Graded derivation modules and algebraic free divisors

Journal of Pure and Applied Algebra 219 (2015) 5442–5466

Contents lists available at ScienceDirect

Journal of Pure and Applied Algebra www.elsevier.com/locate/jpaa

Graded derivation modules and algebraic free divisors ✩ Cleto B. Miranda-Neto Departamento de Matemática, Universidade Federal da Paraíba, 58051-900 João Pessoa, PB, Brazil

a r t i c l e

i n f o

Article history: Received 19 January 2015 Received in revised form 10 April 2015 Available online 4 June 2015 Communicated by A.V. Geramita Dedicated to Bernd Ulrich MSC: Primary: 13N15; 13D02; 13E15; 13A30; 14J70; secondary: 37F75; 13C13; 13C14; 13D40

a b s t r a c t The main purpose of this paper is to furnish new criteria for freeness of (algebraic, homogeneous) divisors, especially by means of the minimal number of generators of certain graded derivation modules. Our approach is based on the description of the graded syzygies of the derivation module in the hypersurface case, which allows us to derive several other applications. We investigate, under certain conditions, the Castelnuovo–Mumford regularity and the Hilbert function of such module, as well as an Eisenbud matrix factorization of the given polynomial. We also obtain the defining ideals of the blowup algebras of the derivation module, as a dual version, in the hypersurface case, of the so-called tangent algebras introduced by Simis, Ulrich and Vasconcelos. Finally, we give an explicit Ulrich ideal and the Hilbert polynomial in the distinguished case of linear free divisors (in the sense of Buchweitz and Mond). © 2015 Elsevier B.V. All rights reserved.

1. Introduction In this work we prove several results on the module of derivations – or algebraic tangent vector fields – of a standard graded algebra of finite type over a field of characteristic zero, essentially in the hypersurface case. Our central result describes the graded syzygies of the derivation module of a hypersurface ring, which allows us to detect a number of features and applications, especially on new criteria for the freeness of divisors, in particular for the class of linear free divisors together with an interpretation in the theory of Ulrich ideals. As a matter of preview, this paper furnishes, under certain conditions on the gradient ideal of the polynomial and essentially as a byproduct of our main technical result, the computation of important numerical invariants of the derivation module, such as the Castelnuovo–Mumford regularity (which is a measure of the complexity of a module) and the postulation number (the least integer from which the Hilbert function becomes the Hilbert polynomial), as well as results about a matrix factorization of the polynomial, in the sense of Eisenbud’s classical theory. Moreover, without restriction on the gradient ideal, we propose and describe a dual version of the so-called tangent algebras considered by Simis, Ulrich and Vasconcelos. ✩

This work was partially supported by CNPq-Brazil (grant 202284/2011-5). E-mail address: [email protected].

http://dx.doi.org/10.1016/j.jpaa.2015.05.026 0022-4049/© 2015 Elsevier B.V. All rights reserved.

C.B. Miranda-Neto / Journal of Pure and Applied Algebra 219 (2015) 5442–5466

5443

Further, we study the celebrated class of (algebraic) free divisors, by giving a new characterization of freeness and various other features; in particular, we consider the property of linear freeness due to Buchweitz and Mond, and we characterize it especially in terms of the Ulrich property of a certain ideal in the hypersurface ring. In the general context of not necessarily principal ideals, we develop an approach on minimal number of generators that turned out to be crucial for our freeness criteria. Before explaining the content of the paper in a little bit more details, we fix the setup and the notations, and give some motivation. Let R = k[x1 , . . . , xn ] be a standard graded polynomial ring over a field k of characteristic zero, and let I ⊂ R be a homogeneous, radical, proper ideal. We consider two distinguished modules of derivations attached to I. First, the logarithmic derivation module TR/k (I) (usually denoted by Derk (−log X), where X = V (I)), which is an idealizer of tangential type since it is formed by the n ∂ elements of the (free) derivation module DR/k = j=1 R ∂xj leaving I invariant (note that TR/k (I) is a graded submodule of DR/k ), and second, the classical module DA/k of the k-derivations of the residue ring A = R/I. They are connected by means of a natural surjection TR/k (I) → DA/k with kernel ID R/k . Such objects appear under various notations and terminologies in the literature, and play a fundamental role in the investigation of tangent vector fields and holomorphic foliations, for instance concerning their singularities, topological aspects and invariant varieties. Also, special attention has been devoted to an important link to algebraic combinatorics, notably on the theory of (free) hyperplane arrangements and its generalizations. Some references are [6,8,9,16,23,25–27,30,31,40,41]. In Section 2, we focus on the principal ideal case I = (f ) ⊂ R, where f is a homogeneous, reduced polynomial of degree d ≥ 2. In the first part, Subsection 2.1, we furnish some preliminaries on derivations and we fix a (standard) grading that will be in force in the entire paper. In Subsection 2.2 we prove our central result (Theorem 2.3), which gives a graded R-free presentation of the graded derivation module DA/k of A = R/(f ) (we point out that a free resolution of the module TR/k (f ) = Derk (−log V (f )), in case f defines a generic hyperplane arrangement, was constructed in [40]). This allows us to exploit ∂f ∂f several aspects of DA/k , assuming that f has a gradient ideal J(f ) = ( ∂x , . . . , ∂x ) with either projective 1 n dimension 1, or projective dimension 2 but no homogeneous minimal syzygy of degree d. First, we compute its Castelnuovo–Mumford regularity (Corollary 2.5), which, in the particular case where f defines a hyperplane arrangement, has been studied by some authors focusing on the module TR/k (f ) quite typically endowed ∂ with the non-standard grading inherited from DR/k where the ∂x ’s are given degree zero (cf. [10, Section 5], j [27]). Second, we consider the Hilbert function and the postulation number of DA/k (Corollary 2.9), and further, in Corollary 2.10, we describe a matrix factorization of f in the sense of [11], which is known to be tightly related to the theory of maximal Cohen–Macaulay A-modules and periodic free resolutions (see also [1,2,7,15,21,39]). Still in Section 2, Subsection 2.3 is concerned with the proposal of a dual version of the so-called tangent algebras of Simis, Ulrich and Vasconcelos (see [33]), by replacing the module of Kähler differentials by its dual, the derivation module DA/k . For instance, in case A = R/(f ), we obtain the defining ideal of the Rees algebra of DA/k (Proposition 2.12(ii)). Some interesting points (linear type property, fiber cone and Cohen–Macaulayness) are discussed in Remarks 2.13, and in Proposition 2.14 it is noticed that if the Rees algebra of DA/k is Cohen–Macaulay then A must be a normal domain (the converse is not true, as we show in Example 2.15(ii)). In Subsection 2.4, we illustrate most of the results given in Section 2, by working out the case of the cuspidal projective plane cubic. Section 3 is devoted to comparing the minimal numbers of generators ν(DA/k ) and ν(TR/k (I)) – the latter is always an upper bound for the former – for any homogeneous, radical ideal I ⊂ R (some results and questions on ν(DA/k ), for certain rings A, are given, e.g., in [18]). This study will be a crucial tool for our characterization of free divisors later in Subsection 4.2. First, we treat the principal ideal case I = (f ). We prove, in Proposition 3.1, that ν(DA/k ) = ν(TR/k (f )) provided J(f ) has no homogeneous minimal syzygy of degree d (a mild condition; see Remark 3.3). As a first application, we compute the Krull dimension of the symmetric algebra of DA/k in case the projective hypersurface defined by f is smooth (Corollary 3.4).

5444

C.B. Miranda-Neto / Journal of Pure and Applied Algebra 219 (2015) 5442–5466

Further, we deal with a not necessarily principal I ⊂ R. Denoting A{q} = R/I q for any given integer q ≥ 1, we prove that ν(DA{q} /k ) = ν(TR/k (I)) for every q ≥ 2. In Proposition 3.6 we notice that the deviation ν(TR/k (I)) − ν(DA/k ) is bounded above by n · ν(I) (this is sharp at least for n = 3; see Example 3.7). The monomial case is discussed in Remark 3.8, where in particular we raise the possibility of a “tangential” characterization of the Stanley–Reisner property. Recall that the reduced polynomial f ∈ R is said to be a free divisor if TR/k (f ) is a free R-module, necessarily of rank n. This condition was first investigated by Saito remarkably in [25] (see also Terao’s paper [34]) within the local complex analytic context, and an extensive literature exists on this concept, in particular providing new examples and freeness criteria as well as fruitful connections to several branches of mathematics such as algebraic combinatorics, algebraic geometry and Lie algebra theory. We refer, for instance, to [8,9,17,23,28,30,34,35,41]. In Section 4, our goal is to contribute to this influential theory. In Subsection 4.1, we apply results from Section 2 in order to collect some features of DA/k when f is a free divisor – free resolutions, Hilbert function, and the non-Cohen–Macaulayness of its Rees algebra. In Theorem 4.2, our main objective, we furnish freeness criteria through a different angle; for instance, we prove that f is free if and only if ν(DA/k ) = n and J(f ) has no homogeneous minimal syzygy of degree d. We begin Subsection 4.3 by noticing that a (probably known) necessary condition for the freeness of f is d ≥ ν(J(f )) (hence d ≥ n in the standard situation where all the partial derivatives of f are linearly independent over k), and in Proposition 4.5 we provide a proof of this fact without resorting to the determinantal nature of J(f ) given by the classical Hilbert–Burch theorem. We then focus on the class of linear free divisors introduced by Buchweitz and Mond in [8] (see also [17]), which by definition must satisfy the ∂f property that the minimal syzygies of the ∂x ’s are linear (consequently, such divisors have degree d = n). j From Proposition 4.7 we derive, in Remark 4.8, that f is a linear free divisor if and only if ν(J(f )) = n and the Jacobian ideal J(f )/(f ) of A has the Ulrich property, which means being a maximal Cohen–Macaulay A-module that is maximally generated in the sense of Ulrich’s paper [36], where the intriguing question and the first results on the existence of such modules – typically called Ulrich modules – were first presented. A natural issue concerns their possible ranks, and in particular whether Ulrich ideals exist. Beautiful papers (cf. [2,1,5,19,21]) have made progress on Ulrich’s question, but quite amazingly the problem remains unanswered in most situations. Finally, in Proposition 4.10, we compute the Hilbert polynomial of the graded module DA/k , where A = k[x1 , . . . , xn ]/(f) and the form f is any linearfree divisor (n is fixed); explicitly, it can be written as n−2 n−1 n P (DA/k , ρ) = (n−1)! α=1 (ρ + α) − β=0 (ρ − β) . General conventions and notations. In this paper, all rings are assumed to be commutative and with 1. If R is a Noetherian ring and I is a proper R-ideal, we write htR (I), or simply ht(I), for its height in R. If ϕ is an m × n matrix with entries in R, we denote by Ij (ϕ) the R-ideal generated by the order j minors of ϕ, for 1 ≤ j ≤ min{n, m}. In particular, I1 (ϕ) is the ideal generated by the entries of ϕ. We write rkR (M ) = r if M is a finitely generated R-module with (generic, constant) rank r in the usual sense: if Q is the total ring of fractions of R, then M ⊗R Q  Qr as Q-modules. If R is local (resp. standard graded over a field), then the minimal number of generators of a finitely generated (resp. graded) R-module M is denoted νR (M ), or simply ν(M ) if there is no risk of ambiguity; of course, this number does not depend on whether we regard M as a module over R or R/I, where I is any R-ideal annihilating M . We write depth(M ) for the depth of M with respect to the (resp. irrelevant) maximal ideal of R, and, in case R is a standard graded polynomial ring over a field, we let pd(M ) stand for the projective dimension of M over R. Finally, given integers p, q,   we set pq = 0 whenever p < q, as usual. The main setup and specific notations that will be in force throughout the paper are introduced in Subsection 2.2, complementing the context of Subsection 2.1. We have used Macaulay [4] for the auxiliary calculations in some of our examples and remarks.

C.B. Miranda-Neto / Journal of Pure and Applied Algebra 219 (2015) 5442–5466

5445

2. Free presentation in the hypersurface case In this section, we produce an explicit graded R-free presentation of the derivation module DA/k of a homogeneous hypersurface ring A = R/(f ), where k is a field of characteristic zero and R is a standard graded polynomial k-algebra. This result allows us to study, under certain conditions on the gradient ideal of f , the Castelnuovo–Mumford regularity of DA/k as well as its Hilbert function and postulation number; also, keeping the same setup, we derive a matrix factorization of f , in the sense of Eisenbud [11]. We furnish, moreover, the defining ideals of the symmetric and Rees algebras of DA/k , dubbed dual tangent algebras herein, being a sort of dual version, in the hypersurface case, of the so-called tangent algebras investigated by Simis, Ulrich and Vasconcelos (see [33]). In the last part of the section, we work out the case of the cuspidal plane cubic. We begin with a few basic preliminaries on (logarithmic) derivations. 2.1. Basics on logarithmic derivations Let R = k[x1 , . . . , xn ], n ≥ 2, be a polynomial ring over a field k with char k = 0 (even though some of the properties presented herein remain valid in arbitrary characteristic). The R-module DR/k formed by ∂ the k-derivations of R is free on the partial derivations ∂x ’s. Given an ideal I ⊂ R, we may consider its j logarithmic derivation module TR/k (I) = {τ ∈ DR/k | τ (I) ⊆ I}, which in the geometric context (I radical and, typically, k = C) is usually denoted by Derk (−log X), where X = V (I). This is an idealizer of tangential type (its elements are ambient vector fields tangent along X \ Xsing ) and clearly has a structure of Lie subalgebra of DR/k . Moreover, rkR (TR/k (I)) = n since I annihilates the cokernel of the inclusion TR/k (I) ⊂ DR/k . Its connection to the derivation module DA/k of the factor ring A = R/I is given by the standard fact below (cf., e.g., [6, Lemma 2.1.2]). Lemma 2.1. DA/k 

TR/k (I) as A-modules. IDR/k

 In this paper, we consider the context where R = l≥0 Rl , with the standard grading deg(xj ) = 1, for every j. We endow the R-module DR/k with the standard Z-grading, that is, for each given ρ ∈ Z, its ρth homogeneous piece is [DR/k ]ρ =

n  j=1

Rρ+1

∂ ∂xj

which clearly vanishes if ρ ≤ −2. Note that, as in the natural Z-grading of the Weyl algebra of R, each ∂ ∂xj has degree −1, so that the well-known isomorphism between DR/k and the R-dual of the standard n graded module of differentials ΩR/k = j=1 R dxj is degree-preserving (warning: this is not the geometric grading typically adopted by the specialists in complex foliation theory, where the partial derivations are given degree zero; such non-standard grading is also used by some authors in the theory of hyperplane arrangements). If I is homogeneous, it is easy to see that TR/k (I) is a graded submodule of DR/k . From Lemma 2.1 above, it follows that the derivation module DA/k of the standard graded k-algebra A = R/I is graded as well. Now we focus on the principal ideal case I = (f ), where f ∈ R \ {0} is homogeneous of degree d ≥ 2. Write

5446

C.B. Miranda-Neto / Journal of Pure and Applied Algebra 219 (2015) 5442–5466

J(f ) =

∂f ∂f ,..., ∂x1 ∂xn

⊂ R,

the gradient ideal or Jacobian ideal of f . Notice that f ∈ J(f ), in virtue of the Euler identity d · f = (f ), n ∂ where = j=1 xj ∂x ∈ [TR/k (f )]0 is the Euler derivation. Also, note that J(f ) is the lifting of the (true) j Jacobian ideal JA/k = J(f )/(f ) of the k-algebra A = R/(f ). 0 We write TR/k (f ) for the module of relations on the partial derivatives of f , realized as the homogeneous submodule 0 TR/k (f ) = {τ ∈ DR/k | τ (f ) = 0} ⊂ TR/k (f ), ∂ ∂ whose degree −1 piece may be non-zero; for instance, if f = xd−1 (y + z) ∈ R = k[x, y, z], then ∂y − ∂z ∈ 0 [TR/k (f )]−1 ⊂ [TR/k (f )]−1 . Lemma 2.2 below is well-known (we give a proof for the sake of completeness). It admits an analogous version in the quasi-homogeneous case, but we shall not stick to such generality as the standard graded 0 setting is enough for our purposes. Note that this lemma readily implies that rkR (TR/k (f )) = n − 1. 0 Lemma 2.2. TR/k (f ) = TR/k (f ) ⊕ R .

Proof. Given τ ∈ TR/k (f ), there is a uniquely determined gτ ∈ R such that τ (f ) = gτ f . This defines an 0 R-linear map η: TR/k (f ) → R by η(τ ) = gτ , whose kernel is TR/k (f ) and which is seen to be surjective: given q ∈ R, we have η((q/d) ) = q, where d = deg(f ). Thus, the R-linear map σ: R → TR/k (f ), σ(h) = (h/d) , 0 0 for h ∈ R, is a splitting of η and hence TR/k (f ) = TR/k (f ) ⊕ image(σ) = TR/k (f ) ⊕ R . 2 2.2. Syzygies, regularity, Hilbert function and matrix factorization Throughout the entire paper, unless explicitly stated otherwise (which will happen only in a few parts), we adopt the following setup and notation, in addition to the preliminaries given in Subsection 2.1 above. First, R = k[x1 , . . . , xn ], n ≥ 2, is a standard graded polynomial ring over a field k with char k = 0, and f ∈ R is a homogeneous, reduced polynomial of degree d ≥ 2, non-degenerate in the sense that ht(J(f )) ≥ 2 ∂f (e.g., if ∂x = 0 for every j). Next, let j · · · −→

s  t=1

Φ

R(−ct ) −→

r 

Ψ

R(−(di + d − 1)) −→ Rn (−(d − 1)) −→ J(f ) −→ 0

i=1

be (the first two steps of) a graded free resolution of J(f ) ⊂ R with respect to its ordered, signed generating ∂f ∂f set { ∂x , . . . , ∂x }, which may be non-minimal in general; for example, if n = 3 and f = x(y − z) ∈ 1 n k[x, y, z], then of course ν(J(f )) = 2 but we only consider the free resolution associated to the generating set {y − z, x, −x}. We regard Ψ and Φ as matrices (with respect to the canonical bases of the free modules involved), with I1 (Φ) ⊆ (x) since Φ can be taken minimal. We denote by V1 , . . . , Vr the homogeneous k-derivations of R 0 corresponding to the column-vectors of Ψ, that is, the Vi ’s generate the graded R-module TR/k (f ) and, in fact, can be taken to form a minimal generating set – even if ν(J(f )) < n, which means I1 (Ψ) = R. Hence 0 r = ν(TR/k (f ))

and from the above presentation we see that the non-negative integers d1 , . . . , dr are the degrees of the homogeneous minimal syzygies of J(f ); equivalently, these numbers satisfy

C.B. Miranda-Neto / Journal of Pure and Applied Algebra 219 (2015) 5442–5466

Vi ∈ [DR/k ]di −1

5447

i = 1, . . . , r

according to the standard grading on DR/k recalled in Subsection 2.1. 2.2.1. Graded free presentation With a view to the construction of a free presentation for the derivation module of A = R/(f ), the key step is to consider the derivations Wj = f

∂ 1 ∂f − ∂xj d ∂xj

j = 1, . . . , n,

which are immediately seen to vanish at f , that is, 0 Wj ∈ TR/k (f )

j = 1, . . . , n.

Therefore, we can find homogeneous polynomials fij ∈ R (which are not uniquely determined, in general) satisfying Wj =

r

fij Vi

j = 1, . . . , n.

i=1

We denote the corresponding r × n matrix by Λ = (fij ). Our result is:

where:

| Λ), Theorem 2.3. Consider the (r + 1) × (s + n) matrix Π = (Φ

is the (r + 1) × s matrix obtained by adding a (bottom) null row to the matrix Φ; • Φ ∂f ∂f

• Λ is the (r + 1) × n matrix obtained by adding a (bottom) row with entries d1 ∂x , . . . , d1 ∂x to the 1 n matrix Λ. Then, the derivation module DA/k of A = R/(f ) has a graded R-free presentation s 

Π

R(−(ct − d)) ⊕ Rn (−(d − 1)) −→

t=1

r 

R(−(di − 1)) ⊕ R −→ DA/k −→ 0

i=1

Proof. By Lemma 2.1 and Lemma 2.2, the A-module DA/k can be generated by the homogeneous derivations V 1 , . . . , V r , (bar indicates reduction modulo f DR/k ). In order to determine the syzygies, let g1 , . . . , gr , r g ∈ R be forms satisfying i=1 gi V i + g = 0. Lifting this relation, we get r

gi Vi + g = f V

i=1

for some V = write

n i=j

∂ hj ∂x ∈ DR/k . Evaluating both sides at f , and using that Vi (f ) = 0 for every i, we can j

1 ∂f 1 V (f ) = hj . d d j=1 ∂xj n

g=

Now, recalling that

5448

C.B. Miranda-Neto / Journal of Pure and Applied Algebra 219 (2015) 5442–5466

f

∂ 1 ∂f = Wj + ∂xj d ∂xj

j = 1, . . . , n

we obtain

fV =

n

hj

j=1

∂ f ∂xj



⎛

⎞ n n 1 ∂f ⎝ ⎠ hj W j + hj hj Wj + g = = d j=1 ∂xj j=1 j=1 n

and therefore r

gi Vi =

i=1

n

hj W j =

j=1

n

hj

 r

j=1

 fij Vi

,

i=1

which means that the r-tuple ⎛ ⎝g1 −

n

hj f1j , . . . , gr −

j=1

n

⎞ hj frj ⎠ ∈ Rr

j=1

r 0 is a syzygy of the R-module TR/k (f ) = i=1 R Vi . On the other hand, by the second step of the given free 0 resolution of the ideal J(f ), the syzygy module of TR/k (f ) is generated by the vectors Φ1 , . . . , Φs ∈ Rr corresponding to the s columns of the matrix Φ. Hence, there exist polynomials p1 , . . . , ps ∈ R such that (g1 , . . . , gr ) =

s

pt Φt +

t=1

n

hj (f1j , . . . , frj ).

j=1

Setting

j = Λ



1 ∂f f1j , . . . , frj , d ∂xj

∈ Rr+1 ,

j = 1, . . . , n

and

t = (Φt , 0) ∈ Rr+1 , Φ

t = 1, . . . , s,

we finally get that the general syzygy U = (g1 , . . . , gr , g) ∈ Rr+1 of DA/k can be expressed as U=

s t=1

t + pt Φ

n

j , hj Λ

j=1

which are seen to be

| Λ), that is, an R-linear combination of the column-vectors of the matrix Π = (Φ

t ’s are syzygies of syzygies of DA/k . Indeed, the Φt ’s are syzygies of the Vi ’s and hence obviously the Φ

j ’s, simply note that, for each j, DA/k . As to the Λ r i=1

fij Vi +

1 ∂f 1 ∂f ∂ = Wj + =f ≡ 0 (mod f DR/k ). d ∂xj d ∂xj ∂xj

Thus, we have shown that Π is an R-free presentation matrix for the module DA/k , built up from the generators V 1 , . . . , V r , . Finally, it is clear that such free presentation is graded with the shifts as stated;

is formed by polynomials of fixed degree d − 1, and recall that the Euler the last row of the matrix Λ

C.B. Miranda-Neto / Journal of Pure and Applied Algebra 219 (2015) 5442–5466

5449

derivation has degree 0, whence the presence of the graded free module Rn (−(d − 1)). As to the free s

we can confirm its shifts through a routine summand t=1 R(−(ct − d)) corresponding to the columns of Φ, verification, by appropriately adapting the shifts of the given free resolution of J(f ) at the step Φ. 2 Remark 2.4. Since DA/k is annihilated by f , an A-free presentation matrix of DA/k can be obtained simply by reducing Π modulo (f ). 2.2.2. Castelnuovo–Mumford regularity If M = 0 is a finitely generated graded R-module, then its Castelnuovo–Mumford regularity, or simply regularity, is a number typically defined in terms of the (non-)vanishing of graded components of certain local cohomology modules of M , but it is well-known that such invariant, denoted reg(M ), admits a concrete description by means of a minimal graded free resolution of M over the polynomial ring R. More precisely, if 0 −→ Fp −→ . . . −→ F0 −→ M −→ 0, bi with Fi = j=1 R(−ai,j ), i = 0, . . . , p, is such a free resolution, and if for each i we define the number ρi = max{ai,j | 1 ≤ j ≤ bi }, then the regularity of M is given by reg(M ) = max{ρi − i | 0 ≤ i ≤ p}, which is a sort of numerical measure of the complexity of M . See [3], [12] and [13] for more on this notion. As a first application of Theorem 2.3, we compute the regularity (in fact, a minimal graded R-free resolution) of the graded module DA/k in a couple of suitable situations. The condition pd(J(f )) = 1 required in part (i) below means precisely that f is a free divisor, an important concept that we shall exploit in details in Section 4. Corollary 2.5. We have: (i) Assume that pd(J(f )) = 1. Then reg(DA/k ) = d − 2 (ii) Assume that pd(J(f )) = 2 and that di = d for every i. Set c = max{ct | 1 ≤ t ≤ s}. Then di ≤ c − d for every i, and reg(DA/k ) = max{d − 2, c − d − 1} Proof. First, notice that the condition pd(J(f )) ≤ 2 forces DA/k to have projective dimension 1 over R. Indeed, if JA/k = J(f )/(f ) is the Jacobian ideal of A = R/(f ), we clearly have pd(JA/k ) = pd(J(f )) ≤ 2, and the Auslander–Buchsbaum formula yields depth(JA/k ) ≥ n − 2. Now recall that, possibly up to a free direct summand, DA/k is the A-module of first-order syzygies of JA/k . Thus, we get depth(DA/k ) ≥ n − 1 and hence pd(DA/k ) = 1 (note that DA/k is a torsion R-module). However, a priori, there is no guarantee for the minimality of the resolution, and we now turn to verify it in each of the particular cases stated in (i) and (ii). 0 (f ) is free (necessarily of rank n −1), so that r +1 = n and hence ν(DA/k ) ≤ n, (i) If pd(J(f )) = 1, then TR/k which must be an equality since rkA (DA/k ) = n − 1 (DA/k cannot be free, by the well-known graded case

and Theorem 2.3 gives in fact a minimal of the Zariski–Lipman conjecture). Therefore, in this case, Π = Λ graded R-free resolution

C.B. Miranda-Neto / Journal of Pure and Applied Algebra 219 (2015) 5442–5466

5450

Λ

0 −→ Rn (−(d − 1)) −→

n−1 

R(−(di − 1)) ⊕ R −→ DA/k −→ 0.

i=1

that d ≥ di + 1 We readily get d − 1 = di − 1 + deg(fij ) if fij = 0, which implies (by the minimality of Λ) for every i. Hence, reg(DA/k ) = max{d1 − 1, . . . , dn−1 − 1, d − 2} = d − 2. (ii) Reading shifts, as given by Theorem 2.3, we obtain that deg(fij ) = d − di if fij = 0, thus forcing di ≤ d whenever the ith row of Λ is non-zero (see Remark 2.6(ii) below). Hence, as by hypothesis di = d for every i, we get I1 (Λ) ⊆ (x1 , . . . , xn ) which means that I1 (Π) ⊆ (x1 , . . . , xn ) since Φ can be taken minimal in the original free resolution of J(f ). Moreover, pd(DA/k ) = 1 since we are supposing that pd(J(f )) = 2. It follows, in this situation, a minimal graded R-free resolution 0 −→

r−n+1 

Π

R(−(ct − d)) ⊕ Rn (−(d − 1)) −→

t=1

r 

R(−(di − 1)) ⊕ R −→ DA/k −→ 0

i=1

and hence reg(DA/k ) = max{d1 − 1, . . . , dr − 1, d − 2, c − d − 1}. Finally, we claim that c − d ≥ di for every i. Let qit ’s be the entries of the r × (r − n + 1) matrix Φ. Given i, we have qit = 0 for some t, and necessarily deg(qit ) ≥ 1. On the other hand, tracking degrees, we can write ct = deg(qit ) + di + d − 1. Therefore c − d ≥ ct − d = deg(qit ) + di − 1 ≥ di , as claimed. 2 Remarks 2.6. (i) As it is clear from the initial portion of the proof above, the condition pd(J(f )) ≤ 2 is, in fact, equivalent to the maximal Cohen–Macaulayness of DA/k over A. Note that this is automatic in case n ≤ 3 – for instance, if f defines a projective plane curve (see Corollary 2.7 below for the regularity in the smooth case). (ii) We point out that, in the proof of Corollary 2.5(ii), it was not shown that in general di ≤ d for every i. The bound di ≤ d is true, for a given i, if the matrix Λ = (fij ) can be chosen so that its ith row has some non-zero entry. This is, as we can expect, the typical situation, but we warn that there exist examples with the rather unpredictable behavior that di > d for some i (hence, for each such i, the ith row of any chosen Λ – not the entire ith row of the corresponding Π – must vanish). One instance is the quintic f = x5 + y 5 − xyz 3 ∈ R = k[x, y, z], for which di = 6 for some i (although no di equals d = 5), corresponding to the minimal generator 15y 4 z 2

∂ ∂ ∂ 0 + 3yz 5 + (25x3 y 3 − z 6 ) ∈ [TR/k (f )]5 . ∂x ∂y ∂z

Further, for a given f , such behavior may occur for more than a single di . An example is the quartic f = x2 y 2 + x2 z 2 + x2 w2 + y 2 z 2 + y 2 w2 + z 2 w2 ∈ k[x, y, z, w], for which (r = 9 and) three of the di ’s are equal to 5. Corollary 2.7. If f ∈ R = k[x, y, z] defines a smooth projective plane curve of degree d ≥ 2, then reg(DA/k ) = 2d − 4. Proof. In this case, J(f ) is generated by an R-sequence of 3 forms and hence pd(J(f )) = 2, with both Ψ and Φ having entries in Rd−1 . Therefore, di = d − 1 for every i, and c = 3(d − 1) so that c − d − 1 = 2(d − 2) ≥ d − 2. Now, Corollary 2.5(ii) yields the result. 2 In most cases it happens that c − d − 1 ≥ d − 2, which in many examples we can check to be an equality. On the other hand, it may also be strict, for instance if f defines a smooth projective plane curve of degree

C.B. Miranda-Neto / Journal of Pure and Applied Algebra 219 (2015) 5442–5466

5451

at least 3, as it is clear from the proof of Corollary 2.7 above. Now, we are going to illustrate the opposite strict inequality c − d − 1 < d − 2. Example 2.8. Take the quartic f = x2 y 2 + x2 z 2 + y 2 z 2 − 2x2 yz − 2xy 2 z − 2xyz 2 ∈ R = k[x, y, z]. Its gradient ideal has a (minimal) graded free resolution of the shape 0 −→ R(−6) −→ R3 (−5) −→ R3 (−3) −→ J(f ) −→ 0 so that c = 6 and hence c − d − 1 = 1 < 2 = d − 2. By Corollary 2.5(ii), we get reg(DA/k ) = 2. Since 2 = 2d − 4, this example shows, in particular, the necessity of the smoothness hypothesis in Corollary 2.7. 2.2.3. Hilbert function and postulation number Let dimk stand for k-vector space dimension, and let H(DA/k , −): Z → Z≥0

ρ → dimk [DA/k ]ρ

be the Hilbert function of DA/k . From the graded R-free resolution of DA/k , described in the proof of Corollary 2.5 in each of the two situations treated therein, we can easily compute H(DA/k , ρ). Indeed, we have shown that such (minimal) R-free resolution is of the form 0 → F1 → F0 → DA/k → 0, so that we get the equality H(DA/k , ρ) = H(F0 , ρ) − H(F1 , ρ), whose right-hand side can be explicitly derived from the graded free modules F0 and F1 in the usual manner (cf., e.g., [12, Corollary 1.2]). Moreover, as we have pointed out in Remark 2.6(i), the module DA/k is Cohen–Macaulay in this case (in particular, its Hilbert function is non-decreasing, by [24, Theorem 1]). Now we are able to determine the so-called postulation number of DA/k , that is, the integer pn(DA/k ) = min{λ ∈ Z | H(DA/k , ρ) = P (DA/k , ρ), ∀ ρ ≥ λ} where P (DA/k , −) stands for the Hilbert polynomial of DA/k . Corollary 2.9. We have: (i) If pd(J(f )) = 1, then pn(DA/k ) = d − n; (ii) If pd(J(f )) = 2 and di = d for every i, then pn(DA/k ) = {max{d − n, c − d − n + 1}. Proof. By [12, Corollary 4.8], and using that depth(DA/k ) = n − 1 in these two situations, we obtain pn(DA/k ) = 2 − n + reg(DA/k ). Combining this fact with our Corollary 2.5, the result follows. 2 As we already have mentioned, the situation (i) is the case where f is a free divisor, which we will revisit in Subsection 4.1 (see also Proposition 4.10). An explicit illustration of (ii) will be given in Subsection 2.4. 2.2.4. Matrix factorization and periodic A-free resolution The theory of matrix factorizations of divisors – together with the key role it plays on the study of maximal Cohen–Macaulay modules and periodic free resolutions – was established by Eisenbud in the classical paper [11]. We briefly recall the basic notion needed here, within our previous setup and notation.

5452

C.B. Miranda-Neto / Journal of Pure and Applied Algebra 219 (2015) 5442–5466

A matrix factorization of the polynomial f ∈ R is a pair (X, Y ) of square matrices of the same size, say q × q, such that XY = f Iq , where Iq is the q × q identity matrix. In this case, since f = 0 we necessarily have Y X = f Iq (see [11, page 51, Remark (1)]). Now, if M is a finitely generated, maximal Cohen–Macaulay module over A = R/(f ), then the Auslander– X Buchsbaum equality yields that M admits an R-free resolution of the form 0 → Rq → Rq → M → 0, where we do not distinguish between regarding X as an element of HomR (Rq , Rq ) or the corresponding matrix with respect to the canonical basis {e1 , . . . , eq } of Rq . Since f annihilates M = coker(X), we have f Rq ⊆ image(X) and hence there are h1X , . . . , hqX ∈ Rq such that f ej = X(hjX ), j = 1, . . . , q, which are uniquely determined since X is a monomorphism. Thus, if Y stands for the matrix whose jth column is hjX , we immediately get that (X, Y ) is a matrix factorization of f . Corollary 2.10. Consider once again the two situations as in the previous corollary, that is, either, pd(J(f )) = 1, or pd(J(f )) = 2 and di = d for every i. We have: Π ∈ Rr+1 such that, if Γ is (i) There exist uniquely determined, effectively computable vectors h1Π, . . . , hr+1 Π the (r + 1) × (r + 1) matrix whose jth column is given by hj , then (Π, Γ) is a matrix factorization of f ; (ii) Assume that I1 (Γ) ⊆ (x), and denote by ΠA and ΓA the reductions of Π and Γ modulo (f ). Then, there is a minimal, 2-periodic infinite A-free resolution Γ

Π

Γ

Π

A A A A · · · −→ Ar+1 −→ Ar+1 −→ Ar+1 −→ Ar+1 −→ Ar+1 −→ DA/k −→ 0

Proof. By the proof of Corollary 2.5, Π defines a minimal R-free resolution of the maximal Cohen–Macaulay A-module DA/k . Therefore, by the above general considerations, there exist uniquely determined vectors Π h1Π , . . . , hr+1 ∈ Rr+1 such that Π(hjΠ ) = f ej

j = 1, . . . , r + 1,

which can be effectively computed since Π is known. This proves (i). The assertion (ii) follows from (i) and the well-known equivalences summarized in [11, Corollary 6.3]. 2 Later in Subsection 2.4 we shall compute explicitly the matrix factorization (Π, Γ) for the cuspidal projective plane cubic f = x3 − y 2 z. 2.3. Dual tangent algebras In geometric terms, the main goal of the paper [33] by Simis, Ulrich and Vasconcelos is to compare the Zariski tangent cone π: TX → X to an affine variety X with the closure of π −1 (X \ Sing(X)) in TX . Working essentially on the algebraic counterparts of these objects, the authors focused on exploiting several features of certain blowup rings related to an algebra A (not necessarily a hypersurface ring) essentially of finite type over a perfect field k, or more directly, related to the module ΩA/k of Kähler k-differentials of A. Such blowup rings, dubbed tangent algebras therein, are SA/k = SA (ΩA/k )

RA/k = RA (ΩA/k )

that is, respectively, the symmetric and Rees algebras of ΩA/k . Now, under a certain dual viewpoint, we propose the study of the following version of tangent algebras: S∗A/k = SA (DA/k )

R∗A/k = RA (DA/k )

C.B. Miranda-Neto / Journal of Pure and Applied Algebra 219 (2015) 5442–5466

5453

that is, respectively, the symmetric and Rees algebras of the A-module DA/k of derivations, which is isomorphic to the dual Ω∗A/k = HomA (ΩA/k , A). We are interested in the structural side of such dual tangent algebras, focusing mainly on the (saturation-type) ideal defining the Rees algebra R∗A/k as a quotient of an appropriate polynomial ring. This seems to be quite a challenging task, since a well-structured free presentation of DA/k – and hence the equations defining S∗A/k – is not known in general, unlike what happens to ΩA/k , which is presented by means of an explicit Jacobian matrix over A. We shall carry out such project in case A is a graded hypersurface ring. While the description of S∗A/k will follow directly from Theorem 2.3, the defining ideal of R∗A/k will demand an auxiliary lemma, which we provide below within a very brief review on blowup algebras of modules. For details on this subject, see [14,20,29,32,37,38]. 2.3.1. A glimpse on Rees algebras of modules Let A be a Noetherian ring and let M be a finitely generated A-module. Recall that, given a free presentation matrix φ of M over A, the symmetric algebra SA (M ) of M can be expressed in the form SA (M ) = A[T1 , . . . , Tν ]/L, where T1 , . . . , Tν are indeterminates over A, and L = I1 ((T1 · · · Tν ) · φ). If M has a (positive) rank, its Rees algebra is RA (M ) = SA (M )/ T , where T ⊂ SA (M ) is the submodule of A-torsion, which in fact turns out to be an ideal of SA (M ). Note that RA (M ) is a graded A-algebra with a grading naturally inherited from SA (M ). The module M is said to be of linear type if T = (0), that is, RA (M ) = SA (M ). For instance, it is clear that free A-modules of finite rank have this property, as in this case the symmetric algebra is a polynomial ring over A. For a Noetherian A-algebra B, we can consider the saturation of a B-ideal I with respect to an A-ideal a,  which is the B-ideal I: a∞ = j≥1 I: aj . Of course, by Noetherianess, we have I: a∞ = I: at for t  0. The result below is, in essence, contained in [29], and furnishes a useful description of the Rees algebra. Lemma 2.11. Let A be a Noetherian ring and let M be a finitely generated A-module with positive rank. Let a ⊂ A be an ideal containing a non-zero-divisor, such that M is of linear type locally on the Zariski open set defined by a. Then RA (M ) = SA (M )/(0): a∞ .

Proof. Since by hypothesis SA (M )p ( SA (Mp )) is Ap -torsionfree for every prime ideal p ⊂ A with p  a, we can apply [29, Lemma 5.2] (take B = SA (M )) in order to conclude that the A-torsion ideal T ⊂ SA (M ) is given by the saturation T = (0): a∞ = {g ∈ SA (M ) | g at = 0, t  0},

as needed. 2 2.3.2. Defining equations in the hypersurface case We return to the setting of Subsection 2.2, where the free presentation matrix Π of DA/k was obtained. Write x = x1 , . . . , xn , so that R = k[x], and let T = T1 , . . . , Tr+1 be a set of r + 1 indeterminates over R. As we are going to show now, the ideal L(f ) := (f, I1 ((T) · Π)) ⊂ k[x, T] plays a crucial role into the explicit description of the dual tangent algebras S∗A/k = SA (DA/k ) and R∗A/k = RA (DA/k ), where A = R/(f ).

C.B. Miranda-Neto / Journal of Pure and Applied Algebra 219 (2015) 5442–5466

5454

Proposition 2.12. We have: k[x, T] L(f ) k[x, T] = L(f ): J(f )∞

(i) S∗A/k = (ii) R∗A/k

Proof. (i) From Theorem 2.3 we immediately get that SR (DA/k ) = R[T]/I1 ((T) · Π). Since f annihilates DA/k , we have DA/k = DA/k ⊗R A and hence, by the base change property of symmetric algebras, S∗A/k = SR (DA/k ) ⊗R A = R[T]/L(f ). (ii) Since A is reduced, its module ΩA/k of Kähler differentials has a well-defined rank, equal to n − 1. The (n −1)th Fitting ideal of ΩA/k is the Jacobian ideal JA/k = J(f )/(f ) ⊂ A (which clearly has positive grade). Therefore, for any prime ideal p ⊂ A not containing JA/k , the Ap -module (ΩA/k )p ( ΩAp /k ) is free, and hence so is its Ap -dual (DA/k )p , which in particular must be of linear type. Now we can apply Lemma 2.11, with a = JA/k , in order to obtain R∗A/k = S∗A/k /(0): J∞ A/k . Since (0): J∞ A/k = we finally get R∗A/k = R[T]/L(f ): J(f )∞ .

L(f ): J(f )∞ ⊂ S∗A/k , L(f )

2

Remarks 2.13. (i) Linear type property. From the result above it follows readily that the module DA/k is of linear type if and only if saturating the ideal L(f ) by J(f ) (in the polynomial ring k[x, T]) causes no effect. It is clear that any linear form in R yields a linear type derivation module, as in this case DA/k is free, but quite intriguingly we have found no example in higher degree. We have tested lots of examples – in both smooth and singular cases (including singular divisors that are free; see the definition in Section 4), as well as in both low and high dimensions – and in all of them we verified that L(f ) : J(f )∞ = L(f ) (equivalently, J(f )R[T] ⊆ P for some P ∈ AssR[T] (S∗A/k ); see Example 2.15(i)), which seems to suggest the question as to whether there exists a non-linear form f ∈ R whose corresponding derivation module DA/k is of linear type. (ii) Fiber cone. Once we have obtained the defining ideal of the Rees algebra R∗A/k , we can compute its special fiber F∗A/k = R∗A/k ⊗A k = R∗A/k /(x)R∗A/k , that is, the fiber cone of DA/k . Such graded k-algebra can be expressed as a residue ring of k[T], and its Krull dimension is the analytic spread of DA/k . Further, we can ask about the geometric role that Proj (F∗A/k ) plays with respect to Proj(A) (here, assume that k is algebraically closed). Our motivation for suggesting this question relies on the nice geometric meaning of the special fiber of the tangent algebra RA/k = RA/k (ΩA/k ) of Simis, Ulrich and Vasconcelos; to wit, if A is a reduced finitely generated standard graded k-algebra, then Proj (RA/k ⊗A k) is the tangential variety to (the embedded) Proj (A) (cf. [29, Proposition 5.17]). (iii) Cohen–Macaulayness. As in the general theory of blowup algebras of finitely generated modules over affine rings, we can regard S∗A/k and R∗A/k as non-negatively graded algebras, having A – or R as well – as their degree zero piece. Now, R∗A/k is Cohen–Macaulay if and only if its depth (with respect to the maximal ideal (x, T) ⊂ k[x, T]) is equal to 2n − 2 = (n − 1) + (n − 1), which is the dimension of A plus the rank of the A-module DA/k , by the dimension formula for Rees algebras (cf. [32, Proposition 2.2]). Of course, depth(R∗A/k ) can be computed through the Auslander–Buchsbaum formula, whenever the projective dimension of L(f ): J(f )∞ over k[x, T] is known. A basic constraint for the Cohen–Macaulayness of R∗A/k is given in Proposition 2.14 below.

C.B. Miranda-Neto / Journal of Pure and Applied Algebra 219 (2015) 5442–5466

5455

Proposition 2.14. If R∗A/k is Cohen–Macaulay, then A is a normal ring (in particular, f is irreducible). Proof. By Serre’s classical criterion, it suffices to check that the graded hypersurface ring A satisfies the so-called R1 condition. If the Rees algebra R∗A/k is Cohen–Macaulay, then the module DA/k must be free locally in height 1, by [32, Corollary 4.3]. Thus, by the well-known Zariski–Lipman conjecture in this case, it follows that A is regular locally in height 1, as needed. 2 Examples 2.15. (i) Consider the smooth projective conic f = x2 + y 2 + z 2 ∈ R = C[x, y, z]. A computation yields that both S∗A/C and R∗A/C are 4-dimensional Cohen–Macaulay rings. Now, denoting R = R[T1 , T2 , T3 , T4 ] and taking the prime ideal P = J(f )R = (x, y, z)R , the defining ideal of R∗A/C in R has the form L(f ): P

∞

= L(f ): P = (L(f ), Q),

Q=

4

Ti2 .

i=1

The fiber cone of DA/C is the hypersurface ring F∗A/C = C[T1 , T2 , T3 , T4 ]/(Q). Further, we can verify that L(f ): (Q) = P, so that P is an associated prime ideal of S∗A/C = R /L(f ) as an R -module. (ii) Now we want to show that the converse of Proposition 2.14 is not true. Consider the Fermat cubic f = x3 + y 3 − z 3 ∈ R = Q[x, y, z]. Clearly, A is a normal domain, but R∗A/Q is not Cohen–Macaulay as we can check that its defining ideal L(f ) : (x2 , y 2 , z 2 )∞ has projective dimension 3 over the ring R[T1 , T2 , T3 , T4 ], and hence the Auslander–Buchsbaum formula yields depth(R∗A/Q ) = 7 −4 = 3, while R∗A/Q is 4-dimensional. As to S∗A/Q , it is Cohen–Macaulay of dimension 4. Later on, in Corollary 3.4, we will derive a formula for the dimension of S∗A/k in the smooth case. In particular, it can be used to confirm that such ring is 4-dimensional in both examples above. 2.4. An explicit example: cuspidal plane cubic Our objective now is to work out an explicit example, thus illustrating the main results and remarks obtained so far. We keep the previous notation. Consider the cuspidal projective plane cubic f = x3 − y 2 z ∈ R = k[x, y, z], where k is a field with char k = 0. Graded R-free resolution of DA/k . The gradient ideal J(f ) = (3x2 , −2yz, −y 2 ) of f has a (minimal) graded free resolution Φ

Ψ

0 −→ R(−5) −→ R2 (−4) ⊕ R(−3) −→ R3 (−2) −→ J(f ) −→ 0 where ⎛

⎞ −y Φ = ⎝ 2z ⎠ 3x2

⎛

2yz Ψ = ⎝ 3x2 0

y2 0 3x2

⎞ 0 y ⎠ −2z

0 In particular, r = 3, s = 1 and c = 5, and the R-module TR/k (f ) is generated by

V1 = 2yz

∂ ∂ + 3x2 ∂x ∂y

V2 = y 2

∂ ∂ + 3x2 ∂x ∂z

V3 = y

∂ ∂ − 2z ∂y ∂z

5456

C.B. Miranda-Neto / Journal of Pure and Applied Algebra 219 (2015) 5442–5466

so that d1 = d2 = 2, d3 = 1. After computing the derivations W1 , W2 , W3 and expressing them as R-linear combinations of the Vi ’s, we get that the matrix Λ = (fij ) can be taken as ⎛

− 13 y

⎜ Λ = ⎝ − 13 z

1 3x

0

0

1 3x 1 2 3y

− 13 yz

0

⎞ ⎟ ⎠

so that, by Theorem 2.3, the module DA/k = (R V1 + R V2 + R V3 + R )(mod f DR/k ) has a graded R-free presentation Π

0 −→ R4 (−2) −→ R2 (−1) ⊕ R2 −→ DA/k −→ 0 where ⎛

−y

⎜ 2z

=⎜

| Λ) Π = (Φ ⎜ 2 ⎝ 3x 0

− 13 y

1 3x

0

− 13 z

0

0

− 13 yz

x2

− 23 yz

1 3x 1 2 3y − 13 y 2

⎞ ⎟ ⎟ ⎟ ⎠

which gives in fact a (minimal) graded R-free resolution of DA/k , as we have noticed in certain generality in the proof of Corollary 2.5(ii). Note that det(Π) = 13 f 2 = 0 (thus vanishing modulo (f ), as expected). Of course, we can multiply the last three columns of Π by 3 in order to make it simpler, but our intention is to keep the format of Theorem 2.3. Regularity and Hilbert function of DA/k . From the above minimal resolution of DA/k we readily obtain reg(DA/k ) = 1 (in accordance with Corollary 2.5(ii)), and its Hilbert function can be written as





2+ρ 1+ρ ρ H(DA/k , ρ) = H(R (−1) ⊕ R , ρ) − H(R (−2), ρ) = 2 · +2· −4· 2 2 2 2

2

4

which by a standard calculation yields that the Hilbert polynomial P (DA/k , ρ) satisfies H(DA/k , ρ) = P (DA/k , ρ) = 6ρ + 2,

ρ ≥ 0.

Note that the postulation number of DA/k is zero (see Corollary 2.9(ii)). Matrix factorization of f . Having computed the matrix Π, we can obtain vectors hjΠ ’s of R4 satisfying Π(hjΠ ) = f ej , j = 1, 2, 3, 4. They are given by h1Π = ( 13 yz, 2yz, 3x2 , 0), h2Π = (− 13 y 2 , y 2 , 0, 3x2 ), h3Π = ( 13 x, 0, y, −2z), h4Π = (0, x, y, z). Thus, denoting by Γ the matrix whose jth column is hjΠ , we can check that Π · Γ = f · I4 that is, (Π, Γ) is a matrix factorization of f , as predicted by Corollary 2.10(i). As a consequence, DA/k has a minimal, 2-periodic A-free resolution as given by Corollary 2.10(ii). Dual tangent algebras. Finally, we investigate the rings S∗A/k and R∗A/k . Introducing 4 = r +1 indeterminates T1 , T2 , T3 , T4 over R, we get that a (minimal) set of generators of the ideal L(f ) = (f, I1 ((T1 T2 T3 T4 ) ·Π)) ⊂ R = R[T1 , T2 , T3 , T4 ] is {f, L1 , L2 , L3 , L4 }, where

C.B. Miranda-Neto / Journal of Pure and Applied Algebra 219 (2015) 5442–5466

L1 = −yT1 + 2zT2 + 3x2 T3 , L3 = xT1 − yzT3 − 2yzT4 ,

5457

L2 = −yT1 − zT2 + 3x2 T4 , L4 = xT2 + y 2 T3 − y 2 T4 .

It follows that the algebra S∗A/k = R /L(f ) is a 4-dimensional Cohen–Macaulay ring since L(f ) has height 3 and projective dimension 2 over R (notice, for computational purposes, that one way to make L(f ) homogeneous is through the grading deg(x) = deg(y) = deg(z) = deg(T3 ) = deg(T4 ) = 1, deg(T1 ) = deg(T2 ) = 2). Now, saturating L(f ) by J(f ) = (x2 , yz, y 2 ) in R , we obtain L(f ): J(f )∞ = L(f ): J(f ) = (L(f ), H1 , H2 ) where H1 = T12 − xz(T3 + 2T4 )2 ,

H2 = T1 T2 + xy(T32 + T3 T4 − 2T42 )

Thus, by Proposition 2.12(ii), the Rees algebra of DA/k can be presented as R∗A/k = R /(L(f ), H1 , H2 ), which is 4-dimensional, and has depth 3 since (L(f ), H1 , H2 ) has projective dimension 3 over R . It follows that R∗A/k is not Cohen–Macaulay, in accordance with Proposition 2.14 since A is non-normal. Further, the fiber cone of DA/k is given by F∗A/k = k[T1 , T2 , T3 , T4 ]/(T12 , T1 T2 ), and note that the corresponding reduced k-algebra (F∗A/k )red is isomorphic to a polynomial ring in 3 indeterminates over k; in particular, DA/k has analytic spread 3. 3. Minimal number of generators In this section, we investigate the minimal number of generators of derivation modules (see, e.g., [18]), by comparing it with the same invariant of the module of logarithmic derivations, for any – not necessarily principal – given R-ideal, the latter being a natural upper bound for the former. Such comparison will lead us, in particular, to a new characterization of free divisors later in Subsection 4.2. We treat first the principal ideal case, and we obtain equality between numbers of generators under a mild condition. Further, we deal with the general case, by checking what happens when we take powers of the ideal. 3.1. The hypersurface case We maintain the setup and notation of Subsection 2.2. By Lemma 2.1, we see that ν(DA/k ) ≤ ν(TR/k (f )), but a priori there is no clue for equality in general. In this regard, Simis and Ulrich have (independently) suggested, by personal conversations, the question as to whether such a bound is always attained. We now prove that this is true under a rather weak condition (see also Remark 3.3 below). Proposition 3.1. If J(f ) has no (homogeneous) minimal syzygy of degree d, then ν(DA/k ) = ν(TR/k (f )). Proof. The argument we need is in fact contained in the proof of Corollary 2.5(ii), but for completeness we write it again as here there is no hypothesis on the projective dimension of J(f ) (except, of course, that 0 it is not allowed to be zero). First note that, by Lemma 2.2, ν(TR/k (f )) = ν(TR/k (f )) + 1 = r + 1, and by Theorem 2.3 we have ν(DA/k ) = r + 1 if and only if I1 (Π) ⊆ (x), which is easily seen to be equivalent to I1 (Λ) ⊆ (x). Using the shifts in the graded R-free presentation of DA/k given therein, we readily get deg(fij ) = d − di whenever fij = 0, and hence d ≥ di in this case (see the discussion in Remark 2.6(ii)).

5458

C.B. Miranda-Neto / Journal of Pure and Applied Algebra 219 (2015) 5442–5466

∂f Since by hypothesis the ∂x ’s admit no homogeneous minimal syzygy of degree d, or equivalently di = d j for every i, it follows that every non-zero fij must be non-constant, and we are done. 2

In other words, the result above says that ν(DA/k ) = r + 1 provided no di equals d (that is, no Vi has degree d − 1). Clearly, if in addition ν(J(f )) = n, then ν(DA/k ) = b1 + 1, where b1 is the first Betti number of J(f ). In the smooth case, we get: Corollary 3.2. If f defines a smooth projective hypersurface, then

n ν(DA/k ) = + 1. 2 Proof. This follows from the above considerations since in this case J(f ) is generated by an R-sequence   of n forms of degree d − 1 (hence, di = d − 1 for every i), and note that b1 = n2 by standard counting of Koszul relations. 2 Remark 3.3 (The open case). We have not been able to prove that ν(DA/k ) = ν(TR/k (f )) in the remaining, non-typical case where there exists i with di = d, that is, some Vi having degree d − 1. We point out that such equality between numbers of generators has been confirmed in each of the few examples we have found in this critical situation. To quote one instance, we take the singular projective plane curve defined by f = x4 + y 4 − xyz 2 ∈ R = k[x, y, z] (in the affine context, it defines a cone over a tacnode). In this example, 0 one of the minimal generators of the module TR/k (f ) is 8y 3 z

∂ ∂ ∂ 0 + 2yz 3 + (16x2 y 2 − z 4 ) ∈ [TR/k (f )]3 . ∂x ∂y ∂z

Even so, we can verify that ν(DA/k ) = 5 = r + 1. Another instance, in higher dimension, is the projective surface given by f = x4 + x3 y + x2 z 2 + y 3 w + zw3 ∈ R = k[x, y, z, w], which has only isolated singularities. 0 In this case, we can check that TR/k (f ) admits a minimal generator of degree 3 = d − 1; moreover, we find ν(DA/k ) = 9 = r + 1, so that the conjectured equality holds once again. It would be interesting to discover whether, and eventually how, the existence of a degree d minimal syzygy of J(f ) is influenced by the nature of the singularities of the hypersurface. Corollary 3.4. If f defines a smooth projective hypersurface, then the symmetric algebra S∗A/k of DA/k has Krull dimension given by dim(S∗A/k )



n = + 1. 2

Proof. In this situation, A is a (normal) graded domain such that the A-module DA/k is free locally at the non-maximal prime ideals of A. By the graded analogue of [37, Corollary 1.2.3], the dimension of S∗A/k satisfies dim(S∗A/k ) = max{ν(DA/k ), 2n − 2}, where we note that the number 2n − 2 equals the dimension of A plus the rank of DA/k (which is also the dimension of the Rees algebra of DA/k ; see Remark 3.2(iii)). The result now follows from Corollary 3.2,   noticing that n2 + 1 ≥ 2n − 2 for every integer n ≥ 2. 2

C.B. Miranda-Neto / Journal of Pure and Applied Algebra 219 (2015) 5442–5466

5459

3.2. The general case Let now I ⊆ (x) = (x1 , . . . , xn ) ⊂ R be any homogeneous proper ideal. Given any integer q ≥ 1, consider the factor ring A{q} = R/I q , where I q is the (ordinary) qth power of I, as well as its derivation module DA{q} /k . We shall assume that the ring A = A{1} is reduced. As we are going to prove now, the function q → ν(DA{q} /k ) stabilizes already for q ≥ 2, and we identify the stable number of generators. Proposition 3.5. If I is radical, then for any integer q ≥ 2: (i) TR/k (I q ) = TR/k (I) (ii) ν(DA{q} /k ) = ν(TR/k (I)) Proof. (i) The inclusion TR/k (I) ⊆ TR/k (I q ) follows from iterated use of Leibniz’s rule. Indeed, for any given τ ∈ TR/k (I) and f1 , . . . , fq ∈ I, we can write  τ

q 

i=1

 fi

=

q j=1

⎛



⎝

⎞ fi ⎠ τ (fj ) ∈ I q .

i=j

Now, in order to prove that TR/k (I q ) ⊆ TR/k (I), recall the general fact, valid in characteristic zero and √ observed by Kaplansky (see [22, page 12]), that TR/k (L) ⊆ TR/k ( L) for any ideal L ⊂ R. Apply this result with L = I q , and use that I is assumed to be radical. (ii) For any given q ≥ 2, we have I q DR/k ⊆ I 2 DR/k ⊆ I TR/k (I) ⊆ (x)TR/k (I), which induces a surjection πq : TR/k (I)/I q DR/k −→ TR/k (I)/(x)TR/k (I) = TR/k (I) ⊗R k. Therefore, using the equality given in (i) above, we obtain that πq maps the module TR/k (I q )/I q DR/k  DA{q} /k onto TR/k (I) ⊗R k, and consequently ν(DA{q} /k ) ≥ ν(TR/k (I) ⊗R k) = ν(TR/k (I)). On the other hand, from the surjection TR/k (I) = TR/k (I q ) −→ DA{q} /k we derive ν(DA{q} /k ) ≤ ν(TR/k (I)), which then must be an equality. 2 A natural question arises as to whether taking the square of I is in fact necessary, that is, we can ask whether it is always true that ν(DA/k ) = ν(TR/k (I)), extending the issue on the hypersurface case considered in the previous subsection (see Remark 3.3). As it turns out, unlike the principal ideal case, we can check that quite typically such equality is not true. First, to the radical ideal I ⊂ R we can attach the “tangential” deviation

5460

C.B. Miranda-Neto / Journal of Pure and Applied Algebra 219 (2015) 5442–5466

t(I) = ν(TR/k (I)) − ν(DA/k ). One way to compute the number ν(TR/k (I)) is through the method described in [23, Section 2]. As to ν(DA/k ), it can be calculated by standard commands provided by well-known computer algebra programs (e.g., [4]), since DA/k is a (second-order) syzygy module – precisely, the kernel of the A-linear map induced by the Jacobian matrix associated to some (in fact, any) set of generators of I. Note that, if we denote t(I) = t(f ) in the principal ideal case I = (f ), the issue considered in Remark 3.3 can be rephrased as whether t(f ) = 0 (the remaining open case, as we have seen, is when J(f ) admits a homogeneous minimal syzygy of degree d). Proposition 3.6. We have t(I) ≤ n · ν(I). Proof. This follows easily from the exact sequence of k-vector spaces n  (I ⊗R k) −→ TR/k (I) ⊗R k −→ DA/k ⊗R k −→ 0

derived from Lemma 2.1.

2

It turns out that the bound above is sharp, at least for n = 3, as shown in Example 3.7 below. Example 3.7. Take the radical ideal I = (f, g) = (x2 − yz, y 2 − xz) ⊂ R = k[x, y, z]. Then, we check that ν(TR/k (I)) = 8 and ν(DA/k ) = 2, so that t(I) = 6 = n · ν(I). Explicitly, the module TR/k (I) is minimally generated by the derivations corresponding to the column-vectors of the matrix ⎛

y2 yz z2

x yz ⎝ y xy z xz

0 f 0

0 0 f

0 g 0

0 0 g

⎞ 0 0⎠ h

where h = z 2 − xy. Reducing the entries modulo I, we easily see that the first and the last columns of the resulting matrix yield generators for DA/k . Going one step further and considering the square I 2 , the module DA{2} /k can be minimally generated by the images, modulo I 2 DR/k , of the derivations read off from the columns of the matrix ⎛

x yz ⎝ y xy z xz

y2 yz z2

y2 y2 yz yz − f xy − h z2

0 3g −4f

0 g 0

⎞ 0 0⎠ g

so that ν(DA{2} /k ) = 8 = ν(TR/k (I)), as predicted by Proposition 3.5. Remark 3.8 (The monomial case). Let I ⊂ (x)2 ⊂ R be a radical monomial ideal, so that A is a Stanley– n ∂ Reisner ring. By [6, Theorem 2.2.1], we can write TR/k (I) = j=1 Ij ∂x , where Ij = I: (I: (xj )) for each j, j n and consequently DA/k  j=1 Ij , where Ij := Ij /I ⊂ A. It follows that t(I) =

n

(ν(Ij ) − ν(Ij )).

j=1

In particular, t(I) = 0 if and only if I ⊆ (x)Ij for every j. Moreover, since xj ∈ Ij for each j and by hypothesis n the initial degree of I is at least 2, we have ν(Ij ) ≥ 1 for every j, and therefore t(I) ≤ ( j=1 ν(Ij )) − n. In n Examples 3.9 below, we illustrate the opposite situations t(I) = 0 and t(I) = ( j=1 ν(Ij )) −n (of course, the

C.B. Miranda-Neto / Journal of Pure and Applied Algebra 219 (2015) 5442–5466

5461

latter means that ν(DA/k ) = n). Finally, based on many experiments with the tangential idealizer TR/k (I) for various classes of graded ideals I ⊆ (x)2 , we raise the question as to whether the class of monomial ideals is the only one (possibly up to projective change of coordinates) such that TR/k (I) – and hence DA/k – splits totally into a direct sum of ideals; this seems to be of interest, as it might lead to a “tangential” characterization of the Stanley–Reisner property. Examples 3.9. (i) If I = (xy, xz) ⊂ R = k[x, y, z], then I1 = (x) and I2 = I3 = (y, z), so that ν(TR/k (I)) = 5. Clearly, I ⊂ (x, y, z)(x) and I ⊂ (x, y, z)(y, z), which by the above considerations yields t(I) = 0, and hence ν(DA/k ) = 5. (ii) Now, consider the (Cohen–Macaulay) ideal I = (xy, xz, yz) ⊂ R = k[x, y, z]. We have I1 = (x, yz), I2 = (y, xz), I3 = (z, xy), hence ν(TR/k (I)) = 6 and each Ij is cyclic, so that ν(DA/k ) = 3. Thus, in this n case, t(I) = 3 = 6 − 3 = ( j=1 ν(Ij )) − n. 4. Algebraic free divisors As in most of the preceding parts, we maintain, in the entire section, the setting of Subsection 2.2. In particular, d ≥ 2 is the degree of the reduced form f , and d1 , . . . , dr are the degrees of the homogeneous 0 minimal syzygies of J(f ), or what amounts to the same, Vi ∈ [TR/k (f )]di −1 for each i. We write (x) = (x1 , . . . , xn ), the homogeneous maximal ideal of R. Recall that the polynomial f ∈ R is said to be a free divisor – we also say that it defines a free hypersurface – if its logarithmic derivation module TR/k (f ) is a free R-module, necessarily of rank n. This condition was first investigated by Saito in [25], originally in the context of complex analytic geometry, and has shown to be of much significance for a variety of branches in mathematics. Free divisors exist in any dimension; for instance, in the case of the normal crossing f = x1 · · · xn , a basis n ∂ for TR/k (f ) is {x1 ∂x , . . . , xn ∂x∂n }. This divisor defines the free hyperplane arrangement j=1 Hj , where Hj 1 is the jth coordinate hyperplane in (n − 1)-dimensional projective space over k. For the class of irreducible free divisors the matter is much more difficult and subtle. Another well-known feature of projective free hypersurfaces of degree at least 2, for n ≥ 3, is that they are necessarily singular. This is an easy consequence of the homological criterion below (cf. [23, Corollary 4.4], [30, Proposition 3.7]). Lemma 4.1. f ∈ R is a free divisor if and only if pd(J(f )) = 1 (equivalently, J(f ) is a codimension 2 perfect ideal). 0 (f ) is the module of first-order Proof. Up to a twist (and, eventually, up to a free direct summand), TR/k syzygies of J(f ). Moreover, in the present context, projective graded R-modules are necessarily free. Therefore, the criterion follows from the structure result given in Lemma 2.2. 2

4.1. Derivation modules of free divisors Combining Lemma 4.1 with some of the results given in Section 2, we can readily establish and collect a number of properties of the derivation module DA/k (with the grading as adopted in Subsection 2.1) in the present context where f is a free divisor: • It has a minimal graded R-free resolution

Λ

0 −→ Rn (−(d − 1)) −→

n−1  i=1

R(−(di − 1)) ⊕ R −→ DA/k −→ 0

C.B. Miranda-Neto / Journal of Pure and Applied Algebra 219 (2015) 5442–5466

5462

so that its Castelnuovo–Mumford regularity is reg(DA/k ) = d − 2 (see Corollary 2.5(i)) and its Hilbert function is given by H(DA/k , ρ) =

n−1 i=1





n − di + ρ n−1+ρ n−d+ρ + −n· n−1 n−1 n−1

Moreover, its postulation number is pn(DA/k ) = d − n (see Corollary 2.9(i)). An explicit computation of the Hilbert polynomial of DA/k will be given in Proposition 4.10 in case f is a linear free divisor in the sense of Buchweitz and Mond; • It has a minimal, 2-periodic A-free resolution

Λ

Γ

Λ

Γ

A A A A · · · −→ An −→ An −→ An −→ An −→ An −→ DA/k −→ 0

A and ΓA are the reductions, modulo (f ), of Λ

(which equals Π in this situation) and Γ (the where Λ matrix described in Corollary 2.10), provided di ≥ 1 for every i, which in this case is easily seen to

Γ) is a matrix factorization of f , and the periodic imply that the entries of Γ lie in (x). The pair (Λ, A-free resolution above is guaranteed by Corollary 2.10(ii). Note that, by Lemma 2.2, we have r + 1 = n since f is free; • Its Rees algebra R∗A/k cannot be Cohen–Macaulay; indeed, since ht(J(f )) = 2 (by Lemma 4.1), the Jacobian ideal JA/k must be of codimension 1 in A, hence A is non-normal, and the assertion follows from Proposition 2.14. Our goal in the sequel is to detect other new properties related to free divisors, especially by means of their associated derivation modules and Jacobian ideals. 4.2. A new characterization ∂f ∂f As in Subsection 2.2, we fix the generators ∂x , . . . , ∂x of the ideal J(f ) ⊂ R. Moreover, we denote 1 n {2} 2 A = R/(f ), following the notation of Subsection 3.2.

Theorem 4.2. The following assertions are equivalent: (i) (ii) (iii) (iv)

f is a free divisor. ν(DA{2} /k ) = n. ν(DA/k ) = n, and J(f ) has no (homogeneous) minimal syzygy of degree d. n−1 ν(DA/k ) = n, DA/k is a maximal Cohen–Macaulay A-module, and d = 1 + i=1 di .

Proof. We first prove that (i) implies (iv). Assume that f is free. The equality ν(DA/k ) = n has been verified in the proof of Corollary 2.5(i), and the maximal Cohen–Macaulayness of DA/k over A follows from 0 Remark 2.6(i) and Lemma 4.1. Now, TR/k (f ) is free (by Lemma 4.1), necessarily of rank r = n −1. Precisely, 0 TR/k (f ) =

n−1 

R Vi 

i=1

n−1 

R(−(di − 1))

i=1

and J(f ) has a Hilbert–Burch graded free resolution 0 0 −→ (TR/k (f ))(−d) 

n−1  i=1

Ψ

R(−(di + d − 1)) −→ Rn (−(d − 1)) −→ J(f ) −→ 0

C.B. Miranda-Neto / Journal of Pure and Applied Algebra 219 (2015) 5442–5466

5463

n−1 so that In−1 (Ψ) = J(f ). Therefore, the degree i=1 di of each (n − 1) × (n − 1) subdeterminant of Ψ must be equal to d − 1. n−1 To see that (iv) implies (iii), simply note that the formula d = 1 + i=1 di forces di ≤ d − 1 for each i (caution: this upper bound cannot be improved to d − 2 in general; see Example 4.3 below). Now, assume (iii). In particular, no di equals d, and then, by Proposition 3.1, the modules DA/k and TR/k (f ) have the same minimal number of generators. On the other hand, Proposition 3.5(ii) yields ν(TR/k (f )) = ν(DA{2} /k ), so that (ii) follows. Finally, assume (ii). By Proposition 3.5(ii), we get ν(TR/k (f )) = n and hence f must be a free divisor since TR/k (f ) has rank n (see Subsection 2.1). 2 n Example 4.3. Consider the arrangement of 2 hyperplanes defined by qn = x1  ∈ R2 , with  = j=1 xj , n ≥ 3. A computation shows that the derivation module DA/k of A = R/(qn ) can be minimally generated by V 1 , . . . , V n−1 , (the bar denotes reduction modulo qn DR/k ), where V1 = x1

∂ ∂ − (x1 + ) , ∂x1 ∂x2

Vj =

∂ ∂ − , ∂x2 ∂xj+1

j = 2, . . . , n − 1

and notice that none of them corresponds to a degree 2 minimal syzygy of J(qn ). Thus, by Theorem 4.2, qn is a free divisor. Further, since V1 has degree zero, we have d1 = 1 = d − 1 and therefore it is not true that di ≤ d − 2 for every i. 4.3. Linear freeness and the Ulrich property Our objective in this part is to characterize linear free divisors in the sense of Buchweitz and Mond. As it will turn out, an explicit Ulrich ideal for such class of hypersurface rings will be detected. Before going into some preliminary notions and the characterization, let us notice that a basic necessary condition for the freeness of f is that d ≥ ν(J(f )) (of course, the typical and standard situation is when ν(J(f )) = n, which simply means that all the partial derivatives of f are linearly independent over k). To see this, use that J(f ) is a perfect ideal of codimension 2 (see Lemma 4.1) and apply the classical Hilbert–Burch structure theorem in order to get J(f ) ⊂ (x)ν(J(f ))−1 , which yields d ≥ ν(J(f )) (because of the Euler identity), as asserted. Moreover, for the sake of completeness, we are able to give a proof of this fact without resorting to the determinantal nature of J(f ); instead, we will apply the natural graded analogue of the standard fact below (cf., e.g., [19, Corollary 1.3] or [36, page 26]). Lemma 4.4. Let S be a Cohen–Macaulay Noetherian local ring with infinite residue field and positive Krull dimension. If M is a Cohen–Macaulay finitely generated S-module with positive rank (hence M is in fact a maximal Cohen–Macaulay S-module) and if e(−) denotes multiplicity, then ν(M ) ≤ e(S) rkS (M ). Proposition 4.5. If f ∈ R is a free divisor, then d ≥ ν(J(f )). Proof. Lemma 2.1 yields the short exact sequence of A-modules 0 −→

TR/k (f ) f DR/k −→ −→ DA/k −→ 0 f TR/k (f ) f TR/k (f )

where the middle term TR/k (f ) ⊗R A is A-free (of rank n) since f is a free divisor. Further,

5464

C.B. Miranda-Neto / Journal of Pure and Applied Algebra 219 (2015) 5442–5466

DR/k f DR/k   JA/k f TR/k (f ) TR/k (f ) where JA/k = J(f )/(f ) ⊂ A and the rightmost isomorphism comes from the fact that TR/k (f ) is the kernel of the natural surjection DR/k → JA/k given by evaluation at f , modulo (f ). It follows that JA/k can be realized as the module of syzygies of DA/k , and hence the homogeneous ideal JA/k ⊂ A is a maximal Cohen–Macaulay A-module since so is DA/k (because f is free). Now we are in a position to apply Lemma 4.4, which gives ν(JA/k ) ≤ e(A)rkA (JA/k ) = deg(f ) · 1 = d. Finally, note that ν(JA/k ) = ν(J(f )), since f ∈ (x)J(f ) in virtue of the Euler identity. 2 For convenience, we consider the following definition: Definition 4.6. We say that a free divisor f ∈ R (of degree d ≥ 2, as before) is minimal if equality holds in Proposition 4.5, that is, d = ν(J(f )). Our aim is to characterize minimal free divisors with a view to the important class of linear free divisors. As it turns out, the theory of Ulrich modules comes into the scene. First, we recall the basic definitions. If equality occurs in Lemma 4.4 (more generally, such equality is ν(M ) = e(M ), which covers the case where M has no well-defined generic rank), then the maximal Cohen–Macaulay S-module M is said to be an Ulrich module – also, maximally generated maximal Cohen–Macaulay module, or linear maximal Cohen–Macaulay module. An S-ideal is said to be an Ulrich ideal if it has the Ulrich property as an S-module. This well-known terminology pays tribute to the influential paper [36] by Ulrich. As introduced in [8], a free divisor f ∈ R is said to be linear if di = 1 for every i, that is, the minimal ∂f ∂f syzygies of ∂x , . . . , ∂x are linear. 1 n We notice the following simple characterization, particularly in terms of the Ulrichness of the Jacobian ideal JA/k of A = R/(f ). Proposition 4.7. The following statements are equivalent: (i) f is a minimal free divisor. (ii) JA/k ⊂ A is an Ulrich ideal. (iii) f is a free divisor and reg(DA/k ) = ν(J(f )) − 2. Proof. From the proof of Proposition 4.5 it is clear that (i) implies (ii). The converse is similar: if JA/k is Ulrich over A, then, again, ν(JA/k ) = deg(f ) · 1 = d and hence ν(J(f )) = d. Moreover, as in particular JA/k is a Cohen–Macaulay module, we get pd(JA/k ) = 1 so that pd(J(f )) = 1, which by Lemma 4.1 means that f is free. The equivalence between (i) and (iii) is immediate from Corollary 2.5(i). 2 Remark 4.8. Adding to the equivalent assertions of Proposition 4.7 the natural condition ν(J(f )) = n, they turn out to be equivalent to: (iv) f is a linear free divisor (in the sense of Buchweitz and Mond). Indeed, let us assume that f is a minimal free divisor (condition (i) above) and ν(J(f )) = n. Then d = n n−1 and, necessarily, di ≥ 1 for every i. Therefore, Theorem 4.2 yields n = d = 1 + i=1 di , which implies that n−1 di = 1 for every i. Conversely, if f is a linear free divisor, then d = 1 + i=1 di = 1 + (n − 1) = n. On the other hand, n = ν(J(f )) since, in particular, no di can be zero. It follows that f is minimal, as wanted.

C.B. Miranda-Neto / Journal of Pure and Applied Algebra 219 (2015) 5442–5466

5465

Thus, we have obtained, in particular, that f is a linear free divisor if and only if ν(J(f )) = n and JA/k is an Ulrich ideal. Examples 4.9. (i) The free quadric qn = x1 (x1 + . . . + xn ) ∈ R = k[x1 , . . . , xn ] (n ≥ 3) considered in Example 4.3 is minimal, arbitrarily non-linear (since ν(J(qn )) = 2 < n), and JA/k = (x2 + . . . + xn , x1 )/(qn ) is an Ulrich ideal of A by Proposition 4.7. (ii) By [17, Theorem 2.8(ii)], the quartic f = y 2 z 2 − 4xz 3 − 4y 3 w + 18xyzw − 27x2 w2 ∈ R = k[x, y, z, w] is an irreducible, linear free divisor. Here, reg(DA/k ) = 2 = n − 2. We finish the paper with the computation of the Hilbert polynomial P (DA/k , ρ) of the graded derivation module of A = R/(f ) in case f ∈ R = k[x1 , . . . , xn ] (n is fixed) is any given linear free divisor. Notice that, since necessarily d = n in this situation, we get pn(DA/k ) = 0 by Corollary 2.9(i). Proposition 4.10. If the form f ∈ R is a linear free divisor, then ⎛ ⎞ n−1 n−2   n ⎝ P (DA/k , ρ) = (ρ + α) − (ρ − β)⎠ (n − 1)! α=1 β=0

which has degree n − 2 = dim(DA/k ) − 1, as expected. In the particular case R = k[x, y, z], we have P (DA/k , ρ) = 6ρ + 3 Proof. Putting di = 1, for every i, in the formula for the Hilbert function H(DA/k , ρ) written in Subsection 4.1, and using that d = n, we obtain P (DA/k , ρ) = n ·



n−1+ρ ρ −n· n−1 n−1

that is, P (DA/k , ρ) = n ·

ρ(ρ − 1) · · · (ρ − (n − 2)) (ρ + n − 1)(ρ + n − 2) · · · (ρ + 1) −n· (n − 1)! (n − 1)!

which gives the result. Note, in particular, that the term ρn−1 can be cancelled. In the special case n = 3, this polynomial is easily seen to be equal to 6ρ + 3. 2 Acknowledgements The author thanks the Department of Mathematics at Purdue University, where this work was written, for hospitality during his 1-year stay as a visiting scholar. He warmly thanks Prof. Bernd Ulrich – to whom this paper is dedicated – for support during this year, especially for fruitful discussions and questions that inspired some parts of this paper. The author has also benefited from stimulating conversations with Prof. Aron Simis on some of the present topics. Finally, he is grateful to the anonymous referee for a careful reading of the manuscript and for detailed comments and suggestions, in particular for bringing to the author’s attention a number of nice references. References [1] J. Backelin, J. Herzog, On Ulrich-modules over hypersurface rings, in: Commutative Algebra, Berkeley, CA, 1987, in: Math. Sci. Res. Inst. Publ., vol. 15, Springer, New York, 1989, pp. 63–68.

5466

C.B. Miranda-Neto / Journal of Pure and Applied Algebra 219 (2015) 5442–5466

[2] J. Backelin, J. Herzog, H. Sanders, Matrix factorizations of homogeneous polynomials, in: Algebra – Some Current Trends, Varna, 1986, in: Lect. Notes Math., vol. 1352, Springer, Berlin, 1988, pp. 1–33. [3] D. Bayer, D. Mumford, What can be computed in algebraic geometry? in: D. Eisenbud, L. Robbiano (Eds.), Computational Algebraic Geometry and Commutative Algebra, Proceedings, Cortona, 1991, Cambridge University Press, 1993, pp. 1–48. [4] D. Bayer, M. Stillman, Macaulay, a system for computation in algebraic geometry and commutative algebra, available via anonymous ftp from math.harvard.edu, 1992. [5] J. Brennan, J. Herzog, B. Ulrich, Maximally generated Cohen–Macaulay modules, Math. Scand. 61 (1987) 181–203. [6] P. Brumatti, A. Simis, The module of derivations of a Stanley–Reisner ring, Proc. Am. Math. Soc. 123 (1995) 1309–1318. [7] R.-O. Buchweitz, G.J. Leuschke, Factoring the adjoint and maximal Cohen–Macaulay modules over the generic determinant, Am. J. Math. 129 (2007) 943–981. [8] R.-O. Buchweitz, D. Mond, Linear free divisors and quiver representations, in: Singularities and Computer Algebra, in: Lond. Math. Soc. Lect. Note Ser., vol. 324, Cambridge Univ. Press, Cambridge, 2006, pp. 41–77. [9] J. Damon, On the legacy of free divisors: discriminants and Morse type singularities, Am. J. Math. 120 (1998) 453–492. [10] H. Derksen, J. Sidman, Castelnuovo–Mumford regularity by approximation, Adv. Math. 188 (2004) 104–123. [11] D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Am. Math. Soc. 260 (1980) 35–64. [12] D. Eisenbud, The Geometry of Syzygies: A Second Course in Algebraic Geometry and Commutative Algebra, Grad. Texts Math., vol. 229, Springer, 2005. [13] D. Eisenbud, S. Goto, Linear free resolutions and minimal multiplicities, J. Algebra 88 (1984) 107–184. [14] D. Eisenbud, C. Huneke, B. Ulrich, What is the Rees algebra of a module? Proc. Am. Math. Soc. 131 (2003) 701–708. [15] V. Ene, Maximal Cohen–Macaulay modules over hypersurface rings, An. Ştiinţ. Univ. ‘Ovidius’ Constanţa, Ser. Mat. 15 (1) (2007) 75–90. [16] E. Esteves, The Castelnuovo–Mumford regularity of an integral variety of a vector field on projective space, Math. Res. Lett. 9 (2002) 1–15. [17] M. Granger, D. Mond, M. Schulze, Free divisors in prehomogeneous vector spaces, Proc. Lond. Math. Soc. 102 (2011) 923–950. [18] R.V. Gurjar, V. Wagh, On the number of generators of the module of derivations and multiplicity of certain rings, J. Algebra 319 (2008) 2030–2049. [19] J. Herzog, M. Kühl, Maximal Cohen–Macaulay modules over Gorenstein rings and Bourbaki-sequences, in: Commutative Algebra and Combinatorics, Kyoto, 1985, in: Adv. Stud. Pure Math., vol. 11, North-Holland, Amsterdam, 1987, pp. 65–92. [20] J. Herzog, A. Simis, W. Vasconcelos, Koszul homology and blowing-up rings, in: Commutative Algebra, Trento, 1981, in: Lect. Notes Pure Appl. Math., vol. 84, Marcel Dekker, New York, 1983, pp. 79–169. [21] J. Herzog, B. Ulrich, J. Backelin, Linear maximal Cohen–Macaulay modules over strict complete intersections, J. Pure Appl. Algebra 71 (1991) 187–202. [22] I. Kaplansky, An Introduction to Differential Algebra, Hermann, Paris, 1957. [23] C.B. Miranda-Neto, Vector fields and a family of linear type modules related to free divisors, J. Pure Appl. Algebra 215 (2011) 2652–2659. [24] T.J. Puthenpurakal, The Hilbert function of a maximal Cohen–Macaulay module, Math. Z. 251 (2005) 551–573. [25] K. Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 27 (1980) 265–291. [26] J.B. Sancho de Salas, Tangent algebraic subvarieties of vector fields, Trans. Am. Math. Soc. 357 (9) (2004) 3509–3523. [27] H. Schenck, Elementary modifications and line configurations in P2 , Comment. Math. Helv. 78 (2003) 447–462. [28] H. Schenck, S. Tohaneanu, Freeness of conic-line arrangements in P2 , Comment. Math. Helv. 84 (2009) 235–258. [29] A. Simis, Remarkable Graded Algebras in Algebraic Geometry, XII ELAM, IMCA, Lima, Peru, 1999. [30] A. Simis, Differential idealizers and algebraic free divisors, in: Commutative Algebra, in: Lect. Notes Pure Appl. Math., vol. 244, Chapman & Hall/CRC, Boca Raton, FL, 2006, pp. 211–226. [31] A. Simis, S. Tohaneanu, Homology of homogeneous divisors, Isr. J. Math. 200 (2014) 449–487. [32] A. Simis, B. Ulrich, W. Vasconcelos, Rees algebras of modules, Proc. Lond. Math. Soc. 87 (2003) 610–646. [33] A. Simis, B. Ulrich, W. Vasconcelos, Tangent algebras, Trans. Am. Math. Soc. 364 (2012) 571–594. [34] H. Terao, Arrangements of hyperplanes and their freeness I, II, J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 27 (1980) 293–320. [35] S. Tohaneanu, On freeness of divisors on P2 , Commun. Algebra 41 (8) (2013) 2916–2932. [36] B. Ulrich, Gorenstein rings and modules with high numbers of generators, Math. Z. 188 (1984) 23–32. [37] W.V. Vasconcelos, Arithmetic of Blowup Algebras, Lond. Math. Soc. Lect. Note Ser., vol. 195, Cambridge Univ. Press, Cambridge, 1994. [38] W.V. Vasconcelos, Integral Closure. Rees Algebras, Multiplicities, Algorithms, Springer Monogr. Math., Springer, New York, 2005. [39] Y. Yoshino, Auslander’s work on Cohen–Macaulay modules and recent development, in: Algebras and Modules I, Trondheim, 1996, in: CMS Conf. Proc., vol. 23, Amer. Math. Soc., Providence, RI, 1998, pp. 179–198. [40] S. Yuzvinsky, A free resolution of the module of derivations for generic arrangements, J. Algebra 136 (1991) 432–438. [41] G.M. Ziegler, Matroid representations and free arrangements, Trans. Am. Math. Soc. 320 (1990) 525–541.