain of system of parameters such that each two consecutiveonly differ by one element. Thus it suffices to show that if ( † ) holds for x, x , . . . , x d , thenit holds for y, x , . . . , x d . By the discussion in the first paragraph we can replace x by x t for t ≫ x, x , . . . , x d ) ⊆ ( y, x , . . . , x d ), and thus by a change of variables wemay assume x = yz . Thus it is enough to prove that if ( † ) holds for yz, x , . . . , x d , then itholds for y, x , . . . , x d .From now on we use x − to denote x , . . . , x d . For each M ∈ { M n } , we have ℓ ( M/ ( yz, x − ) M ) − χ (( yz, x − ) , M ) = e (( yz, x − ) , M ) ≤ e ( yz, M/ ( x − ) M ) = ℓ ( M/ ( yz, x − ) M ) − ℓ (Ann M/x − M yz ) . Thus ℓ (Ann M/x − M yz ) ≤ χ (( yz, x − ) , M ). Since we assume ( † ) holds for ( yz, x − ), we havelim n −→ ∞ ℓ (Ann M n /x − M n yz ) ν ( M n ) ≤ ℓ ( R/ ( yz, x − )) · lim n −→ ∞ χ (( yz, x − ) , M n ) ℓ ( M n / ( yz, x − ) M n ) = 0 . Since Ann M n /x − M n y and Ann M n /x − M n z are submodules of Ann M n /x − M n yz , we have(2.5.1) lim n −→ ∞ ℓ (Ann M n /x − M n y ) ν ( M n ) = 0 , and lim n −→ ∞ ℓ (Ann M n /x − M n z ) ν ( M n ) = 0 . t this point, we look at the long exact sequence of the Koszul homology:0 −→ H d (( yz, x − ) , M ) −→ H d − ( x − , M ) yz −→ H d − ( x − , M ) −→ H d − (( yz, x − ) , M ) −→−→ H d − ( x − , M ) yz −→ H d − ( x − , M ) −→ H d − (( yz, x − ) , M ) −→−→ · · · · · ·−→ H ( x − , M ) yz −→ H ( x − , M ) −→ H (( yz, x − ) , M ) −→ Ann
M/x − M yz −→ . Recall that if N is any finitely generated R -module and w ∈ R is such that ℓ ( N/wN ) < ∞ ,then e ( w, N ) = ℓ ( N/wN ) − ℓ (Ann N w ). Thus taking the alternating sum of lengths andmultiplicities of the above long exact sequence, we get:(2.5.2) d − X j =1 ( − j − e ( yz, H j ( x − , M )) = χ (( yz, x − ) , M ) − ℓ (Ann M/x − M yz ) . The same argument shows that(2.5.3) d − X j =1 ( − j − e ( y, H j ( x − , M )) = χ (( y, x − ) , M ) − ℓ (Ann M/x − M y ) , and(2.5.4) d − X j =1 ( − j − e ( z, H j ( x − , M )) = χ (( z, x − ) , M ) − ℓ (Ann M/x − M z ) . Since we assume ( † ) holds for ( yz, x − ), applying (2.5.2) for each M ∈ { M n } shows that (2.5.5) lim n −→ ∞ P d − j =1 ( − j − e ( yz, H j ( x − , M n )) ν ( M n ) ≤ ℓ ( R/ ( yz, x − )) · lim n −→ ∞ χ (( yz, x − ) , M n ) ℓ ( M n / ( yz, x − ) M n ) = 0 . By (2.5.1), (2.5.3), (2.5.4) and the non-negativity of χ , we have(2.5.6) lim n −→ ∞ P d − j =1 ( − j − e ( y, H j ( x − , M n )) ν ( M n ) = lim n −→ ∞ χ (( y, x − ) , M n ) ν ( M n ) ≥ , and(2.5.7) lim n −→ ∞ P d − j =1 ( − j − e ( z, H j ( x − , M n )) ν ( M n ) = lim n −→ ∞ χ (( z, x − ) , M n ) ν ( M n ) ≥ . Finally, we recall that e ( yz, N ) = e ( y, N ) + e ( z, N ) for any finitely generated R -module N such that ℓ ( N/ ( yz ) N ) < ∞ . Thus by (2.5.5), we know that0 = lim n −→ ∞ P d − j =1 ( − j − e ( yz, H j ( x − , M n )) ν ( M n )= lim n −→ ∞ P d − j =1 ( − j − e ( y, H j ( x − , M n )) ν ( M n ) + lim n −→ ∞ P d − j =1 ( − j − e ( z, H j ( x − , M n )) ν ( M n ) . To see this, we can complete R and N . Let V be a coefficient ring of R and we can view N as a moduleover the regular ring A = V [[ y, z ]] with ℓ ( N/ ( yz ) N ) < ∞ . Since the multiplicity is the same as the Eulercharacteristic computed over A , the desired formula follows from the additivity of χ A ( − , N ) applied to theshort exact sequence 0 −→ A/y −→ A/yz −→ A/z −→ ut by (2.5.6) and (2.5.7), the last two limits above are both non-negative, hence they areboth zero. But then by (2.5.6) and (2.5.7) again, we havelim n −→ ∞ χ (( y, x − ) , M n ) ν ( M n ) = lim n −→ ∞ χ (( z, x − ) , M n ) ν ( M n ) = 0 . Therefore by Lemma 2.4, ( † ) holds for the system of parameters ( y, x − ). The last conclusionfollows from Lemma 2.4. This finishes the proof. (cid:3) We need the following important consequence of the above proposition.
Corollary 2.6.
Let ( R, m ) be a local ring of dimension d with an infinite residue field andlet { M n } be a weakly lim Cohen–Macaulay sequence. Then lim n −→ ∞ e ( m , M n ) ν ( M n ) ≥ . Proof.
Let z = z , . . . , z d be a minimal reduction of m . Since { M n } is weakly lim Cohen–Macaulay, by Proposition 2.5 we know that χ ( z, M n ) = o ( ν ( M n )). Thereforelim n −→ ∞ e ( m , M n ) ν ( M n ) = lim n −→ ∞ e ( z, M n ) ν ( M n ) = lim n −→ ∞ ℓ ( M n / ( z ) M n ) ν ( M n ) ≥ . (cid:3) Finally, we introduce lim Ulrich and weakly lim Ulrich sequence.
Definition 2.7.
Let ( R, m ) be a local ring of dimension d . A sequence of finitely generated R -modules { U n } of dimension d is called lim Ulrich (resp. weakly lim Ulrich ) if it is limCohen–Macaulay (resp. weakly lim Cohen–Macaulay) andlim n −→ ∞ e ( m , U n ) ν ( U n ) = 1 . The following is the main result of this section.
Theorem 2.8.
Let ( R, m ) −→ ( S, n ) be a flat local extension of local rings such that R is adomain and dim R = dim S . Suppose R admits a weakly lim Ulrich sequence { U n } . Then e ( R ) ≤ e ( S ) .Proof. We can replace S by S [ t ] n S [ t ] to assume S has an infinite residue field. Since R is adomain and { U n } is a weakly lim Ulrich sequence, we have: e ( R ) = lim n −→ ∞ e ( m , U n )rank R U n = lim n −→ ∞ ν R ( U n )rank R U n = lim n −→ ∞ ν S ( U n ⊗ R S )rank R U n ≤ lim n −→ ∞ e ( n , U n ⊗ R S )rank R U n = e ( S )where the only ≤ follows from Corollary 2.6, because { U n ⊗ R S } is a weakly lim Cohen–Macaulay sequence over S by Lemma 2.3. (cid:3) We end this section with a proposition which follows from more general results in [BHM].As this work is still in the stage of preparation, we give the proof of the proposition for thesake of completeness. roposition 2.9. Let ( R, m ) be a local domain of dimension d and let { M n } be a sequenceof finitely generated modules of dimension d . Suppose H j m ( M n ) has finite length for all n and all j < d . Then { M n } is a lim Cohen–Macaulay sequence (and hence a weakly limCohen–Macaulay sequence) if ℓ ( H j m ( M n )) = o (rank R M n ) for all j < d .Proof. Let x = x , . . . , x d be a system of parameters of R . We have H i ( x, M n ) = h − i ( K • ( x, R ) ⊗ R M n ) = h − i ( K • ( x, R ) ⊗ R R Γ m ( M n )) . Therefore we have a spectral sequence: H j + i ( x, H j m ( M n )) ⇒ H i ( x, M n ) . If j = d , then j + i > d when i ≥
1. So for all i ≥ ℓ ( H i ( x, M n )) ≤ d − X j =0 ℓ (cid:0) H j + i ( x, H j m ( M n )) (cid:1) ≤ d − X j =0 d · ℓ ( H j m ( M n )) = o (rank R M n ) . This implies that { M n } is lim Cohen–Macaulay by Lemma 2.4. (cid:3) Main result for graded rings
In this section we prove our main results. We start with a Segre product constructionwhich will play a crucial role in our construction of weakly lim Ulrich sequence.
Setting 3.1.
We fix an infinite field k of characteristic p > q = p e (eventually wewill let e −→ ∞ so one should think q being very large). We consider W nq := k [ x , y ] k [ x , y ]( q ) · · · k [ x n , y n ](( n − q ) , which is a rank one module over the ring T n = k [ x , y ] k [ x , y ] · · · k [ x n , y n ] . We note that T n is a standard graded ring of dimension n + 1: the degree j part is spannedby monomials whose total degree in x i and y i is j for each 1 ≤ i ≤ n . Hence T n is module-finite over A n = k [ z , z , . . . , z n +1 ] where z , . . . , z n +1 are general homogeneous degree oneelements in T n . We will view W nq as a graded module over A n that sits in non-negativedegrees (because k [ x , y ] only lives in non-negative degrees). We abuse notations a bit andlet m denote the homogeneous maximal ideal of A n . Since W nq is torsion-free and reflexive,we have H m ( W nq ) = H m ( W nq ) = 0.The next lemma on the degrees and dimensions of local cohomology modules of W nq iselementary. In fact, since W nq is explicitly described, precise dimensions of each degree of itslocal cohomology modules can be computed (geometrically, this simply corresponds to thesheaf cohomology of O P ( t ) ⊠ O P ( t + q ) ⊠ · · · ⊠ O P ( t + ( n − q ) on a product of projectivelines when t, q vary). We are not interested in the precise formulas so we state what we need. Lemma 3.2.
With notations as in Setting 3.1, we have(a) For each ≤ j ≤ n , H j m ( W nq ) sits in degrees − ( j − q − , . . . , − ( j − q .(b) H n +1 m ( W nq ) sits in degrees ≤ − ( n − q − . c) Fix a negative integer − r , then as q −→ ∞ , dim k H m ( W nq ) − r , dim k H m ( W nq ) − q − r , . . . , dim k H n +1 m ( W nq ) − ( n − q − r grow like o ( q n ) while dim k H n +1 m ( W nq ) − ( n + t ) q − r grows like o ( q n +1 ) for each fixed t ≥ .Proof. We use induction on n , the case n = 1 is obvious. Now suppose the lemma is provenfor n −
1. Since W nq = W n − q k [ x n , y n ](( n − q ), it follows from the Kunneth formula forlocal cohomology (see [GW78]) that(3.2.1) ( H j m ( W nq ) = H j m ( W n − q ) k [ x n , y n ](( n − q )) , for all j ≤ nH n +1 m ( W nq ) = H n m ( W n − q ) H m ( k [ x n , y n ](( n − q )) . Note that we are abusing notations a bit here and simply use m to denote the homogeneousmaximal ideal over the corresponding ring. Also note that we are ignoring terms that are 0coming from the inductive hypothesis when applying the Kunneth formula.From (3.2.1), part ( a ) and ( b ) are clear by the inductive hypothesis. For example, when j = n , H n m ( W n − q ) lives in degree ≤ − ( n − q − k [ x n , y n ](( n − q ) lives in degree ≥ − ( n − q , which shows that H n m ( W nq ) lives in degree − ( n − q − , · · · , − ( n − q . Toestablish part ( c ), we note that by (3.2.1) and the induction hypothesis, for j ≤ n ,dim k H j m ( W nq ) − ( j − q − r = dim k H j m ( W n − q ) − ( j − q − r · dim k ( k [ x n , y n ]) ( n +1 − j ) q − r = o ( q n − ) · (( n + 1 − j ) q − r + 1) = o ( q n ) . For the top local cohomology, again by (3.2.1) and the induction hypothesis,dim k H n +1 m ( W nq ) − ( n + t ) q − r = dim k H n m ( W n − q ) − ( n + t ) q − r · dim k H m ( k [ x n , y n ]) − ( t +1) q − r = o ( q n ) · (( t + 1) q + r − . This gives o ( q n ) for t = − o ( q n +1 ) for t ≥ (cid:3) The following immediate consequence is what we will need in the sequel. We adopt thefollowing notation: if M is a Z -graded module, then M a (mod q ) := ⊕ i ∈ Z M a + iq Corollary 3.3.
With notations as in Setting 3.1, for any fixed negative integer − r and any ≤ j ≤ n , dim k H j m ( W nq ) − r (mod q ) = o ( q n ) as q −→ ∞ . Proof.
This follows directly from part ( a ) and ( c ) of Lemma 3.2. (cid:3) Remark . We caution the reader that the degree range in Lemma 3.2 and Corollary 3.3are important: it is not true that dim k H j m ( W nq ) t = o ( q n ) for every t and every 0 ≤ j ≤ n .Now we state and prove our main result on weakly lim Ulrich sequence. Theorem 3.5.
Let ( R, m ) be a standard graded domain over an infinite F -finite field k ofcharacteristic p > (localized at the homogeneous maximal ideal). Then R admits a weaklylim Ulrich sequence.Proof. Let dim R = d . We will assume d ≥ d = 1, then it is easy to see that m N is a Ulrich module for N ≫ R trivially admitsa weakly lim Ulrich sequence). Since R is standard graded and k is infinite, there existshomogeneous degree one elements z , . . . , z d of R that forms a minimal reduction of m . We dentify the subring A := k [ z , . . . , z d ] with the ring A d − as in Setting 3.1. Thus we have asequence of finitely generated modules { W d − q } over A where q = p e . We will show that thefollowing sequence: U e := F e ∗ (cid:0) ( R ⊗ A W d − q ) − q ) (cid:1) is a weakly lim Ulrich sequence over R .Note that the R -module structure on U e is well-defined: under the e -th Frobenius push-forward, x ∈ R acts as x q so elements in R ⊗ A W d − q of degree ≡ − q ) are preservedunder the R -action. Also note that we take the degree ≡ − q ) in the definition of U e just for simplicity: in fact the proof will show that any fixed negative integer − r will work(on the other hand, non-negative integers will not work!).We first consider the special case that R is Cohen–Macaulay: the proof in this case issubstantially less technical while revealing the idea behind the construction. It is worth topoint out that in this case, we can actually show that { U e } is lim Ulrich. However, we alsopoint out that the individual U e is not Cohen–Macaulay.
The case R is Cohen–Macaulay. Since R is Cohen–Macaulay and is a graded module-finite extension of the polynomial ring A , we know R ∼ = ⊕ si =1 A ( − a i ) as a graded A -modulewhere s = rank A R and a i ≥ i . Thus we have U e ∼ = ⊕ si =1 F e ∗ ( W d − q ( − a i ) − q ) ) ∼ = ⊕ si =1 F e ∗ (( W d − q ) − − a i (mod q ) )as graded A -modules. Recall that W d − q is a rank one module over T d − , anddim k ( T d − ) t = ( t + 1) d − = t d − + o ( t d − ) , thus the multiplicity of T d − as an A -module is ( d − e ( z, W d − q ) = e ( m A , W d − q ) = ( d − . It follows that rank A W d − q = ( d − − r , the rank of( W d − q ) − r (mod q ) as a module over the q -th Veronese subring of A is equal to ( d − A ( q ) denote the q -th Veronese subring of A , we claim ( W d − q ) − r (mod q ) ⊗ A ( q ) Frac( A ) = W d − q ⊗ A Frac( A ): W d − q is torsion free over A , every homogeneous element of the latter tensorproduct can be written as wx , where w ∈ W d − q and x ∈ A , we can pick y ∈ A such thatdeg w + deg y ≡ − r (mod q ) since A is generated in degree one, thus wx = wyxy so it is in theformer tensor product).Thus the rank of F e ∗ (( W d − q ) − r (mod q ) ) over A ( q ) is equal to ( d − q d + α where α = log p [ k : k p ]. Therefore, since rank A ( q ) A = q , for every fixed negative integer − r , we have(3.5.1) rank A F e ∗ (( W d − q ) − r (mod q ) ) = ( d − q d + α − . To show { U e } is (weakly) lim Cohen–Macaulay, by Proposition 2.9 it is enough to provethat for every fixed negative integer − r and each j ≤ d − ℓ (cid:0) H j m ( F e ∗ (( W d − q ) − r (mod q ) )) (cid:1) = o (cid:0) rank A F e ∗ (( W d − q ) − r (mod q ) ) (cid:1) = o ( q d + α − ) . But since H j m ( F e ∗ (( W d − q ) − r (mod q ) )) = F e ∗ ( H j m ( W d − q ) − r (mod q ) ) and under the Frobenius push-forward F e ∗ , the lengths get multiplied by p α , (3.5.2) follows from Corollary 3.3. inally, to show { U e } is (weakly) lim Ulrich, we note that e ( m , U e ) = e ( z, U e ) = s X i =1 e ( m A , F e ∗ (( W d − q ) − − a i (mod q ) ))= s X i =1 rank A F e ∗ (( W d − q ) − − a i (mod q ) ) = ( d − sq d + α − by (3.5.1). On the other hand, since R ⊗ A W d − q lives in non-negative degrees, m [ q ] · ( R ⊗ A W d − q ) lives in degree ≥ q . Therefore by the definition of U e , we know that ν R ( U e ) ≥ dim k F e ∗ (cid:0) ( R ⊗ A W d − q ) q − (cid:1) . However, by the definition of W d − q as in Setting 3.1, for every fixed negative integer − r , weknow that(3.5.3) dim k ( W d − q ) q − r = ( q − r + 1)(2 q − r + 1) · · · (( d − q − r + 1) = ( d − q d − + o ( q d − ) . Therefore, since a i ≥
0, we havedim k F e ∗ (cid:0) ( R ⊗ A W d − q ) q − (cid:1) = s X i =1 dim k F e ∗ (cid:0) ( W d − q ) q − − a i (cid:1) = ( d − sq d + α − + o ( q d + α − ) . Putting the above together, we havelim e −→ ∞ e ( m , U e ) ν R ( U e ) ≤ e ( m , U e )dim k F e ∗ (cid:0) ( R ⊗ A W d − q ) q − (cid:1) = 1 . Since we know the above limit is always ≥ { U e } is a (weakly) lim Cohen–Macaulay), this shows the above limit is equal to 1 andhence { U e } is a (weakly) lim Ulrich sequence. The general case.
To handle the general case we first observe that our argument in theCohen–Macaulay case proves that for every fixed negative integer − r , F e ∗ (( W d − q ) − r (mod q ) )is a lim Cohen–Macaulay sequence over A (see (3.5.1), (3.5.2), and Proposition 2.9). Inparticular, we have (dropping F e ∗ results in dividing lengths by q α )(3.5.4) ℓ ( W d − q ) − r (mod q ) ( z q )( W d − q ) − r (mod q ) ! = ( d − q d − + o ( q d − ) . On the other hand, we know that dim k ( W d − q ) q − r = ( d − q d − + o ( q d − ) by (3.5.3) andthat ( W d − q ) q − r ∩ ( z q )( W d − q ) − r (mod q ) = 0 for degree reason (recall that W d − q only lives innon-negative degrees). This together with (3.5.4) imply that(3.5.5) dim k (cid:16)(cid:0) W d − q / ( z q ) W d − q (cid:1) − r (mod q ) , = q − r (cid:17) = o ( q d − ) . We now prove that { U e } is a weakly lim Cohen–Macaulay sequence. Let s = rank A R . Wehave a degree-preserving short exact sequence(3.5.6) 0 −→ ⊕ si =1 A ( − b i ) −→ R −→ C −→ here C has dimension less than d (note that b i ≥ i ). The rank of U e over A is thesame as the rank of F e ∗ (cid:0) (( ⊕ si =1 A ( − b i )) ⊗ A W d − q ) − q ) (cid:1) ∼ = ⊕ si =1 F e ∗ (( W d − q ) − − b i (mod q ) )over A . Therefore by (3.5.1), we still haverank A U e = e ( z, U e ) = ( d − sq d + α − . Thus to show { U e } is weakly lim Cohen–Macaulay, it is enough to show ℓ ( U e / ( z ) U e ) ≤ ( d − sq d + α − + o ( q d + α − ) by Lemma 2.4 (applied to x = z ). Dropping F e ∗ , this comes downto prove that(3.5.7) ℓ ( R ⊗ A W d − q ) − q ) ( z q )( R ⊗ A W d − q ) − q ) ! ≤ ( d − sq d − + o ( q d − ) . From (3.5.6), we obtain an exact sequence: ⊕ si =1 ( W d − q ) − − b i (mod q ) ( z q )( ⊕ si =1 ( W d − q ) − − b i (mod q ) ) −→ ( R ⊗ A W d − q ) − q ) ( z q )( R ⊗ A W d − q ) − q ) −→ ( C ⊗ A W d − q ) − q ) ( z q )( C ⊗ A W d − q ) − q ) −→ . By (3.5.4), in order to establish (3.5.7) it is enough to show that ℓ ( C ⊗ A W d − q ) − q ) ( z q )( C ⊗ A W d − q ) − q ) ! = o ( q d − ) . Since C is a finitely generated graded A -module of dimension less than d and lives in non-negative degrees, it has a graded filtration by ( A/P i )( − c i ), where P i are nonzero homogeneousprime ideals of A and c i ≥
0. So it is enough to prove that for any fixed homogeneous primeideal P ⊆ A and any c ≥
0, we have ℓ ( W d − q ) − − c (mod q ) ( P W d − q ) − − c (mod q ) + ( z q )( W d − q ) − − c (mod q ) ! = o ( q d − ) . At this point, we invoke (3.5.5). Thus in order to establish the above, it is enough to showthat dim k ( W d − q /P W d − q ) q − − c = o ( q d − ) . Fix 0 = z ∈ P of degree a >
0. Since W d − q /zW d − q ։ W d − q /P W d − q and W d − q is torsion-free, we know thatdim k ( W d − q /P W d − q ) q − − c ≤ dim k ( W d − q /zW d − q ) q − − c = dim k ( W d − q ) q − − c − dim k ( W d − q ) q − − c − a = o ( q d − )where the last equality follows from (3.5.3). This completes the proof of (3.5.7) and hencewe have established that { U e } is weakly lim Cohen–Macaulay.Finally, we prove that { U e } is weakly lim Ulrich. Again since R ⊗ A W d − q only lives innon-negative degrees, m [ q ] · ( R ⊗ A W d − q ) lives in degree ≥ q . Thus by the definition of U e ,we know that ν R ( U e ) ≥ dim k F e ∗ (cid:0) ( R ⊗ A W d − q ) q − (cid:1) . Thus it remains to show that(3.5.8) dim k ( R ⊗ A W d − q ) q − ≥ ( d − sq d − + o ( q d − ) , ecause this then implies that dim k F e ∗ (cid:0) ( R ⊗ A W d − q ) q − (cid:1) ≥ ( d − sq d + α − + o ( q d + α − ) while e ( m , U e ) = e ( z, U e ) = ( d − sq d + α − . To establish (3.5.8) is tricker as R ⊗ A W d − q is nolonger a direct sum of graded shifts of W d − q . We need the following claim. Claim 3.6.
Let M be a finitely generated graded A -module which sits in non-negative degrees.Then for any fixed negative integer − r and any i ≥ , we have ℓ (Tor Ai ( M, W d − q ) − r (mod q ) ) = ℓ ( h − i ( M ⊗ L A W d − q ) − r (mod q ) ) = o ( q d − ) . Proof of Claim.
Since all the lower local cohomology modules of W d − q have finite length, W d − q is Cohen–Macaulay on the punctured spectrum of A . Since A is regular, this means W d − q is finite free on the punctured spectrum of A and hence Tor Ai ( M, W d − q ) has finitelengths for all i ≥
1. A simple spectral sequence argument shows that h − i ( M ⊗ L A W d − q ) = h − i ( R Γ m ( M ⊗ L A W d − q )) = h − i ( M ⊗ L A R Γ m ( W d − q )) for all i ≥ Aj + i ( M, H j m ( W d − q )) ⇒ h − i ( M ⊗ L A W d − q ) . Next we consider a minimal graded finite free resolution of M over A :(3.6.1) 0 −→ ⊕ l A ( − a nl ) −→ · · · −→ ⊕ l A ( − a l ) −→ ⊕ l A ( − a l ) −→ n = pd A M and all the a ij are non-negative integers (since M lives in non-negativedegrees). If j ≤ d −
1, then using the above free resolution to compute Tor Aj + i ( M, H j m ( W d − q )),we see that ℓ (Tor Aj + i ( M, H j m ( W d − q )) − r (mod q ) ) ≤ X l dim k H j m ( W d − q ) − r − a i + j,l (mod q ) = o ( q d − )by Corollary 3.3. But if j = d , then j + i ≥ d + 1 so Tor Aj + i ( M, H j m ( W d − q )) = 0 since A isregular of dimension d . Therefore all the E -contributions of h − i ( M ⊗ L A W d − q ) − r (mod q ) havelengths o ( q d − ). This completes the proof of the claim. (cid:3) Now we return to the proof of the theorem, the short exact sequence (3.5.6) induces:Tor A ( C, W d − q ) q − −→ ⊕ si =1 W d − q ( − b i ) q − −→ ( R ⊗ A W d − q ) q − −→ ( C ⊗ A W d − q ) q − −→ . It follows thatdim k ( R ⊗ A W d − q ) q − ≥ s X i =1 dim k ( W d − q ) q − − b i − dim k Tor A ( C, W d − q ) q − = ( d − sq d − + o ( q d − )where the last equality follows from (3.5.3) and Claim 3.6. This completes the proof of(3.5.8) and hence { U e } is a weakly lim Ulrich sequence, as desired. (cid:3) Remark . We suspect the sequence { U e } constructed in Theorem 3.5 is in fact lim Ulrichbeyond the Cohen–Macaulay case. We hope to investigate this in future work. On the otherhand, we also believe that the weakly lim Ulrich condition may be more flexible to workwith (and easier to construct in practice). heorem 3.8. Let ( R, m ) −→ ( S, n ) be a flat local extension of local rings. Suppose ( R, m ) is a standard graded ring over a perfect field k (localized at the homogeneous maximal ideal).Then e ( R ) ≤ e ( S ) .Proof. Since every minimal prime of R is homogeneous, by the same argument as in [Ma17,Lemma 2.2], we may assume ( R, m ) is a standard graded domain and dim R = dim S . We canfurther assume that k is infinite and F -finite by replacing R and S by R [ t ] m R [ t ] and S [ t ] n S [ t ] .The conclusion in characteristic p > k is F -finite).Next we suppose k has characteristic 0 and R −→ S is a counter-example to the theorem.Then b R −→ b S is a flat local extension with e ( b R ) > e ( b S ). Applying the argument in [Ma17,Lemma 5.1], we may assume k ∼ = R/ m ∼ = S/ n is algebraically closed and b R −→ b S is module-finite (note that R is still standard graded over k ). Now applying the reduction procedurein [Ma17, Subsection 5.1] , there exists a pointed ´etale extension R ′ of R m and a finiteflat extension S ′ of R ′ such that e ( R ) = e ( R ′ ) > e ( S ). But then by inverting elements ifnecessary, we may assume that we have R −→ R ′′ = (cid:18) R [ x ] f (cid:19) g −→ S ′′ such that R ′′ is standard ´etale over R near a maximal ideal m ′′ lying over m , R ′′ −→ S ′′ is finiteflat with a maximal ideal n ′′ ∈ S ′′ lying over m ′′ , and that e ( m , R ) = e ( m ′′ , R ′′ ) > e ( n ′′ , S ′′ ).We can reduce this set up to characteristic p ≫ R κ −→ R ′′ κ −→ S ′′ κ with n ′′ κ a maximal ideal of S ′′ κ lying over the homogeneous maximal ideal m κ of R κ , such that( R κ ) m κ −→ ( S ′′ κ ) n ′′ κ is flat and e (( R κ ) m κ ) > e (( S ′′ κ ) n ′′ κ ) (note that R κ −→ R ′′ κ is always flat since f is a monic polynomial in x ). Thus we arrive at a counter-example (with ( R κ , m κ ) standardgraded over an F -finite field κ ) in characteristic p >
0, which is a contradiction. (cid:3)
Lastly, we mention that in [BHM], it is proven that every complete local domain of charac-teristic p > F -finite residue field admits a lim Cohen–Macaulay sequence { F e ∗ R } ,which follows from standard methods in tight closure theory [HH90]. The results of thispaper suggest the following challenging question, in which a positive answer would settleLech’s conjecture in characteristic p > F -finite (then the equalcharacteristic 0 case follows by [Ma17, Section 5]). Question . Does every complete local domain of characteristic p > F -finiteresidue field admit a lim Ulrich sequence, or at least a weakly lim Ulrich sequence? Acknowledgements:
Many ideas of this manuscript originate from [BHM], I would liketo thank Bhargav Bhatt and Mel Hochster for initiating this collaboration. In particular, Ithank Mel Hochster for valuable discussions on various weak notions of lim Cohen–Macaulaysequences. I also thank Ray Heitmann and Bernd Ulrich for their comments on a previousdraft of this manuscript. The author is supported in part by NSF Grant DMS In [Ma17], we are not assuming R is the completion of a finite type algebra therefore we choose a completeregular local ring A inside R and descend data to the Henselization of the localization of a polynomial ring,while here R is finite type (in fact standard graded) over k so we can run the same argument over R , thecounter-example then descends to the Henselization of R m and thus to a pointed ´etale extension of R m . eferences [BHM] B. Bhatt, M. Hochster, and L. Ma : Lim Cohen-Macaulay sequence , in preparation.[Eis95]
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Department of Mathematics, Purdue University, West Lafayette, IN 47907
E-mail address : [email protected]@purdue.edu