Ebook: Computational Commutative and Non-Commutative Algebraic Geometry

A NATO Advanced Research Workshop "Computational Commutative and Non-Commutative Algebraic Geometry" was held in Chishinau, Moldova from 6 to 11 June, 2004. The financial support was offered by the NATO grant PST.ARW.980392. The workshop was organized by Gerhard Pfister and Svetlana Cojocaru with assistance of members of the Organizing Committee: Jürgen Herzog, Lorenzo Robbiano, Victor Ufnarovski. There were 25 lectures given by the participants and open discussions on the subjects presented. The central theme of this workshop was the interplay between commutative and noncommutative algebraic geometry, with its theoretical and computational aspects. The Scientific Program emphasizes current trends in Commutative and Non-Commutative Algebraic Geometry and Algebra. The contributors to this volume review the state of the art and present new evolution and progress reflecting the topics discussed in the lectures. The volume addresses in the first place researchers and graduate students. The editors would like to thank the contributors to this volume. Thanks are due to the NATO Public Diplomacy Division for their generous financial support. Special thanks to Tatiana Verlan, who had the task to correct and prepare the manuscript.

December 2004, Svetlana Cojocaru, Gerhard Pfister, Victor Ufnarovski

In this paper we study the non‐commutative scheme structure of the set of iso‐classes of simple modules on a finitely generated k‐algebra, k an algebraically closed field. We introduce the notion of geometric algebra and, referring to [12], we prove that these algebras are determined by the structure of this non‐commutative scheme. We consider natural completions of these schemes, adding indecomposable modules at infinity. This leads to a notion of correspondence on plane curves, which we explore to some degree. We end the paper with a sketch of how to relate global invariants of the algebra, like cyclic homology, to corresponding invariants of the scheme of simple finite‐dimensional modules.

In this paper we present new algorithms for non‐commutative Gröbner ready algebras, which enable one to perform advanced operations with ideals and modules. In spite of the big interest in algorithmic treatment of related problems, preimage of ideal and central character decomposition were not discussed before.

An important algotithm for computation of the kernel of a homomorphism of left modules is described in the form, optimized for performance. We present these algorithms together with their implementation in computer algebra system SINGULAR:PLURAL and detailed applications.

We describe algebraically the rank two stable vector bundles on the simple node Proj R⊂P²_k, with R=k[y₁,y₂,y₃]/(f),k an algebraically closed field with characteristic zero, f=y³₁−y²₁y₃−y²₂y₃.

Let (C,0) be a germ of a plane curve singularity defined by f∊C{x,y}, if we denote by Σ the sum semigroup of the semigroup of values of C, we want to decide whether f is irreducible or not just by computing the contact with the Polar in any direction h and comparing it with c(Σ)+(f,h)−1.

One generalization of Shubert's varieties is presented based on use of two towers of “general”(very natural) form with no subspace from one of the towers belonging totally to some subspace from another tower. These varieties are proved to be reducible, their general positions are described. The number of general positions and dimensions of corresponding orbits are calculated also.

The technique of Bourbaki sequences is used in [4] in order to study the graded maximal Cohen‐Macaulay modules of rank two over the singularity x³₁+x³₂+x³₃+x³₄. In the second section of this paper we complete a small gap in the classification of these modules which are non–orientable. In the last section we give a brief survey on the classification of the graded non–orientable maximal Cohen–Macaulay modules of rank two over x³₁+x³₂+x³₃+x³₄.

Recent implementations of algorithms for resolution of singularities have not only shown that it is feasible to implement these algorithms, but also that the performance allows the use of desingularisation as a tool in more complex computational tasks. In this article, we want to follow this idea and outline some rather direct applications of the algorithmic resolution of singularities such as the intersection matrix of the exceptional divisors, spectral numbers of a hypersurface singularity or the Denef‐Loeser zeta function.

Classical and new results on finite free resolutions and on Betti‐numbers are presented.

We describe an algorithm to compute the integral closure Ī of an ideal I in a polynomial ring R:=k[x₁,…,x_n] over a field. It is based on the known fact that Ī is encoded in the integral closure of the Rees algebra [Rscr ](I):=⌖_k≥0I^kt^k⊆R[t] of I and on an algorithm to compute the normalization of an affine domain. In particular we explain an implementation of this algorithm in the computer algebra system SINGULAR [11]. As an application we mention the connection of the integral closure of ideals to the study of Whitney equisingularity.

We generalize the notion and various characterizations of a modular stratum in the base space of a miniversal deformation of an isolated singularity to other deformation functors. In particular, its descriptions as the flattening of associated Tjurina modules can be used to compute with the help of the computer algebra system SINGULAR non‐trivial examples. The splitting into multi‐germs for ordinary modular deformations as well as for modular deformations with fat point section is discussed.

We give some applications of a classical theorem (Oskar Perron 1927) on algebraic dependence to Bézout's type estimations.

We present a geometric way of getting matrix factorizations for graded maximal Cohen‐Macaulay modules of rank one over the graded ring

k[X₀,X₁,X₂]/(f)

where f is a homogenous cubic defining a smooth curve in the projective plane P²(k).

The aim of the article is to describe a new feature in the computer algebra package BERGMAN – an interface shell that simplifies the usage of the system. We start from a brief description of BERGMAN and problems arising in its interface. Then we show how the shell solves those problems. At last we describe in more details how to use the shell and comment its additional possibilities and future development.

In this paper we describe an efficient involutive algorithm for constructing Gröbner bases of polynomial ideals. The algorithm is based on the concept of involutive monomial division which restricts the conventional division in a certain way. In the presented algorithm a reduced Gröbner basis is the internally fixed subset of an involutive basis, and having computed the later, the former can be output without any extra computational costs. We also discuss some accounts of experimental superiority of the involutive algorithm over Buchberger's algorithm.

Based on an explicit description of the idealization of a graded submodule of a graded free module, we examine the behaviour of Gröbner bases and minimal homogeneous systems of generators under this process. Then we show how one can idealize a homogeneous presentation. Using this theory, we present a unified description of several strategies for computing minimal homogeneous presentations and minimal graded free resolutions, in particular of the vertical and horizontal strategies. We obtain at a simple and compact algorithm for computing minimal graded free resolutions degree‐by‐degree which lends itself well to further optimizations.

We consider different methods to reduce the number of critical pairs necessary for the constructing and checking SAGBI bases.

The article consists of two sections. The first one is a survey of results on lattices of module topologies issued before. The second one contains a new construction of a topology preceding the given metrizable topology in the lattice of all group topologies admitting a basis of neighbourhoods of zero consisting of subgroups on an abelian group of period p².

In the present paper the problem of finding of invariant GL(2,R)‐ntegrals is considered on example of two autonomous polynomial differential systems of the first order. The methods of commutative and non‐commutative algebra, as the elements of modern group analysis of differential equations, are used in the work.

The well‐known problem of the description of finite projective planes is studied. It is known that for prime power order n there exists a projective plane of order n. In this work the method of description of all projective planes of fixed prime power order n is demonstrated.