Two differential themes in characteristic zero

Two differential themes in characteristic zero This is also dedicated to Wolmer Vasconcelos on his sixtieth fifth birthday

Aron Simis2 Departamento de Matem´atica, CCEN Universidade Federal de Pernambuco 50740-540 Recife, PE, Brazil

Foreword As a commutative algebraist one has time and again found oneself intrigued by the way in which differentials and derivations play their frequently unexpected act in ring and module theory (for instances of my own bewilderment or ignorance see [1], [2], [3], [5], [7], [8],[10], [11], [12],[13], [14],[16], [17]). At the jeopardy of loosing any readers left, I will allow myself yet another perplexity. This paper is divided in two sections, as conceived from the title. Both deal with a finitely generated k-algebra A, where k is a field mainly of characteristic zero. The first section is concerned with k-derivations of A. There are as of today withstanding questions concerning the structure of the module Derk (A) of k-derivations of A, such as the Zariski– Lipman conjecture - not yet settled in dimension two - and its natural extended versions such as the homological Zariski–Lipman conjecture of Herzog–Vasconcelos. It appears to me that some of the difficulty inherent to these questions has to do with the lack of resiliency in taking derivations with extended values. In this section we will mainly assume that A is a finitely generated k-subalgebra of a polynomial ring B := k[t] = k[t1 , . . . , td ]. Besides the A-module of k-derivations of A, it is natural in this context to consider the A-module Derk (A, B) of A with values in B. True, one can consider a similar derivation module for any ring extension A ⊂ B, but one will lack the possibility of taking partial derivatives with respect to parameters which is, after all, the old tool in classical differential geometry (cf. [1]). Clearly, there is a natural inclusion of A-modules Derk (A) ⊂ Derk (A, B). The latter is naturally a B-module and we will be interested in determining its structure. As mentioned above, the main tool is the jacobian matrix Θ(g) of g with respect to t, where g is a set of generators of A over k. A preliminary basic result about the rank of this matrix is the fact that rank Θ(g) = dim A when k is a field of characteristic zero. 0

AMS 1980 Mathematics Subject Classification (1985 Revision). Primary 13H10, 14C17; Secondary 13C40, 13D10, 13N05, 13C15, 14B07, 14E25, 14M05, 14M25. 2 Partially supported by a grant from CNPq and PRONEX

1

This equality was possibly familiar in its essence to Jacobi and his contemporaries (one inequality in the homogeneous case was proved by E. Noether [9] and a rather special case was given later in [20, Lemma 3]). It seems to be well-known to most people, however I could not trace in the more recent literature a complete proof in the present generality. The inequality rank Θ(g) ≤ dim A follows from a basic property of the module of K¨ahler differentials of the algebra A, which may seem quite natural as this module is in many aspects a tautological module of A. The reverse inequality in all generality was shown to me by P. Brumatti, to whom I express my thanks. We include a tiny subsection on derivations of degree zero - such as the Euler derivation ϵ. We ask how often is the case that such derivations are substantially many in the sense that the submodule Derk (A) 0 they generate has the same rank as the entire module of derivations. Interestingly enough, questions of this sort play a role in algebraic vector field theory (cf. [21], [5]). Alas, easy examples show that quite often Derk (A) 0 = ϵA (≃ A). The simplest example is the homogeneous coordinate ring of a smooth plane curve of degree at least 3. A thorough analysis of this problem is being reported in [5]. On the bright side, the original question may have an affirmative answer for rings with sufficient symmetries – an arbitrary isolated singularity (even a complete intersection) may not have the required symmetries. The symmetries may rather stem from the very nature of the ring in question, such as certain determinantal structures. Thus, ideals generated by subdeterminants of generic matrices or symmetric generic matrices will have in fact Derk (A) 0 = Derk (A). Similar behaviour is displayed by generic pfaffians. Of course this touches the philosophy that complete intersections are not the typical examples of Cohen– Macaulay or Gorenstein rings: geometrically, it is the fact that they have too high a degree; combinatorially, their lack of basic symmetries. The second section hinges on differentials. Let k be a field (preferrably of characteristic zero) and let R be a polynomial ring over k. Let I ⊂ R be a homogeneous ideal and let µ D := ker (R/I ⊗k R/I → R/I), where µ denotes the multiplication map. This ideal is called the diagonal ideal of R/I ⊗k R/I. The main point here is to tie up the symbolic form ring of I (i.e., the associated graded ring of the filtration of symbolic powers I (r) ) to the tangent star algebra (i.e., the associated graded ring of the filtration of powers of D). The main idea is to bridge up between these two rings using certain polarization map. We state this as a question, then derive a (conjectural) consequence which seems to be quite surprising, namely, that if I is homogeneous and generated in fixed degree, and if D is of linear type then I is normally torsionfree (in particular, normal). Frankly, I do not consider this consequence a lot credible. However, bridging up by means of polarization does look quite enticing.

2

1

Polynomial-valued derivations of sub-polynomial algebras

As in the foreword, let A be a finitely generated k-subalgebra of a polynomial ring B := k[t] = k[t1 , . . . , td ]. We fix generators of A as k-subalgebra of B, say, g = {g1 , . . . , gn } which we may assume to belong to the ideal (t)k[t]. Fix a presentation ideal I (necessarily prime) of A with respect to g. Thus, k[X]/I ≃ A under the k-algebra homomorphism Xj 7→ gj , where X = {X1 , . . . , Xn }. We denote by Θ(g) the jacobian matrix of g with respect to t. Further, if f is a set of polynomials in k[X], the symbol Θ(f ) (resp. Θ(f )(g)) will denote the jacobian matrix of f with respect to X (resp. further evaluated on the elements of g). Clearly, Θ(f )(g) is the same matrix that results from taking Θ(f ) modulo the ideal I and then using the isomorphism k[X]/I ≃ A. Moreover, the rank of Θ(f )(g) is the same regardless from whether it is considered as a matrix over the subring A = k[g] ⊂ B = k[t] or over the ambient polynomial ring B (take subdeterminants to see this).

1.1

The rank of the Jacobian matrix

Retaining the notation introduced above, one has. Proposition 1.1 If k is a field of characteristic zero then dim A = rank Θ(g). Proof. Let as above A ≃ k[X]/I, where X = {X1 , . . . , Xn }. Picking a set of generators f = {f1 , . . . , fm } of I and applying the chain rule of derivatives to its elements yields a short complex of free k[t]-modules k[t]m

Θ(f )(g)∗

Θ(g)∗

−→ k[t]n −→ k[t]d ,

from which follows that rank Θ(g) ≤ n − rank Θ(f )(g). On the other hand, using the isomorphism A ≃ k[X]/I, the matrix Θ(f )(g) fits in the well-known fundamental exact sequence of differentials Am

Θ(f )(g)∗

−→ An −→ Ω(A/k) → 0.

Since rank Ω(A/k) = dim A, it follows immediately that rank Θ(g) ≤ dim A. To show inequality in the other direction, after convenient reordering, let g′ = g1 , . . . , gr be a transcendence basis of A over k. In particular, dim A = r = dim k[g′ ]. Obviously, rank Θ(g) ≥ rank Θ(g′ ). Therefore, the required inequality follows from the proposition in special case where A is generated by algebraically independent elements over k. Thus, we assume that g is algebraically independent over k and will argue by induction on the difference d − n, where t = {t1 , . . . , td }. If n = d, one is to show that det(Θ(g)) ̸= 0. For every i, 1 ≤ i ≤ n, let g(i) denote the augmented set {g1 , . . . , gn , ti }. By assumption,

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each such set is algebraically dependent over k, so let Fi ∈ k[T1 , . . . , Tn+1 ] stand for a nonzero ∂Fi polynomial relation thereof of least possible degree. Since char k = 0, (g(i)) ̸= 0. ∂Tn+1 Letting Θ(g(i)) denote the Jacobian matrix of g(i), since Fi vanishes at g(i), the chain rule of derivatives yields ( ) ∂Fi ∂Fi (g(i)), . . . , (g(i)) · Θ(g(i)) = 0, (1) ∂T1 ∂Tn+1 for each i, 1 ≤ i ≤ n. Now clearly, Θ(g(i)) is the matrix obtained from Θ(g) by further stacking it with the row vector ei , where ei = (0, . . . , |{z} 1 , . . . , 0). Therefore, (1) implies (

)

i

∂Fi ∂Fi ∂Fi (g(i)), . . . , (g(i)) · Θ(g) = − (g(i)) · ei , ∂T1 ∂Tn ∂Tn+1

(2)

∂Fi (g(i)) ̸= 0, for each i, 1 ≤ i ≤ n, this implies that the ∂Tn+1 k(t)-linear map of k(t)n defined by Θ(g) is invertible. Therefore, det(Θ(g)) ̸= 0 and we are through. for each i, 1 ≤ i ≤ n. Since

Now let d − n ≥ 1. Pick a polynomial h ∈ k[t1 , . . . , td ] such that {g, h} is algebraically independent over k. By the inductive hypothesis, Θ(g, h) is of rank n + 1, hence the ideal of (n + 1)-minors In+1 (Θ(g, h)) ⊂ k[t1 , . . . , td ] is nonzero. Since the latter is contained in the ideal In (Θ(g)), the latter is nonzero as well, hence Θ(g) has rank at least n = dim A, as desired.

1.2

The structure of the B-module Derk (A, B)

We retain the previous notation. Theorem 1.2 Let k be a field of characteristic zero and let A = k[g] ( B = k[t] be as above. Assume that the height of the B-ideal Ir (Θ(g)) is at least 2, where r = dim A (3 ). Then Derk (A, B) = Im(Θ(g)), where Im(Θ(g)) denotes the B-submodule of B n generated by the columns of the matrix Θ(g). Proof. We first argue that the chain rule for derivatives implies an inclusion Im(Θ(g)) ⊂ Derk (A, B). Indeed, Derk (A, B) is the kernel of the map B n → B m represented by the evaluated matrix Θ(f )(g) (after sufficient identification, e.g., HomA (An , B) ≃ B n ). To see this, one applies HomA ( , B) to the fundamental exact sequence A 3

m Θ(f )

∗

−→ An → ΩA/k → 0,

One must also assume that Im(Θ(g)) : Ir (Θ(g))∞ = Im(Θ(g))

4

∗

and uses the identification HomA (Θ(f ) , B) = Θ(f )(g)B , where the matrix on the right hand side is the evaluated jacobian matrix considered as a matrix over B = k[t]. Note that in particular rank (Derk (A, B)) = n − rank (Θ(f )(g)B ) = n − rank (Θ(f )(g)) = n − (n − rank (ΩA/k )) = rank (ΩA/k ) = dim A,

(3)

from previous observation. Therefore, rank (Θ(g)) = dim A = rank (Derk (A, B)) by Proposition 1.1 and by (3), respectively. Now, we have an inclusion of B-modules Im(Θ(g)) ⊂ Derk (A, B) of the same rank. Moreover, Derk (A, B) is reflexive (being a second syzygy over a polynomial ring) and the inclusion is equality locally in codimension one due to the assumption ht Ir (Θ(g)) ≥ 2 with r = rank (Θ(g)). Therefore, the inclusion is actually an equality (4 ). Corollary 1.3 Let k be a field of characteristic zero and let A = k[g] ⊂ B = k[t] be of maximal dimension (i.e., dim A = d = dim B). Assume that ht Id (Θ(g)) ≥ 2, where d = dim B (5 ). Then Derk (A, B) ≃ B d and coker(Θ(f (g)B )) has projective dimension 2 as B-module. Proof. From Theorem 1.2 it follows immediately that coker(Θ(f (g)B )) has projective dimension at most 2. Since B is a polynomial ring and Id (Θ(g)) ̸= B, the projective dimension is exactly 2. Example 1.4 Let A = k[(t)2 ] be the 2-Veronesean ring, with #t = 3. Then C := Im(Θ(f (g)B )) is torsionfree of rank 3. Moreover, it is locally free in the punctured spectrum and its dual is free (since the jacobian matrix of the 2 × 2 minors of a 3 × 3 generic symmetric matrix is symmetric itself). Therefore, C is an example of an ideal module in the terminology of [18] and is the module of sections of a rank 2 vector bundle over P2 . Example 1.5 If the condition ht Ir (Θ(g)) ≥ 2 fails to hold things will go wrong at the very start in codimension one on B, so there will be no hope to get the required equality Derk (A, B) = Im(Θ(g)). For example, take t = t and g = {t3 , t4 , t5 } (to get a truly homogeneous case, take the homogeneous coordinate ring of any singular monomial curve in P3 ). Since the previous footnote assumption Im(Θ(g)) : Ir (Θ(g))∞ = Im(Θ(g)) means that the cokernel of Im(Θ(g)) is torsionfree 5 Again assume that Im(Θ(g)) : Ir (Θ(g))∞ = Im(Θ(g)) 4

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1.3

Remark on derivations of degree zero

Here one assumes that A ≃ k[X]/I with I homogeneous and turn back to the ordinary 2 module of derivations Derk (A). We assume throughout ∑ that I ⊂ (X) . Note that Derk (A) is a graded submodule of the graded free A-module i A ∂/∂X ∑i . In order to conform with the fact that the latter is the A-dual of the free A-module i A dXi of differentials, the vectors ∂/∂Xi are assumed to have degree 0.∑ Therefore, a homogeneous vector of Derk (A) of degree m in this gradation is of the form i vi (X) ∂/∂Xi , where every vi (X) is a form of degree m + 1 in the standard grading of k[X]. The submodule Derk (A) 0 ⊂ Derk (A) generated by the derivations of degree 0 is of special interest. Clearly, Derk (A) 0 ̸= (0) because the Euler derivation ϵ has degree 0. We may right off pose the following Question 1.6 When is rank (Derk (A)) = rank (Derk (A) 0 ) ? Example 1.7 Let A ≃ k[X]/I be the face ring of a simplicial complex ∆ (with ∆ = core(∆)) on a set of vertices corresponding to the variables X. Here Derk (A) splits into the direct sum of submodules isomorphic to the ideals I : (I : Xi ), for 1 ≤ i ≤ n (cf. [3]). Clearly, Xi ∈ I : (I : Xi ) always holds. Therefore, Derk (A) admits a “diagonal” submodule in degree 0. To see that this submodule has rank r = rank (Derk (A)) = dim A, one takes the product of the complementary variables to the ones generating a minimal prime of I of least possible height. The resulting monomial has as support a facet, hence cannot be a multiple of a non-face. Thus, it yields a nonzero r × r determinant of the diagonal part of the matrix whose column vectors generate Derk (A). Therefore, the conjecture is true for this case. A special case is that of A ≃ k[X]/I where I is the edge ideal of a simple graph on a set of vertices corresponding to the variables X. In this case, the ideal I : Xi is generated by the variables corresponding to the vertices which are connected to Xi . Therefore, I : (I : Xi ) is generated by the squarefree monomials Xi1 · · · Xir satisfying the following property: for every vertex Xj connected to Xi there is a vertex Xij with ij ∈ Supp (Xi1 · · · Xir ) such that Xij is connected to Xj . It follows that the variables contained in I : (I : Xi ) correspond to those vertices that are adjacent to every vertex adjacent to Xi . One wonders whether such vertices afford any meaningful graph theoretic gadget.

2 2.1

Relating two associated graded rings Symbolic form ring

Let R = k[X1 , . . . , Xn ] be a polynomial ring over a field k and let I ⊂ R be an ideal. Let r ≥ 0. The rth symbolic power of I is the ideal I (r) := {f ∈ R | ∃s ∈ R \ P (∀P ∈ Ass (R/I)), with sf ∈ I r }. 6

The rth infinitesimal power of I is the ideal I := {f ∈ R |

∂αf ∈ I, ∀α, |α| ≤ r − 1}. ∂Xα

The basic result tying up these two concepts is: Theorem 2.1 [4, 3.9] If char k = 0 and I is a radical ideal then I (r) = I , ∀r ≥ 0. Henceforth one assumes that char k = 0 and I is a radical ideal. The symbolic associated graded ring or symbolic form ring is the graded ring ∑ ∑ gr(I) (R) := I (r) /I (r+1) = I /I . r≥0

r≥0

The following result, which is yet another facet of Theorem 2.1, was pointed out in [13]. Proposition 2.2 ([13]) (char k = 0) Let I ⊂ R be a radical ideal. There is an injection of R/I-algebras ¯ gr (R) → (R/I)[dX1 , . . . , dXn ]. d: (4) (I) induced by the assignment f 7→ d(r) (f ): =

∑ u1 +...+un

1 ∂rf u1 un u1 un dX1 . . . dXn , u ! . . . u ! ∂X . . . ∂X n n 1 =r 1

(5)

for f ∈ I (r) . It may be suggestive to refer to this distinctive embedding d¯ of the torsion free R/Ialgebra gr(I) (R) into the polynomial ring (R/I)[dX1 , . . . , dXn ] as the differential embedding of gr(I) (R) .

2.2

The tangent star algebra

The following notions have been largely treated in [17] and [8] (cf. also [6] for a geometric background). µ Let D := ker (R/I ⊗k R/I → R/I), where µ denotes the multiplication map. This ideal is called the diagonal ideal of R/I ⊗k R/I (or of R/I). ¯ = R/I. Set R Definition 2.3 The∑ tangent star algebra (resp. Zariski tangent algebra) is the associated := S(D/D2 ). := r≥0 Dr /Dr+1 (resp. the symmetric algebra SR/k graded ring TR/k ¯ ¯

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These are graded R/I-algebras equipped with a canonical graded surjective homomorphism SR/k → TR/k ¯ ¯ . One says that R/I is starlike linear if the preceding map is an isomorphism. In general, the two algebras may even fail to have the same dimension (cf. [17]). One knows that D/D2 ≃ Ωk (R/I), the module of K¨ahler differentials of R/I over k. Using this identification, in the sequel we will denote by J ⊂ R[dX1 , . . . , dXn ] the inverse image of the kernel of the surjective R/I-algebra map (R/I)[dX1 , . . . , dXn ] → TR/k obtained ¯ by composing the surjective map ∑ (R/I)[dX1 , . . . , dXn ] ≃ S( (R/I) dXi ) → S(Ωk (R/I)) = SR/k ¯ i

→ TR/k ≃ R[dX1 , . . . , dXn ]/(I, J). with the canonical surjection SR/k ¯ ¯ ¯ . Thus, one has TR/k ¯ The ideal J = (I, J)/IR[dX1 , . . . , dXn ] ⊂ (R/I)[dX1 , . . . , dXn ] will be called the differential presentation ideal of TR/k ¯ . Note ≃ R[dX1 , . . . , dXn ]/(I, df1 , . . . , dfm ), where I = (f1 , . . . , fm ), where ¯ ∑ that SR/k df = i (∂f /∂Xi ) dXi , the ordinary differential of the polynomial f . Therefore, I is starlike linear if and only if J = (df1 , . . . , dfm ). Note the important fact that if I is a homogeneous ideal generated by forms of a fixed degree, then J is a bihomogeneous ideal in the sets of variables X, dX. This will be used below.

2.3

Polarization

Here is a suitable version of polarization that will be used to tie up between the two versions of associated graded rings. Quite generally, let B be any commutative ring. One is given a sequence of elements z = {z1 , . . . , zm } of B and, correspondingly, indeterminates T = {T1 , . . . , Tm } over B. The structure of free B-module of the polynomial ring B[T] will be object of emphasis here. Definition 2.4 The assignment { ∑ αj −1 α1 αm 7→ αm Tα = T1α1 · · · Tm · · · Tm αj ̸=0 αj zj T1 · · · Tj T0 = 1 7→ 1

if some αj ̸= 0 otherwise

extends, by B-linearity, to an B-module endomorphism λz of B[T], called the z-polarization of B[T]. We refer to [12] for the main properties of this notion. By iterating λz , one obtains the (i) (0) z-polarization of order i, denoted λz . We also set λz = Id.

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Let as before I ⊂ R[X1 , . . . , Xn ] be an ideal. We will apply polarization as above with ¯ = R/I, T = dX and z = X. To simplify, denote λ = λX . B=R (0) As a matter of consistency, for every homogeneous generator f of I, d(1) (f ) = λX (df ) where in the left-hand side d(1) (f ) is, modulo I, an algebra generator of gr(I) (R), while in the right-hand side df is a bihomogeneous generator of J. Assume that I is a homogeneous ideal generated in a fixed degree. Along with the ideal ¯ ¯ ¯ J¯ ⊂ R[dX 1 , . . . , dXn ], one may consider the R-subalgebra AJ¯ ⊂ R[dX1 , . . . , dXn ] generated ¯ by a set of (X, dX)-bihomogeneous generators of J. We are now ready to state a question that ties up the notions discussed in the last three subsections. Question 2.5 Let I ⊂ R[X1 , . . . , Xn ] be generated by homogeneous polynomials of the ¯ as same degree and let J¯ ⊂ R[dX ¯ 1 , . . . , dXn ] be the differential presentation ideal of TR/k (r) (r+1) (r) ¯ defined previously. Then, for every r ≥ 0 and for every f ∈ I /I , where f ∈ I is a homogeneous polynomial, there exists a (X, dX)-bihomogeneous polynomial F ∈ AJ¯ and (i) an index i ≥ 0 such that d¯(r) (f¯) = λX (F¯ ). Technically involved as it is, in an informal way the question asks whether, as a subalgebra of (R/I)[dX1 , . . . , dXn ], the symbolic form ring is generated by polarizations of suitable order of certain defining equations of the tangent star algebra. Corollary 2.6 (To the question) Let I ⊂ R be a homogeneous radical ideal generated in a fixed degree. If I is starlike linear then it is normally torsion free (in particular, it is a normal ideal). Proof. Since I is supposed to be starlike linear, we know that the differential presentation of TR/k is generated by df , for a set of homogeneous generators f of I, where ¯ denotes ¯ ¯ ]. Let g ∈ I (r) \ I r , with r least residue modulo IR[dX1 , . . . , dXn ]. Therefore, AJ¯ = R[df possible (i.e., I (s) = I s , ∀s < r). If the answer to the main question above is affirmative, (i) ¯ ] such that d¯(r) (¯ there is a bihomogeneous F ∈ R[df g ) = λX (F¯ ), for some index i ≥ 0. Now, λ acts as derivation on the differentials, namely, λ((df )s ) = sf (df )s−1 . Since d(r) (g) is a polynomial of degree r in dX and since df are linear in dX, it follows that d(r) (g) ∈ d(r) (I r ). Say, d(r) (g) = d(r) (f ) for some f ∈ I r . Since the map d¯(r) is injective by Proposition 2.2, one must have g − f ∈ I r , i.e., g ∈ I r . This gives a contradiction, so the equality I (s) = I s holds throughout. As a tiny evidence for both the conjectured question and the corollary, one can look at certain ideals generated by forms of degree two. The first meaningful case is an ideal generated by squarefree monomials of degree two – a so called edge ideal (since their generators correspond to the edges of a simple graph). We give two basic examples.

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Example 2.7 Let n be an odd integer and let I = (X1 X2 , X2 X3 , . . . , Xn−1 Xn , Xn X1 ) (the edge ideal of an odd cycle). Now, it can be shown that gr(I) (R) is generated by I/I 2 and I (

n+1 ) 2

/I (

n+3 ) 2

as an R/I-

( n+1 ) 2

n+3

algebra. The only fresh generator is the product of the variables X1 · · · Xn ∈ I \ I ( 2 ). On the other hand, direct inspection shows that the product dX1 · · · dXn is a generator of the differential presentation ideal of TR/k ¯ . A calculation yields that d(

n+1 ) 2

(X1 · · · Xn ) = (

n−1 ( n−1 ) )! λX 2 (dX1 · · · dXn ). 2

Thus, the question is affirmative for this example. Example 2.8 Let I be the edge ideal of a bipartite graph. It is known that such an ideal is starlike linear only if the corresponding graph is bipartite ([15, Corollary 1.2]) - it is reasonable to conjecture that the converse statement holds as well. On the other hand, such a graph is bipartite if and only if the corresponding edge ideal is normally torsionfree ([19, Theorem 5.9]). Therefore, the conjectured corollary holds in this case. The answer to the question is also affirmative for all bipartite graphs provided the converse above is true. Another evidence comes from the ideal of maximal minors of certain generalized Hankel matrices. Definition 2.9 Let n ≥ 3, r ≥ 2 be integers. An r-step Hankel 2 × n matrix over a ring R is a matrix of the form ( ) a1 a2 . . . an , ar+1 ar+2 . . . ar+n for a given sequence of elements a1 , . . . , an+r ∈ R. Proposition 2.10 Let R = k[X] = k[X1 , . . . , Xr+n ], a polynomial ring over a field k, and consider the r-step Hankel 2 × n matrix (X) whose entries are the elements of X in the given order. Let I = I2 (X). The following conditions are equivalent: (i) I is starlike linear (ii) r ≥ n − 2 (iii) I is normally torsionfree.

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Proof. We show that both (i) and (iii) are separately equivalent to (ii). First, (i) ⇔ (ii). The direction (ii) ⇒ (i) is proved in [15, Proposition 2.6]. For the other direction, suppose that r ≤ n − 3 and consider the 3 × 3 matrix   X1 X2 X2 M =  Xr+1 Xr+2 Xr+3  . X2r+1 X2r+2 X2r+3 This makes sense since 2r + 3 ≤ r + n is our standing assumption. It is easy to see that the 2 × 2 minors of this matrix belong to I. Therefore, the result follows immediately from [17, Proposition 4.11]. Now deal with (iii) ⇔ (ii). First, assume that r ≤ n − 3. We claim that D ∈ I (2) , where D := det(M). Since clearly D ̸∈ I 2 , we will be done for one direction. But in fact it is well known that D multiplies each entry of M into I 2 . Since I is a prime ideal, any such entry lies outside I. For the reverse direction, one notes that r ≥ n − 2 implies that the r-step Hankel matrix is nearly generic, hence the symbolic powers of its ideal of maximal minors behave as in the generic case, namely, for this size of matrix, all symbolic powers are ordinary powers. Remark 2.11 Let dM denote the matrix M above read in the differential variables dX. Then, in the complementary case r ≤ n − 3, D := det(dM) is a generator of the differential presentation ideal of TR/k which does not belong to the presentation ideal of SR/k ¯ ¯ , while (2) 2 D ∈ I \ I , as shown in the proof of Proposition 2.10. By direct calculation, one sees that d¯(2) (D) = λX (D). This is an additional evidence to the main question. Remark 2.12 The “converse” of the corollary is false, as one sees by taking I a complete intersection such that R/I has not the expected star dimension. By the criterion for expected star dimension in [17], the first instance would be a (radical) complete intersection generated by three polynomials in 5 variables (defining a projective surface in P4 ). An easy example is I = (X1 X2 , X3 X4 , (X1 + X2 + X3 + X4 ) X5 ). The defining equations of TR/k ¯ are of rather large degree and with many terms.

References [1] P. Brumatti, P. Gimenez and A. Simis, On the Gauss algebra associated to a rational map Pd → Pn , J. Algebra 207 (1998), 557–571. [2] P. Brumatti, Y. Lequain, D. Levcovitz and A. Simis, A note on the Nakai conjecture, Proc. Amer. Math. Soc., 130 (2002), 15–21.

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[3] P. Brumatti and A. Simis, The module of derivations of a Stanley–Reisner ring, Proc. Amer. Math. Soc. 123 (1995), 1309-1318. [4] D. Eisenbud, Commutative Algebra (with a view toward Algebraic Geometry), Graduate Texts in Mathematics, vol. 150, Springer-Verlag, 1995. [5] E. Esteves, A. Simis and B. Ulrich, On the module of tangent vector fields (provisory title), in preparation. [6] L. van Gastel, Excess intersections and a correspondence principle, Invent. Math. 103 (1991), 197–221. [7] G. Giang and A. Simis, Higher relative primitive ideals, Proc. Amer. Math. Soc. 129 (2001), 647–655. [8] G. Kennedy, A. Simis and B. Ulrich, Specialization of Rees algebra with a view toward tangent star algebras, in Commutative Algebra, Proceedings,ICTP, Trieste, September 1992, World Sc., Singapore 1994, 130–139. [9] E. Noether, [10] F. Russo, and A. Simis, On birational maps and Jacobian matrices, Compositio Math. 126 (2001), 335–358. ´ [11] A. Simis, Algebraic aspects of tangent cones, in XII Escola de Algebra, Proceedings of Diamantina D. Avritzer and M. Spira, Eds.), Brazil, July 1992, Matem´atica Contemporˆanea, vol. 7, 1994, pp. 71–127. [12] A. Simis, On the jacobian module associated to a graph, Proc. Amer. Math. Soc., 126 (1998), 989–997. [13] A. Simis, Effective computation of symbolic powers by jacobian matrices, Comm. in Algebra 24 (1996), 3561–3565. [14] A. Simis, K. Smith and B. Ulrich, An algebraic proof of Zak’s inequality for the dimension of the Gauss image, Math. Z., to appear. [15] A. Simis and B. Ulrich, Joins of varieties defined by quadrics, preprint, 2002. [16] A. Simis, B. Ulrich and W. Vasconcelos, Jacobian dual fibrations, Amer. J. Math. 115 (1993), 47–75. [17] A. Simis, B. Ulrich and W. Vasconcelos, Tangent star cones , J. reine angew. Math., 483 (1997), 23–59.

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[18] A. Simis, B. Ulrich and W. Vasconcelos, Rees algebras of modules, Proc. London Math. Soc., to appear. [19] A. Simis, W. Vasconcelos and R. Villarreal, The ideal theory of graphs, J. Algebra 167(2) (1994), 389–416. [20] B. Segre, Bertini forms and Hessian matrices, J. London Math. Soc., 26 (1951), 164– 176. [21] M. G. Soares, The Poincar´e problem for hypersurfaces invariant by one-dimensional foliations, Invent. Math. 128 (1997), 495–500. ´ tica, CCEN, Universidade Federal de Pernambuco, Departamento de Matema ´ ria, 50740-540 Recife, PE, Brazil ([email protected]) Cidade Universita

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