When does a perturbation of the equations preserve the normal cone
aa r X i v : . [ m a t h . A C ] J a n WHEN DOES A PERTURBATION OF THE EQUATIONSPRESERVE THE NORMAL CONE
PHAM HUNG QUY AND NGO VIET TRUNG
Abstract.
Let ( R, m ) be a local ring and I, J two arbitrary ideals of R . Letgr J ( R/I ) denote the associated ring of
R/I with respect to J , which correspondsto the normal cone in geometry. The main result of this paper shows that if I = ( f , ..., f r ), where f , ..., f r is a J -filter regular sequence, there exists a number N such that if f ′ i ≡ f i mod J N and I ′ = ( f ′ , ..., f ′ r ), then gr J ( R/I ) ∼ = gr J ( R/I ′ ). If J is an m -primary ideal, this result implies a long standing conjecture of Srinivas andTrivedi on the invariance of the Hilbert-Samuel function under small perturbations,which has been solved recently by Ma, Quy and Smirnov [23]. As a byproduct, theArtin-Rees number of I and I ′ with respect to J are the same. Furthermore, we giveexplicit upper bounds for the smallest number N with the above property. These re-sults solve two problems raised by Ma, Quy and Smirnov in [23] even for this generalsetting. There are other interesting consequences on the invariance of the Achilles-Manaresi function, which plays a similar role as the Hilbert-Samuel function, therelation type, the Castelnuovo-Mumford regularity, the Cohen-Macaulayness andthe Gorensteiness of the Rees algebra of R/I with respect to J under small pertur-bation of I . We also prove a converse of the main result showing that the condition I being generated by a J -filter regular sequence is the best possible for its validity.The main result can be also extended to perturbations with respect to filtrationsof ideals. As a consequence, if R is a power series ring, f , ..., f r is a filter regularsequence, and f ′ i is the n -jet of f i for n ≫
0, then I and I ′ have the same ini-tial ideal with respect to any Noetherian monomial order. A special case of thisconsequence was a conjecture of Adamus and Seyedinejad [2] on approximations ofanalytic complete intersection singularities. Introduction
Let I = ( f , ..., f r ) be an ideal in a local ring ( R, m ). We call an ideal I ′ = ( f ′ , ..., f ′ r )a small perturbation of I if f ′ i ≡ f i mod m n for n ≫
0. It is of great interest toknow which properties of
R/I are preserved by
R/I ′ under small perturbation. Forinstance, when studying the singularities of an analytic space we often want to replacethe defining power series by their n -jets for some n ≫
0. This problem have beenstudied by many mathematicians, mainly with the aim to find necessary or sufficientconditions for
R/I ∼ = R/I ′ (see e.g. [12, 13, 18, 24]). Mathematics Subject Classification.
Primary 13A30, 13D10; Secondary 13H15, 13P10,14B12.
Key words and phrases.
Perturbation, singularity, normal cone, filter regular sequence, associ-ated graded ring, Rees algebra, initial ideal, Artin-Rees number, Hilbert-Samuel function, extendeddegree, filtration, monomial order.The authors are partially supported by Vietnam National Foundation for Science and TechnologyDevelopment (NAFOSTED) under grant numbers 101.04-2020.10 and 101.04-2019.313. n 1996, Srinivas and Trivedi [32] showed that the Hilbert-Samuel function of R/I with respect to an m -primary ideal J does not change under small perturbationsof I if R is a generalized Cohen-Macaulay ring and f , ..., f r is part of a system ofparameters. They conjectured that the same is true if R is an arbitrary local ringand f , ..., f r is a filter-regular sequence, which means that f , ..., f r is locally a regularsequence on the punctured spectrum. The conjecture of Srinivas and Trivedi has beenrecently solved by Ma, Quy and Smirnov [23]. Actually, Srinivas and Trivedi [32] aswell as Ma, Quy and Smirnov [23] proved that gr J ( R/I ) ∼ = gr J ( R/I ′ ), which impliesthat the Hilbert-Samuel functions of R/I and
R/I ′ with respect to J are the same.This isomorphism is of great interest because Spec(gr J ( R/I )) is the normal cone andProj(gr J ( R/I )) is the exceptional fiber of the blow-up of
R/I along J .Let us call the least number N such that if I ′ = ( f ′ , ..., f ′ r ) with f ′ i ≡ f i mod J N ,then gr J ( R/I ) ∼ = gr J ( R/I ′ ), the perturbation index of gr J ( R/I ). The proof of Ma,Quy and Smirnov [23] is based on two deep results of Huneke and Trivedi on theheight of ideals [20] and of Eisenbud on approximation of complexes [17], which doesnot reveal how the perturbation index could be estimated. This problem is crucialfor applications, for instance, if we want to approximate an analytic singularity by analgebraic singularity. In fact, Ma, Quy and Smirnov asked the following question:
Problem 1.1. [23, Question 1.1] Does there exist an explicit upper bound for theperturbation index of gr J ( R/I )?We have found a new approach to the conjecture of Srinivas and Trivedi, whichallows us to find conditions for gr J ( R/I ) ∼ = gr J ( R/I ′ ) when J is an arbitrary ideal and I ′ is a J -adic perturbation of I , and to solve Problem 1.1 even in this general setting.Pertubations with respect to a not necessarily m -primary ideal were studied before byCutkosky and Srinivasan [12, 13], M¨ohring and van Straten [24], who gave conditionsfor R/I ∼ = R/I ′ , when R is a power series ring and J is the Jacobian ideal of I . Ourmain result is the following theorem. Theorem 1.2.
Let J be an arbitrary ideal of a local ring R . Let I = ( f , ..., f r ) ,where f , ..., f r is an J -filter regular sequence. There exists a well determined number N such that if f ′ i ≡ f i mod J N , i = 1 , ..., r , and I ′ = ( f ′ , ..., f ′ r ) , then f ′ , ..., f ′ r is an J -filter regular sequence and gr J ( R/I ) ∼ = gr J ( R/I ′ ) . Recall that a sequence of elements f , ..., f r in R is J -filter regular if f i p for anyassociated prime p J of ( f , ..., f i − ), i = 1 , ...., r , i.e. f , ..., f r is locally a regularsequence outside the locus of J . Besides the usual filter-regular sequences, whichplays an important role in the theory of generalized Cohen-Macaulay rings, J -filterregular sequences appear in several other topics such as finiteness of local cohomologymodules [6, 14], d -sequences [19, 35], and Macaulayfication [21, 11].The number N of Theorem 1.2 is given in terms of the Artin-Rees numbers andthe colon ideals of the sequence f , ..., f r (see Theorem 3.5 for details). The proofof Theorem 1.2 is based on the relationships between gr J ( R/I ), the initial ideal of I in gr J ( R ) and the Artin-Rees number of I with respect to J . Along the proof ofTheorem 1.2, we also show that the Artin-Rees numbers of I and I ′ with respect to J are the same under small J -adic perturbations, which confirms another problemraised by Ma, Quy and Smirnov in [23]. or an arbitrary ideal, Achilles and Manaresi [1] introduced a bivariate numericalfunction which plays a similar role as the Hilbert-Samuel function of an m -primaryideal. It is asymptotically a polynomial function whose normalized leading coeffi-cients gives a multiplicity sequence, which generalizes the Hilbert-Samuel multiplic-ity. Similar to the conjecture of the Srinivas and Trivedi, Theorem 1.2 implies thatthe Achilles-Manaresi function of R/I with respect to J does not change under small J -adic perturbations of I (Corollary 3.7).Let ℜ J ( R/I ) be the Rees algebra of
R/I with respect to J . Note that Proj( ℜ J ( R/I ))is the blow-up of
R/I along J . It is known that the structure of ℜ J ( R/I ) is stronglyrelated to that of gr J ( R/I ). Using Theorem 1.2 we are able to show that the relationtype, the Castelnuovo-Mumford, the Cohen-Macaulayness and the Gorensteiness of ℜ J ( R/I ) are preserved under small J -adic perturbations (Corollary 3.8).We also obtain the following result, which provides a converse to Theorem 1.2.Notice that Theorem 1.2 implies gr J ( R/ ( f , ..., f i )) ∼ = gr J ( R/ ( f ′ , ..., f ′ i )) for all i =1 , ..., r . Theorem 1.3.
Let J be an arbitrary ideal and f , ..., f r a sequence of elements in alocal ring R . There exists a number N such that gr J ( R/ ( f , ..., f i )) ∼ = gr J ( R/ ( f ′ , ..., f ′ i )) for any sequence f ′ , ..., f ′ i with f ′ i ≡ f i mod J N , i = 1 , ..., r , if and only if f , ..., f r isan J -filter regular sequence. In particular, gr J ( R/ ( f )) ∼ = gr J ( R/ ( f ′ )) for small J -adic perturbations f ′ of f ifand only if f is a J -filter regular element. This result shows that the condition of I being generated by a J -filter regular sequence is not far from the best possible for theinvariance of gr J ( R/I ) under small J -adic perturbations.Given an m -primary ideal J , we call the least number N such that if I ′ = ( f ′ , ..., f ′ r )with f ′ i ≡ f i mod J N , then R/I and
R/I ′ have the same Hilbert-Samuel function withrespect to J , the Hilbert perturbation index of R/I . In general, an upper bound for theHilbert perturbation index of
R/I can always be used to bound the perturbation indexof gr J ( R/I ) (Proposition 4.7). If R is a Cohen-Macaulay ring, Srinivas and Trivedi[33] gave an upper bound for the Hilbert perturbation index of R/I in terms of themultiplicity. If R is not Cohen-Macaulay, one can not bound the perturbation indexsolely in terms of the multiplicity (Example 4.1). If R is a generalized Cohen-Macaulayring, Quy and V.D. Trung [28] gave an upper bound for the Hilbert perturbationindex of R/I in terms of the homological degree, a generalization of the multiplicity.However, their proof can not be extended to the general case, when R is an arbitrarylocal ring. Using Theorem 1.2 we are able to give a general upper bound for theperturbation index of gr J ( R/I ) in terms of any extended degree, which includes thehomological degree as a special case (Theorem 4.5).Finally, as an application of our approach we study the more general problem:
Problem 1.4.
Let F = { J n } n ≥ be a filtration of ideals in R . Let I = ( f , ..., f r ).When does there exists a number N such that I and I ′ have the same initial idealwith respect to F for any ideal I ′ = ( f ′ , ..., f ′ r ) with f i ≡ f ′ i mod J N ? e are inspired by a recent conjecture of Adamus and Seyedinejad [2] which statesthat if I = ( f , ..., f r ) is a complete intersection ideal in a convergent power series ringand I ′ = ( f ′ , ..., f ′ r ), where f ′ i is the n -jet of f i for some n ≫ i = 1 , ..., r , then I and I ′ have the same initial ideals with respect to the degree lexicographic monomialorder. A solution to this conjecture can be deduced from the afore mentioned resultof Srinivas and Trivedi. A direct proof has been given by Adamus and Patel [3, 4].In general, the initial ideal with respect to any Noetherian monomial order can beviewed as the initial ideal with respect to a Noetherian filtration. Initial ideals withrespect to filtrations also appear in Arnold’s classification of hypersurface singularities[5, 8]. For an arbitrary Noetherian filtration F , we show that there exists a number N such that ( f , ..., f i ) and ( f ′ , ..., f ′ i ) have the same initial ideal with respect to F for any sequence f ′ , ..., f ′ r with f i ≡ f ′ i mod J N , i = 1 , ..., r , if and only if f , ..., f r isa J -filter regular sequence (Theorem 5.6 and Theorem 5.8). This gives a satisfactoryanswer to Problem 1.4.The paper is organized as follows. In Section 2 we prepare basic facts on the initialideal and the Artin-Rees number. In Section 3 we prove Theorem 1.2 and Theorem1.3. Actually, they follow from Theorem 3.5 and Theorem 3.11, respectively. InSection 4 we give bounds for the perturbation index in terms of the extended degree.Perturbations with respect to filtrations of ideals are dealt with in Section 5. Forunexplained terminology we refer the reader to [9].2. Initial ideal and Artin-Rees number
Let ( R, m ) be a local ring. Let I and J be arbitrary ideal in R . Setgr J ( R/I ) := M n ≥ ( J n + I ) / ( J n +1 + I ) , which is the associated graded ring of R/I with respect to the ideal J + I/I .For every element f = 0 in R we denote by o ( f ) the order of f with respect tothe J -adic filtration, which is the largest number n such that f ∈ J n , and by f ∗ the initial element of f in gr J ( R ), which is the residue class of f in J o ( f ) /J o ( f )+1 . Forconvenience, we set 0 ∗ = 0. Let in( I ) denote the initial ideal of I in gr J ( R ), which isgenerated by the the elements f ∗ , f ∈ I .The following properties of initial ideals are more and less known. Lemma 2.1. gr J ( R/I ) ∼ = gr J ( R ) / in( I ) . Proof.
It is easily seen that in( I ) = L n ≥ ( J n ∩ I + J n +1 ) /J n +1 for all n ≥
0. Therefore,gr J ( R/I ) = M n ≥ J n + IJ n +1 + I ∼ = M n ≥ J n J n ∩ ( J n +1 + I )= M n ≥ J n J n ∩ I + J n +1 = gr J ( R ) / in( I ) . (cid:3) Lemma 2.2.
Let K be an ideal in I . Let in( I/K ) denote the initial ideal of I/K in gr J ( R/K ) . Then in( I/K ) = in( I ) / in( K ) . roof. We havein(
I/K ) = M n ≥ J n ∩ I + J n +1 + KJ n +1 + K = M n ≥ J n ∩ I + J n +1 ( J n ∩ I + J n +1 ) ∩ ( J n +1 + K )= M n ≥ J n ∩ I + J n +1 J n ∩ K + J n +1 = in( I ) / in( K ) . (cid:3) By Artin-Rees lemma we know that there exist a number c such that J n ∩ I = J n − c ( J c ∩ I )for all n ≥ c . The least number c with this property is called the Artin-Rees number of I with respect to J . We denote this number by ar J ( I ). It is easily seen that ar J ( I )is the least number c such that J n +1 ∩ I = J ( J n ∩ I ) for all n ≥ c .For any graded ideal Q , we denote by Q n the n -th graded component of Q and by d ( Q ) the maximum degree of the elements of a graded minimal generating set of Q . Proposition 2.3. ar J ( I ) = d (in( I )) .Proof. Let S = gr J ( R ). Note that S is a standard graded ring, i.e. S is generated by S over S . Then d (in( I )) is the least integer c such that in( I ) n +1 = S in( I ) n for all n ≥ c . We have in( I ) n = ( J n ∩ I + J n +1 ) /J n +1 . Therefore, d (in( I )) is the least integer c such that J n +1 ∩ I + J n +2 = J ( J n ∩ I + J n +1 ) = J ( J n ∩ I ) + J n +2 for all n ≥ c . For all n ≥ ar J ( I ), we have J n +1 ∩ I = J ( J n ∩ I ) . Hence, ar J ( I ) ≥ d (in( I )).To prove ar J ( I ) ≤ d (in( I )), it suffices to show that J n +1 ∩ I = J ( J n ∩ I ) for n ≥ d (in( I )). Let f ∈ J n +1 ∩ I . Then f ∈ J ( J n ∩ I ) + J n +2 . Write f = g + f forsome g ∈ J ( J n ∩ I ) and f ∈ J n +2 . Since f, g ∈ I , we have f ∈ J n +2 ∩ I ⊆ J ( J n +1 ∩ I ) + J n +3 ⊆ J ( J n ∩ I ) + J n +3 . Hence, f = g + f ∈ J ( J n ∩ I )+ J n +3 . Continuing like that we have f ∈ J ( I ∩ J n )+ J n + t for all t ≥
1. By Krull’s intersection theorem, this implies f ∈ J ( J n ∩ I ). Thus, J n +1 ∩ I ⊆ J ( J n ∩ I ). Since J n +1 ∩ I ⊇ J ( J n ∩ I ), we get J n +1 ∩ I = J ( J n ∩ I ) for n ≥ d (in( I )). The proof is now complete. (cid:3) Remark 2.4.
We can also characterize the Artin-Rees number ar J ( I ) in terms of the Rees algebra ℜ J ( R ) := ⊕ n ≥ J n . Let Q = ⊕ n ≥ ( J n ∩ I ), which is a graded ideal of ℜ J ( R ). Note that ℜ J ( R ) is a standard graded ring. Then d ( Q ) is the least number c such that J n +1 ∩ I = J ( J n ∩ I ) for n ≥ c . Therefore, ar J ( I ) = d ( Q ).The estimation of the maximal degree of the minimal generators of a graded idealis in general difficult. It often leads to the computation of the Castelnuovo-Mumfordregularity, which is defined as follows. Let S be a standard graded algebra overa commutative ring. Let H iS + ( S ) denote the i -th local cohomology module of S with respect to the graded ideal S + of elements of positive degree and set a i ( S ) := ax { n | H iS + ( S ) n = 0 } with the convention a i ( S ) = −∞ if H iS + ( S ) = 0. Then onedefines reg( S ) := max { a i ( S ) + i | i ≥ } . We refer the reader to [36] for basic facts on the Castelnuovo-Mumford regularity ofstandard graded algebras over a commutative ring. Note that other references usuallydeal with standard graded rings over a field or an artinian local rings.The following bound for the Artin-Rees number was obtained by combining re-sults of Planas-Vilanova on a uniform Artin-Rees property [26] and of Ooishi on theCastelnuovo-Mumford regularity of the Rees algebra [25] in [23, Theorems 2.7 and2.8]. This result can be easily explained by means of initial ideal.
Corollary 2.5. ar J ( I ) ≤ reg(gr J ( R/I )) + 1 .Proof.
Let A = R/J . Let gr J ( R ) = A [ X ] /Q be a representation of gr J ( R ), where A [ X ] is a polynomial ring over A and Q a homogeneous ideal in A [ X ]. Let P be theideal of A [ X ] such that in( I ) = P/Q . By Proposition 2.3, ar J ( I ) = d (in( I )) ≤ d ( P ) . It is well known that d ( P ) ≤ reg P = reg( A [ X ] /P ) + 1. By Lemma 2.1, gr J ( R/I ) ∼ =gr J ( R ) / in( I ) = A [ X ] /P. Therefore, ar J ( R/I ) ≤ reg(gr J ( R/I )) + 1. (cid:3)
The following fact will be useful when dealing with the Artin-Rees number of idealsin quotient rings.
Lemma 2.6.
Let K be an ideal in I . Let ¯ J = ( J + K ) /K and ¯ I = I/K . Then ar ¯ J ( ¯ I ) ≤ ar J ( I ) . Proof.
For n ≥ ar J ( I ) we have( J n +1 + K ) ∩ I = J n +1 ∩ I + K = J ( J n ∩ I ) + K = ( J + K )(( J n + K ) ∩ I ) + K, This can be rewritten as ¯ J n +1 ∩ ¯ I = ¯ J ( ¯ J n ∩ ¯ I ). Hence, ar ¯ J ( ¯ I ) ≤ ar J ( I ). (cid:3) For a sequence of elements f , ..., f r in R we will use the shorter notations in( f , ..., f r )and ar J ( f , ..., f r ) instead of in(( f , ..., f r )) and ar J (( f , ..., f r )).3. Perturbation of filter-regular sequences
Let ( R, m ) be a local ring and J an arbitrary ideal of R . We call a sequence ofelements f , ..., f r in R J -filter regular if f i p for any associated prime p J of( f , ..., f i − ), i = 1 , ...., r . If J = m , we uses the notion filter regular instead of m -filterregular. This notion is due to [35] and has its origin in the theory of generalizedCohen-Macaulay rings [31]. In fact, if R is a generalized Cohen-Macaulay ring, everysystem of parameters is a filter-regular sequence. The converse is true if R is a quotientof a Cohen-Macaulay ring.An elements f ∈ R is called J -filter regular if f is a J -filter regular sequence ofone element, i.e. f p for any associated prime p J of R . It is easy seen that thiscondition is satisfied if and only if there exists a number n such that J n (0 : f ) = 0.Note that a J -filter regular element needs not belong to a system of parameters.Let I be an ideal of R . Let ¯ f denote the residue class of f in R/I and ¯ J = ( J + I ) /I .For convenience we say that f is a J -filter regular element in a quotient ring R/I if f is a ¯ J -filter regular element. It is clear that f , ..., f r is a J -filter regular sequenceif and only if f i is a J -filter regular element in R/ ( f , ..., f i − ), i = 1 , ..., r . Unlike aregular sequence, The property of being a J -filter regular sequence is not permutable,i.e. it depends on the order of the elements in the sequence. The following examplesgive large classes of J -filter regular sequences. Example 3.1. (1) Let J = Q d − i =0 Ann( H i m ( R )). Then every system of parameters of R is a J -filterregular sequence. This follows from [30, Proposition 4]. If R is equidimensionaland admits a dualizing complex, V ( J ) is just the non-Cohen-Macaulay locus of R [30, Theorem 3].(2) Let f , ..., f r be a d -sequence, i.e. ( f , ..., f i − ) : f i f t = ( f , ..., f i − ) : f i for all t ≥ i , and i = 1 , ..., r [19]. Let J = ( f , ..., f r ). Then f , ..., f r is a J -filter regularsequence [35]. A special case of d -sequence is the p -standard sequence, which wasused to construct Macaulayfication of Noetherian schemes [21, 11].In the following we will prove Theorem 1.2. The proof consists of several steps.First, we have to study the perturbation of a J -filter regular element.For any R -module M , we define a J ( M ) := inf { n | J n M = 0 } . In some sense, a J ( M ) is the nilpotency index of the annihilator of M with respect to J . Notice that a J ( M ) = ∞ if J n M = 0 for all n ≥
0. By this notation, f is a regularelement if and only if a J (0 : f ) = 0 and f is a J -filter regular element if and only if a J (0 : f ) < ∞ . Proposition 3.2.
Let f be a J -filter regular element. Set c = max { a J (0 : f ) , ar J ( f ) + 1 } . Let f ′ = f + ε , where ε is an arbitrary element in J c . Then (i) f ′ is a J -filter regular element with f ′ = 0 : f . (ii) in( f ′ ) = in( f ) .Proof. We will first show (ii). For that it suffices to show that ( gf ) ∗ = ( gf ′ ) ∗ forany element g ∈ R . If gf = 0, g ∈ f . Since J c (0 : f ) = 0, gε = 0. Hence, gf ′ = g ( f + ε ) = 0. In this case, we have ( gf ) ∗ = ( gf ′ ) ∗ = 0. Therefore, we mayassume that gf = 0.Let n = o ( gf ). If n ≥ o ( gε ), we have n ≥ o ( ε ) ≥ c > ar J ( f ) . Therefore, gf ∈ ( f ) ∩ J n = J n − c +1 ( J c − ∩ ( f )) = f J n − c +1 ( J c − : f ) . It follows that g ∈ J n − c +1 + (0 : f ). Hence, gε ∈ εJ n − c +1 + ε (0 : f ) ⊆ J n +1 + J c (0 : f ) = J n +1 . Thus, o ( gε ) ≥ n + 1, which contradicts the assumption n ≥ o ( gε ). So we have o ( gf ) = n < o ( gε ), which implies ( gf ′ ) ∗ = ( g ( f + ε )) ∗ = ( gf ) ∗ .As a consequence, ( gf ′ ) ∗ = 0 if and only if ( f g ) ∗ = 0. This means gf ′ = 0 ifand only if gf = 0. Therefore, g ∈ f ′ if and only if g ∈ f . So we obtain : f ′ = 0 : f . Since f is J -filter regular, a J (0 : f ′ ) = a J (0 : f ) ≤ ∞ . Hence, f ′ is also J -filter regular. So we obtain (i). (cid:3) Next, we have to study the perturbation of a J -filter regular sequence of two ele-ments. For that we need the following lemma. Lemma 3.3.
For any pair of elements f , f ∈ R , there exists an isomorphism ( f ) : f ( f ) + 0 : f ∼ = ( f ) : f ( f ) + 0 : f . Proof.
For every x ∈ ( f ) : f we have f x = f y for some element y ∈ ( f ) : f . If f x = f y ′ for another element y ′ , then y − y ′ ∈ f . Sending x to y induces a map ϕ : ( f ) : f → ( f ) : f ( f ) + 0 : f . It is clear that ϕ is surjective. It remains to show that ker( ϕ ) = ( f ) + 0 : f . Wehave x ∈ ker( ϕ ) if and only if f x = f y for some element y ∈ ( f ) + 0 : f . Since f y ∈ f ( f ), this condition is satisfied if and only if f x = f f z for some element z if and only if x − f z ∈ f or, equivalently x ∈ ( f ) + 0 : f . (cid:3) Proposition 3.4.
Let f , f be a J -filter regular sequence. Let a = a J (0 : f ) and a = a J (( f ) : f / ( f ) + 0 : f ) . Set c = max { a + a , ar J ( f ) , ar J ( f , f ) } + 1 Let f ′ = f + ε , where ε is an arbitrary element in J c . Then (i) f ′ , f is a J -filter regular sequence with a J (0 : f ′ ) = a and a J (( f ′ ) : f / ( f ′ )) ≤ a . (ii) in( f ′ , f ) = in( f , f ) .Proof. By Proposition 3.2(i), f ′ is a J -filter regular element with 0 : f ′ = 0 : f .Hence, a J (0 : f ′ ) = a . To prove (i) it suffices to show that a J (( f ′ ) : f / ( f ′ )) ≤ a because the finiteness of this invariant implies that f is a J -filter regular element in R/ ( f ′ ).By the exact sequence0 → ( f ′ ) + 0 : f ( f ′ ) → ( f ′ ) : f ( f ′ ) → ( f ′ ) : f ( f ′ ) + 0 : f → a J (( f ′ ) : f / ( f ′ )) ≤ a J (( f ′ ) + 0 : f / ( f ′ )) + a J (( f ′ ) : f / ( f ′ ) + 0 : f ) . First, we have( f ′ ) + 0 : f ( f ′ ) ∼ = 0 : f ( f ′ ) ∩ (0 : f ) ∼ = 0 : f f ′ (0 : f ′ f ) = 0 : f f ′ (0 : f f ) . We will show that f ′ (0 : f f ) = f (0 : f f ). By Nakayama’s lemma, it suffices toprove that f ′ (0 : f f ) ⊆ f (0 : f f ) and f (0 : f f ) ⊆ f ′ (0 : f f ) + m f (0 : f f ). ince f ′ = f + ε , we only need to show that ε (0 : f f ) ⊆ m f (0 : f f ). Note that ε ∈ J a + a +1 and J a (0 : f ) = 0. Then ε (0 : f f ) ⊆ J a +1 (0 : f ). We have0 : f f (0 : f f ) = 0 : f ( f ) ∩ (0 : f ) = ( f ) + 0 : f ( f ) ⊆ ( f ) : f ( f ) . Since J a (( f ) : f / ( f )) = 0, J a (0 : f /f (0 : f f )) = 0. Hence, J a +1 (0 : f ) ⊆ J f (0 : f f ). So we get f ′ (0 : f f ) = f (0 : f f ). Therefore,( f ′ ) + 0 : f ( f ′ ) ∼ = 0 : f f (0 : f f ) ⊆ ( f ) : f ( f ) . From this it follows that(3.2) a J (( f ′ ) + 0 : f / ( f ′ )) ≤ a J (( f ) : f / ( f )) = a . By Lemma 3.3, ( f ′ ) : f ( f ′ ) + 0 : f ∼ = ( f ) : f ′ ( f ) + 0 : f ′ = ( f ) : f ′ ( f ) + 0 : f We shall see that ( f ) : f ′ = ( f ) : f .Let ¯ R = R/ ( f ). Since J a (( f ) : f ) ⊆ ( f ) ⊆ ( f ) + 0 : f , we have J a (( f ) : f ) ⊆ ( f ) + 0 : f by Lemma 3.3. From this it follows that J a + a (( f ) : f ) ⊆ ( f ) + J a (0 : f ) = ( f ) . Therefore, f is a J -filter regular element in ¯ R with a J ¯ R (0 ¯ R : f ) = a J (( f ) : f / ( f )) ≤ a + a . By Lemma 2.6, ar J ¯ R ( f ¯ R ) ≤ ar J ( f , f ). These facts show that we can apply Propo-sition 3.2 to the elements f , f ′ in the ring ¯ R = R/ ( f ).By Proposition 3.2(i), ( f ) : f ′ = ( f ) : f . Hence( f ′ ) : f ( f ′ ) + 0 : f ∼ = ( f ) : f ( f ) + 0 : f , which implies(3.3) a J (( f ′ ) : f / ( f ′ ) + 0 : f ) ≤ a J (( f ) : f / ( f )) = a . Combining (3.1)-(3,3) we obtain a J (( f ′ ) : f / ( f ′ )) ≤ a , which completes the proofof (i).By Proposition 3.2(ii) we have in( f ′ ¯ R ) = in( f ¯ R ). By Lemma 2.2, this impliesin( f ′ , f ) / in( f ) = in( f , f ) / in( f ) . Therefore, in( f ′ , f ) = in( f , f ). So we obtain (ii). (cid:3) Now we can use induction to study the perturbation of an arbitrary J -filter regularsequence. The outcome is the following Theorem 3.5, which is the main result ofthis paper. What we really have to prove is Theorem 3.5(ii). However, the inductivehypothesis needs Theorem 3.5(i). Theorem 3.5(iii) and Theorem 3.5(iv) are justconsequences of Theorem 3.5(ii). They give affirmative answers to two problemsraised by Ma, Quy and Smirnov. The first problem asks for an explicit upper boundfor the perturbation index of gr J ( R/I ) [23, Question 1], while the second problem tates that the Artin-Rees number ar J ( I ) is preserved under small perturbation [23,Paragraph before Corollary 3.6]. Theorem 3.5.
Let J be an arbitrary ideal. Let I = ( f , ..., f r ) , where f , . . . , f r is a J -filter regular sequence. Let a i = a J (( f , ..., f i − ) : f i / ( f , ..., f i − )) for i = 1 , ..., r . Set N = max { a + 2 a + · · · + 2 r − a r , ar J ( f ) , ..., ar J ( f , ..., f r ) } + 1 . Let f ′ i = f i + ε i , where ε i is an arbitrary elements in J N , i = 1 , ..., r , and I ′ =( f ′ , ..., f ′ r ) . Then (i) f ′ , . . . , f ′ r is a J -filter regular sequence with a J (( f ′ , ..., f ′ i − ) : f ′ i / ( f ′ , ..., f ′ i − )) ≤ i − a i for i = 1 , ..., r. (ii) in( I ′ ) = in( I ) . (iii) gr J ( R/I ′ ) ∼ = gr J ( R/I ) . (iv) ar J ( I ′ ) = ar J ( I ) .Proof. We only need to prove (i) and (ii) because (iii) and (iv) follows from (ii) byLemma 2.1 and Proposition 2.3. The case r = 1 has been already proved in Proposi-tion 3.2.For r ≥
2, we will first show that a J (( f , ..., f i ) : f / ( f , ..., f i )) ≤ a + · · · + a i , i = 1 , ..., r . Notice that ( f , ..., f i ) = 0 if i = 1.The case i = 2 has been proved in the proof of Proposition 3.4. For i >
2, we set m = a + · · · + a i − . By induction we may assume that a J (( f , ..., f i − ) : f / ( f , ..., f i − )) ≤ m. Then J m (( f , ..., f i − ) : f ) ⊆ ( f , ..., f i − ) . By Lemma 3.3 we have( f , ..., f i − ) : f i ( f , ..., f i − ) + ( f , ..., f i − ) : f i ∼ = ( f , ..., f i ) : f ( f , ..., f i ) + ( f , ..., f i − ) : f . Since J a i (( f , ..., f i − ) : f i ) ⊆ ( f , ..., f i − ), this implies J a i (( f , ..., f i ) : f ) ⊆ ( f , ..., f i ) + ( f , ..., f i − ) : f . Therefore, J m + a i (( f , ..., f i ) : f ) ⊆ ( f , ..., f i ) + J m (( f , ..., f i − ) : f ) = ( f , ..., f i ) , which implies a J (( f , ..., f i ) : f / ( f , ..., f i )) ≤ m + a i = a + · · · + a i . As a consequence, f is a J -filter regular element in ¯ R := R/ ( f , ..., f i ).Let i = 1 , ..., r −
1. Since f i +1 is a J -filter regular element in R/ ( f , ..., f i ), f , f i +1 is a J -filter regular sequence in ¯ R with a J (( f , ..., f i ) : f i +1 /f , ..., f i )) = a i +1 . By Lemma 2.6, ar J ¯ R ( f ¯ R ) ≤ ar J ( f , ..., f i ) and ar J ¯ R (( f , f i +1 ) ¯ R ) ≤ ar J ( f , ..., f i +1 ).Hence, N ≥ max { a + · · · + a i +1 , ar J ¯ R ( f ¯ R )) , ar J ¯ R (( f , f i +1 ) ¯ R ) } + 1 . pplying Proposition 3.4(i) to the sequence f , f i +1 in ¯ R , we obtain that f i +1 is a J -filter regular element in R/ ( f ′ , f , ..., f i ) with a J (( f ′ , f , ..., f i ) : f i +1 / ( f ′ , f , ..., f i )) ≤ a i +1 . From this it follows that f ′ , f , ..., f r is a J -filter regular sequence. By Proposi-tion 3.4(ii), we have in(( f ′ , f i +1 ) ¯ R ) = in(( f , f i +1 ) ¯ R ). Hence in( f ′ , f , ..., f i +1 ) =in( f , f , ..., f i +1 ) by Lemma 2.6. By Proposition 2.3, this impliesar J ( f ′ , f , ..., f i +1 ) = ar J ( f , f , ..., f i +1 ) . Set R ′ = R/ ( f ′ ). Using induction on r , we may assume that(i’) f ′ , ..., f ′ r is a J -filter regular sequence in R ′ with a J (( f ′ , ..., f ′ i − ) R ′ : f ′ i / ( f ′ , ..., f ′ i − ) R ′ ) = 2 i − a i for i = 2 , ..., r .(ii’) in(( f ′ , ..., f ′ r ) R ′ ) = in(( f , ..., f r ) R ′ ).By Proposition 3.2(i), f ′ is a J -filter regular element with 0 : f ′ = 0 : f . Therefore,(i’) implies that f ′ , ..., f ′ r is a J -filter regular sequence with a J (( f ′ , ..., f ′ i − ) : f ′ i / ( f ′ , ..., f ′ i − )) = 2 i − a i for i = 1 , ..., r. By Lemma 2.2, (ii’) impliesin( f ′ , f ′ , ..., f ′ r ) = in( f ′ , f , ..., f r ) . Set S = R/ ( f , ..., f r ). We have shown above that f is a J -filter regular element in S with a J (( f , ..., f r ) : f ) ≤ a + · · · + a r . By Lemma 2.6, ar J ( f S ) ≤ ar J ( f , ..., f r ).Applying Proposition 3.2(ii) to this case, we get in( f ′ S ) = in( f S ). By Lemma 2.2,this implies in( f ′ , f , ..., f r ) = in( f , f , ..., f r ) . So we obtain in( f ′ , f ′ , ..., f ′ r ) = in( f , f , ..., f r ) . The proof is now complete. (cid:3)
The upper bound for the perturbation number of Theorem 3.5 becomes especiallysimple if the ideal is a complete intersection, when a = · · · = a r = 0. Moreover,since regular sequences are permutable, we do not need Proposition 3.4 and hence thenumbers ar J ( f , ..., f i ) for i < r . Corollary 3.6. [10, Theorem 3.6]
Let J be an arbitrary ideal. Let I = ( f , ..., f r ) ,where f , . . . , f r is a regular sequence. Set N = ar J ( f , ..., f r ) + 1 . Let f ′ i = f i + ε i ,where ε i is an arbitrary elements in J N , i = 1 , ..., r , and I ′ = ( f ′ , ..., f ′ r ) . Then f ′ , . . . , f ′ r is a regular sequence and in( I ′ ) = in( I ) .Proof. Note that an element f is regular if and only if a J (0 : f ) = 0. ApplyingProposition 3.2 to the element f in R/ ( f , ..., f r ), we get that f ′ is a regular elementin R/ ( f , ..., f r ) and in( f ′ , f , ..., f r ) = in( f , ..., f r ). As a consequence, f , . . . , f r , f ′ is a regular sequence and ar J ( f ′ , f , ..., f r ) = ar J ( f , f , ..., f r ) by Proposition 2.3.Using induction, we may assume that f ′ , ..., f ′ r is a regular sequence in R/ ( f ′ ) andin( f ′ , f ′ , ..., f ′ r ) = in( f ′ , f ..., f r ). Therefore, in( I ′ ) = in( I ). (cid:3) et G := gr m (gr J ( R )) , which is a standard bigraded algebra with bigraded compo-nents G uv = m u J v + J v +1 m u +1 J v + J v +1 . for u, v ≥
0. Note that G uv is a module of finite length. Let h ( r, s ) = r X u =0 s X v =0 ℓ ( G uv ) , where ℓ ( · ) denotes length. It is well-known that for r and s sufficiently large, h ( r, s )is a polynomial of degree d = dim R . If we write this polynomial in the form d X i =0 c i ( J )( d − i )! i ! r d − i s i + terms of lower degree,then c ( J ) , ..., c d ( J ) are nonnegative integers. Achilles and Manaresi [1] call them themultiplicity sequence of R with respect to J . If J is an m -primary ideal, c i ( J ) = 0 for i = 0 and c d ( J ) = e ( J, R ), the Hilbert-Samuel multiplicity of R with respect to J . If J is not m -primary, the multiplicity sequence plays a similar role as the multiplicity inthe m -primary case [1, 27]. For this reason, we may consider h ( r, s ) as a generalizationof the Hilbert-Samuel function and call it the Achilles-Manaresi function of R withrespect to J . Corollary 3.7.
Let J and I be as in Theorem 3.5. The Achilles-Manaresi functionof R/I with respect to J does not change under small J -adic perturbations of I .Proof. Since the Achilles-Manaresi function of
R/I with respect to J is defined by theassociated graded ring gr m (gr J ( R/I )) of gr J ( R/I ) with respect to m , the conclusionfollows from Theorem 3.5(iii). (cid:3) Now we will use Theorem 3.5 to study the impact of small perturbation to the Reesalgebra.
Corollary 3.8.
Let J be an arbitrary ideal. Let I = ( f , ..., f r ) , where f , . . . , f r isa J -filter regular sequence. The following invariants and properties of ℜ J ( R/I ) arepreserved under small J -adic perturbations of I : (1) the relation type, (2) the Castelnuovo-Mumford regularity, (3) the Cohen-Macaulayness, (4) the Gorensteinness.Proof. By [26, Theorem 2] and [25, Lemma 4.8], the relation type and the Castelnuovo-Mumford regularity of ℜ J ( R/I ) and gr J ( R/I ) are equal. By [37, Theorem 1.1] and[38, Theorem 1.1], the Cohen-Macaulayness and the Gorensteiness of ℜ J ( R/I ) dependsolely on the behavior of the local cohomology modules of gr J ( R/I ). Therefore, theconclusion follows from Theorem 3.5(iii). (cid:3)
We can not weaken the condition f , ..., f r being a J -filter regular sequence inTheorem 3.5. By Theorem 3.5(ii), this condition implies in( f ′ , ..., f ′ i ) = in( f , ..., f i ) or any sequence f ′ , ..., f ′ r with f ′ i − f i ∈ J N , i = 1 , ..., r . We shall see that the converseof this implication also holds. For this we need the following observation. Lemma 3.9.
Let f be a J -filter regular element. Let c = ar J ( f ) + 1 . Then f + ε isa J -filter regular element for every ε ∈ J c .Proof. Suppose that (0 : f ) ∩ J n = 0. Let g ∈ (0 : ( f + ε )) ∩ J n . Then gf = − gε ∈ J n + c = J n +1 ( J c − ∩ ( f )) = J n +1 f ( J c − : f ) . Therefore, g ∈ J n +1 ( J c − : f ) + (0 : f ) ⊆ J n +1 + (0 : f ) . Hence, g ∈ ( J n +1 + (0 : f )) ∩ J n = J n +1 + (0 : f ) ∩ J n = J n +1 . Thus, g ∈ (0 : ( f + ε )) ∩ J n +1 . Proceeding as above, we get g ∈ (0 : ( f + ε )) ∩ J n + m forall m ≥
0. By Krull’s intersection theorem, g = 0. Therefore, (0 : ( f + ε )) ∩ J n = 0,which implies that f + ε is a J -filter regular element. (cid:3) The following example shows that the condition f ∈ J c for c = ar J ( f ) + 1 of Lemma3.9 does not imply in( f ) = in( f + ε ) as in Proposition 3.2(ii). Example 3.10.
Let k [[ X, Y ]] be a power series ring in two variables over a field k , R = k [[ X, Y ]] / ( XY, Y ) = k [[ x, y ]], J = m , and f = x . Then ar m ( x ) = 1. Put ε = y .It is easy to check that gr m ( R ) = k [ X, Y ] / ( XY, Y ) = k [ x ∗ , y ∗ ], in( x ) = ( x ∗ ) andin( x + y ) = ( x ∗ ) + ( y ∗ ) . Theorem 3.11.
Let J be an arbitrary ideal and f , ..., f r a sequence of elements in R . There exists a number N such that in( f ′ , ..., f ′ i ) = in( f , ..., f i ) for any sequence f ′ , ..., f ′ r with f ′ i − f i ∈ J N , i = 1 , ..., r , if and only if f , ..., f r is a J -filter regular sequence.Proof. Assume that there exists a number N such that in( f ′ , ..., f ′ i ) = in( f , ..., f i ) if f ′ i − f i ∈ J N , i = 1 , ..., r . By [15, Lemma of Appendix], there exists ε i ∈ J N suchthat f ′ i = f i + ε i p for all associated primes p J of ( f , ..., f i − ), i = 1 , ..., r .Therefore, f ′ i is a J -filter regular element in R/ ( f , ..., f i − ). Without restriction wemay assume that N ≥ ar J ( f , ..., f i ) + 1. Since in( f , ..., f i − , f ′ i ) = in( f , ..., f i ), wehave ar J ( f , ..., f i − , f ′ i ) = ar J ( f , ..., , f i ) < N . Let ¯ R = R/ ( f , ..., f i − ). By Lemma2.6, ar J ( f ′ i ¯ R ) ≤ ar J ( f , ..., f i − , f ′ i ) < N . By Lemma 3.9 (applied to the element f ′ i inthe ring ¯ R ), the condition f i − f ′ i ∈ J N implies that f i is a J -filter regular elementin ¯ R . This holds for all i = 1 , ..., r . Therefore, f , ..., f r is a J -filter regular sequence.The converse of this conclusion follows from Theorem 3.5. (cid:3) By Proposition 2.3, gr J ( R/I ) ∼ = gr J ( R/I ′ ) if and only if in( I ) = in( I ′ ). There-fore, Theorem 1.3 follows from Theorem 3.11. In particular, we have the followingcriterion in the case r = 1, which shows that the condition I being generated by a J -filter regular sequence is not far from being a necessary and sufficient condition forgr J ( R/I ) ∼ = gr J ( R/I ′ ) under small perturbations I ′ of I . Corollary 3.12.
Let J be an arbitrary ideal and f an element in R . There exists anumber N such that gr J ( R/ ( f )) ∼ = gr J ( R/ ( f ′ )) for any element f ′ with f ′ − f ∈ J N if and only if f is a J -filter regular element. . Perturbation index and extended degree
Throughout this section, let ( R, m ) be a local ring and J a m -primary ideal. Forany finitely generated R -module M we denote by e ( J, M ) the multiplicity of M withrespect to J .If R is a Cohen-Macaulay ring and I an ideal generated by a regular sequence,Trivedi [34, Corollary 5] gave an upper bound for the perturbation index of gr m ( R/I )in terms of e ( m , R/I ). If R is an arbitrary local ring, it is easy to see that there is noupper bound for the perturbation index of gr m ( R/I )which depends only on e ( m , R/I ). Example 4.1.
Let k [[ X, Y, Y ]] be a power series ring in three variables over a field k , R = k [[ X, Y, Z ]] / ( XZ, Y Z, Z n +2 ) = k [[ x, y, z ]], J = m , and f = x . Then R is ageneralized Cohen-Macaulay ring, x a filter regular element and e ( m , R/ ( x )) = 1. It iseasy to check that gr m ( R ) = k [ X, Y, Z ] / ( XZ, Y Z, Z n +2 ) = k [ x ∗ , y ∗ , z ∗ ], in( x ) = ( x ∗ )and in( x + z n ) = ( x ∗ ) + ( z ∗ ) n +1 . By Lemma 2.1, gr m ( R/ ( x )) ∼ = gr m ( R ) / ( x ∗ ) andgr m ( R/ ( x + z n )) ∼ = gr m ( R ) / (( x ∗ ) + ( z ∗ ) n +1 ). Therefore, there is a surjective map fromgr m ( R/ ( x )) to gr m ( R/ ( x + z n )), whose kernel is not zero in degree ≥ n + 1. From thisit follows that the perturbation index of gr m ( R/ ( x )) is at least n + 1.We can give an upper bound for the perturbation index of gr J ( R/I ) in terms ofthe following generalization of the multiplicity, which was introduced by Doering,Gunston and Vasconcelos [16] for J = m and by Linh [22] for any m -primary ideal J . Definition 4.2. An extended degree is a number D ( J, M ) assigned to every finitelygenerated R -module M which satisfies the following conditions:(1) D ( J, M ) = D ( J, M/L )+ ℓ ( L ), where L is the maximal submodule of M havingfinite length,(2) D ( J, M ) ≥ D ( J, M/xM ) for a generic element x ∈ J ,(3) D ( J, M ) = e ( J, M ) if M is a Cohen-Macaulay module. Remark 4.3.
A prototype of an extended degree is the homological degree introducedby Vasconcelos [39]. Let R be a homomorphic image of a Gorenstein ring S . For afinitely generated R -module M with dim M = d , the homological degree of M isdefined recursively by setting hdeg( J, M ) := ℓ ( J, M ) if d = 0 andhdeg( J, M ) := e ( J, M ) + d − X i =0 (cid:18) d − i (cid:19) hdeg( J, Ext dim S − iS ( M, S ))if d >
0, where e ( J, M ) denotes the multiplicity of M with respect to J . Note thatdim(Ext dim S − iS ( M, S )) < d for i = 0 , ..., d −
1. If R is not a homomorphic image of aGorenstein ring, we define hdeg( J, M ) := hdeg(
J, M ⊗ ˆ R ) , where ˆ R denotes the m -adic completion of R . In particular, if M is a generalizedCohen-Macaulay module, we havehdeg( J, M ) = e ( J, M ) + d − X i =0 (cid:18) d − i (cid:19) ℓ ( H i m ( M )) , where H i m ( M ) denotes the i -th local cohomology module of M . ur bound for the perturbation index is based on the following upper bound for theCastelnuovo-Mumford regularity of the associated graded ring in terms of an arbitraryextended degree. Proposition 4.4.
Let J be an m -primary ideal of R . Let M be a finitely generated R -module and d := dim( M ) . Then reg(gr J ( M )) ≤ (2 d ! − D ( J, M ) d ! − − . Proof. If d = 0, M is of finite length and P n ≥ ℓ ( J n M/J n +1 M ) = ℓ ( M ). Sincegr J ( M ) n = J n M/J n +1 M = 0 for n ≫
0, reg(gr J ( M )) is the largest integer n suchthat ℓ ( J n M/J n +1 M ) = 0. From this it follows that reg(gr J ( M )) ≤ ℓ ( M ) −
1. ByDefinition 4.2(1), D ( J, M ) = ℓ ( M ). Therefore,(i) reg(gr J ( M )) ≤ D ( J, M ) − d = 0.By [22, Theorem 1.1], we have(ii) reg(gr J ( M )) ≤ D ( J, M ) − d = 1,(iii) reg(gr J ( M )) ≤ ( d − D ( J, M ) d − − − d ≥ J ( M )) ≤ (2 d ! − D ( J, M ) d ! − − d ≥ (cid:3) Theorem 4.5.
Let ( R, m ) be an arbitrary local ring and d = dim R . Let J be anarbitrary m -primary ideal of R . Let I = ( f , ..., f r ) , where f , ..., f r is a filter-regularsequence in R . Let m = max { d ! , r } and D := max { D ( J, R/ ( f , ..., f i )) | i = 0 , ..., r } . Set N := 2 m − D d ! − . Then gr J ( R/I ) ∼ = gr J ( R/I ′ ) for all ideals I ′ = ( f ′ , ..., f ′ r ) , where f ′ , ..., f ′ r is a sequence of elements in R such that f ′ i − f i ∈ J N , i = 1 , ..., r .Proof. Since f , ..., f r is a filter-regular sequence, ( f , ..., f i − ) : f i / ( f , ..., f i − ) is offinite length, i = 1 , ..., r . Consider the chain( f , ..., f i − ) : f i ( f , ..., f i − ) ⊇ J ( f , ..., f i − ) : f i ( f , ..., f i − ) ⊇ J ( f , ..., f i − ) : f i ( f , ..., f i − ) ⊇ · · · . By Nakayama’s lemma, if J n ( f , ..., f i − ) : f i ( f , ..., f i − ) = J n +1 ( f , ..., f i − ) : f i ( f , ..., f i − ) , then J n ( f , ..., f i − ) : f i ( f , ..., f i − ) = 0. Therefore, a J (( f , ..., f i − ) : f i / ( f , ..., f i − )) ≤ ℓ (( f , ..., f i − ) : f i / ( f , ..., f i − )) . By Definition 4.2(1), ℓ (( f , ..., f i − ) : f i / ( f , ..., f i − )) ≤ D ( J, R/ ( f , ..., f i − )) ≤ D. From this it follows that r X i =1 i − a J (( f , ..., f i − ) : f i / ( f , ..., f i − )) ≤ (1 + 2 + · · · + 2 r − ) D = (2 r − D ≤ N. Note that dim R/ ( f , ..., f i ) ≤ d , i = 1 , ..., r . By Corollary 2.5 and Proposition 4.4,ar J (( f , ..., f i )) ≤ reg(gr J ( R/ ( f , ..., f i )) + 1 ≤ (2 d ! − D ( J, R/ ( f , ..., f i )) d ! − − . herefore,max { ar J ( R/ ( f ) + 1 , ..., ar J ( R/ ( f , ..., f r ) + 1 } ≤ (2 d ! − D d ! − − ≤ N. Now we only need to apply Theorem 3.5(iii) to obtain the conclusion. (cid:3)
Remark 4.6.
The bound for the perturbation index of gr J ( R/I ) in Theorem 4.5 isfar from the best possible as one can see from the proof. We do not know whetherthe complexity for such a bound is polynomial in D or not. If J = m , one can use[29, Theorem 3.3] to get a slightly better bound for the perturbation index.If gr J ( R/I ) ∼ = gr J ( R/I ′ ), then ℓ ( I + J n /I + J n +1 ) = ℓ ( I ′ + J n /I ′ + J n +1 ) for all n ≥
0. Hence
R/I and
R/I ′ share the same Hilbert-Samuel function: ℓ ( R/I + J n ) = ℓ ( R/I ′ + J n ) . Thus, the Hilbert perturbation index is always less than or equal to the perturbationindex of gr J ( R/I ).We can derive from the Hilbert perturbation index an upper bound for the pertur-bation index of gr J ( R/I ) by involving the Artin-Rees number.
Proposition 4.7.
Let ( R, m ) be an arbitrary local ring and J an m -primary ideal of R . Let I = ( f , ..., f r ) , where f , ..., f r is a filter-regular sequence in R . Let N =max { p, ar J ( I ) + 1 } , where p is the Hilbert perturbation index of R/I with respect to J .Then gr J ( R/I ) ∼ = gr J ( R/I ′ ) for all ideals I ′ = ( f ′ , ..., f ′ r ) , where f ′ , ..., f ′ r is a sequenceof elements in R such that f ′ i − f i ∈ J N , i = 1 , ..., r .Proof. Assume that f ′ i − f i ∈ J ar J ( I )+1 , i = 1 , ..., r . Then there exists an epimorphism ϕ from gr J ( R/I ) to gr J ( R/I ′ ) [23, Lemma 3.2]. From this it follows that ℓ ( I + J n /I + J n +1 ) ≥ ℓ ( I ′ + J n /I ′ + J n +1 )for all n ≥
0. Therefore, ℓ ( R/I + J n ) ≥ ℓ ( R/I ′ + J n ). Now, it is clear that ϕ becomesan isomorphism if ℓ ( R/I + J n ) = ℓ ( R/I ′ + J n ) for all n ≥
0, which holds if f ′ i − f i ∈ J m , i = 1 , ..., r . (cid:3) If R is a Cohen-Macaulay ring, J = m , and f , ..., f r is part of a system of parame-ters, there is a linear bound for the Hilbert perturbation index in terms of e ( m , R/I )found by Srinivas and Trivedi [33, Proposition 1]. This bound was used by Trivedi[34, Corollary 5] to give a bound for the perturbation index of gr J ( R/I ). Recently,Quy and V.D. Trung [28, Theorem 1.3] have been able to extend the linear boundof Srinivas and Trivedi for generalized Cohen-Macaulay rings. Using this bound, weobtain the following explicit upper bound for the perturbation index of gr J ( R/I ). Corollary 4.8.
Let ( R, m ) be a generalized Cohen-Macaulay ring and d = dim R . Let I = ( f , ..., f r ) , where f , ..., f r is part of a system of parameters in R , and s = d − r .Set N := (2 s ! + 1) hdeg( m , R/I ) s ! − + 1 . Then gr m ( R/I ) ∼ = gr m ( R/I ′ ) for all ideals I ′ = ( f ′ , ..., f ′ r ) , where f ′ , ..., f ′ r is a sequenceof elements in R such that f ′ i − f i ∈ J N , i = 1 , ..., r . roof. Let p be the Hilbert perturbation index of R/I with respect to m . By [28,Theorem 1.3], p ≤ s ! hdeg( m , R/I ) + ( s + 1) d − X i =0 (cid:18) d − i (cid:19) ℓ ( H i m ( R )) + 1 . By the definition of the homological degree, d − X i =0 (cid:18) d − i (cid:19) ℓ ( H i m ( R )) ≤ hdeg( m , R/I ).Hence, p ≤ ( s ! + s + 1) hdeg( m , R/I ) + 1 ≤ (2 s ! + 1) hdeg( m , R/I ) + 1 . By Corollary 2.5 and Proposition 4.4, we havear m ( I ) ≤ reg(gr m ( R/I )) + 1 ≤ (2 s ! −
1) hdeg( m , R/I ) s ! − . Hence, max { p, ar J ( I ) + 1 } ≤ (2 s ! + 1) hdeg( m , R/I ) s ! − + 1. Therefore, the conclusionfollows from Proposition 4.7. (cid:3) Inspired of the results of the linear bounds in [33, Proposition 1] and [28, Theorem1.3] we raise the following questions, to which we are unable to give an answer.
Question 4.9.
Let ( R, m ) be an arbitrary local ring and J an m -primary ideal. Let I = ( f , ..., f r ), where f , ..., f r is a filter-regular sequence. Does there exist a linearupper bound for the Hilbert perturbation index of R/I with respect to J in terms ofthe extended degree D ( J, R/I )? Question 4.10.
Is the perturbation index of gr J ( R/I ) equal to the Hilbert pertur-bation index of
R/I with respect to J ?5. Perturbation with respect to a filtration
Let R be a convergent power series ring over the field R or C or a power series ringin several variables over a field. Let < be a monomial order , i.e. a total order on theset of monomials of R that satisfies the conditions: 1 < g for all monomials g ∈ R and g < g implies hg < hg for all monomials g , g , h ∈ R [7]. For every powerseries f ∈ R , let in < ( f ) denote the smallest monomial of f with respect to < . Let I be an ideal of R . The initial ideal in < ( I ) of I is the ideal generated by the initialmonomials in < ( f ), f ∈ I . Let j n ( f ) denotes the the n -jet of f , i.e. the polynomialpart of degree n of f .This section is motivated by the following problem. Problem 5.1.
Let I = ( f , ..., f r ), where f , ..., f r is a regular sequence in R . Let < be an arbitrary monomial order. Does there exist a number N such that for n ≥ N , j n ( f ) , ..., j n ( f n ) is a regular sequence and in < ( I ) = in < ( I ′ ), where I ′ =( j n ( f ) , ..., j n ( f n ))?If < is the degree lexicographic monomial order, this was a conjecture of Adamusand Syedinejad [2, Conjecture 3.7]. A solution of this conjecture can be deducedfrom the invariance of Hilbert-Samuel functions under small perturbations, which wasproved by Srinivas and Trivedi [32]. In fact, it is well known that the Hilbert-Samuel unction of R/I is the same as that of R/ in( I ). By [2, Lemma 3.2(i)], in < ( I ) ⊆ in < ( I ′ )for n ≫
0. Therefore, if
R/I and
R/I ′ share the same Hilbert-Samuel function,then R/ in < ( I ) and R/ in < ( I ′ ) share the same Hilbert-Samuel function, which impliesin < ( I ) = in < ( I ′ ) for n ≫
0. A direct proof of this conjecture were given by Adamusand Patel [3, 4], also by using Hilbert-Samuel functions. For an arbitrary monomialorder < , one can not solved Problem 5.1 by means of the invariance of Hilbert-Samuelfunctions under small perturbations. In fact, R/I and R/ in < ( I ) need not have thesame Hilbert function because a monomial order need not be a refinement of thedegree order.We shall give a positive answer to Problem 5.1 for any Noetherian monomial order.Recall that a monomial order < is Noetherian if for every monomial f there are onlyfinitely many monomials g with g < f . We can view the ideal in < ( I ) with respect toa Noetherian monomial order as the initial ideal of I with respect to a filtration ofideals, and we will show that the initial ideal with respect to such a filtration doesnot change under small perturbations.From now on, let R be a local ring. A filtration F of ideals in R is a sequence ofideals { J n } n ≥ which satisfy the following conditions for all m, n ≥ J = R ,(2) J m ⊆ J n if m > n ,(3) J m J n ⊆ J m + n .A filtration F = { J n } n ≥ is Noetherian if the Rees algebra ℜ F ( R ) := L n ≥ J n isNoetherian or, equivalently, if there exists a number c such that J n +1 = P ci =1 J i J c − i for all n ≥ c [9, Proposition 5.4.3]. We can approximate a Noetherian filtration by anadic filtration. Lemma 5.2.
Let F = { J n } n ≥ be a Noetherian filtration of ideals. Let δ be themaximum degree of the elements of a minimal homogeneous generating set of ℜ F ( R ) .Then J nδ ⊆ J n for all n ≥ .Proof. The case n = 0 is trivial. For n > J ( n − δ ⊆ J n − . Since ℜ F ( R ) is generated by homogeneous elements of degree ≤ δ , J nδ = P δi =1 J nδ − i J i forall n ≥
1. Note that J nδ − i ⊆ J ( n − δ and J i ⊆ J , i = 1 , ..., δ . Then J nδ ⊆ J ( n − δ J ⊆ J n . (cid:3) To each filtration F one attaches the associated graded ringgr F ( R ) := M n ≥ J n /J n +1 . If F is a Noetherian filtration, it follows from Lemma 5.2 and Krull’s intersectiontheorem that T n ≥ J n = 0 . Therefore, if f = 0, there is a unique number n such that f ∈ J n \ J n +1 . In this case, one defines the initial element of f as the residue class f ∗ of f in J n /J n +1 ⊂ gr F ( R ). For convenience, we set 0 ∗ = 0. Remark 5.3.
For a Noetherian monomial order < in a power series ring R , we canlist the monomials in an increasing sequence 1 = g < g < g < · · · . Let J n bethe ideals generated by all monomials ≥ g n . Then F := { J n } n ≥ is a filtration ofideals. In fact, since 1 = g < g < · · · < g m = g m g < g m g < · · · < g m g n , we ave g m + n < g m g n for all m, n ≥
0. It is clear that gr F ( R/I ) is isomorphic to thepolynomial ring in the same number of variables as R . Let m be the maximal ideal of R . For each n ≥
0, there exist numbers t and m such that F n ⊆ m t ⊆ F m . Therefore,for all ideals I of R , \ n ≥ ( J n + I ) = \ t ≥ ( m t + I ) = I by Krull’s intersection theorem. By [9, Proposition 5.4.5], this together with theNoetherian property of gr F ( R/I ) imply that F is a Noetherian filtration. Now, forevery power series f , we can define the initial element of f with respect to F , whichis clearly the initial monomial of f with respect to < .Let I be an arbitrary ideal of the local ring R . We denote by in F ( I ) the initial ideal of I in gr F ( R ) generated by the initial elements f ∗ , f ∈ I . Setgr F ( R/I ) := M n ≥ ( J n + I ) / ( J n +1 + I ) , which is the associated graded ring of R/I with respect to the induced filtration of F in R/I . Similarly as Lemma 2.1, we have the following relationship between gr F ( R/I )and in F ( I ). Lemma 5.4.
Let F be a Noetherian filtration of ideals. Then gr F ( R/I ) = gr F ( R ) / in F ( I ) . Inspired by Theorem 3.5(ii) we will study the invariance of the initial ideal in F ( I )under small perturbations with respect to the filtration F . For that we need to extendthe notion of Artin-Rees number.If F is a Noetherian filtration, the ideal Q := L n ≥ J n ∩ I of ℜ F ( R ) is finitelygenerated. Let c = d ( Q ), the maximum degree of the elements of a homogeneousminimal generating set of Q . It is easy to see that J n ∩ I = c X t =1 J n − t ( J t ∩ I )for all n ≥ c . We call d ( Q ) the Artin-Rees number of I with respect to F , and wedenote it by ar F ( I ).We have the following formula for ar F ( I ) in terms of the initial ideal in F ( I ). Proposition 5.5.
Let F be a Noetherian filtration of ideals. Then ar F ( I ) = d (in F ( I )) . Proof.
Note that d (in( I )) is the least number c such that for all n ≥ c , in( I ) n isgenerated by graded elements of in( I ) of degree ≤ c and thatin( I ) n = ( J n ∩ I + J n +1 ) /J n +1 Then d (in( I )) is the least integer c such that J n ∩ I + J n +1 = c X t =1 J n − t ( J t ∩ I ) + J n +1 or all n ≥ c . For c = ar F ( I ), we have J n ∩ I = P ct =1 J n − t ( J t ∩ I ) for n ≥ c . Hence,ar F ( I ) ≥ d (in( I )).To prove ar F ( I ) ≤ d (in( I )) let f be an arbitrary element of J n ∩ I , n ≥ d (in( I )).Then f ∈ P ct =1 J n − t ( J t ∩ I ) + J n +1 . Write f = g + f for some g ∈ P ct =1 J n − t ( J t ∩ I )and f ∈ J n +1 . Since f, g ∈ I , we have f ∈ J n +1 ∩ I ⊆ c X t =1 J n +1 − t ( J t ∩ I ) + J n +2 ⊆ c X t =1 J n − t ( J t ∩ I ) + J n +2 . Hence, f = g + f ∈ P ct =1 J n − t ( J t ∩ I ) + J n +2 . Continuing like that we have f ∈ P ct =1 J n − t ( J t ∩ I ) + J m for all m ≥ n . By Lemma 5.2, there exists δ such that J mδ ⊆ J mδ ⊆ J m for m ≥
1. Therefore, f ∈ P ct =1 J n − t ( J t ∩ I ) + J m for all m ≥ n .By Krull’s intersection theorem, this implies f ∈ P ct =1 J n − t ( J t ∩ I ). Thus, J n ∩ I ⊆ P ct =1 J n − t ( J t ∩ I ). Clearly, J n ∩ I ⊇ P ct =1 J n − t ( J t ∩ I ). So we have J n ∩ I = c X t =1 J n − t ( J t ∩ I )for n ≥ d (in( I )), which implies ar F ( I ) ≤ d (in( I )). (cid:3) The above properties of the initial ideal allow us to prove a similar result as Theorem3.5 for perturbations with respect to a Noetherian filtration.
Theorem 5.6.
Let F = { J n } n ≥ be a Noetherian filtration of ideals in a local ring R .Let δ be the maximum degree of the element of a minimal homogeneous generating setof the Rees algebra ℜ F ( R ) . Let I = ( f , ..., f r ) , where f , . . . , f r is a J -filter regularsequence in R . Set a i = a J (( f , ..., f i − ) : f i / ( f , ..., f i − )) for i = 1 , ..., r , and N := max { ( a + 2 a + · · · + 2 r − a r + 1) δ, ar F ( f ) + 1 , ..., ar F ( f , ..., f r ) + 1 } . Let f ′ i = f i + ε i , where ε i is an arbitrary elements in J N , i = 1 , ..., r , and I ′ =( f ′ , ..., f ′ r ) . Then f ′ , . . . , f ′ r is a J -filter regular sequence and in F ( I ′ ) = in F ( I ) . The appearance of δ in Theorem 5.6 comes from the approximation of F by the J -adic filtration. In fact, if ε ∈ J N , then ε ∈ J a +2 a + ··· +2 r − a r +11 by Corollary 5.2.Similar to Corollary 3.6 we have the following consequence. Corollary 5.7.
Let F = { J n } n ≥ be a Noetherian filtration of ideals in a localring R . Let I = ( f , ..., f r ) , where f , . . . , f r is a regular sequence in R . Set N =ar F ( f , ..., f r ) + 1 . Let f ′ i = f i + ε i , where ε i is an arbitrary elements in J N , i = 1 , ..., r ,and I ′ = ( f ′ , ..., f ′ r ) . Then f ′ , . . . , f ′ r is a regular sequence and in F ( I ′ ) = in F ( I ) . Theorem 5.3 has the following converse, which is similar to Theorem 3.11.
Theorem 5.8.
Let F = { J n } n ≥ be a Noetherian filtration of ideals in a local ring R .Let f , ..., f r be elements in R . There exists a number N such that in F ( f ′ , ..., f ′ i ) = in F ( f , ..., f i ) for any sequence f ′ , ..., f ′ i with f ′ i − f i ∈ J N , i = 1 , ..., r , if and only if f , ..., f r is a J -filter regular sequence. e leave the reader to check the proofs of the above results following the argumentsfor similar results in Sections 2 and 3.Applying Corollary 5.7 to the filtration described in Remark 5.3 we immediatelyobtain the following solution to Problem 5.1. Corollary 5.9.
Let R be a power series ring over a field or a convergent power seriesring over R or C in several variables. Let I = ( f , ..., f r ) , where f , . . . , f r is a regularsequence in R . For any Noetherian monomial order in R , there exist a number N such that for n ≥ N , j n ( f ) , ..., j n ( f n ) is a regular sequence and in < ( I ) = in < ( I ′ ) ,where I ′ = ( j n ( f ) , ..., j n ( f n )) . Acknowledgement . This work was initiated during a research stay of the authorsat Vietnam Institute for Advanced Study in Mathematics (VIASM) in 2020.
References [1] R. Achilles and M. Manaresi, Multiplicities of a bigraded ring and intersection theory, Math.Ann. 309 (1997), 573–591.[2] J. Adamus and H. Seyedinejad, Finite determinacy and stability of flatness of analytic mappings,Canad. J. Math. 69 (2017), no. 2, 241–257.[3] J. Adamus and A. Patel, On finite determinacy of complete intersection singularities, Preprint,arXiv:1705.08985.[4] J. Adamus and A. Patel, Equisingular algebraic approximation of real and complex analyticgerms, Preprint, arXiv:1910.11498.[5] V. I. Arnold, Normal forms of functions near degenerate critical points, Russian Math. Surveys29ii (1975), 10–50, Reprinted in Singularity Theory, London Math. Soc. Lecture Notes Vol. 53(1981), 91–131.[6] J. Asadollahi and P. Schenzel, Some results on associated primes of local cohomology modules,Japanese J. Mathematics 29 (2003), 285–296.[7] T. Becker, Standard bases and some computations in rings of power series, J. Symbolic Comput.10 (1990), no. 2, 165–178.[8] Y. Boubakri, G.M. Greuel, and T. Markwig, Normal forms of hypersurface singularities inpositive characteristic, Mosc. Math. J. 11 (2011), no. 4, 657–683.[9] W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics39, Cambridge, 1993.[10] L. Chiantini, On form ideals and Artin-Rees condition, Manuscripta Math. 36 (1981/82), no. 2,125–145.[11] N.T. Cuong and D.T. Cuong, Local cohomology annihilators and Macaulayfication, Acta Math.Vietnam. 42 (2017), 37–60.[12] S. D. Cutkosky and H. Srinivasan, An intrinsic criterion for isomorphism of singularities. Amer.J. Math. 115 (1993), 789–821.[13] S. D. Cutkosky and H. Srinivasan, Equivalence and finite determinancy of mappings, J. Algebra188 (1997), 16–57.[14] H. Dao and P.H. Quy, On the associated primes of local cohomology, Nagoya Math. J. 237(2020), 1–9.[15] E. D. Davis, Ideals of the principal class, R -sequences and a certain monoidal transformation,Pacific J. Math. 20, No. 2 (1967), 197–205.[16] L. R. Doering, T. Gunston and W. Vasconcelos, Cohomological degrees and Hilbert functionsof graded modules, Amer. J. Math. 120 (1998), 493–504.[17] D. Eisenbud, Adic approximation of complexes, and multiplicities, Nagoya Math. J. (1974),61–67.[18] G-M. Greuel and T.H. Pham, Finite determinacy of matrices and ideals, J. Algebra 530 (2019),195–214.
19] C. Huneke, Theory of d -sequences and powers of ideals, Adv. in Math. 46 (1982), 249–279.[20] C. Huneke and V. Trivedi, The height of ideals and regular sequences, Manus. Math. , 137–142(1997).[21] T. Kawasaki, On Macaulayfication of Noetherian schemes, Trans. Amer. Math. Soc. 352 (2000),2517–2552.[22] C. H. Linh, Upper bound for the Castelnuovo-Mumford regularity of associated graded modules,Comm. Algebra 33 (2005), 1817–1831.[23] L. Ma, P.H. Quy, and I. Smirnov, Filter regular sequence under small perturbations, Math.Ann. 378 (2020), 243–254.[24] K. M¨ohring and D. van Straten, A criterion for the equivalence of formal singularities, AmericanJ. Math. 124 (2002), 1319–1327.[25] A. Ooishi, Genera and arithmetic genera of commutative rings, Hiroshima Math. J. 17 (1987),47–66.[26] F. Planas-Vilanova, The strong uniform Artin-Rees property in codimension one, J. ReineAngew. Math. 527 (2000), 185–201.[27] C. Polini, N.V. Trung, B. Ulrich, and J. Validashti, Multiplicity sequence and integral depen-dence, Math. Ann. 378 (2020), 951–969.[28] P. H. Quy and V. D. Trung, Small perturbations in generalized Cohen-Macaulay local rings,Preprint 2020, arXiv:2004.08873.[29] M. E. Rossi, N. V. Trung and G. Valla, Castelnuovo-Mumford regularity and extended degree,Trans. Amer. Math. Soc. 355 (2003), 1773–1786.[30] P. Schenzel, Dualizing complexes and systems of parameters, J. Algebra 58 (1979), 495–501.[31] P. Schenzel, N. V. Trung, and N. T. Cuong, Verallgemeinerte Cohen-Macaulay-Moduln, Math.Nachr. 85 (1978), 57–73.[32] V. Srinivas and V. Trivedi, The invariance of Hilbert functions of quotients under small pertur-bations, J. Algebra (1996), 1–19.[33] V. Srinivas and V. Trivedi, A finiteness theorem for the Hilbert functions of complete intersectionlocal rings, Math. Z. (1997), 543–558.[34] V. Trivedi, Hilbert functions, Castelnuovo-Mumford regularity and uniform Artin-Rees numbers,Manuscripta Math. 94 (1997), 485–499.[35] N. V. Trung, Absolutely superficial sequences, Math. Proc. Cambridge Phil. Soc. 93 (1983),35–47.[36] N. V. Trung, Castelnuovo-Mumford regularity and related invariants, in: Commutative algebraand combinatorics, Ramanujan Math. Soc. Lect. Notes 4, Ramanujan Math. Soc., 2007, 157–180(arXiv:1907.11427).[37] N. V. Trung and S. Ikeda, When is the Rees algebra Cohen-Macaulay? Comm. Algebra 17(1989), no. 12, 2893–2922.[38] N. V. Trung, D. Q. Viet and S. Zarzuella, When is the Rees algebra Gorenstein? J. Algebra175 (1995), 137–156.[39] W. V. Vasconcelos, The homological degree of a module, Trans. Amer. Math. Soc. 350 (1998),1167–1179. Department of Mathematics, FPT University, Hanoi, Vietnam
Email address : [email protected] International Centre for Research and Postgraduate Training, Institute of Math-ematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi,Vietnam
Email address : [email protected]@math.ac.vn