NOETHER RESOLUTIONS IN DIMENSION 2

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Isabel Bermejo, Eva García-Llorente, Ignacio García-Marco, Marcel Morales

To cite this version:

Isabel Bermejo, Eva García-Llorente, Ignacio García-Marco, Marcel Morales. NOETHER RESOLUTIONS IN DIMENSION 2. Journal of Algebra, Elsevier, 2017, 482, pp.398-426.

�10.1016/j.jalgebra.2017.03.026�. �hal-01502923�

NOETHER RESOLUTIONS IN DIMENSION 2

ISABEL BERMEJO, EVA GARCÍA-LLORENTE, IGNACIO GARCÍA-MARCO, AND MARCEL MORALES

A^BSTRACT. LetR:=K[x1, . . . , xⁿ]be a polynomial ring over an infinite fieldK, and letI⊂R be a homogeneous ideal with respect to a weight vectorω = (ω1, . . . , ωⁿ) ∈ (Z⁺)ⁿ such that dim (R/I) = d. In this paper we study the minimal graded free resolution ofR/IasA-module, that we call the Noether resolution ofR/I, wheneverA :=K[xn−d+1, . . . , xn]is a Noether nor- malization ofR/I. Whend= 2andIis saturated, we give an algorithm for obtaining this reso- lution that involves the computation of a minimal Gröbner basis ofIwith respect to the weighted degree reverse lexicographic order. In the particular case whenR/Iis a2-dimensional semigroup ring, we also describe the multigraded version of this resolution in terms of the underlying semi- group. Whenever we have the Noether resolution ofR/I or its multigraded version, we obtain formulas for the corresponding Hilbert series ofR/I, and whenI is homogeneous, we obtain a formula for the Castelnuovo-Mumford regularity ofR/I. Moreover, in the more general setting thatR/Iis a simplicial semigroup ring of any dimension, we provide its Macaulayfication.

As an application of the results for2-dimensional semigroup rings, we provide a new upper bound for the Castelnuovo-Mumford regularity of the coordinate ring of a projective monomial curve. Finally, we describe the multigraded Noether resolution and the Macaulayfication of either the coordinate ring of a projective monomial curveC ⊆PⁿKassociated to an arithmetic sequence or the coordinate ring of any canonical projectionπ^r(C)ofCtoPⁿK⁻¹.

Keywords: Graded algebra, Noether normalization, semigroup ring, minimal graded free resolution, Cohen- Macaulay ring, Castelnuovo-Mumford regularity.

1. INTRODUCTION

Let R := K[x1, . . . , xn] be a polynomial ring over an infinite field K, and let I ⊂ R be a weighted homogeneous ideal with respect to the vector ω = (ω₁, . . . , ω_n) ∈ (Z⁺)ⁿ, i.e., I is homogeneous for the grading deg_ω(xi) = ωi. We denote byd the Krull dimension of R/I and we assume that d ≥ 1. Suppose A := K[xn−d+1, . . . , xn] is a Noether normalization of R/I, i.e., A ֒→ R/I is an integral ring extension. Under this assumption R/I is a finitely generated A-module, so to study the minimal graded free resolution ofR/I asA-module is an interesting problem. Set

F : 0−→ ⊕v∈BpA(−s_p,v)−→ · · ·^ψp −→ ⊕^ψ1 v∈B0A(−s_0,v)−→^ψ0 R/I −→0

this resolution, where for alli ∈ {0, . . . , p} Bi denotes some finite set, andsi,v are nonnegative integers. This work concerns the study of this resolution ofR/I, which will be called theNoether resolution of R/I. More precisely, we aim at determining the sets Bi, the shifts si,v and the morphismsψ_i.

One of the characteristics of Noether resolutions is that they have shorter length than the min- imal graded free resolution of R/I as R-module. Indeed, the projective dimension of R/I as A-module isp = d−depth(R/I), meanwhile its projective dimension of R/I asR-module is n−depth(R/I). Studying Noether resolutions is interesting since they contain valuable informa- tion aboutR/I. For instance, since the Hilbert series is an additive function, we get the Hilbert series ofR/I from its Noether resolution. Moreover, wheneverI is a homogeneous ideal, i.e., homogeneous for the weight vector ω = (1, . . . ,1), one can obtain the Castelnuovo-Mumford regularity ofR/I in terms of the Noether resolution asreg(R/I) = max{si,v−i |0 ≤ i ≤ p, v ∈ Bi}.

1

In Section 2 we start by describing in Proposition 1 the first step of the Noether resolution of R/I. By Auslander-Buchsbaum formula, the depth ofR/I equalsd−p. Hence,R/I is Cohen- Macaulay if and only if p = 0 or, equivalently, if R/I is a free A-module. This observation together with Proposition 1, lead to Proposition 2 which is an effective criterion for determining whetherR/I is Cohen-Macaulay or not. This criterion generalizes [Bermejo & Gimenez (2001), Proposition 2.1]. IfR/I is Cohen-Macaulay, Proposition 1 provides the whole Noether resolution of R/I. When d = 1 and R/I is not Cohen-Macaulay, we describe the Noether resolution of R/I by means of Proposition 1 together with Proposition 3. Moreover, whend = 2andxn is a nonzero divisor ofR/I, we are able to provide in Theorem 1 a complete description of the Noether resolution ofR/I. All these results rely in the computation of a minimal Gröbner basis ofIwith respect to the weighted degree reverse lexicographic order. As a consequence of this, we provide in Corollary 1 a description of the weighted Hilbert series in terms of the same Gröbner basis.

WheneverI is a homogeneous ideal, as a consequence of Theorem 1, we obtain in Corollary 2 a formula for the Castelnuovo-Mumford regularity ofR/I which is equivalent to the one provided in [Bermejo & Gimenez (2000), Theorem 2.7].

In section 3 we study Noether resolutions whenR/I is a simplicial semigroup ring, i.e., when- everI is a toric ideal andA = K[xn−d+1, . . . , xn]is a Noether normalization ofR/I. We recall thatI is a toric ideal ifI =IA withA ={a1, . . . , an} ⊂N^dandai = (ai1, . . . , aid)∈N^d; where IA denotes the kernel of the homomorphism ofK-algebrasϕ : R → K[t1, . . . , td];xi 7→ t^ai = t^a₁ⁱ¹· · ·t^a_d^idfor alli∈ {1, . . . , n}. If we denote byS ⊂ N^dthe semigroup generated bya1, . . . , an, then the image of ϕ is K[S] := K[t^s|s ∈ S] ≃ R/IA. By [Sturmfels (1996), Corollary 4.3], IA is multigraded with respect to the grading induced byS which assignsdeg_S(xi) = ai for all i∈ {1, . . . , n}. Moreover, wheneverAis a Noether normalization ofK[S]we may assume with- out loss of generality that an−d+i = wn−d+iei for all i ∈ {1, . . . , d}, where ωn−d+i ∈ Z⁺ and {e1, . . . , ed}is the canonical basis ofN^d. In this setting we may consider amultigraded Noether resolutionofK[S], i.e., a minimal multigraded free resolution ofK[S]asA-module:

0−→ ⊕s∈SpA·s−→ · · ·^ψp −→ ⊕^ψ1 s∈S0A·s−→^ψ0 K[S]−→0,

whereSiare finite subsets ofS for alli∈ {0, . . . , p}andA·sdenotes the shifting ofAbys ∈ S. We observe that this multigrading is a refinement of the grading given byω = (ω1, . . . , ωn)with ωi := Pd

j=1aij ∈ Z⁺; thus, IA is weighted homogeneous with respect to ω. As a consequence, whenever we get the multigraded Noether resolution or the multigraded Hilbert series ofK[S], we also obtain its Noether resolution and its Hilbert series with respect to the weight vectorω.

A natural and interesting problem is to describe combinatorially the multigraded Noether res- olution of K[S] in terms of the semigroup S. This approach would lead us to results for sim- plicial semigroup ringsK[S] which do not depend on the characteristic of the field K. In gen- eral, for any toric ideal, it is well known that the minimal number of binomial generators of IA

does not depend on the characteristic of K (see, e.g., [Sturmfels (1996), Theorem 5.3]), but the Gorenstein, Cohen-Macaulay and Buchsbaum properties of K[S] depend on the characteristic ofK (see [Hoa (1991)], [Trung & Hoa (1986)] and [Hoa (1988)], respectively). However, in the context of simplicial semigroup rings, these properties do not depend on the characteristic ofK (see [Goto et al. (1976)], [Stanley (1978)] and [García-Sánchez & Rosales (2002)], respectively).

These facts give support to our aim of describing the whole multigraded Noether resolution of K[S]in terms of the underlying semigroupS for simplicial semigroup rings.

The results in section 3 are the following. In Proposition 5 we describe the first step of the multigraded Noether resolution of a simplicial semigroup ringK[S]. As a byproduct we recover in Proposition 6 a well-known criterion forK[S]to be Cohen-Macaulay in terms of the semigroup.

Whend= 2, i.e.,IAis the ideal of an affine toric surface, Theorem 2 describes the second step of the multigraded Noether resolution in terms of the semigroupS. Whend = 2, from Proposition 5 and Theorem 2, we derive the whole multigraded Noether resolution of K[S] by means of S and, as a byproduct, we also get in Corollary 3 its multigraded Hilbert series. WheneverIA a is

2

homogeneous ideal, we get a formula for the Castelnuovo-Mumford regularity ofK[S]in terms ofS, see Remark 1.

Given an algebraic variety, the set of points whereX is not Cohen-Macaulay is the non Cohen- Macaulay locus. Macaulayfication is an analogous operation to resolution of singularities and was considered in Kawasaki [Kawasaki (2000)], where he provides certain sufficient conditions forX to admit a Macaulayfication. For semigroup rings Goto et al. [Goto et al. (1976)] and Trung and Hoa [Trung & Hoa (1986)] proved the existence of a semigroupS^′ satisfyingS ⊂ S^′ ⊂S¯, where S¯denotes the saturation of S and thusK[ ¯S]is the normalization ofK[S], such that we have an exact sequence:

0−→K[S]−→K[S^′]−→K[S^′\ S]−→0

withdim(K[S^′\S])≤dim(K[S])−2. In this setting,K[S^′]satisfies the conditionS2of Serre, and is called theS2-fication ofK[S]. Moreover, whenS is a simplicial semigroup, [Morales (2007), Theorem 5] proves that this semigroup ringK[S^′]is exactly the Macaulayfication ofK[S]; indeed, he proved that K[S^′] is Cohen-Macaulay and the support of K[S^′ \ S] coincides with the non Cohen-Macaulay locus ofK[S]. In [Morales (2007)], the author provides an explicit description of the Macaulayfication ofK[S] in terms of the system of generators of IA provided K[S]is a codimension2simplicial semigroup ring. Section 4 is devoted to study the Macaulayfication of any simplicial semigroup ring. The main result of this section is Theorem 4, where we entirely describe the Macaulayfication of any simplicial semigroup ring K[S]in terms of the set S⁰, the subset ofSthat provides the first step of the multigraded Noether resolution ofK[S].

In sections 5 and 6 we apply the methods and results obtained in the previous ones to cer- tain dimension 2 semigroup rings. More precisely, a sequence m1 < · · · < mn determines the projective monomial curve C ⊂ Pⁿ_K parametrically defined by xi := s^mit^mn−mi for all i ∈ {1, . . . , n− 1}, xn = s^mn, xn+1 := t^mn. If we set A = {a1, . . . , an+1} ⊂ N² where ai := (mi, mn−mi), an := (mn,0)andan+1 := (0, mn), it turns out that the homogeneous coor- dinate ring ofC isK[C] :=K[x₁, . . . , x_n+1]/I_AandA=K[x_n, x_n+1]is a Noether normalization ofR/I_A.

The main result in Section 5 is Theorem 5, where we provide an upper bound on the Castelnuovo- Mumford regularity ofK[C], where C is a projective monomial curve. The proof of this bound is elementary and builds on the results of the previous sections together with some classical results on numerical semigroups. It is known that reg(K[C]) ≤ mn − n + 1 after the work [Gruson et al. (1983)]. In our case, [L’vovsky (1996)] obtained a better upper bound, indeed if we set m0 := 0 he proved thatreg(K[C]) ≤ max1≤i<j≤n{mi −mi−1 +mj −mj−1} −1. The proof provided by L’vovsky is quite involved and uses advanced cohomological tools, it would be interesting to know if our results could yield a combinatorial alternative proof of this result. Even if L’vovsky’s bound usually gives a better estimate than the bound we provide here, we easily construct families such that our bound outperforms the one by L’vovsky.

Also in the context of projective monomial curves, wheneverm₁ < · · ·< m_n is an arithmetic sequence of relatively prime integers, the simplicial semigroup ringR/I_A has been extensively studied (see, e.g., [Molinelli & Tamone (1995), Li et. al (2012), Bermejo et al. (2017)]) and the multigraded Noether resolution is easy to obtain. In Section 6, we study the coordinate ring of the canonical projections of projective monomial curves associated to arithmetic sequences, i.e., the curvesC^r whose homogeneous coordinate rings areK[S^r] =R/IAr, whereA^r := A \ {ar} andS^r ⊂ N² the semigroup generated byA^rfor all r ∈ {1, . . . , n−1}. In Corollary 5 we give a criterion for determining when the semigroup ring K[S^r] is Cohen-Macaulay; whenever it is not Cohen-Macaulay, we get its Macaulayfication in Corollary 6. Furthermore, in Theorem 7 we provide an explicit description of their multigraded Noether resolutions. Finally, in Theorem 8 we get a formula for their Castelnuovo-Mumford regularity.

2. NOETHER RESOLUTION. GENERAL CASE

Let R := K[x1, . . . , xn] be a polynomial ring over an infinite field K, and let I ⊂ R be a ω-homogeneous ideal, i.e., a weighted homogeneous ideal with respect to the vector ω = (ω1, . . . , ωn) ∈ (Z⁺)ⁿ. We assume that A := K[xn−d+1, . . . , xn] is a Noether normalization ofR/I, whered := dim(R/I). In this section we study the Noether resolution ofR/I, i.e., the minimal graded free resolution ofR/I asA-module:

(1) F : 0−→ ⊕v∈BpA(−sp,v)−→ · · ·^ψp −→ ⊕^ψ1 v∈B0A(−s0,v)−→^ψ0 R/I −→0,

where for alli∈ {0, . . . , p} Bⁱ is a finite set of monomials, andsi,v are nonnegative integers.

In order to obtain the first step of the resolution, we will deal with the initial ideal of I + (xn−d+1, . . . , xn)with respect to the weighted degree reverse lexicographic order>ω.

We recall that>ω is defined as follows:x^α >ω x^β if and only if

• deg_ω(x^α)>deg_ω(x^β),or

• deg_ω(x^α) = deg_ω(x^β)and the last nonzero entry ofα−β ∈Zⁿis negative.

For every polynomial f ∈ R we denote by in (f) the initial term of f with respect to >ω. Analogously, for every idealJ ⊂R,in (J)denotes its initial ideal with respect to>ω.

Proposition 1. LetB⁰ be the set of monomials that do not belong to in (I + (xn−d+1, . . . , xn)) Then,

{x^α+I |x^α ∈ B⁰}

is a minimal set of generators ofR/I asA-module and the shifts of the first step of the Noether resolution (1) are given bydeg_ω(x^α)withx^α ∈ B0.

Proof. Since A is a Noether normalization of R/I we have that B⁰ is a finite set. Let B⁰ = {x^α1, . . . , x^αk}. To prove that B := {x^α1 +I, . . . , x^αk +I} is a set of generators of R/I as A-module it suffices to show that for every monomialx^β := x^β₁¹· · ·x^β_n−d^n−d ∈/ in (I), one has that x^β+I ∈R/Ican be written as a linear combination of{x^α1+I, . . . , x^αk+I}. Since{x^α1+ (I+ (xn−d+1, . . . , xn)), . . . , x^αk+ (I+ (xn−d+1, . . . , xn))}is aK-basis ofR/(I+ (xn−d+1, . . . , xn)), we have thatg := x^β−Pk

i=1λix^αi ∈I + (xn−d+1, . . . , xn)for someλ1, . . . , λk ∈ K. Then we deduce thatin (g) ∈ in (I + (xn−d+1, . . . , xn))which is equal toin (I) + (xn−d+1, . . . , xn), and thusin (g) ∈ in (I). Sincex^β ∈/ in (I)andx^αi ∈/ in (I)for all i ∈ {1, . . . , k}, we conclude that g = 0andx^β +I = (Pk

i=1λix^αi) +I. The minimality ofBcan be easily proved.

WhenR/I is a free A-module or, equivalently, when the projective dimension of R/I as A- module is0and henceR/I is Cohen-Macaulay, Proposition 1 provides the whole Noether reso- lution ofR/I. In Proposition 2 we characterize the Cohen-Macaulay property forR/I in terms of the initial idealin (I)previously defined. This result generalizes [Bermejo & Gimenez (2001), Theorem 2.1], which applies forI a homogeneous ideal.

Proposition 2. Let A = K[xn−d+1, . . . , xn] be a Noether normalization ofR/I. Then, R/I is Cohen-Macaulay if and only ifxn−d+1, . . . , xndo not divide any minimal generator ofin (I).

Proof. We denote by{e_v|v in B0}the canonical basis of⊕v∈B0A(−deg_ω(v)). By Proposition 1 we know thatψ0 :⊕^v∈B0A(−deg_ω(v))−→R/I is the morphism induced byev 7→v+I ∈R/I. By Auslander-Buchsbaum formula,R/I is Cohen-Macaulay if and only ifψ0is injective.

(⇒) By contradiction, we assume that there existsα = (α1, . . . , αn) ∈ Nⁿ such that x^α = x^α₁¹· · ·x^α_nⁿ is a minimal generator of in (I) and thatαi > 0 for somei ∈ {n−d + 1, . . . , n}. Setu :=x^α₁¹· · ·x^α_n−d^n−d, sincein(I + (xn−d+1, . . . , xn)) = in(I) + (xn−d+1, . . . , xn), we have that u ∈ B0. We also setx^α′ := x^α_n−d+1^n−d+1· · ·x^α_nⁿ ∈ A andf the remainder ofx^α modulo the reduced Gröbner basis of I with respect to >ω. Then x^α −f ∈ I and every monomial in f does not belong toin (I). As a consequence,f =Pt

i=1cix^βi, whereci ∈ K andx^βi =vix^β′i withvi ∈ B0

2

and x^β′i ∈ A for all i ∈ {1, . . . , t}. Hence, x^α′eu −Pt

i=1cix^β′ievi ∈ Ker(ψ0) and R/I is not Cohen-Macaulay.

(⇐)Assume that there exists a nonzero g ∈ Ker(ψ₀), namely, g = P

v∈B0g_ve_v ∈ Ker(ψ₀) withgv ∈ Afor allv ∈ B⁰. Then,P

v∈B0gvv ∈I. We writein (g) = cx^αuwithc∈ K, x^α ∈ A andu ∈ B⁰. Sincexn−d+1, . . . , xn do not divide any minimal generator of in (I), we have that

u∈in (I), a contradiction.

WhenR/I has dimension1, its depth can be either0or1. Whendepth(R/I) = 1, thenR/I is Cohen-Macaulay and the whole Noether resolution is given by Proposition 1. WhenR/I is not Cohen-Macaulay, then its depth is0and its projective dimension asA-module is1. In this setting, to describe the whole Noether resolution it remains to determineB1,ψ1 and the shiftss1,v ∈Nfor allv ∈ B1. In Proposition 3 we explain how to obtainB1 andψ1 by means of a Gröbner basis of Iwith respect to>ω.

Considerχ1 :R −→ Rthe evaluation morphism induced by xi 7→ xi fori ∈ {1, . . . , n−1}, x_n7→1.

Proposition 3. LetR/Ibe1-dimensional ring of depth0. LetLbe the idealχ₁(in(I))·R. Then, B1 =B0∩L

in the Noether resolution (1) ofR/I and the shifts of the second step of this resolution are given bydeg_ω(ux^δ_n^u),whereu∈ B¹andδu := min{δ|ux^δ_n∈in(I)}.

Proof. For everyu=x^α₁¹· · ·x^α_n−1ⁿ⁻¹ ∈ B⁰∩L, there existsδ∈ Nsuch thatux^δ_n∈ in(I); letδu be the minimum of all suchδ. Considerpu ∈R the remainder ofux^δ_n^u modulo the reduced Gröbner basis ofI with respect to>ω. Thus ux^δ_n^u −pu ∈ I is ω-homogeneous and every monomial x^β appearing inpu does not belong toin(I), then by Proposition 1 it can be expressed asx^β =vx^β_nⁿ, whereβn ≥0andv ∈ B0. Moreover, sinceux^δ_n^u >ω x^β, thenβn ≥δu andu >ω v. Thus, we can write

pu = X

v∈B0 u>ωv

x^δ_n^umuvv,

withmuv =cx^αuv ∈A=K[xn]a monomial (possibly0) for allv ∈ B0,u >ω v.

Now we denote by{ev|v in B0} the canonical basis of⊕v∈B0A(−deg_ω(v))and consider the graded morphismψ0 :⊕v∈B0A(−deg_ω(v))−→ R/I induced byev 7→ v+I ∈ R/I. The above construction yields that

hu :=x^δ_n^u(eu− X

v∈B0 u>ωv

muvev)∈Ker(ψ0)

for allu∈ B⁰∩L. We will prove thatKer(ψ0)is a freeA-module with basis C :={h_u|u∈ B0 ∩L}.

Firstly, we observe that theA-module generated by the elements ofCis free due to the triangular form of the matrix formed by the elements of C. Let us now take g = P

v∈B0 gvev ∈ Ker(ψ0) withg_v ∈ A, we assume thatg ∈ ⊕v∈B0A(−deg_ω(v))isω-homogeneous and, thus,g_v is either 0 or a monomial of the formcx^β_n^v with c ∈ K and β_v ∈ N for all v ∈ B0. We consider ψ¯₀ :

⊕^v∈B0A(−deg_ω(v))−→Rthe monomorphism ofA-modules induced byev 7→v. Sinceψ0(g) = 0, then the polynomialg^′ := ¯ψ0(g) = P

u∈B0guu ∈ I and in(g^′) = cx^γ_nw for some w ∈ B⁰, γ ∈ N and c ∈ K. Since in(g^′) ∈ in(I), we get that w ∈ B⁰ ∩L and γ ≥ δw. Hence, g1 := g−cx^γ−δ_n−1^whw ∈ Ker(ψ0). Ifg1 is identically zero, theng ∈ ({hu|u∈ B0 ∩L}). Ifg1 is not zero, we have that0 6= in( ¯ψ0(g1)) < in( ¯ψ0(g))and we iterate this process withg1 to derive

that{hu|u∈ B0∩L}generatesKer(ψ0).

The rest of this section concernsIa saturated ideal such thatR/I is2-dimensional and it is not Cohen-Macaulay (and, in particular, depth(R/I) = 1). We assume that A = K[x_n−1, x_n]is a Noether normalization ofR/I and we aim at describing the whole Noether resolution ofR/I. To achieve this it only remains to describeB¹,ψ1and the shiftss1,v ∈Nfor allv ∈ B¹. In Proposition 4 we explain how to obtainB¹ andψ1 by means of a Gröbner basis ofI with respect to>ω. Since K is an infinite field,I is a saturated ideal andAis a Noether normalization ofR/I, one has that xn+τ xn−1is a nonzero divisor onR/I for allτ ∈K but a finite set. Thus, by performing a mild change of coordinates if necessary, we may assume thatxnis a nonzero divisor onR/I.

Now considerχ:R −→Rthe evaluation morphism induced byxi 7→xifori∈ {1, . . . , n−2}, xi 7→1fori∈ {n−1, n}.

Proposition 4. LetR/I be 2-dimensional, non Cohen-Macaulay ring such thatxn is a nonzero divisor. LetJ be the idealχ(in(I))·R. Then,

B¹ =B⁰∩J

in the Noether resolution (1) ofR/I and the shifts of the second step of this resolution are given bydeg_ω(ux^δ_n−1^u ),whereu∈ B¹andδu := min{δ|ux^δ_n−1 ∈in(I)}.

Proof. Sincexn is a nonzero divisor of R/I andI is a ω-homogeneous ideal, then xn does not divide any minimal generator ofin (I). As a consequence, for everyu=x^α₁¹· · ·x^α_n−2ⁿ⁻² ∈ B0∩J, there exists δ ∈ N such that ux^δ_n−1 ∈ in(I); by definition, δu is the minimum of all such δ.

Considerpu ∈ R the remainder ofux^δ_n−1^u modulo the reduced Gröbner basis ofI with respect to

>ω. Thenux^δ_n−1^u −pu ∈ I is ω-homogeneous and every monomial x^β appearing inpu does not belong toin(I), then by Proposition 1 it can be expressed asx^β =vx^β_n−1ⁿ⁻¹x^β_nⁿ, whereβ_n−1, β_n≥0 andv ∈ B0. Moreover, we have thatux^δ_n−1^u >_ω x^β which implies that eitherβ_n ≥1, orβ_n = 0, βn−1 ≥δu andu >ω v. Thus, we can write

pu = X

v∈B0 u>ωv

x^δ_n−1^u fuvv+ X

v∈B0

xnguvv, withfuv ∈K[xn−1]for allv ∈ B0,u >ω v andguv ∈Afor allv ∈ B0.

Now we denote by{ev|v in B0}the canonical basis of⊕v∈B0A(−deg_ω(v))and consider the graded morphismψ0 :⊕v∈B0A(−deg_ω(v))−→ R/I induced byev 7→ v+I ∈ R/I. The above construction yields that

hu :=x^δ_n−1^u eu− X

v∈B0 u>ωv

x^δ_n−1^u fuvev − X

v∈B0

xnguvev ∈Ker(ψ0)

for allu∈ B0∩J. We will prove thatKer(ψ0)is a freeA-module with basis C :={hu|u∈ B⁰∩J}.

Firstly, we prove that the A-module generated by the elements of C is free. Assume that P

u∈B0∩Jquhu = 0 where qu ∈ A for all u ∈ B⁰ ∩ J and we may also assume that xn does not divideqv for somev ∈ B⁰ ∩J. We consider the evaluation morphismτ induced byxn 7→ 0 and we get thatP

u∈B0∩Jτ(q_u)τ(h_u) =P

u∈B0∩Jτ(q_u) (x^δ_n−1^u e_u+P

v∈B0

u>ω v x^δ_n−1^u f_uve_v) = 0, which implies thatτ(qu) = 0for allu∈ B⁰∩J and, hence,xn |qufor allu∈ B⁰∩J, a contradiction.

Let us takeg =P

v∈B0gvev ∈Ker(ψ0)withgv ∈ A, we assume thatg ∈ ⊕^v∈B0A(−deg_ω(v)) is ω-homogeneous and, thus, gv is either 0 or a ω-homogeneous polynomial for all v ∈ B0. We may also suppose that there exists v ∈ B0 such that xn does not divide gv. We consider ψ¯0 : ⊕v∈B0A(−deg_ω(v)) −→ R the monomorphism of A-modules induced byev 7→ v. Since ψ0(g) = 0, then the polynomialg^′ := ¯ψ0(g) = P

u∈B0guu ∈ I andin(g^′) = cx^γ_n−1w for some w∈ B0 and somec∈K, which implies thatw∈ B0∩J. By definition ofδw we get thatγ ≥δw, henceg1 := g−cx^γ−δ_n−1^whw ∈ Ker(ψ0). Ifg1 is identically zero, theng ∈ ({hu|u ∈ B0 ∩J}).

2

Ifg₁ is not zero, we have that0 6= in( ¯ψ₀(g₁)) < in( ¯ψ₀(g))and we iterate this process with g₁ to derive that{h_u|u∈ B0∩J}generatesKer(ψ₀).

From Propositions 1 and 4 and their proofs, we can obtain the Noether resolutionF ofR/I by means of a Gröbner basis of I with respect to >_ω. We also observe that for obtaining the shifts of the resolution it suffices to know a set of generators ofin (I). The following theorem gives the resolution.

Theorem 1. LetR/I be a2-dimensional ring such thatxn is a nonzero divisor. We denote byG be a Gröbner basis ofIwith respect to>ω. Ifδu := min{δ|ux^δ_n−1 ∈in (I)}for allu∈ B¹, then

F : 0−→ ⊕u∈B1A(−deg_ω(u)−δ_uω_n−1)−→ ⊕^ψ1 v∈B0A(−deg_ω(v))−→^ψ0 R/I −→0, is the Noether resolution ofR/I, where

ψ0 : ⊕^v∈B0A(−deg_ω(v)) → R/I, ev 7→ v+I

and ψ₁ : ⊕u∈B1A(−deg_ω(u)−δ_uω_n−1) −→ ⊕v∈B0A(−deg_ω(v)), eu 7→ x^δ_n−1^u eu−P

v∈B0fuvev

wheneverP

v∈B0fuvv withfuv ∈Ais the remainder of the division ofux^δ_n−1^u byG. From this resolution, we can easily describe the weighted Hilbert series ofR/I.

Corollary 1. LetR/I be a2-dimensional ring such thatxn is a nonzero divisor, then its Hilbert series is given by:

HSR/I(t) = P

v∈B0t^degω(v)−P

u∈B1t^{degω(u)+δuwn−1} (1−t^ωn−1)(1−t^ωn)

In the following example we show how to compute the Noether resolution and the weighted Hilbert series of the graded coordinate ring of a surface inA⁴_K.

Example 1. LetIbe the defining ideal of the surface ofA⁴_Kparametrically defined byf1 :=s³+ s²t, f2 :=t⁴+st³, f3 :=s², f4 :=t² ∈K[s, t]. UsingSINGULAR [Decker et al. (2015)],COCOA [Abbott et al. (2015)]orMACAULAY 2 [Grayson & Stillman (2015)]we obtain that wheneverK is a characteristic0field, the polynomials{g1, g2, g3, g4}constitute a minimal Gröbner basis of its defining ideal with respect to>ω withω = (3,4,2,2), whereg1 := 2x2x²₃ −x²₁x4 +x³₃x4 − x²₃x²₄, g2 := x⁴₁ −2x²₁x³₃ +x⁶₃ −2x²₁x²₃x4 −2x⁵₃x4+x⁴₃x²₄, g3 := x²₂ −2x2x²₄ −x3x³₄ +x⁴₄ and g4 := 2x²₁x2 −x²₁x3x4+x⁴₃x4−3x²₁x²₄−2x³₃x²₄+x²₃x³₄.In particular,

in (I) = (x2x²₃, x⁴₁, x²₂, x²₁x2).

Then, we obtain that

• B⁰ ={u1, . . . , u6}withu1 := 1, u2 :=x1, u3 :=x2, u4 :=x²₁, u5 :=x1x2, u6 :=x³₁,

• J = (x2, x⁴₁)⊂K[x1, x2, x3, x4], and

• B1 ={u3}.

Sincex3divides a minimal generator ofin (I), by Proposition 2 we deduce thatR/I is not Cohen- Macaulay. We computeδ3 = min{δ|u3x^δ₃ ∈ in (I)}and get thatδ3 = 2and thatr3 =−x4u4+ (x³₃x4−x²₃x²₄)u1 is the remainder of the division ofu3x²₃ byG. Hence, following Theorem 1, we obtain the Noether resolution orR/I:

F : 0−→A(−8)−→^ψ A⊕A(−3)⊕A(−4)⊕

⊕A(−6)⊕A(−7)⊕A(−9) −→R/I −→0,

whereψ is given by the matrix









−x³₃x4 +x²₃x²₄ 0 x²₃ x4

0 0









Moreover, by Corollary 1, we obtain that the weighted Hilbert series ofR/I is HS_R/I(t) = 1 +t³+t⁴+t⁶+t⁷−t⁸ +t⁹

(1−t²)² .

If we consider the same parametric surface over an infinite field of characteristic2, we obtain that{x²₁ +x³₃ +x²₃x4, x²₂+x3x³₄ +x⁴₄}is a minimal Gröbner basis ofI with respect to>ω, the weighted degree reverse lexicographic order withω = (3,4,2,2). Then we have that

B⁰ ={v1 := 1, v2 :=x1, v3 :=x2, v4 :=x1x2},

andB1 =∅, soR/Iis Cohen-Macaulay. Moreover, we also obtain the Noether resolution ofR/I F^′ : 0−→A⊕A(−3)⊕A(−4)⊕A(−7)−→R/I −→0

and the weighted Hilbert series ofR/I is

HS_R/I(t) = 1 +t³+t⁴+t⁷ (1−t²)² .

To end this section, we consider the particular case whereI is standard graded homogeneous, i.e.,ω = (1, . . . ,1). In this setting, we obtain a formula for the Castelnuovo-Mumford regularity ofR/I in terms ofin (I)or, more precisely, in terms ofB0 andB1. This formula is equivalent to that of [Bermejo & Gimenez (2000), Theorem 2.7] providedxnis a nonzero divisor ofR/I.

Corollary 2. LetR/I be a2-dimensional standard graded ring such thatxnis a nonzero divisor.

Then,

reg (R/I) = max{deg(v),deg(u) +δu −1|v ∈ B0, u∈ B1} In the following example we apply all the results of this section.

Example 2. LetK be a characteristic zero field and let us consider the projective curveC ofP⁴_K parametrically defined by:

x₁ =s³t⁵−st⁷, x₂ =s⁷t, x₃ =s⁴t⁴, x₄ =s⁸, x₅ =t⁸.

A direct computation withSINGULAR, COCOA orMACAULAY 2yields that a minimal Gröbner basis G of the defining ideal I ⊂ R = K[x1, . . . , x5] of C with respect to the degree reverse lexicographic order consists of10elements and that

in (I) = (x⁴₁, x⁴₂, x³₁x3, x1x3x²₄, x²₁x2, x1x²₂, x1x2x3, x²₂x3, x²₁x4, x²₃).

Then, we obtain that the setB⁰ is the following

B0 ={u1:= 1, u2 :=x1, u3 :=x2, u4 :=x3, u5 :=x²₁, u6 :=x1x2, u7:=x²₂, u8 :=x1x3, u9 :=x2x3, u10:=x³₁, u11:=x³₂, u12 :=x²₁x3}

and the idealJ is

J = (x²₁, x1x3, x²₃, x²₂x3, x⁴₂)⊂R.

Thus,B1 ={u5, u8, u10, u12}. Fori ∈ {5,8,10,12}we computeδi, the minimum integer such thatuix^δ₄ⁱ ∈ in (I)and get thatδ4 = δ10 =δ12 = 1andδ8 = 2. If we setri the remainder of the division ofuix^δ₄ⁱ for alli∈ {4,8,10,12}, we get that

• r4 =−x4x²₅b1+ 2x4x5b4+x5b6+x5b7,

2

• r₈ =x²₄x₅b₃+x₅b₁₁,

• r₁₀=x²₄x₅b₂+ 3x₄x₅b₈+ (x²₅−x₄x₅)b₉, and

• r₁₂=x²₄x²₅b₁+x₄x₅b₆+x²₅b₇.

Hence, we obtain the following minimal graded free resolution ofR/I

F : 0−→A(−3)⊕A³(−4)−→^ψ A⊕A³(−1)A⁵(−2)⊕A³(−3)−→R/I −→0, whereψ is given by the matrix





















x4x²₅ 0 0 −x²₄x²₅ 0 0 −x²₄x5 0

0 0 0 0

−2x4x5 −x²₄x5 0 0

x4 0 0 0

−x₅ 0 0 −x₄x₅

−x₅ 0 0 −x²₅ 0 x²₄ −3x₄x₅ 0 0 0 x4x5−x²₅ 0

0 0 x4 0

0 −x5 0 0

0 0 0 x4





















Moreover, the Hilbert series ofR/I is

HS_R/I(t) = 1 + 3t+ 5t² + 2t³−3t⁴ (1−t)² . andreg(R/I) = max{3,4−1}= 3.

3. NOETHER RESOLUTION. SIMPLICIAL SEMIGROUP RINGS

This section concerns the study of Noether resolutions in simplicial semigroup ringsR/I, i.e., wheneverI = IA withA = {a1, . . . , an} ⊂ N^d andan−d+i = wn−d+iei for all i ∈ {1, . . . , d}, where{e1, . . . , ed}is the canonical basis ofN^d. In this setting,R/IA is isomorphic to the semi- group ringK[S], whereS is the simplicial semigroup generated by A. WhenK is infinite, IA

is the vanishing ideal of the variety given parametrically byxi := t^ai for alli∈ {1, . . . , n}(see, e.g., [Villarreal (2015)]) and, hence, K[S] is the coordinate ring of a parametric variety. In this section we study the multigraded Noether resolution of K[S] with respect to the multigrading deg_S(xi) =ai ∈ S; namely,

F : 0−→ ⊕s∈SpA·s−→ · · ·^ψp −→ ⊕^ψ1 s∈S0A·s−→^ψ0 K[S]−→0.

where Sⁱ ⊂ S for all i ∈ {0, . . . , p}. We observe that this multigrading is a refinement of the grading given byω = (ω1, . . . , ωn)withωi := Pd

j=1aij ∈ Z⁺; thus, IA isω-homogeneous and the results of the previous section also apply here.

Our objective is to provide a description of this resolution in terms of the semigroup S. We completely achieve this goal whenK[S]is Cohen-Macaulay (which includes the cased= 1) and also whend = 2.

For any value ofd ≥1, the first step of the resolution corresponds to a minimal set of generators ofK[S]asA-module and is given by the following well known result.

Proposition 5. LetK[S]be a simplicial semigroup ring. Then,

S0 ={s∈ S |s−ai ∈ S/ for alli∈ {n−d+ 1, . . . , n}}.

Moreover, ψ0 : ⊕s∈S0A·s −→ K[S]is the homomorphism of A-modules induced byes 7→ t^s, where{es|s∈ S0}is the canonical basis of⊕s∈S0A·s.

Proposition 5 gives us the whole multigraded Noether resolution ofK[S]whenK[S]is Cohen- Macaulay.

In [Goto et al. (1976), Theorem 1] (see also [Stanley (1978), Theorem 6.4]), the authors provide a characterization of the Cohen-Macaulay property ofK[S]. In the following result we are proving an equivalent result that characterizes this property in terms of the size of S0. The proof shows how to obtain certain elements of Ker(ψ₀) and this idea will be later exploited to describe the whole resolution whend= 2andK[S]is not Cohen-Macaulay.

Proposition 6. LetS be a simplicial semigroup as above. Set D :=Qd

i=1ωn−d+i

/[Z^d : ZS], where [Z^d : ZS] denotes the index of the group generated by S in Z^d. Then, K[S] is Cohen- Macaulay⇐⇒ |S0|=D.

Proof. By Auslander-Buchsbaum formula we deduce thatK[S] is Cohen-Macaulay if and only if ψ0 is injective, whereψ0 is the morphism given in Proposition 5. We are proving that ψ0 is injective if and only if|S0| =D. We define an equivalence relation onZ^d,u∼ v ⇐⇒ u−v ∈ Z{ωn−d+1e1, . . . , ωned}. This relation partitions ZS into D = [ZS : Z{ωn−d+1e1, . . . , ωned}] equivalence classes. Since

Z^d/ZS ≃ Z^d/Z{ωn−d+1e1, . . . , ωned}

/(ZS/Z{ωn−d+1e1, . . . , ωned}), we get thatD=Qd

i=1ωn−d+i

/[Z^d:ZS]. Moreover, the following two facts are easy to check:

for every equivalence class there exists an elementb∈ S0, andS =S0+N{ω_n−d+1e₁, . . . , ω_ne_d}. This proves that|S⁰| ≥D.

Assume that |S⁰| > D, then there exist u, v ∈ S⁰ such that u ∼ v or, equivalently, u+ Pd

i=1λiωn−d+iei = v +Pd

i=1µiωn−d+iei for some λi, µi ∈ N for all i ∈ {1, . . . , d}. Thus x^λ_n−d+1¹ · · ·x^λ_n^deu −x^µ_n−d+1¹ · · ·x^µ_n^dev ∈Ker(ψ0)andψ0 is not injective.

Assume now that |S⁰| = D, then for everys1, s2 ∈ S⁰, s1 6= s2, we have that s1 6∼ s2. As a consequence, an element ρ ∈ ⊕^s∈S0A· s is homogeneous if and only if it is a monomial, i.e., ρ = cx^αes for some c ∈ K, x^α ∈ A and s ∈ S⁰. Since the image by ψ0 of a monomial is another monomial, then there are no homogeneous elements inKer(ψ0)different from0, soψ0is

injective.

From now on suppose thatK[S]is a 2-dimensional non Cohen-Macaulay semigroup ring. In this setting, we consider the set

∆ :={s ∈ S |s−an−1, s−an ∈ Sands−an−an−1 ∈ S}/ .

The set∆or slight variants of it has been considered by other authors (see, e.g., [Goto et al. (1976), Stanley (1978), Trung & Hoa (1986)]). We claim that∆has exactly|S0| −Delements. Indeed, if we consider the equivalence relation ∼ of Proposition 6, then ∼ partitions ZS in D classes C1, . . . , CD and it is straightforward to check that|∆∩Ci|=|S0∩Ci| −1for alli∈ {1, . . . , D}. From here, we easily deduce that|∆| = |S0| −D. Hence, a direct consequence of Proposition 6 is that∆is nonempty becauseK[S]is not Cohen-Macaulay. Furthermore, as Theorem 2 shows, the set∆is not only useful to characterize the Cohen-Macaulay property but also provides the set of shifts in the second step of the multigraded Noether resolution ofK[S].

Theorem 2. LetK[S]be a2-dimensional semigroup ring and let

∆ ={s∈ S |s−an−1, s−an∈ S ands−an−an−1 ∈ S}/ , as above. Then,S¹ = ∆.

Proof. SetB⁰ the monomial basis ofR/(in(IA), xn−1, xn), wherein (IA)is the initial ideal ofIA

with respect to>ω. For everyu=x^α₁¹· · ·x^α_nⁿ ∈ B1we setδu ≥1the minimum integer such that ux^δ_n−1^u ∈ in(IA). Considerpu ∈R the remainder ofux^δ_n−1^u modulo the reduced Gröbner basis of

2

I_A with respect to>_ω, thenux^δ_n−1^u −p_u ∈ I_A. Since I_A is a binomial ideal, we get thatp_u =x^γ for some(γ₁, . . . , γ_n)∈Nⁿ. Moreover, the conditionx^α > x^γand the minimality ofδ_uimply that γ_n >0andγ_n−1 = 0, sox^γ =v_ux^γn^vu withv_u ∈ B0. As we proved in Proposition 4, if we denote by{ev|v ∈ B0}the canonical basis of⊕v∈B0A(−deg_S(v))and hu := x^δ_n−1^u eu−x^γn^vuevu for all u∈ B1, thenKer(ψ0)is theA-module minimally generated byC :={hu|u ∈ B1}.Let us prove that

{deg_S(hu)|u∈ B¹}={s ∈ S |s−an−1, s−an ∈ Sands−an−1−an ∈ S}/ .

Takes= deg_S(hu)for someu∈ B¹, thens= deg_S(hu) = deg_S(u)+δuan−1 = deg_S(vu)+γvuan. Sinceδu, γvu≥1, we get that boths−an−1, s−an∈ S. Moreover, ifs−an−1−an=Pn

i=1δiai ∈ S, thenx^δ_n−1^u−1u−x^λxn+1 ∈IA, which contradicts the minimality ofδu.

Take nows∈ Ssuch thats−a_n−1, s−a_n∈ Sands−a_n−1−a_n∈ S/ . Sinces−a_n−1, s−a_n ∈ S, there existss^′, s^′′ ∈ S0andγ1, γ2, λ1, λ2 ∈Nsuch thats−an =s^′+γ1an−1+γ2anands−an+1 = s^′′+λ₁a_n−1+λ₂a_n. Observe thatγ₂ = 0, otherwises−a_n−1−a_n =s^′+γ₁a_n−1+(γ₂−1)a_n ∈ S, a contradiction. Analogouslyλ1 = 0. Take u, v ∈ B⁰ such thatdeg_S(u) = s^′ anddeg_S(v) = s^′′. We claim thatu ∈J and thatδu =γ1. Indeed,f := ux^γ_n−1¹ −vx^λ_n² ∈ IA andin(f) = ux^γ_n−1¹ , so u∈ B¹. Moreover, if there existsγ^′ < δu, thens−an−1−an ∈ S, a contradiction.

One of the interests of Proposition 6 and Theorem 2 is that they describe multigraded Noether resolutions of dimension2semigroup rings in terms of the semigroupS and, in particular, they do not depend on the characteristic of the fieldK.

Now we consider the multigraded Hilbert Series ofK[S], which is defined by HSK[S](t) = X

s∈S

t^s = X

s=(s1,...,sd)∈S

t^s₁¹· · ·t^s_d^d,

Whend = 2, from the description of the multigraded Noether resolution of K[S] we derive an expression of its multigraded Hilbert series in terms ofS⁰ andS¹.

Corollary 3. LetK[S]be a dimension2semigroup ring. The multigraded Hilbert series ofK[S] is:

HSK[S](t) = P

s∈S0t^s−P

s∈S1t^s (1−t^ω₁ⁿ⁻¹)(1−t^ω₂ⁿ).

Remark 1. WhenK[S]is a two dimensional semigroup ring andS is generated by the setA = {a1, . . . , an} ⊂N², if we setω = (ω1, . . . , ωn)∈Nⁿwithωi :=ai,1+ai,2for alli∈ {1, . . . , n}, thenIAisω-homogeneous, as observed at the beginning of this section. The Noether resolution of K[S]with respect to this grading is easily obtained from the multigraded one. Indeed, it is given by the following expression:

F : 0−→ ⊕(b1,b2)∈S1A(−(b1+b2))−→ ⊕^ψ1 (b1,b2)∈S0A(−(b1+b2))−→^ψ0 K[S]−→0.

In addition, the weighted Hilbert series of K[S] is obtained from the multigraded one by just considering the transformationt^α₁¹t^α₂² 7→t^α1+α2.

Whenω1 = · · · = ωn, thenIA is a homogeneous ideal. In this setting, the Noether resolution with respect to the standard grading is

F : 0−→ ⊕^(b1,b2)∈S1A(−(b1+b2)/ω1)−→ ⊕^{ψ1 (b1,b2)∈S0}A(−(b1+b2)/ω1)−→^ψ0 K[S] −→0.

Thus, the Castelnuovo-Mumford regularity ofK[S]is (2) reg (K[S]) = max

b₁+b₂

ω1 |(b₁, b₂)∈ S0

∪

b₁+b₂

ω1 −1|(b₁, b₂)∈ S1

. Moreover, the Hilbert series ofK[S]is obtained from the multigraded Hilbert series by just con- sidering the transformationt^α₁¹t^α₂² 7→t^{(α1+α2)/ω1}.