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Let (R, $$\mathfrak{m}$$ ) be a standard graded K-algebra over a field K. Then R can be written as S/I, where I ⊆ (x1,…, xn)2 is a graded ideal of a polynomial ring S = K[x1,…, xn]. Assume that n ⩽ 3 and I is a strongly stable monomial ideal. We study the symmetric algebra SymR(Syz1( $$\mathfrak{m}$$ )) of the first syzygy module Syz1( $$\mathfrak{m}$$ ) of $$\mathfrak{m}$$ . When the minimal generators of I are all of degree 2, the dimension of SymR (Syz1( $$\mathfrak{m}$$ )) is calculated and a lower bound for its depth is obtained. Under suitable conditions, this lower bound is reached.

We extend the concept of Grobner bases to relative Grobner bases for ideals in and modules over quotient rings of a polynomial ring over a field. We develop a “relative” variant of both Buchberger’s criteria for avoiding reductions to zero and Schreyer’s theorem for a Grobner basis of the syzygy module. As main contribution, we then introduce the novel notion of relative involutive bases and present an algorithm for their explicit construction. Finally, we define the new notion of relatively quasi-stable ideals and exploit it for the algorithmic determination of coordinates in which finite relative Pommaret bases exist.

Let $${\mathrm{Syz}}_1(\mathfrak {m})$$ be the first syzygy of the graded maximal ideal $$\mathfrak {m}$$ of a polynomial ring $$K[x_1,\ldots ,x_n]$$ over a field K. The multiplicity and (Castelnuovo–Mumford) regularity of the symmetric algebra $${\mathrm{Sym}}({\mathrm{Syz}}_1(\mathfrak {m}))$$ are estimated by using the theory of s-sequences. It is proved that the multiplicity of $${\mathrm{Sym}}({\mathrm{Syz}}_1(\mathfrak {m}))$$ is 1 when $$n\ge 5$$ , and $$n-2$$ is an upper bound for its regularity. In virtue of Grobner bases, this bound is shown to be reached provided $$n\le 5$$ .

Let I be a monomial ideal in the polynomial ring S generated by elements of degree at most d. In this paper, it is shown that, if the i-th syzygy of I has no elements of degrees j,…,j+(d−1) (where j≥i+d), then (i+1)-th syzygy of I does not have any element of degree j+d. Then we give several applications of this result, including an alternative proof for Green–Lazarsfeld index of the edge ideals of graphs as well as an alternative proof for Froberg’s theorem on classification of square-free monomial ideals generated in degree 2 with linear resolution. Among all, we deduce a partial result on subadditivity of the syzygies for monomial ideals.

Let R0 be a Noetherian local ring and R a standard graded R0-algebra with maximal ideal 𝔪 and residue class field 𝕂 = R/𝔪. For a graded ideal I in R we show that for k ≫ 0: (1) the Artin-Rees number of the syzygy modules of Ik as submodules of the free modules from a free resolution is constant, and thereby the Artin-Rees number can be presented as a proper replacement of regularity in the local situation; and (2) R is a polynomial ring over the regular R0, the ring R/Ik is Golod, its Poincaré-Betti series is rational and the Betti numbers of the free resolution of 𝕂 over R/Ik are polynomials in k of a specific degree. The first result is an extension of the work by Swanson on the regularity of Ik for k ≫ 0 from the graded situation to the local situation. The polynomiality consequence of the second result is an analog of the work by Kodiyalam on the behaviour of Betti numbers of the minimal free resolution of R/Ik over R.

Let $R = K[X_1, ..., X_n]$ be a polynomial ring over some field $K$. In this paper, we prove that the $k$-th syzygy module of the residue class field $K$ of $R$ has Stanley depth $n-1$ for $\lfloor n/2 \rfloor \leq k < n$, as it had been conjectured by Bruns et. al. in 2010. In particular, this gives the Stanley depth for a whole family of modules whose graded components have dimension greater than $1$. So far, the Stanley depth is known only for a few examples of this type. Our proof consists in a close analysis of a matching in the Boolean algebra., minor corrections, 11 pages, 4 figures

In this paper we introduce an algorithm for computing the Stanley depth of a finitely generated multigraded module $M$ over the polynomial ring $\mathbb{K}[X_1, \ldots, X_n]$. As an application, we give an example of a module whose Stanley depth is strictly greater than the depth of its syzygy module. In particular, we obtain complete answers for two open questions raised by Herzog. Moreover, we show that the question whether $M$ has Stanley depth at least $r$ can be reduced to the question whether a certain combinatorially defined polytope $\mathscr{P}$ contains a $\mathbb{Z}^n$-lattice point., 19 pages. Section 5.3 of the first version has been withdrawn. Several typos have been also corrected in this second version. Accepted for publication in Mathematics of Computation

Suppose that $M$ is a finitely-generated graded module of codimension $c\geq 3$ over a polynomial ring and that the regularity of $M$ is at most $2a-2$ where $a\geq 2$ is the minimal degree of a first syzygy of $M$. Then we show that the sum of the betti numbers of $M$ is at least $\beta_0(M)(2^c + 2^{c-1})$. In addition, if $c \geq 9$ then for each $1\leq i\leq \lceil c/2\rceil$, we show $\beta_i(M)\geq 2\beta_0(M){c \choose i}$., Comment: 14 pages, 2 figures

A new algorithm, which combines the GVW algorithm with the Mora normal form algorithm, is presented to compute the standard bases of ideals in a local ring. Since term orders in local ring are not well-orderings, there may not be a minimal signature in an infinite set, and we can not extend the GVW algorithm from a polynomial ring to a local ring directly. Nevertheless, when given an anti-graded order in R and a term-over-position order in Rm that are compatible, we can construct a special set such that it has a minimal signature, where R , Rm are a local ring and a R -module, respectively. That is, for any given polynomial v0 in R, the set consisting of signatures of pairs (u,v)in Rm x R has a minimal element, where the leading power products of v and v0 are equal. In this case, we prove a cover theorem in R , and use three criteria (syzygy criterion, signature criterion and rewrite criterion) to discard useless J-pairs without any reductions. Mora normal form algorithm is also extended to do regular top-reductions in Rm x R, and the correctness and termination of the algorithm are proved. The proposed algorithm has been implemented in the computer algebra system Maple, and experiment results show that most of J-pairs can be discarded by three criteria in the examples.

Let Syz1(𝔪) be the first syzygy of the graded maximal ideal 𝔪 of a polynomial ring K[x1,…, xn] over a field K. Using the theory of s-sequences, the dimension and depth of the symmetric algebra Sym(Syz1(𝔪)) are calculated. As a conclusion, Sym(Syz1(𝔪)) is not Cohen–Macaulay for any n ≥ 4.

Estimates are obtained for the degrees of minimal syzygies of quotient algebras of polynomial rings. For a class that includes Koszul algebra in almost all characteristics, these degrees are shown to increase by at most 2 from one syzygy module to the next one. Even slower growth is proved if, in addition, the algebra satisfies Green and Lazarsfeld's condition N_q with q > 1., 19 pages

Given a standard graded polynomial ring $R=k[x_1,...,x_n]$ over a field $k$ of characteristic zero and a graded $k$-subalgebra $A=k[f_1,...,f_m]\subset R$, one relates the module $\Omega_{A/k}$ of K\"ahler $k$-differentials of $A$ to the transposed Jacobian module $\mathcal{D}\subset \sum_{i=1}^n R dx_i$ of the forms $f_1,...,f_m$ by means of a {\em Leibniz map} $\Omega_{A/k}\rar \mathcal{D}$ whose kernel is the torsion of $\Omega_{A/k}$. Letting $\fp$ denote the $R$-submodule generated by the (image of the) syzygy module of $\Omega_{A/k}$ and $\fz$ the syzygy module of $\mathcal{D}$, there is a natural inclusion $\fp\subset \fz$ coming from the chain rule for composite derivatives. The main goal is to give means to test when this inclusion is an equality -- in which case one says that the forms $f_1,...,f_m$ are {\em polarizable}. One surveys some classes of subalgebras that are generated by polarizable forms. The problem has some curious connections with constructs of commutative algebra, such as the Jacobian ideal, the conormal module and its torsion, homological dimension in $R$ and syzygies, complete intersections and Koszul algebras. Some of these connections trigger questions which have interest in their own., Comment: 20 pages. Minor changes after referee's report and updated bibliography

Let k be a field. For each pair of positive integers ( n , N ) , we resolve Q = R / ( x N , y N , z N ) as a module over the ring R = k [ x , y , z ] / ( x n + y n + z n ) . Write N in the form N = a n + r for integers a and r, with r between 0 and n − 1 . If n does not divide N and the characteristic of k is fixed, then the value of a determines whether Q has finite or infinite projective dimension. If Q has infinite projective dimension, then value of r, together with the parity of a, determines the periodic part of the infinite resolution. When Q has infinite projective dimension we give an explicit presentation for the module of first syzygies of Q. This presentation is quite complicated. We also give an explicit presentation for the module of second syzygies for Q. This presentation is remarkably uncomplicated. We use linkage to find an explicit generating set for the grade three Gorenstein ideal ( x N , y N , z N ) : ( x n + y n + z n ) in the polynomial ring k [ x , y , z ] . The question “Does Q have finite projective dimension?” is intimately connected to the question “Does k [ X , Y , Z ] / ( X a , Y a , Z a ) have the Weak Lefschetz Property?”. The second question is connected to the enumeration of plane partitions. When the field k has positive characteristic, we investigate three questions about the Frobenius powers F t ( Q ) of Q. When does there exist a pair ( n , N ) so that Q has infinite projective dimension and F ( Q ) has finite projective dimension? Is the tail of the resolution of the Frobenius power F t ( Q ) eventually a periodic function of t (up to shift)? In particular, we exhibit a situation where the tail of the resolution of F t ( Q ) , after shifting, is periodic as a function of t, with an arbitrarily large period. Can one use socle degrees to predict that the tail of the resolution of F t ( Q ) is a shift of the tail of the resolution of Q?

Let K be a field and let S = K[x_1, ..., x_n] be a polynomial ring. Consider a homogenous ideal I in S. Let t_i denote reg(Tor_i (S/I, K)), the maximal degree of an ith syzygy of S/I. We prove bounds on the numbers t_i for i > n/2 purely in terms of the previous t_i. As a result, we give bounds on the regularity of S/I in terms of as few as half of the numbers t_i. We also prove related bounds for arbitrary modules. These bounds are often much smaller than the known doubly exponential bound on regularity purely in terms of t_1., 11 pages

Let F. be a any free resolution of a Z^n-graded submodule of a free module over the polynomial ring K[x_1, ..., x_n]. We show that for a suitable term order on F., the initial module of the p'th syzygy module Z_p is generated by terms m_ie_i where the m_i are monomials in K[x_{p+1}, ..., x_n]. Also for a large class of free resolutions F., encompassing Eliahou-Kervaire resolutions, we show that a Gr\"obner basis for Z_p is given by the boundaries of generators of F_p. We apply the above to give lower bounds for the Stanley depth of the syzygy modules Z_p, in particular showing it is at least p+1. We also show that if I is any squarefree ideal in K[x_1, ..., x_n], the Stanley depth of I is at least of order the square root of 2n., Comment: 13 pages

Standard bases of ideals of the polynomial ring R[X] = R[x1, …, xk] over a commutative Artinian chain ring R that are concordant with the norm on R have been investigated by D. A. Mikhailov, A. A. Nechaev, and the author. In this paper we continue this investigation. We introduce a new order on terms and a new reduction algorithm, using the coordinate decomposition of elements from R. We prove that any ideal has a unique reduced (in terms of this algorithm) standard basis. We solve some classical computational problems: the construction of a set of coset representatives, the finding of a set of generators of the syzygy module, the evaluation of ideal quotients and intersections, and the elimination problem. We construct an algorithm testing the cyclicity of an LRS-family LR(I), which is a generalization of known results to the multivariate case. We present new conditions determining whether a Ferre diagram \(\mathcal{F}\) and a full system of \(\mathcal{F}\)-monic polynomials form a shift register. On the basis of these results, we construct an algorithm for lifting a reduced Grobner basis of a monic ideal to a standard basis with the same cardinality.

Let I = ⟨ f 1 , f 2 , f 3 ⟩ I = \langle f_1 , f_2 , f_3\rangle be a bigraded ideal in the bigraded polynomial ring k [ s , u ; t , v ] k[s, u; t, v] . Assume that I I has codimension 2. Then Z = V ( I ) ⊂ P 1 × P 1 Z = \mathbb {V}(I) \subset \mathbf {P}^{1} \times \mathbf {P}^{1} is a finite set of points. We prove that if Z Z is a local complete intersection, then any syzygy of the f i f_i vanishing at Z Z , and in a certain degree range, is in the module of Koszul syzygies. This is an analog of a recent result of Cox and Schenck (2003).

Janet’s algorithm to create normal forms for systems of linear pdes is outlined and used as a tool to construct resolutions for finitely generated modules over polynomial rings over fields as well as over rings of linear differential operators with coefficients in a differential field. The main result is that a Janet basis for a module allows to read off a Janet basis for the syzygy module. Two concepts are introduced: The generalized Hilbert series allowing to read off a basis (over the ground field) of the modules, once the Janet basis is constructed, and the Janet graph, containing all the relevant information connected to the Janet basis. In the context of pdes, the generalized Hilbert series enumerates the free Taylor coefficients for power series solutions. Rather than presenting Janet’s algorithm as a powerful computational tool competing successfully with more commonly known Grobner basis techniques, it is used here to prove theoretical results.

It is folklore (see for example Theorem 7.3.3 in [68]) that if V is a valuation domain, then V[X 1, . . . ,X k ] (k ∈ ℕ) is coherent: that is, syzygy modules of finitely-generated ideals of V[X 1, . . . ,X k ] are finitely-generated. The proof in the above-mentioned reference relies on a profound and difficult result published in a huge paper by Gruson and Raynaud [75]. There is nevertheless no known general algorithm for this remarkable result, and it seems difficult to compute the syzygy module even for small polynomials.

In this paper we prove parts of a conjecture of Herzog giving lower bounds on the rank of the free modules appearing in the linear strand of a graded $k$-th syzygy module over the polynomial ring. If in addition the module is $\mathbb{Z}^n$-graded we show that the conjecture holds in full generality. Furthermore, we give lower and upper bounds for the graded Betti numbers of graded ideals with a linear resolution and a fixed number of generators., 15 pages

We investigate, for quotients of the non-commutative polynomial ring, a property that implies finiteness of Grobner bases computation, and examine its connection with Noetherianity. We propose a Grobner bases theory for our factor algebras, of particular interest for one-sided ideals, and show a few applications, e.g. how to compute (one-sided) syzygy modules.

Let S = k[x1,…,xn] be a polynomial ring, and let ωS be its canonical module. First, we will define squarefreeness for N n-graded S-modules. A Stanley–Reisner ring k[Δ] = S/IΔ, its syzygy module Syzi(k[Δ]), and ExtiS(k[Δ], ωS) are always squarefree. This notion will simplify some standard arguments in the Stanley–Reisner ring theory. Next, we will prove that the i-linear strand of the minimal free resolution of a Stanley–Reisner ideal IΔ ⊂ S has the “same information” as the module structure of ExtiS(k[Δ∨ ], ωS), where Δ∨ is the Alexander dual of Δ. In particular, if k[Δ] has a linear resolution, we can describe its minimal free resolution using the module structure of the canonical module of k[Δ∨ ], which is Cohen–Macaulay in this case. We can also give a new interpretation of a result of Herzog and co-workers, which states that k[Δ] is sequentially Cohen–Macaulay if and only if IΔ∨ is componentwise linear.

Let S be a polynomial ring over a field. For a graded S-module generated in degree at most P, the Castelnuovo-Mumford regularity of each of (i) its nth symmetric power, (ii) its nth torsion-free symmetric power and (iii) the integral closure of its nth torsion-free symmetric power is bounded above by a linear function in n with leading coefficient at most P. For a graded ideal I of S, the regularity of I' is given by a linear function of n for all sufficiently large n. The leading coefficient of this function is identified. Let S = k[xl,... , Xd] be a polynomial ring over a field k with its usual grading, i.e., each xi has degree 1, and let m denote the maximal graded ideal of S. Let N be a finitely generated non-zero graded S-module. The Castelnuovo-Mumford regularity of N, denoted reg(N), is defined to be the least integer m so that, for every j, the Jth syzygy of N is generated in degrees F -> o N --> 0 where Fi = 1 S(-aij) for some integers ai -which we will refer to as the twists of Fi. Then, reg(N) < maxij {aij i} with equality holding if the resolution is minimal. For other equivalent definitions and properties of this invariant, see [Snb]. For a graded ideal I in S, the behaviour of the regularity of I' as a function of n has been of some interest. If I defines a smooth complex projective variety, it is shown in [BrtEinLzr, Proposition 1] using the Kawamata-Viehweg vanishing theorem that reg(In) < Pn + Q where P is the maximal degree of a minimal generator of I and Q is a constant expressed in terms of the degrees of generators of I. In [GrmGmgPtt, Theorem 1.1] and in [Chn, Theorem 1] it is shown that if dim(R/I) < 1, then reg(In) < n. reg(I) for all n E N. In [Chn, Conjecture 1], this is conjectured to be true for an arbitrary graded ideal. Supporting this conjecture is the result of [Swn, Theorem 3.6] that reg(In) < Pn for some constant P and for all n E N. The method of proof makes it difficult to explicitly identify such a constant. For monomial ideals, such a P is explicitly calculated in [SmtSwn, Theorem 3.1] and improved upon in [HoaTrn, Corollary 3.2]. We show that with S and N as above, the regularity of Symn(N) and related modules is bounded above by a linear function of n with leading coefficient at Received by the editors October 28, 1997 and, in revised form, April 15, 1998. 1991 Mathematics Subject Classification. Primary 13D02; Secondary 13D40. (?1999 American Mathematical Society

Let K be a field of characteristic 0 and consider exterior algebras of finite dimensional K-vector spaces. In this short paper we exhibit principal quadric ideals in a family whose Castelnuovo-Mumford regularity is unbounded. This negatively answers the analogue of Stillman's Question for exterior algebras posed by I. Peeva. We show that these examples are dual to modules over polynomial rings that yield counterexamples to a conjecture of J. Herzog on the Betti numbers in the linear strand of syzygy modules., 7 pages - Complete rewrite of previous draft with new section and shorter proofs

For a polyhedral subdivision Δ of a region in R d , the r -differentiable piecewise polynomial functions on Δ form a module over the polynomial ring R [ x 1 , ..., x d ]. We study to what extent freeness and homological dimension of these modules is determined by the topological and combinatorial properties of Δ . We obtain a complete classification of homological dimension for three classes of complexes: when d ≤ 2, for simplicial Δ when r = 0, and for complexes whose dual graphs contain at most one cycle. In order to prove the latter result, we introduce the syzygy module of the dual graph of Δ .

Signature-based algorithms, including F5, F5C, G2V and GVW, are efficient algorithms for computing Grobner bases in commutative polynomial rings. In this paper, we present a signature-based algorithm to compute Grobner bases in solvable polynomial algebras which include usual commutative polynomial rings and some non-commutative polynomial rings like Weyl algebra. The generalized Rewritten Criterion (discussed in Sun and Wang, ISSAC 2011) is used to reject redundant computations. When this new algorithm uses the partial order implied by GVW, its termination is proved without special assumptions on computing orders of critical pairs. Data structures similar to F5 can be used to speed up this new algorithm, and Grobner bases of syzygy modules of input polynomials can be obtained from the outputs easily. Experimental data show that most redundant computations can be avoided in this new algorithm.

We present two modular algorithms for computing a generating set of the syzygy module of a given sequence of elements in R l , where R is a polynomial ring or a Weyl algebra over Q.

We study bounds for the Castelnuovo-Mumford regularity of homogeneous ideals in a polynomial ring in terms of the number of variables and the degree of the generators. In particular our aim is to give a positive answer to a question posed by Bayer and Mumford, by showing that the known upper bound in characteristic zero holds true also in positive characteristic. We first analyze Giusti's proof, which provides the result in characteristic 0, giving some insight on the combinatorial properties needed in that context. For the general case we provide a new argument which employs Bayer and Stillman criterion for detecting regularity., Comment: 12 pages

When Zharkov and Blinkov ([ZB93]) applied the classical ideas of involutive systems originating from the theory of partial differential equations (c.f. [Ja29], [Po78]) to the computation of Grobner bases (c.f. [Bu65], [BW93]) their theory seemed to be a rather marginal concept. But due to the opportunity of gaining a faster version for one of the most frequently applied algorithms the method came into the focus of computer algebra research (c.f. [Ap95], [GB95], [GS95], [Ma95]). It turned out that Pommaret bases are not only of interest for fast implementations (c.f. [ZB93]) but that they are also a point of contact of different theories which were investigated intensively for a long time. So, the theory of Pommaret bases enables the exchange of useful ideas between the theories as well as it benefits itself from the relationships. A certain similarity of the Zharkov/Blinkov method and the Kandri-Rody/Weispfenning closure technique motivates the study of commutative polynomial rings from a non-commutative point of view. The theory of Pommaret bases can be presented in an algebraic way using the Grobner theory of graded structures. Here we will present the straight forward generalization of Pommaret bases to the class of algebras of solvable type. Under the non-commutative grading most calculations are pushed back to the free non-commutative polynomial ring. This provides a link to the theory of term rewriting and the Zharkov/Blinkov method appears as an application of the prefix reduction/saturation technique of Madlener and Reinert (c.f.[MR93]) with a restricted saturation. The restricted saturation has its natural origin in the syzygy theory and heavily improves the termination behaviour in the particular case of Pommaret bases. So, it seems to be worth to investigate the effect of splitting the saturation step also for similar term rewriting problems.

In this section we shall examine some further uses of the homological tools. After preliminaries on minimal free resolutions, we shall use the Koszul complex to prove the Hilbert syzygy theorem and the basic results of Auslander-Buchsbaum and Serre about regular local rings and polynomial rings: the characterization of regular local rings as the rings of finite global dimension, and the consequences that localizations of regular local rings are regular and that regular local rings are factorial. We also derive some important relations of depth to homology in the Auslander-Buchsbaum formula and the formula connecting the vanishing of Ext i R (M, N) with the depth of the annihilator of M on N. This “explains” in a certain sense the relations of depth and the homology of the Koszul complex that we saw in Chapter 18.

In this paper we introduce four conjectures refining a very useful result of Mark Green's on the linear part of the free resolution of a graded module over a polynomial ring. We prove a number of special cases of the conjectures, including all cases pertaining to modules over a polynomial ring in at most four variables. In a later paper we will give a complete classification of the nets of skew-symmetric transformations on a 5-dimensional vector space over an algebraically closed field, from which we deduce that our conjectures are true in five variables as well.

Let A be a noetherian AS-regular Koszul quiver algebra (if A is commutative, it is essentially a polynomial ring), and grA the category of finitely generated graded left A-modules. Following Jørgensen, we define the Castelnuovo–Mumford regularity reg(M•) of a complex M•∈Db(grA) in terms of the local cohomologies or the minimal projective resolution of M•. Let A! be the quadratic dual ring of A. For the Koszul duality functor G:Db(grA)→Db(grA!), we have reg(M•)=max{i∣Hi(G(M•))≠0}. Using these concepts, we interpret results of Martinez-Villa and Zacharia concerning weakly Koszul modules (also called componentwise linear modules) over A!. As an application, refining a result of Herzog and Römer, we show that if J is a monomial ideal of an exterior algebra E=⋀〈y1,…,yd〉, d≥3, then the (d−2)nd syzygy of E/J is weakly Koszul.

A different proof of the fact that the first syzygy module of minors of certain size defined by a generic matrix over a commutative ring with identity is generated by linear relations of certain type is given. The proof utilizes the Grobner basis theory and the division algorithm over the polynomial ring Z [ X ] that the author developed in a separate paper.

In this paper we will define analogs of Grobner bases for R -subalgebras and their ideals in a polynomial ring R [ x 1 ,..., x n ] where R is a noetherian integral domain with multiplicative identity and in which we can determine ideal membership and compute syzygies.The main goal is to present and verify algorithms for constructing these Grobner basis counterparts.As an application, we will produce a method for computing generators for the first syzygy module of a subset of an R [ x 1 ,..., x n ] where each coordinate of each syzygy must be an element of the subalgebra.

In this paper we classify the 3-dimensional vector spaces (nets) of 4 × 4 and of 5 × 5 alternating matrices over an algebraically closed field. We apply the second classification to check our "linear syzygy conjectures" for modules over the polynomial ring of dimension 5.

Let S = k[x, ,..., x,] be a polynomial ring over a field and let A4 = @,*-a, M, be a finitely generated graded module; in the most interesting case A4 is an ideal of S. For a given natural number p, there is a great interest in the question: Can M be generated by (homogeneous) elements of degree

The graded algebragrvR relative to a valuationv of the quotient field of a noetherian local domainR centered atR is anS-gradedK-algebra whereS is the value semigroup ofv andK the residue field ofR. We show how the semigroup theoretical properties ofS allow to describe a minimal system of homogeneous generators forgrvR and to obtain anS-graded minimal resolution ofgrvR asA[v]-module,A[v] being certain associated polynomial ring. We derive a formula to obtain the number of generators of a fixed degree (inS) for each syzygy module and so to compute them in combinatorial terms.