Equigenerated ideals of analytic deviation one
aa r X i v : . [ m a t h . A C ] S e p Equigenerated ideals of analytic deviation one
Zaqueu Ramos Aron Simis Abstract
The overall goal is to approach the Cohen–Macaulay property of the special fiber F ( I )of an equigenerated homogeneous ideal I in a standard graded ring over an infinite field.When the ground ring is assumed to be local, the subject has been extensively lookedat. Here, with a focus on the graded situation, one introduces two technical conditions,called respectively, analytical tightness and analytical adjustment , in order to approachthe Cohen–Macaulayness of F ( I ). A degree of success is obtained in the case where I in addition has analytic deviation one, a situation looked at by several authors, beingessentially the only interesting one in dimension three. Naturally, the paper has someapplications in this case. Introduction
Let R denote a standard graded algebra over a field. The main focus in this work is onthe behavior of the special fiber F ( I ) of an equigenerated homogeneous ideal I ⊂ R ofanalytic deviation one, in its tight relation with the associated graded ring gr I ( R ) and,quite often, the Rees algebra R R ( I ). A main target is the Cohen–Macaulayness of F ( I ), avenerable subject studied by many authors, specially in the case when the ground ring islocal. Here one approaches the problem presuming that the equigeneration fact ought toyield additional output, a deed of a more recent vintage among experts.Throughout this introduction, let I ⊂ R be a d -equigenerated ideal, i.e., I = ([ I ] d ). Twotechnical conditions are central in this work, as related to the Cohen–Macaulayness of F ( I ).The first is the notion of an analytically tight sequence in power n , namely, letting ℓ ( I ) denotethe analytic spread of I , this is a sequence of ℓ ( I ) forms in I d having a regularity behaviorwith respect to powers of I . Traces of the nature of such sequences are spread out in a hugenumber of important papers – see [1], [4], [9], [14], [15], [22], [26], [27], among uncountablemany others. We make special note of [27] for the close relationship of its notion of sequencesof regular type and the present notion of analytically tight sequences. However, though thetechnology becomes similar in various passages, the overall objectives are quite different,since here the focus is on the special fiber. By and large the present notion is light enoughto handle and does not involve introducing a priori superficial sequences, as we believe isunnecessary for understanding the special fiber of an equigenerated ideal. The idea of usingthis concept has a considerable degree of success in the case of an ideal of analytic deviationone in that its existence is guaranteed asymptotically along the powers provided the ideal isunmixed. Now, a good deal of previous work assumes as standing hypothesis that the givenideal is generically a complete intersection. Under such a hypothesis, equigenerated ideals AMS Mathematics Subject Classification (2010 Revision). Primary 13A02, 13A30, 13D02, 13H10,13H15; Secondary 14E05, 14M07, 14M10, 14M12. Key Words and Phrases : plane reduced points, special fiber, perfect ideal of height two, Rees algebra,associated graded ring, Cohen–Macaulay. Under a post-doc fellowship from INCTMAT/Brazil (88887.373066/2019-00) Partially supported by a CNPq grant (302298/2014-2).
1f analytic spread one admit analytically tight sequences in any power, a phenomenon thatmay explain the success of this popular generic constraint.Perhaps a true novelty is the use of a second notion, that of an adjusted set L of formsin I d and, more particularly, that of an analytically adjusted such set. The concept is statedin terms of the number of generators of LI and at first sight looks simplistic. Neverthelessit has the merit of fooling quadratic relations. The latter, as known by the experts, is one ofthe nearly insurmountable obstacles in the theory, usually dribbled by assuming a G s typeof condition. As the first notion above, it has a strong bearing to properties of minimalreductions of I .Instead of boring the reader with a tedious long introduction, we now briefly describethe contents of each section.The first section is a quick recap of the main algebras to be used in the work, with areminder of the special behavior of the special algebra in the equigenerated case. Here,the first few batches concern general aspects of any equigenerated homogeneous ideal, soit can help tracking steps in directions other than the one undertaken here. It includes afew additional aspects of the so-called condition G s of Artin–Nagata in order to stress someslightly bypassed behavior. The first of these is a more precise reformulation, in terms of thiscondition, of the Valla dimension of the symmetric algebra – a result essentially obtainedin the work of Huneke–Rossi ([17]). We give some easy consequences of this reformulation.A seemingly overshadowed result concerns a lower bound for the initial defining degree ofthe fiber F ( I ) of a height two perfect ideal in terms of the G s condition, a result essentiallydue to Tchernev ([25]).The second section is the core of the paper. It focuses on properties of F ( I ) as relatedto gr I ( R ), in particular when the latter is Cohen–Macaulay. Thus, one proves that, foran equigenerated ideal I of analytic deviation one, if gr I ( R ) is Cohen–Macaulay then thesame property for F ( I ) is dictated by the existence of analytically tight sequences in anypower. This result may be previously known in the case where I is moreover genericallya complete intersection, but we could not find it otherwise in this exact form. As to thenotion of analytical adjustment, it is naturally related to properties of the special fiber F ( I ).Thus, for example, a subset of I d which is regular sequence over F ( I ) is adjusted. Anotherimportant example is that of any linearly independent subset of I d when I is a syzygeticideal or a perfect ideal of height two satisfying condition G . A basic natural appearance ofan analytically adjusted set is as minimal generators of a minimal reduction under certainconditions. By and large, if F ( I ) is Cohen–Macaulay, any minimal set of generators of aminimal reduction J ⊂ I is analytically adjusted. This is a consequence of the freeness ofthe corresponding Noether normalization. It is natural to search for conditions under whicha converse statement holds. At the moment we could only obtain such a result in dimensionthree, treated in the last section.When dim R = 3, some of the results become particularly enhanced. This is the contentof the last part of the paper. Here both analytical tightness and analytical adjustment fallin one and the same basket to characterize Cohen–Macaulayness of F ( I ). The section isalso focused on the reduction number of the ideal I and the multiplicity of its special fiber,with a side about the rational map defined by the linear system of I in its generating degree.2 Sprouts of the G s condition Let R be a Noetherian ring and let I ⊂ R denote an ideal of R .We consider in this work the following graded algebras associated to the pair ( R, I ) : • The Rees algebra R R ( I ) = L i ≥ I i t i ⊂ R [ t ] of I. • The associated graded ring gr I ( R ) = R R ( I ) /I R R ( I ) of I. These algebras are related by a natural R -algebra surjection of graded R -algebras R R ( I ) ։ gr I ( R ) . This allows comparison in many directions. Thus, for example, in many situations theirdimensions differ by one.In this work we assume throughout that ( R, m ) is a standard graded algebra over a field k with maximal irrelevant ideal m = ([ R ] ). In this case, if I ⊂ m is a homogeneous idealthen dim gr I ( R ) = dim R , while dim R R ( I ) = dim R + 1 provided grade I ≥ special fiber cone (or the fiber cone ) of I to be F R ( I ) := R R ( I ) / m R R ( I ) , and the analytic spread of I , denoted by ℓ ( I ), to be the (Krull) dimension of F R ( I ).Let I be a homogeneous ideal of R minimally generated by forms f , . . . , f r of the samedegree d – one then says that I is d -equigenerated for short.Consider the bigraded k -algebra R ⊗ k k [ y , . . . , y r ] = R [ y , . . . , y r ] , where k [ y , . . . , y r ] isa polynomial ring over k and the bigrading is given by bideg([ R ] ) = (1 ,
0) and bideg( y i ) =(0 , t ) = ( − d, R R ( I ) = R [ It ] inherits a bigraded structure over k . One has a bihomogeneous R -homomorphism S −→ R R ( I ) ⊂ R [ t ] , y i f i t. Thus, thebigraded structure of R R ( I ) is given by R R ( I ) = M c,n ∈ Z [ R R ( I )] c,n and [ R R ( I )] c,n = [ I n ] c + nd t n . One is often interested in the R -grading of the Rees algebra. Thus, one sets, namely,[ R R ( I )] c = L ∞ n =0 [ R R ( I )] c,n , and of particular interest is[ R R ( I )] = ∞ M n =0 [ I n ] nd u n = k [[ I ] d u ] ≃ k [[ I ] d ] = ∞ M n =0 [ I n ] nd ⊂ R. Clearly, R R ( I ) = [ R R ( I )] ⊕ (cid:16)L c ≥ [ R R ( I )] c (cid:17) = [ R R ( I )] ⊕ m R R ( I ). Therefore, one gets k [[ I ] d ] ≃ [ R R ( I )] ≃ R R ( I ) / m R R ( I ) = F R ( I ) (1)as graded k -algebras.In the sequel, for simplicity, we often write I d = [ I ] d if no confusion arises.Consider in addition a free R -module presentation of I : R m ϕ −→ R r → I → . (2)The following notion was introduced in [2, Section 2] (see also [18, Definiton 1.3]):3 efinition 1.1. Let R be a Noetherian ring and let I ⊂ R be an ideal. Given an integer s ≥
1, one says that I satisfies condition G s if µ ( I p ) ≤ ht p for every prime p ∈ V ( I ) withht p ≤ s − . Remark 1.2.
As is known, this sort of condition and its analogues have an impact oncertain dimension theoretic aspects. It is more effective under additional hypotheses, suchas when R is Cohen–Macaulay and/or when I has a rank, in which case its effectiveness isexpressed in terms of the ideals of minors of a matrix ϕ of a presentation of I as in (2).Throughout this part, R will denote a Noetherian ring of finite Krull dimension d . Inthis context, the most notable cases of the above condition have s = d and s = d + 1,the second also denoted G ∞ . Here we intend to consider lower cases of s , to see how theyimpact in a few situations.The symmetric algebra above is very sensitive to it (see, e.g., [24, Section 7.2.3 and ff]).The next result is an encore of [17, Corollary 2.13] in the present terminology. The conclu-sion of the statement was originally conceived by G. Valla, which spurred the subsequentwork by C. Huneke and M. E. Rossi.For the present purpose, recall that a Noetherian ring of finite Krull dimension is saidto be equicodimensional if all its maximal ideals have the same height (= dim R ). Proposition 1.3. (The Valla dimension)
Let R be a Noetherian ring of finite Krull di-mension d and let I ⊂ R be an ideal generated by r elements. Suppose that the followingconditions hold: (i) R is equicodimensional and grade I ≥ . (ii) The number of generators of I locally in height d − is at most r − . (iii) I satisfies condition G d − .Then dim S R ( I ) = max { d + 1 , r } . Proof.
The proof, based on the Huneke–Rossi formula, follows the same pattern with themore or less obvious adaptation. Thus, first since grade I ≥ P ∈ Spec R of maximal dimension one has I P , and hence dim R/P + µ ( I P ) = dim R +1 = d + 1, so dim S R ( I ) ≥ d + 1. On the other hand, dim S R ( I ) ≥ r ([17, Corollary 2.8]).Therefore, it suffices to show the upper bound dim S R ( I ) ≤ max { d + 1 , r } . For I P there is nothing to prove, so assume that I ⊂ P . If P is a maximal ideal, equicodimen-sionality takes care since dim R/P = 0 and µ ( I P ) ≤ r . If P has height d − R/P ≤
1. Finally, for P of height at most d − G d − assumption takes over. Remark 1.4.
It is curious that a small piece of the proof in [17, Corollary 2.13] dependson knowing that the Noetherian base ring R is equicodimensional. This hypothesis isunderstated in [17, Corollary 2.13] possibly because the authors assume throughout that R is a ring with sufficient conditions for equicodimensionality to hold. Since it has beenshown afterwards that these hypotheses are superfluous for the main dimension formula, soit seemed that all the corollaries would dispense as well with any extra hypotheses on R (unless otherwise explicit in the statements).4articular cases of interest in this work are as follows: Corollary 1.5.
Let I ⊂ R denote an ideal of grade ≥ in a Noetherian equicodimensionalring R of dimension . Let I be generated by r ≥ elements. Then the following areequivalent: (i) µ ( I P ) ≤ r − for every height prime P ∈ Spec R (ii) dim S R ( I ) = max { , r } . Proof. (i) ⇒ ) (ii). This implication follows from Proposition 1.3 and does not require thebound r ≥ ⇒ ) (i). Let ht P = 2. Then dim R/P = 1 since R is equicodimensional. Therefore,1 + µ ( I P ) = dim R/P + µ ( I P ) ≤ dim S R ( I ) = max { , r } = r, hence µ ( I P ) ≤ r − Example 1.6.
The following simple example illustrates the above corollary. Let R = k [ x, y, z ] ( k a field) and let I = ( x, y ) ∩ ( x, z ) , a 4-generated ideal. Then the number ofgenerators of I locally at the minimal primes ( x, y ) and ( x, z ) does not drop, and in fact,as it turns out, dim S R ( I ) = 5 > Corollary 1.7.
Let R be a Noetherian equicodimensional, equidimensional catenary ringane let ϕ denote an n × ( n − matrix with entries in R . Suppose that dim R := d < n and let L = I ( y .ϕ ) denote a presentation ideal of S R ( I ) . If ϕ satisfies condition G d − and I ( ϕ ) has height d then ht L = d ( maximal possible ) . Proof.
Since n > d , by Proposition 1.3 one has dim S R ( I ) = n . By the assumptions on R ,one has ht L = d + n − n = d .If I = ( f , . . . , f r ) ⊂ R is an equigenerated ideal in a standard graded ring over a field k then, by a previous identification, the kernel Q of the surjection of graded k -algebras k [ y , . . . , y r ] → k [ f t, . . . , f r t ] , y i → f i t is referred to as the presentation ideal of F R ( I ). Obviously, QR [ y , . . . , y r ] ⊂ J , where thelatter is the presentation ideal of R R ( I ) on R [ y , . . . , y n ]. In the sequel we will omit thesubscript R in the notation of the special algebra to avoid confusion with the ground field k over which it is naturally an algebra.As an immediate consequence of [25, Theorem 5.1] we have Proposition 1.8.
Let R be a Noetherian Cohen-Macaulay ring and I an ideal of R suchthat pd I ≤ . If I satisfies G s then S R ( I ) i ≃ I i for each ≤ i ≤ s − . In particular, A i = { } for each ≤ i ≤ s − . A consequence of Proposition 1.8 is
Corollary 1.9.
Let R be a standard graded Cohen–Macaulay ring over a field and let I bean equigenerated perfect homogeneous ideal of R of height . If I satisfies property G s then indeg( Q ) ≥ s. s = dim R , which is a typical assumption in many known results,the knowledge of smaller cases can also be useful. One situation is checking the birationalityof a rational map via the main criterion of [6]. Next is one example. Example 1.10.
Let I ⊂ R = k [ x , x , x , x ] be the height two perfect ideal generated bythe maximal minors of the matrix ϕ = x h x x h x x x h x x h x x h x x d , where d ≥ h , . . . , h are forms of degree d. Clearly, I has one-less than maximal linear rank. Besides, without some calculation, onecannot decide whether ℓ ( I ) = 4, that is, if the map F defined by the maximal ideals of ϕ is finite. Thus, [6, Theorem 3.2] is far from applicable to deduce that the map is birationalonto the image. Instead, we go about the birationality without deciding a priori if the mapis finite. Namely, a straightforward verification shows that a prime ideal P containing I ( ϕ )contains x , x d +33 , x x , hence, ht I ( ϕ ) ≥ . Thus, the ideal I satisfies G .On the other hand, using the three rightmost linear syzygies of ϕ we obtain a submatrixof the Jacobian dual matrix of F as follows: ψ = y y y y y y y y , where the y i ’s are new variables over k . By Corollary 1.9, the 3-minor y ( y y − y ) of ψ does not vanish modulo Q. Hence, ψ has maximal rank over Q . By [6, Theorem 2.18 (b)], F is birational onto its image. Note, however, that I does not satisfy G since x appearsat most on two columns, forcing I ( ϕ ) to have height 3 only – in addition, when d ≥ G would imply that deg( F ) = d ≥
2, the product of the syzygy degrees (see [3, Corollary 3.2]).
As a piece of additional notation, let I ⊂ R be a homogeneous ideal in a standard gradedring R over a field and set m := ( R + ) ⊂ R . The image of an element f ∈ I in I/I ⊂ gr I ( R )will be denoted by f o . In addition, set gr I ( R ) + := ( I/I )gr I ( R ) for the ideal generated inpositive degrees.Similarly, the image of an element f ∈ I in I/ m I ⊂ F ( I )) will be denoted by f ∗ .However, in this case, the notation will become obsolete throughout due to the identificationof graded k -algebras F ( I ) ≃ k [ I d ] = L n ≥ [ I n ] nd ⊂ R established in (1), whereby f ∗ i getsidentified with f i ∈ [ I ] d ⊂ k [ I d ].We will use the following lemmata, certainly known in one way or another. Proofs aregiven for completeness. 6 emma 2.1. Let R denote a standard graded ring over an infinite field k and let I ⊂ R bea d -equigenerated homogeneous ideal. Let J ⊂ I be a minimal reduction. One has: (i) F ( I ) ≃ k [ I d ] is Cohen–Macaulay if and only if k [ I d ] is a free k [ J d ] -module via thenatural extension k [ J d ] ⊂ k [ I d ] . (ii) If F ( I ) ≃ k [ I d ] is Cohen–Macaulay then { } ∪ B ∪ . . . ∪ B r − is a free basis of k [ I d ] over k [ J d ] , where B n is the lifting of a homogeneous basis of the k -vector space [ I n /J I n − ] nd and r denotes the reduction number of I . Proof. (i) This is the well-known argument, as applied to the graded Noether normalization k [ J d ] ⊂ k [ I d ].(ii) Since J I r = I r +1 , it is clear that the stated set generates k [ I d ] as k [ J d ]-module. Bythe Krull–Nakayama lemma, this set is a minimal set of generators, hence it must be a freebasis. Lemma 2.2.
With gr I ( R ) + as above, one has: (1) If R is Cohen–Macaulay or a domain ( equidimensional catenary suffices ) then ht gr I ( R ) + ≤ ht I. (2) If gr I ( R ) is Cohen–Macaulay then ht gr I ( R ) + ≥ ht I . (3) If both R and gr I ( R ) are Cohen–Macaulay then ht gr I ( R ) + = ht I . Proof. (1) and (2) come out from the isomorphism gr I ( R ) / gr I ( R ) + ≃ R/I , while (3) isthe conjunction of (1) and (2).
Lemma 2.3.
Let R denote a standard graded ring over an infinite field and let I ⊂ R bea d -equigenerated homogeneous ideal of height g . Let f , . . . , f g ∈ I d be a maximal regularsequence in I such that ( f , . . . , f g ) ∩ I n = ( f , . . . , f g ) I n − . (3) for every n ≥ . Then: (i) f ∗ , . . . , f ∗ g is a regular sequence in F ( I ); in particular, depth F ( I ) ≥ g. (ii) ( f , . . . , f g ) P is a minimal reduction of I P for every prime P ⊃ I with ht ( P ) = ht ( I ) . (iii) The sequence f , . . . , f g can be extended to a sequence generating a minimal reductionof I. Proof. (i) Having identified F ( I ) ≃ k [ I d ] = L n ≥ [ I n ] nd ⊂ R , the assertion is merely aproperty of the graded inclusion k [ I d ] ⊂ R , whereby one has to check that, for 0 ≤ i ≤ g − f i +1 is a regular element on L n ≥ [ I n / ( f , . . . , f i ) I n − ] nd . Thus, let there be given n and h ∈ [ I n ] nd such that hf i +1 ∈ ( f , . . . , f i )[ I n ] nd ⊂ ( f , . . . , f i ) R .Since { f , . . . , f i +1 } is a regular sequence in R , then h ∈ ( f , . . . , f i ) R , hence h ∈ [( f , . . . , f i ) R ∩ I n ] nd . Then, by [28, Corollary 2.7], h ∈ [( f , . . . , f i ) I n − ] nd , as was tobe shown. 7ii) Localizing (3) at a prime P ⊃ I gives( f , . . . , f g ) P ∩ I nP = ( f , . . . , f g ) P I n − P . If, moreover, ht P = ht I then, since ht ( f , . . . , f g ) = ht I , it follows that I P is containedin the radical of ( f , . . . , f g ) P . Therefore, I nP ⊂ ( f , . . . , f g ) P for some n >>
0, and hence, I nP = ( f , . . . , f g ) P I n − P . This shows that ( f , . . . , f g ) P is a reduction of I P , hence a minimalone since any minimal reduction is minimally generated by ℓ ( I P ) ≥ ht I P = g elements.(iii) Since k is infinite, and, by (i), f , . . . , f g is a homogeneous regular sequence in k [ I d ],then it is a subset of a homogeneous system of parameters of k [ I d ]. The latter generatesa minimal reduction of the irrelevant maximal ideal of k [ I d ]. By a well-known fact, thisreduction lifts to a minimal reduction of I extending the R -regular sequence f , . . . , f g .By [28], one knows that the hypotheses of the previous lemma imply (actually, areequivalent to the assertion) that the residues f o , . . . , f og constitute a regular sequence ingr I ( R ). The next lemma gives a condition on gr I ( R ) for as to when such sequences exist atall. Lemma 2.4.
Let R denote a standard graded ring over an infinite field and let I ⊂ R be a d -equigenerated homogeneous ideal of height g . If the associated graded ring gr I ( R ) is Cohen-Macaulay then there exist forms f , . . . , f g ∈ I d such that f o , . . . , f og is a regularsequence in gr I ( R ); if, in addition, R is Cohen–Macaulay, then gr I ( R ) + has grade g . Proof.
It is basically a prime avoidance argument. To wit, letting I = ( f , . . . , f r ), one hasht ( f o , . . . , f or ) = dim gr I ( R ) − dim gr I ( R )( f o , . . . , f or ) , since gr I ( R ) is Cohen–Macaulay= dim R − dim R/I ≥ g. Now, since gr I ( R ) is Cohen–Macaulay then ht P = 0 for every associated prime P of gr I ( R ) . Since ht ( f o , . . . , f or ) ≥ g > f o , . . . , f or ) is not contained in any of these primes. Byprime avoidance, a sufficiently general k -linear combination f of f o , . . . , f or is such that f o is not contained in any P , hence is gr I ( R )-regular. Let f ∈ I d be any lifting of f .The general inductive is similar, whereby one assumes the existence of f , . . . , f i ∈ I d , for1 ≤ i ≤ g −
1, such that f o , . . . , f oi is a regular sequence in gr I ( R ). Since gr I ( R ) / ( f o , . . . , f oi )is Cohen-Macaulay then again,ht ( f o , . . . , f or )gr I ( R ) / ( f o , . . . , f oi ) = g − ≥ , hence, by prime avoidance, there exists a form f i +1 ∈ I d such that f oi +1 is regular ongr I ( R ) / ( f o , . . . , f oi ).The supplementary assertion is a consequence of Lemma 2.2 (3). Remark 2.5.
Although Lemma 2.4 claims the existence of some appropriate d -forms fillingthe request, it is clear from the proof that by choosing general forms in I d the result is equallyobtainable.The following example, which we don’t claim to be a ‘smallest’ or ‘simplest’, shows thateven general forms may not work if gr I ( R ) is not Cohen–Macaulay.8 xample 2.6. Let I = ( z , yz , y z , xy z , x y z , x y ) ⊂ R = k [ x, y, z ], a height twoperfect ideal of analytic deviation one. A calculation with [11] gives that both gr I ( R ) and F ( I ) are almost Cohen–Macaulay (i.e., both have depth 2). If f , f ∈ I are generalforms, an additional computation outputs ( f , f ) ∩ I ( f , f ) I (alternatively, the gradeof gr I ( R ) + is 1). Corollary 2.7.
Let R denote a standard graded ring over an infinite field and let I ⊂ R bea d -equigenerated homogeneous ideal. Suppose that I is equimultiple ( that is, ht I = ℓ ( I )). If gr I ( R ) is Cohen–Macaulay then so is the special fiber F ( I ) . Remark 2.8.
In the last corollary, if R is Cohen–Macaulay, the assumption that gr I ( R ) isCohen–Macaulay can be weakened by just assuming that there exist ht I − I d whose images in gr I ( R ) constitute a regular sequence.The role of the Castelnuovo–Mumford regularity of a standard graded ring A over afield, denoted reg( A ), has been largely treated – see [8] and [27], ‘classically’ tailored, and[19], of a more recent vintage. Lemma 2.9.
Let A denote a standard graded domain over an algebraically closed field k such that reg( A ) ≤ . Then either A is Cohen–Macaulay with minimal multiplicity or else depth A ≤ dim A − . Proof.
We may assume that reg( A ) = 1 as otherwise A is a polynomial ring over k . Then,writing A = S/I , where S is a polynomial ring over k , A has a 2-linear resolution over S .Suppose as if it were, that depth A = dim A −
1. Then, by [12, Theorem 1.5] one has e ( A ) = p !( p − − b p ( p − < p, where p denotes the homological dimension of A over S and b p is the last Betti number of theresolution. On the other hand, p = edim A − depth A . Therefore, e ( A ) < edim A − dim A +1 = ecod A + 1. Since A is a domain over an algebraically closed field, this contradicts awell-known result ([8]). It follows that A is Cohen–Macaulay and, necessarily, has minimalmultiplicity. Example 2.10.
The following well-known example shows that the above result does nothold in general if A is not a domain: I = ( x , x ) ∩ ( x , x ) ⊂ R = k [ x , x , x , x ]. Heredepth R/I = 1 = dim
R/I − R/I = 1.Next is an application in our context. Recall that the analytic deviation of an ideal I (in a local or standard graded ring) is the difference ℓ ( I ) − ht I . Corollary 2.11.
Let R denote a standard graded domain over an algebraically closed fieldand let I ⊂ R be a d -equigenerated homogeneous ideal of analytic deviation one. If gr I ( R ) is Cohen–Macaulay and has regularity at most one, then F ( I ) is Cohen-Macaulay withminimal multiplicity. Proof.
By [20] (see also [27, Theorem 1.1]), reg gr I ( R ) = reg R ( I ), and by [22, Theorem 1.3and Theorem 3.2] or [9, Section 1], reg F ( I ) ≤ reg R ( I ) . We may assume that reg F ( I ) = 1as otherwise there is nothing to prove. Since dim F ( I ) = ℓ ( I ) = ht I + 1 by the analyticdeviation assumption, and depth F ( I ) ≥ ht I , it follows that depth F ( I ) ≥ dim F ( I ) − R is a domain and F ( I ) ≃ k [ I d t ] ⊂ R [ It ] ⊂ R [ t ], then F ( I ) is adomain. Therefore, the result is a consequence of the previous lemma.9 .2 Analytically tight sequences of forms Let R denote a standard graded ring over an infinite field and let I ⊂ R be a d -equigeneratedhomogeneous ideal. We introduce the main notion of this section: Definition 2.12.
Let I ⊂ R be a d -equigenerated homogeneous ideal in a standard gradedring over an infinite field. Set ℓ = ℓ ( I ) (analytic spread) and µ = µ ( I ) (minimal number ofgenerators). We say that a sequence of forms f , . . . , f ℓ ∈ I d is analytically tight in power n if [(( f , . . . , f ℓ − ) : f ℓ ) ∩ I n ] nd = [( f , . . . , f ℓ − ) ∩ I n ] nd . (4)The sequence f , . . . , f ℓ ∈ I d is analytically tight if it is analytically tight in power n forevery n ≥ . A few properties of this notion are:1. If f , . . . , f ℓ ∈ I d is analytically tight in power n then it is so in any power ≥ n .2. If an equality as (4) holds for every subset f , . . . , f i , with 0 ≤ i ≤ ℓ , then we wouldbe speaking of a fully tight sequence in power n .3. A fully tight sequence satisfying in addition the Valabrega–Valla condition( f , . . . , f i ) ∩ I n ⊂ ( f , . . . , f i ) I n − is a sequence of regular type n − Theorem 2.13.
Let R denote a standard graded ring over an infinite field and let I ⊂ R be a d -equigenerated homogeneous ideal of analytic deviation one and height g . Let f , . . . , f g , f g +1 be forms in I d such that (a) The images of f , . . . , f g in gr I ( R ) form a regular sequence. (b) J = ( f , . . . , f g +1 ) is a reduction of I .Then the following are equivalent: (i) f , . . . , f g , f g +1 analytically tight. (ii) The special fiber F ( I ) is Cohen-Macaulay. Proof. (i) ⇒ (ii) By Lemma 2.3 (i), f , . . . , f g is a regular sequence in F ( I ) = k [ I d ] = M n ≥ [ I n ] nd , regarded as elements of [ I ] d . Thus, we are to prove that f g +1 ∈ [ I ] d is a non-zero-divisor of L n ≥ [ I n / ( f , . . . , f i ) I n − ] nd . 10o, let h ∈ [ I n ] n d be such that hf g +1 = 0 in L n ≥ [ I n / ( f , . . . , f i ) I n − ] nd . This meansthat h ∈ [(( f , . . . , f g ) I n : f g +1 ) ∩ I n ] n d ⊂ [(( f , . . . , f g ) : f g +1 ) ∩ I n ] n d . Therefore, h ∈ [( f , . . . , f g ) ∩ I n ] n d because f , . . . , f g , f g +1 is analytically tight= [( f , . . . , f g ) I n − ] n d because f o , . . . , f og is a regular sequence in gr I ( R ) . . (ii) ⇒ (i) Since J = ( f , . . . , f g +1 ) is a reduction of I then f , . . . , f g +1 , regarded aselements of [ I ] d , is a system of parameters of F ( I ) = k [ I d ], hence a regular sequence of F ( I ) since the latter is assumed to be Cohen-Macaulay.The proof is similar to the argument in the previous item, only changing sense. Thus,for the essential inclusion to be seen, pick h ∈ [(( f , . . . , f g ) : f g +1 ) ∩ I n ] nd . Then hf g +1 ∈ [( f , . . . , f g ) ∩ I n +1 ] ( n +1) d = [( f , . . . , f g ) I n ] ( n +1) d , where the equality results from assumption (a) via [28, Corollary 2.7]. Since f g +1 is anon-zero-divisor of L n ≥ [ I n / ( f , . . . , f i ) I n − ] nd , one has h ∈ [( f , . . . , f g ) I n − ] nd = [( f , . . . , f g ) ∩ I n ] nd because f o , . . . , f og is a regular sequence in gr I ( R ) . In the next proposition sufficient conditions are established for a sequence to be analyt-ically tight in power n for every n >>
0. The proof is an adaptation of an argument in [15,Remark 2.1].
Proposition 2.14.
Let R denote a standard graded ring over an infinite field k and let I ⊂ R be an unmixed d -equigenerated homogeneous ideal of analytic deviation one andheight g . Suppose that { f , . . . , f g , f g +1 } ⊂ [ I ] d generates a reduction of I , with f , . . . , f g aregular sequence in R . One has: (i) f , . . . , f g , f g +1 is analytically tight in sufficiently high powers. (ii) If, moreover, ( f , . . . , f g ) ∩ I n = ( f , . . . , f g ) I n − , for every n ≥ . Then f , . . . , f g , f g +1 is analytically tight in powers n ≥ n , where n = max { r ( I P ) | P ∈ Min(
R/I ) } + 1 . Inparticular, in the case where I is generically a complete intersection, f , . . . , f g , f g +1 is analytically tight in all powers. Proof. (i) By degree reasons, it suffices to show that there is an integer n such that(( f , . . . , f g ) : f g +1 ) ∩ I n = ( f , . . . , f g ) ∩ I n , for every n ≥ n . To prove the essential inclusion set Ass ( R/ ( f , . . . , f g )) = { P , . . . , P r , Q , . . . , Q s } ,where Ass ( R/I ) = { P , . . . , P r } . Since I is assumed to be unmixed, ht P i = ht Q j = g for all i, j . Write ( f , . . . , f g ) = N ∩ N , where N is the intersection of the primary compo-nents of ( f , . . . , f g ) corresponding to { P , . . . , P s } and N is the intersection of the primarycomponents of ( f , . . . , f g ) corresponding to { Q , . . . , Q s } . Since √ I = √ N , there exists aninteger n such that I n ⊂ N for every n ≥ n . Pick h ∈ (( f , . . . , f g ) : f g +1 ) ∩ I n with n ≥ n . Then hf g +1 ∈ N ∩ N and h ∈ I n . Since p ( f , . . . , f g , f g +1 ) = √ I and I Q j for j = 1 , . . . , s, forcefully f g +1 / ∈ Q j for j = 1 , . . . , s. Thus, h ∈ N ∩ I n ⊂ N ∩ N = ( f , . . . , f g ), as required.11ii) As in the proof of item (i), set Ass ( R/ ( f , . . . , f g )) = { P , . . . , P r , Q , . . . , Q s } ,where Ass ( R/I ) = { P , . . . , P r } and ( f , . . . , f g ) = N ∩ N , where N is the intersectionof the primary components of ( f , . . . , f g ) corresponding to { P , . . . , P s } and N is theintersection of the primary components of ( f , . . . , f g ) corresponding to { Q , . . . , Q s } . Bya similar token as in the proof of (i), it is sufficient to prove that I n ⊂ N for n =min { r ( I P ) | P ∈ Min(
R/I ) } + 1.Now, by assumption, ( f , . . . , f g ) ∩ I n = ( f , . . . , f g ) I n − , for every n ≥ . In particular,for every P ∈ Ass (
R/I ) = Min(
R/I ) , the images of f , . . . , f g in R P is a regular sequenceand ( f , . . . , f g ) P ∩ I nP = ( f , . . . , f g ) P I n − P for every n ≥ . Thus, the images of f , . . . , f g in gr I P ( R P ) form a regular sequence. Inparticular, depth gr I P ( R P ) = dim gr I P ( R P ) = g, that is, gr I P ( R P ) is Cohen-Macaulay.Hence, by [26, Theorem 1.2], the reduction number r ( I P ) of I P is independent of theminimal reduction. By Lemma 2.3 (ii), ( f , . . . , f g ) P is a minimal reduction of I P , hence r ( I P ) ≤ n − I n P = ( f , . . . , f g ) P I n − P ⊂ ( f , . . . , f g ) P = N P . But, since I n P ⊂ N P for every P ∈ Ass (
R/N ) = Ass (
R/I ), we have I n ⊂ N as desired.To see the supplementary assertion, if I is generically a complete intersection then I P = ( f , . . . , f g ) P over any minimal prime P of I . Hence, in this case, the argument showsthat one can take n = 1 . Remark 2.15.
Recall again that, by [28, Corollary 2.7], the additional hypothesis of item(ii), along with the standing assumption of item (a), makes it equivalent to requiring thatthe images of f , . . . , f g in gr I ( R ) are a regular sequence. Computationally, this angle ispreferable. Corollary 2.16.
Let R denote a standard graded Cohen–Macaulay ring over an infinite fieldand let I ⊂ R be an equigenerated homogeneous ideal satisfying the following conditions: (i) I is unmixed and generically a complete intersection. (ii) I has analytic deviation one. (iii) gr I ( R ) is Cohen–Macaulay.Then F ( I ) is Cohen–Macaulay. An example of the above corollary in dimension three is given by the ideal generatedby the ( m − m ≥ R defining a central generic arrangementThis is a special case of the main result in [21] (see also [10, Proposition 4.1]). Proposition 2.17.
Let R denote a standard graded ring over an infinite field and let I ⊂ R be a d -equigenerated homogeneous ideal of analytic deviation one and of height g . (i) If gr I ( R ) is Cohen–Macaulay, there exists a sequence of forms f , . . . , f g , f g +1 in I d such that f o , . . . , f og is a regular sequence in gr I ( R ) and J = ( f , . . . , f g +1 ) is a minimalreduction of I. Suppose, moreover, that I is unmixed. If R and R R ( I ) are Cohen–Macaulay locallyat the minimal primes of I , then any sequence as in (i) is analytically tight in powers ≥ ℓ ( I ) − . In particular, this is the case when R is regular locally at the minimalprimes of R and gr I ( R ) is Cohen–Macaulay locally at these primes. Proof. (i) This is an immediate consequence of Lemma 2.4 and Lemma 2.3.(ii) Since both R P and R R P ( I P ) are Cohen–Macaulay for every P ∈ Min(
R/I ), then [1,Theorem 5.1] yields r ( I P ) ≤ dim R P − g − P . Hence, in the notationof Proposition 2.14, n = max { r ( I P ) | P ∈ Min(
R/I ) } + 1 ≤ g = ℓ ( I ) − . On the other hand, since f o , . . . , f og is a regular sequence in gr I ( R ), then it satisfies theValabrega–Valla condition ( f , . . . , f g ) ∩ I n = ( f , . . . , f g ) I n − , for every n ≥
1. Since I is assumed to be unmixed, Proposition 2.14 (ii) says that the sequence f , . . . , f g , f g +1 isanalytically tight in powers ≥ ℓ ( I ) − F ( I ) orgr I ( R ). Computational evidence suggests that perfect height two ideals in dimension threemay be an adequate source. Next is an example to keep this in mind. Example 2.18.
Let I = ( z , yz , xyz , xy z , xy z , x y z, x y ) ⊂ R = k [ x, y, z ], a perfectheight two ideal with ℓ ( I ) = 3. For general f , f , f ∈ I , a calculation with [11] gives[(( f , f ) : f ) ∩ I n ] n = [( f , f ) ∩ I n ] n , for n = 1, hence for every n ≥ { f , f , f } is analytically tight. However,although r ( f ,f ,f ) ( I ) = reg( F ( I )) = 2, the special fiber F ( I ) is not Cohen-Macaulay. Inconfront with Theorem 2.13, one derives that the associated graded ring is not Cohen–Macaulay – actually, depth gr I ( R ) = depth gr I ( R ) + = 1. Some of the ideas of the previous subsection can be stated in terms of certain other expectedequalities.
Definition 2.19.
Let I ⊂ R be a d -equigenerated homogeneous ideal in a standard gradedring over an infinite field. Set µ = µ ( I ) (minimal number of generators). A set { f , . . . , f l } ⊂ I d of k -linearly independent forms will be said to be l - adjusted if µ ( IJ ) = lµ ( I ) − (cid:0) l (cid:1) , where J = ( f , . . . , f l ).If l = ℓ ( I ) is the analytic spread of I , an l -adjusted set will also be said to be analyticallyadjusted . Remark 2.20. (1) Clearly, for any choice of such indices, one always has the inequality µ ( IJ ) ≤ lµ ( I ) − (cid:0) l (cid:1) . We observe that a natural candidate for an analytically adjusted set ofgenerators is one such that J ⊂ I is a minimal reduction – recall that, in the equigeneratedcase, one can take a reduction J ⊂ I that is a subset of a minimal set of generators of I .132) An advantage of equigeneration is that one essentially counts vector space dimen-sions, hence µ ( I ) ≥ µ ( IJ ) in the setup of Definition 2.19. Therefore, the present notionis somewhat stronger than the Freiman kind of lower bound inequality (see [13, Theorem1.9]). The same paper gives examples where any reasonable expectation may go wrong if I is not equigenerated, even in the monomial case.Next are two situations in which the above condition is satisfied. Proposition 2.21.
Let I ⊂ R be a d -equigenerated homogeneous ideal in a standard poly-nomial ring over an infinite field. Suppose that the special fiber F ( I ) admits no quadraticpolynomial relations – e.g., height two perfect ideals satisfying G and, more generally,syzygetic equigenerated ideals. Then, any set of k -linearly independent forms f , . . . , f l ⊂ I d is l -adjusted. Proof.
Since f , . . . , f l ∈ I d are k -linearly independent, one can extend them to a minimalset { f , . . . , f µ } of generators of I . The ideal ( f , . . . , f l ) I is a 2 d -equigenerated homogeneousideal generated by the following set of forms of degree 2 d : B = { f f j | ≤ j ≤ µ ( I ) } ∪ { f f j | ≤ j ≤ µ ( I ) } ∪ . . . ∪ { f l f j | i ≤ j ≤ µ ( I ) } It suffices to prove that this set is k -linearly independent. Indeed, if so then the set B is abasis of the k -vector space [( f , . . . , f l ) I ] d . In particular, B is a minimal set of homogneousgenerators of ( f , . . . , f l ) I . Since B has µ (( f , . . . , f l ) I ) = lµ ( I ) − (cid:0) l (cid:1) elements, we arethrough.But a nonzero k -linear combination of the elements of B yields a nonzero quadraticpolynomial relation of the special fiber F ( I ), which contradicts the assumption. Proposition 2.22.
Let R denote a standard graded ring over an infinite field k and let I ⊂ R be a d -equigenerated homogeneous ideal. If f , . . . , f l ∈ [ I ] d are forms such that f , . . . , f l is a regular sequence in k [ I d ] then, for any ≤ i ≤ l , the subset { f , . . . , f i } is i -adjusted. In particular, if the special fiber F ( I ) is Cohen-Macaulay then every set of forms { f , . . . , f ℓ } ⊂ I d generating a minimal reduction J of I is analytically adjusted. Proof.
First extend the regular sequence f , . . . , f l to a full k -vector basis { f , . . . , f l , f l +1 , . . . , f µ ( I ) } of I d . Clearly, µ (( f , . . . , f i ) I ) = dim k [( f , . . . , f i ) I ] d . As already pointed out earlier,the inequality dim k [( f , . . . , f i ) I ] d ≤ iµ ( I ) − (cid:0) i (cid:1) is immediate because [( f , . . . , f i ) I ] d has an obvious set of generators with iµ ( I ) − (cid:0) i (cid:1) elements. To prove the reverse in-equality we induct on i. For i = 1 the result is clear. Now, for 1 < i ≤ l let B de-note a k -vector base of [( f , . . . , f i − ) I ] d . A vanishing k -linear combination of elements in B ∪ { f i , f i f i +1 , . . . , f i f µ ( I ) } can be written as f i ( λ i f i + · · · + λ µ ( I ) f µ ( I ) ) = k -linear combination of the elements of B ∈ ( f , . . . , f i − ) I Thus, since f i is F ( I ) / ( f , . . . , f i − )-regular we have λ i f i + · · · + λ µ ( I ) f µ ( I ) = 0, hence λ i = · · · = λ µ ( I ) = 0 because f i , . . . , f µ ( I ) are k -linearly independent. Consequently, one has that14 ∪ { f i , f i f i +1 , . . . , f i f µ ( I ) } is a k -linearly independent subset of [( f , . . . , f i ) I ] d . Since itsnumber of elements is dim k [( f , . . . , f i − ) I ] d + µ ( I ) − i +1, by induction, dim k [( f , . . . , f i ) I ] d ≥ iµ ( I ) − (cid:0) i (cid:1) as desired.If F ( I ) is moreover Cohen–Macaulay, then any minimal set of generators of a min-imal reduction is a regular sequence. Still, an independent proof can be given as fol-lows. Clearly, the k -vector space [ J I ] d is spanned by the set B := { f i f j | ≤ i < j ≤ ℓ } ∪ { f h i , . . . , f ℓ h i | ≤ i ≤ s } of cardinality ℓµ ( I ) − (cid:0) ℓ (cid:1) .One claims that B is k -linearly independent. Indeed, let X ≤ i Let R denote a standard graded ring over an infinite field and let I ⊂ R be a d -equigenerated homogeneous ideal such that (a) I is unmixed of analytic deviation one. (b) For any minimal prime P of R/I the reduction number of I P is at most one.If f , f , . . . , f g +1 is a sequence of forms in I d satisfying the properties in Proposition 2.17(i) , then the following are equivalent: (i) F ( I ) is Cohen-Macaulay. (ii) f , f , . . . , f g +1 is analytically adjusted. (iii) f , f , . . . , f g +1 is analytically tight. Proof. The equivalence (i) ⇔ (iii) is a consequence of Theorem 2.13, while the implication(i) ⇒ (ii) is a consequence of Proposition 2.22. Therefore, it remains to prove that (ii) ⇒ (i).By Lemma 2.3 (i), { f , . . . , f g } ⊂ [ I ] d is a regular sequence in L n ≥ [ I n ] nd . We will bethrough if f g +1 ∈ [ I ] d is regular on L n ≥ [ I n / ( f , . . . , f g ) I n − ] nd . For this, let h ∈ [ I n ] nd besuch that hf g +1 ∈ [( f , . . . , f g ) I n ] ( n +1) d . We separate the argument in two cases: Case n = 1 : We can extend the set { f , . . . , f g +1 } to a basis { f , . . . , f µ ( I ) } of the k -vectorspace I d . Since { f , . . . , f g +1 } is analytically adjusted then the set B = { f i f j | ≤ i ≤ j ≤ g + 1 } ∪ { f i f j | ≤ i ≤ g + 1 , g + 2 ≤ j ≤ µ ( I ) } is a basis of the k -vector space [ J I ] d , where J = ( f , . . . , f g +1 ) . On the other hand, since { f , . . . , f g } is a regular sequence in F ( I ) = k [ I d ], it is g -adjusted by Proposition 2.22, i.e., µ (( f , . . . , f g +1 ) I ) = gµ ( I ) − (cid:0) g (cid:1) , and hence, the set B ′ = { f i f j | ≤ i ≤ j ≤ g } ∪ { f i f j | ≤ i ≤ g, g + 1 ≤ j ≤ µ ( I ) } ⊂ B 15s a basis of the k -vector space [( f , . . . , f g ) I ] d . Since n = 1, we have hf g +1 ∈ [( f , . . . , f g ) I ] d by hypothesis.Writing h = λ f + · · · + λ g +1 f g +1 + · · · + λ µ ( I ) f µ ( I ) , then hf g +1 = λ f f g +1 + · · · + λ g +1 f g +1 + · · · + λ µ ( I ) f µ ( I ) f g +1 ∈ [( f , . . . , f g ) I ] d implies that λ g +1 f g +1 + · · · + λ µ ( I ) f µ ( I ) f g +1 is a k -linear combination of the elements of B ′ . But, B = B ′ ∪ { f g +1 , . . . , f µ ( I ) f g +1 } and B ′ ∩ { f g +1 , . . . , f µ ( I ) f g +1 } = ∅ . Therefore, it mustbe the case that λ g +1 = · · · = λ µ ( I ) = 0 . Hence, h ∈ [( f , . . . , f g )] d . Case n ≥ : By assumption, hf g +1 ∈ [( f , . . . , f g ) I n ] ( n +1) d , i.e., h ∈ [(( f , . . . , f g ) : f g +1 ) ∩ I n ] nd . By Proposition 2.14 (ii), assumption (b) implies that f , . . . , f g +1 is an analyticallytight sequence in powers ≥ 2, hence h ∈ [( f , . . . , f g ) ∩ I n ] nd = [( f , . . . , f g ) I n − ] nd , where the equality is the Valabrega–Valla condition, as a consequence of f o , . . . , f og being aregular sequence in gr I ( R ). The following is a consequence of the last part of the previous section: Corollary 3.1. Let R denote a standard graded ring of dimension three over an infinitefield and let I ⊂ R be a perfect d -equigenerated homogeneous ideal of height and analyticspread . Suppose that, for any height two prime P , the reduction number of I P is at mostone. For any sequence of forms f , f , f in I d satisfying the properties in Proposition 2.17(i) , the following are equivalent: (i) F ( I ) is Cohen-Macaulay. (ii) f , f , f is analytically adjusted. (iii) f , f , f is analytically tight. Proof. It is a reading of Theorem 2.23 in dimension three. Remark 3.2. We note that the assumption on the local reduction number is satisfied whenboth R and R ( I ) are Cohen–Macaulay locally at the minimal primes of R/I .For height two equigenerated ideals, not necessarily unmixed, one can file the followingresult: Theorem 3.3. Let R denote a -dimensional standard graded Cohen–Macaulay domainover an infinite field and let I ⊂ R be a perfect d -equigenerated homogeneous ideal of height . Suppose that: (i) µ ( I ) ≥ and ℓ ( I ) = 3 . (ii) The Rees algebra of I is Cohen–Macaulay. The defining ideal of F ( I ) is generated in degrees ≥ .Then: (a) F ( I ) is Cohen–Macaulay and the reduction number of I is . (b) The defining ideal of F ( I ) is equigenerated in degree and the minimal graded freeresolution of F ( I ) is -linear. (c) The multiplicity of F ( I ) is (cid:0) m − (cid:1) , where m := µ ( I ) = dim k ( I d ) . Proof. (a) The assumption in (ii) implies that the associated graded ring gr I ( R ) is Cohen–Macaulay. Then, by Lemma 2.4 and Lemma 2.3 (iii), there exist forms f , f , f in I d suchthat f o , f o is a regular sequence in gr I ( R ) and J = ( f , f , f ) is a minimal reduction of I. By Proposition 2.21, condition (iii) implies that { f , f , f } is an analytically adjusted set.Then the implication (ii) ⇒ (i) of Theorem 2.23 (b) gives that F ( I ) is Cohen–Macaulay aswell.Since F ( I ) is Cohen–Macaulay, the reduction number r ( I ) is independent on the minimalreduction and coincides with the Castelnuovo-Mumford regularity reg( F ( I )) (see, e.g., [10,Proposition 1.2]). On the other hand, since I is minimally generated by at least fourelements, condition (iii) implies that reg( F ( I )) ≥ . But by (ii), r ( I ) ≤ ℓ ( I ) − F ( I )) = 2, then (iii) implies that the defining ideal of F ( I ) is equigeneratedin degree 3. In particular, the minimal graded free resolution of F ( I ) is 3-linear.(c) Since the minimal graded free resolution of F ( I ) is 3-linear, this follows immediatelyfrom [16, Theorem 1.2]. Remark 3.4. Suitable versions of Theorem 3.3 for arbitrary number of variables may bestated. Both results may profit from the still standing Simis–Vasconcelos question as towhether, for an equigenerated homogeneous ideal I in a standard polynomial ring overa field, if the Rees algebra of I is Cohen–Macaulay then so is the special fiber of I (seeCorollary 2.7 for the equimultiple case, valid under weaker hypothesis).The following calculation is certainly well-known. A proof is given for the reader’sconvenience. Lemma 3.5. Let I ⊂ R = k [ x, y, z ] be a d -equigenerated homogeneous perfect ideal of height with minimal graded free resolution → N − M i =1 R ( − d − m i ) ϕ −→ R ( − d ) N → R → R/I → . Then e ( R/I ) = d + P N − i =1 m i . Proof. The Hilbert series of R/I calculated from the given resolution is H R/I ( t ) = B R/I ( t )(1 − t ) B R/I ( t ) = 1 − N t d + N − X i =1 t d + m i . By a well-known formula (see, e.g., [24, Corollary 7.4.12]) e ( R/I ) = 12 ∂ B R/I ( t ) ∂ t (1) = − d ( d − N + P N − i =1 ( d + m i )( d + m i − − d ( d − N + P N − i =1 ( d ( d − 1) + (2 d − m i + m i )2= − d ( d − N + d ( d − N − 1) + (2 d − P N − i =1 m i + P N i =1 m i − d ( d − 1) + (2 d − d + P N − i =1 m i d + P N − i =1 m i . Corollary 3.6. Let I ⊂ R = k [ x, y, z ] be a d -equigenerated homogeneous perfect ideal ofheight that is generically complete intersection, but not a complete intersection. Then: (a) ℓ ( I ) = 3 . (b) The defining ideal of F ( I ) is generated in degrees ≥ .Moreover, if the Rees algebra R ( I ) is Cohen-Macaulay then the following statements areequivalent: (i) I is linearly presented. (ii) The rational map F : P → P N − defined by the linear system I d is birational onto itsimage. Proof. (a) This follows as usual from [5].(b) This comes out of Corollary 1.9 since I necessarily satisfies condition G .Now, we head out to the supplementary assertion.(i) ⇒ (ii) By (a), this is a case of [6, Theorem 3.2].(ii) ⇒ (i) Let 0 → N − M i =1 R ( − d − m i ) ϕ −→ R ( − d ) N → I → I , where m i ≥ i . Since I is genericallya complete intersection, [23, Theorem 6.6(b)] gives e ( F ( I )) · deg( F ) = d − e ( R/I ) . (6)18pplying the formula of Corollary 3.5 yields e ( F ( I )) · deg( F ) = d − d + P N − i =1 m i d − P N − i =1 m i N − X i =1 m i + 2 X ≤ i Among the hypotheses of Theorem 3.6 used in the supplementary part, (a)can be dispensed with by instead adding to item (i) the hypothesis that the entries of thesyzygy matrix of I generate the maximal ideal of R . This detour is afforded by [7, Theorem2.4 (a)], of which the implication (ii) ⇒ (i) is a sort of weak converse.The following simple examples illustrate some aspects of the above results. Example 3.8. Let I = ( x , xy, xz, yz ) ⊂ R = k [ x, y, z ].(1) It is easy to see that I is non-perfect since x ( x, y, z ) ⊂ I and that F ( I ) is definedby an obvious degree 2 binomial equation – in particular, none of parts (b) and (c) ofTheorem 3.3 holds.Computation with [11] gives:(2) R R ( I ) is Cohen–Macaulay.(3) r ( I ) = 1, hence part (a) of Theorem 3.3 does not hold either. Example 3.9. Let I = ( x − y , xy, xz, yz ) ⊂ R = k [ x, y, z ].(1) It is easy to see that I is non-perfect (e.g., x ( x, y, z ) ⊂ I ) and a complete intersectionat its unique minimal prime ( x, y ).Computation with [11] gives: 192) I has maximal linear rank, but is not linearly presented; in particular, I defines abirational map onto the image, thus showing that perfectness is essential for the implication(ii) ⇒ (i) of Corollary 3.6.(3) R R ( I ) is not Cohen–Macaulay, and yet r ( I ) = 2.(4) F ( I ) is defined by a degree 3 equation, hence both parts (b) and (c) of Theorem 3.3hold. Question 3.10. One wonders whether the following generalization of the supplementarypart of Corollary 3.6 holds: under the same hypotheses, except that I is not assumed to begenerically a complete intersection, can one replace ‘linear presentation’ by ‘sub-maximallinear rank’ ?Note that a perfect ideal of height 2 has sub-maximal rank (= N − 2) if and only if itis generated by linear syzygies plus one single syzygy of degree ≥ 2. The following simpleexample shows that the answer to the question is negative if the Rees algebra of I is notassumed to be Cohen–Macaulay. Example 3.11. Let I be the ideal generated by the maximal minors of the following matrix z x z y z 00 0 x z y It is easy to see that I is not generically a complete intersection (hence, does not satisfy G ). A calculation with [11] shows that F ( I ) is Cohen–Macaulay and generated in degree 3.This implies that the discussed rational map is birational onto the image, while the linearrank is only 2. 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Simis, Commutative Algebra , De Gruyter Graduate, Berlin–Boston, 2020. 4, 13, 18[25] A. B. Tchernev, Torsion freeness of symmetric powers of ideals, Trans. Amer. Math.Soc. (2007), 3357–3367. 2, 5[26] N. V. Trung, Reduction exponent and degree bound for the defining equations of gradedrings, Proc. Amer. Math. Soc. (1987), 229–236. 1, 12[27] N. V. Trung, The Castelnuovo regularity of the Rees algebra and the associated gradedring, Trans. Amer. Math. Soc. (1998), 2813–2832. 1, 9, 10[28] P. Valabrega and G. Valla, Form rings and regular sequences, Nagoya Math. J. (1978), 93–101. 7, 8, 11, 12 Addresses: Zaqueu Ramos Departamento de Matem´atica, CCETUniversidade Federal de Sergipe49100-000 S˜ao Cristov˜ao, Sergipe, Brazil e-mail : [email protected] Aron Simis Departamento de Matem´atica, CCENUniversidade Federal de Pernambuco50740-560 Recife, PE, Brazil e-maile-mail