Commutative Algebra, Singularities and Computer Algebra

Frontmatter

Association for Flag Configurations

Abstract

We extend the classical notion of association from point configurations in projective spaces to flag configurations.

Ciprian S. Borcea

Gröbner Bases and Determinantal Ideals

An introduction

Abstract

We give an introduction to the theory of determinantal ideals and rings, their Grobner bases, initial ideals and algebras, respectively. The approach is based on the straightening law and the Knuth-Robinson-Schensted correspondence. The article contains a section treating the basic results about the passage to initial ideals and algebras.

Winfried Bruns, Aldo Conca

Bounds for Castelnuovo-Mumford Regularity in Terms of Degrees of Defining Equations

Abstract

We present here some of our (often shared) work on Castelnuovo-Mumford regularity, give few applications to Gröbner basis theory and show some examples.

Marc Chardin

The Computer Algebra Package Bergman: Current State

Abstract

Bergman is a special-purpose system for computations in commutative and purely non-commutative graded algebra.

Bergman is mainly intended to be a powerful instrument for calculating Gröbner basis in several situations: commutative and non-commutative algebras, modules over them.

Besides Gröbner bases it provides some facilities to calculate appropriate invariants of the algebras and modules, such as the Hilbert series, and (in the non-commutative case only), the Poincar¨¦ series, Anick’s resolution and the Betti numbers.

Jörgen Backelin, Svetlana Cojocaru, Victor Ufnarovski

Monomialization and Ramification of Valuations

A survey

Abstract

We discuss our recent proof of monomialization of extensions of algebraic local rings in possibly transcendental extensions of algebraic functions fields of characteristic 0. We also present some applications of this theory. We discuss our generalization of the classical theory of ramification of local Dedekind domains to general valuations, which is joint work with Olivier Piltant.

Steven Dale Cutkosky

Hyperplane Arrangements, M-Tame Polynomials and Twisted Cohomology

Abstract

A new relation between a class of complex polynomials with a good behavior at the infinity studied by A. Nemethi and A. Zaharia and the cohomology groups of affine complex hyperplane arrangement complements is introduced and explored.

This approach gives in particular new upper-bounds for the dimension of the twisted cohomology groups of line arrangement complements in the complex affine plane.

Alexandru Dimca

A Note on the Intersection of Veronese Surfaces

Abstract

Motivated by our study (elsewhere) of linear syzygies of homogenous ideals generated by quadrics and their intersections to subvarieties of the ambient projective space, we investigate in this note possible zero-dimensional intersections of two Veronese surfaces in P⁵.

The case of two Veronese surfaces in P⁵ meeting in 10 simple points appears also in work of Coble, Conner and Reye in relation to the 10 nodes of a quartic symmetroid in P³, and we provide here a modern account for some of their results.

David Eisenbud, Klaus Hulek, Sorin Popescu

Rank One Maximal Cohen-Macaulay Modules Over Singularities of Type Y 1 3 + Y 2 3 + Y 3 3 + Y 4 3

Abstract

We describe, by matrix factorizations, the rank one graded maximal Cohen-Macaulay modules over the hypersurface Y ₁ ³+Y ₂ ³+ Y ₃ ³ + Y ₄ ³.

Viviana Ene, Dorin Popescu

Towards a Theory of Gorenstein

M-Primary Integrally Closed Ideals

Abstract

Let A be a Noetherian local ring with the maximal ideal m and d = dim A. The set Χ _A of Gorenstein m-primary integrally closed ideals in A is explored in this paper. If k = A/m is alge- braically closed and d≥2, then Χ_A is infinite. In contrast, for each field k which is not algebraically closed and for each integer d ≥ 0, there exists a Noetherian complete equi-characteristic local integral domain A with dim A = d such that (1) the normalization of A is regular, (2) Χ_A = {m}, and (3) k=A/m. When d = 1, ΧA is finite if and only if A/p is not a DVR for any p E Min A, where A denotes the m-adic completion. The list of elements in Χ_A is given, when A is a one-dimensional Noetherian complete local integral domain.

Shiro Goto, Futoshi Hayasaka, Satoe Kasuga

Universal Gröbner Bases, Integer Programming and Finite Graphs

Abstract

The universal Gröbner basis of a zero-dimensional lattice ideal arising in the algebraic study of a family of integer programs associated with the incidence matrix of a finite graph will be described explicitly.

Hidefumi Ohsugi, Tomonori Kitamura, Takayuki Hibi

Torsion in Tensor Powers and Flatness

Abstract

We prove a result connecting the torsion-freeness of the symmetric powers and the flatness of an algebra over a noetherian ring containing ℚ.

Cristodor Ionescu

Basic Tools for Computing in Multigraded Rings

Abstract

In this paper we study ℤ^m-gradings on the polynomial ring>P = K[x₁,...,x_nover a field K which are suitable for developing algorithms which take advantage of the full amount of homogeneity contained in a given problem. After introducing and characterizing weakly positive and positive gradings, we provide the basic properties of Macaulay bases and multihomogenization with respect to such gradings as well as the connection between these notions. Finally, we formulate the multihomogeneous version of the Buchberger algorithm for computing homogeneous Grobner bases and minimal homogeneous systems of generators.

Martin Kreuzer, Lorenzo Robbiano

A Problem in Group Theory Solved by Computer Algebra

Abstract

 It is briefly explained how to characterize the class of finite solvable groups by 2-variable identities, a result obtained by T. Bandman, G.-M. Greuel, R Grunewald, B. Kunyavski, E.Plotkin, and the author. The description uses algebraic gemetry (the Hasse-Weil Theorem) and computer algebra (Groebner basis)

Gerhard Pfister

On Curves of Small Degree on a Normal Rational Surface Scroll

Abstract

Let C cP ^r _K denote a curve lying on a normal rational surface scroll S. Suppose that degC ¡Ü2r ¡ª 1. Then there is a classification of C into three types. These are distinguished by their arithmetical genus, their Hartshorne-Rao module and their homological behavior. The classification is done by computations of the cohomology of certain divisors on the surface scroll. Finally several illustrating examples are discussed.

Peter Schenzel

On Sagbi Bases and Resultants

Abstract

A resultant-type identity for univariate polynomials is proved and applied to characterization of SAGBI bases of subalgebras, generated by two polynomials. Besides a new condition for polynomials f(x) and g(x) to form a SAGBI basis, expressed in terms of field extensions is derived.

Anna Torstensson, Victor Ufnarovski, Hans Öfverbeck

Modules of G-Dimension Zero over Local Rings with the Cube of Maximal Ideal Being Zero

Abstract

Let (R,m) be a commutative Noetherian local ring with m³ = (0). We give a condition for R to have a non-free module of G-dimension zero. We shall also construct a family of non- isomorphic indecomposable modules of G-dimension zero with parameters in an open subset of pro- jective space. We shall finally show that the subcategory consisting of modules of G-dimension zero over R is not necessarily a contravariantly finite subcategory in the category of finitely generated R-modules.

Yuji Yoshino