Multigraded Algebras Research Articles

Multigraded algebras and multigraded linear series

AbstractThis paper is devoted to the study of multigraded algebras and multigraded linear series. For an ‐graded algebra , we define and study its volume function , which computes the asymptotics of the Hilbert function of . We relate the volume function to the volume of the fibers of the global Newton–Okounkov body of . Unlike the classical case of standard multigraded algebras, the volume function is not a polynomial in general. However, in the case when the algebra has a decomposable grading, we show that the volume function is a polynomial with nonnegative coefficients. We then define mixed multiplicities in this case and provide a full characterization for their positivity. Furthermore, we apply our results on multigraded algebras to multigraded linear series. Our work recovers and unifies recent developments on mixed multiplicities. In particular, we recover results on the existence of mixed multiplicities for (not necessarily Noetherian) graded families of ideals and on the positivity of the multidegrees of multiprojective varieties.

When are multidegrees positive?

Let k be an arbitrary field, P=Pkm1×k⋯×kPkmp be a multiprojective space over k, and X⊆P be a closed subscheme of P. We provide necessary and sufficient conditions for the positivity of the multidegrees of X. As a consequence of our methods, we show that when X is irreducible, the support of multidegrees forms a discrete algebraic polymatroid. In algebraic terms, we characterize the positivity of the mixed multiplicities of a standard multigraded algebra over an Artinian local ring, and we apply this to the positivity of mixed multiplicities of ideals. Furthermore, we use our results to recover several results in the literature in the context of combinatorial algebraic geometry.

Computing noncommutative Hilbert series

We propose methods for computing the Hilbert series of multigraded right modules over the free associative algebra. In particular, we compute such series for noncommutative multigraded algebras. Using results from the theory of regular languages, we provide conditions when the methods are effective and hence the Hilbert series have a rational sum. Efficient variants of the methods are also developed for the truncations of infinite-dimensional algebras which provide approximations of possibly irrational Hilbert series. Moreover, we provide a characterization of the finite-dimensional algebras in terms of the nilpotency of a key matrix involved in the computations. Finally, we present a well-tested and complete implementation for the computation of graded and multigraded Hilbert series which has been developed in the kernel of the computer algebra system Singular (for the details, see preprint[1]).

Multigraded Hilbert series of noncommutative modules

In this paper, we propose methods for computing the Hilbert series of multigraded right modules over the free associative algebra. In particular, we compute such series for noncommutative multigraded algebras. Using results from the theory of regular languages, we provide conditions when the methods are effective and hence the sum of the Hilbert series is a rational function. Moreover, a characterization of finite-dimensional algebras is obtained in terms of the nilpotency of a key matrix involved in the computations. Using this result, efficient variants of the methods are also developed for the computation of Hilbert series of truncated infinite-dimensional algebras whose (non-truncated) Hilbert series may not be rational functions. We consider some applications of the computation of multigraded Hilbert series to algebras that are invariant under the action of the general linear group. In fact, in this case such series are symmetric functions which can be decomposed in terms of Schur functions. Finally, we present an efficient and complete implementation of (standard) graded and multigraded Hilbert series that has been developed in the kernel of the computer algebra system Singular. A large set of tests provides a comprehensive experimentation for the proposed algorithms and their implementations.

Nagata type statements

Nagata solved Hilbert’s 14-th problem in 1958 in the negative. The solution naturally led him to a tantalizing conjecture that remains widely open after more than half a century of intense efforts. Using Nagata’s theorem as starting point, and the

Postulation and reduction vectors of multigraded filtrations of ideals

We study the relationship between postulation and reduction vectors of admissible multigraded filtrations $\mathcal{F}= \{\mathcal{F} (\underline{n})\}_{\underline{n} \in \mathbb{Z} ^s}$ of ideals in Cohen-Macaulay local rings of dimension at most two. This is enabled by a suitable generalization of the Kirby-Mehran complex. An analysis of its homology leads to an analogue of Huneke's fundamental lemma which plays a crucial role in our investigations. We also clarify the relationship between the Cohen-Macaulay property of the multigraded Rees algebra of $\mathcal{F} $ and reduction vectors with respect to complete reductions of $\mathcal{F} $.

M-vector analogue for the cd-index

A well-known conjecture of McMullen, proved by Billera, Lee and Stanley, describes the face numbers of simple polytopes. The necessary and sufficient condition is that the toric g-vector of the polytope is an M-vector, that is, the vector of dimensions of graded pieces of a standard graded algebra A. Recent work by Murai, Nevo and Yanagawa suggests a similar condition for the coefficients of the cd-index of a Gorenstein* poset P. The coefficients of the cd-index are conjectured to be the dimensions of graded pieces in a standard multigraded algebra A. We prove the conjecture for simplicial spheres and we give numerical evidence for general shellable spheres. In the simplicial case we construct the multi-graded algebra A explicitly using lattice paths.

HOMOGENEOUS PBW DEFORMATION FOR ARTIN–SCHELTER REGULAR ALGEBRAS

AbstractWe introduce a method named homogeneous PBW deformation that preserves the regularity and some other homological properties for multigraded algebras. The method is used to produce Artin–Schelter regular algebras without the hypothesis on grading.

Polyhedral divisors and torus actions of complexity one over arbitrary fields

We show that the presentation of affine T-varieties of complexity one in terms of polyhedral divisors holds over an arbitrary field. We also describe a class of multigraded algebras over Dedekind domains. We study how the algebra associated with a polyhedral divisor changes when we extend the scalars. As another application, we provide a combinatorial description of affine G-varieties of complexity one over a field, where G is a (not necessarily split) torus, by using elementary facts on Galois descent. This class of affine G-varieties is described via a new combinatorial object, which we call (Galois) invariant polyhedral divisor.

Mixed multiplicities of multigraded modules

Let S be a finitely generated standard multigraded algebra over an Artinian local ring A , and let M be a finitely generated multigraded S -module. This paper answers the question when mixed multiplicities of M are positive and characterizes them in terms of lengths of A -modules. As an application, we get interesting results on mixed multiplicities of ideals, and recover some early results.

Calculating generators of multigraded algebras

We present an algorithm to find generators of the multigraded algebra A associated to an arbitrary p-divisor D on some variety Y. A modified algorithm is also presented for the case where Y admits a torus action. We demonstrate our algorithm by computing generators for the Cox ring of the smooth del Pezzo surface of degree 5.

Poincaré series of the multigraded algebras of SL 2-invariants

Let $\mathcal{C}_{\mathbi{d}},$ $\mathcal{I}_{\mathbi{d}},$ $\mathbi{d}{=}(d_1,d_2,..., d_n)$ be the algebras of join covariants and joint invariants of the $n$ binary forms of degrees $d_1,d_2,..., d_n.$ Formulas for computation of the multivariate Poincar\'e series $\mathcal{P}(\mathcal{C}_{\mathbi{d}},z_1,z_2,...,z_n,t)$ and $\mathcal{P}(\mathcal{I}_{\mathbi{d}},z_1,z_2,...,z_n)$ are found.

On (FC)-sequences and Mixed Multiplicities of Multi-graded Algebras

Let $S=\bigoplus_{n_1,\ldots, n_s \geq 0}S_{(n_1,\ldots, n_s)}$ be a finitely generated standard multi-graded algebra over a Noetherian local ring $A$. This paper investigates the positivity of mixed multiplicities of $S$ and characterizes them in terms of Hilbert-Samuel multiplicities. As an application, we get some results on the mixed multiplicities of ideals that covers the main results in [Vi] and [TV].

On the Multiplicity of Multigraded Modules Over Artinian Local Rings

Let $S$ be a finitely generated standard multigraded algebra over an Artinian local ring $A$; $M$ a finitely generated multigraded $S$-module. This paper first investigates the relationship between the multiplicity and mixed multiplicities of $M$. Next, we give some applications to multigraded fiber cones.

Hilbert functions of multigraded algebras, mixed multi- plicities of ideals and their applications

Hilbert functions of multigraded algebras, mixed multi- plicities of ideals and their applications

Toric degeneration of branching algebras

For each classical symmetric pair ( G , H ) , there is a naturally defined multi-graded algebra A ( G , H ) , called the branching algebra for ( G , H ) , which encodes the branching rule from G to H. This algebra has a natural family of subalgebras, depending on integer parameters. For a certain range of the parameters, the subalgebras have a particularly simple structure and are called stable branching algebras. In this paper, we show that the stable branching algebras for eight out of the ten families of classical symmetric pairs are flat deformations of the semigroup algebras of explicitly described lattice cones.

Non-associative Gröbner bases

The theory of Gröbner basis for ideals can be applied in the non-associative, noncommutative free magma algebra K { X } with unit freely generated by a set X over a field K . In this article we introduce a class of admissible orders on the magma freely generated by X which is denoted by Mag ( X ) and some special admissible orders on Mag ( X ) . We prove that the reduced Gröbner basis of a multigraded ideal I in the multigraded algebra K { X } consists of the reduced multihomogeneous polynomials with multidegrees ( α ) ∈ N n . We obtain a generalization for the Hilbert series of the multigraded algebra A = K { X } / J of residue classes modulo multigraded ideal J generated by multihomogeneous polynomials in the non-associative free magma algebra of tree polynomials K { X } , where X is a multigraded set of variables. It relates H A to G X , the generating series in n variables for X , and G Γ , the generating series of the reduced Gröbner basis Γ of J . Let I alt ( X ) be the alternator ideal generated by alternators in the free magma algebra K { x , y , z } , then we obtain the elements of multidegree (2, 1, 1) in the reduced Gröbner basis Γ of I alt ( X ) w.r.t. the admissible order degree first factor on Mag ( X ) . We consider the Cayley algebra O and the admissible order degree first factor on Mag ( X ) , where X = { i , j , ℓ } , with fix order i < j < ℓ . Then we obtain the reduced Gröbner basis Γ of the ideal J generated by all relations in the Cayley algebra O w.r.t. this admissible order. Also we obtain the generating series of G O ( t ) and G Γ ( t ) .

Reductions and Mixed Multiplicities of Ideals#

The notion of (FC)-sequences was built in Viet [Viet, D. Q. (2000). Mixed multiplicities of arbitrary ideals in local rings. Comm. Algebra 28(8):3803–3821] containing important information for multiplicities and reductions of ideals [see Viet, D. Q. 2000; Viet, D. Q. (2003a). On some properties of (FC)-sequences of ideals in local rings. Proc. Am. Math. Soc. 131:45–53; Viet, D. Q. (2003b). Sequences determining mixed multiplicities and reductions of ideals. Comm. Algebra 31(10):5047–5069; Viet, D. Q. (2003c). On the mixed multiplicity and the multiplicity of blow-up rings of equimultiple. J. Pure Appl. Algebra 183:313–327]. In this paper, we will construct generalized joint reductions generated by maximal weak-(FC)-sequences of a set of arbitrary ideals. We showed that the mixed multiplicity of ideals and the mixed multiplicity of their reductions are the same. The paper also determines the vanishing and non-vanishing of mixed multiplicities and as an application of above results we compute mixed multiplicities and the multiplicity of multi-graded Rees algebras in some particular cases.

Filter-regularity and cohen-macaulay multigraded rees algebras

Filter-regularity and cohen-macaulay multigraded rees algebras

On the multiplication map of a multigraded algebra

Given a multigraded algebra $A$, it is a natural question whether or not for two homogeneous components $A_u$ and $A_v$, the product $A_{nu}A_{nv}$ is the whole component $A_{nu+nv}$ for $n$ big enough. We give combinatorial and geometric answers to this question.