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2002 | Book

Rings, Ideals and Standard Bases Read chapter Read first chapter

A Singular Introduction to Commutative Algebra

Authors: Gert-Martin Greuel, Gerhard Pfister

Publisher: Springer Berlin Heidelberg

Included in: Professional Book Archive

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About this book

In theory there is no difference between theory and practice. In practice there is. Yogi Berra A SINGULAR Introduction to Commutative Algebra offers a rigorous intro­ duction to commutative algebra and, at the same time, provides algorithms and computational practice. In this book, we do not separate the theoretical and the computational part. Coincidentally, as new concepts are introduced, it is consequently shown, by means of concrete examples and general proce­ dures, how these concepts are handled by a computer. We believe that this combination of theory and practice will provide not only a fast way to enter a rather abstract field but also a better understanding of the theory, showing concurrently how the theory can be applied. We exemplify the computational part by using the computer algebra sys­ tem SINGULAR, a system for polynomial computations, which was developed in order to support mathematical research in commutative algebra, algebraic geometry and singularity theory. As the restriction to a specific system is necessary for such an exposition, the book should be useful also for users of other systems (such as Macaulay2 and CoCoA) with similar goals. Indeed, once the algorithms and the method of their application in one system is known, it is usually not difficult to transfer them to another system.

Table of Contents

Frontmatter
1. Rings, Ideals and Standard Bases
Abstract
The concept of a ring is probably the most basic one in commutative and non-commutative algebra. Best known are the ring of integers ℤ and the polynomial ring K[x] in one variable x over a field K.
Gert-Martin Greuel, Gerhard Pfister
2. Modules
Abstract
Module theory may, perhaps, best be characterized as linear algebra over a ring. While classical commutative algebra was basically ideal theory, modules are in the centre of modern commutative algebra as a unifying approach. Formally, the notion of a module over a ring is the analogue of the notion of a vector space over a field, in the sense that a module is defined by the same axioms, except that we allow ring elements as scalars and not just field elements. Just as vector spaces appear naturally as the solution sets of systems of linear equations over a field, modules appear as solution sets of such systems over a ring. However, contrary to vector spaces, not every module has a basis and this makes linear algebra over a ring much richer than linear algebra over a field.
Gert-Martin Greuel, Gerhard Pfister
3. Noether Normalization and Applications
Abstract
Integral extension of a ring means adjoining roots of monic polynomials over the ring. This is an important tool for studying affine rings, and it is used in many places, for example, in dimension theory, ring normalization and primary decomposition. Integral extensions are closely related to finite maps which, geometrically, can be thought of as projections with finite fibres plus some algebraic conditions. We shall give a constructive introduction with explicit algorithms to these subjects.
Gert-Martin Greuel, Gerhard Pfister
4. Primary Decomposition and Related Topics
Abstract
It is well-known that every integer is a product of prime numbers, for instance 10 = 2.5. This equation can also be written as an equality of ideals, 〈10〉 =〈2〉 ∩ 〈5〉 in the ring ℤ. The aim of this section is to generalize this fact to ideals in arbitrary Noetherian rings.
Gert-Martin Greuel, Gerhard Pfister
5. Hilbert Function and Dimension
Abstract
The Hilbert function of a graded module associates to an integer n the dimension of the n-th graded part of the given module. For sufficiently large n, the values of this function are given by a polynomial, the Hilbert polynomial. To show this, we use the Hilbert-Poincaré series, a formal power series in t with coefficients being the values of the Hilbert function. This power series turns out to be a rational function.
Gert-Martin Greuel, Gerhard Pfister
6. Complete Local Rings
Abstract
For certain applications the local rings K[x]〈x〉, x = (x1, ... , x n ), are not “sufficiently local”. As explained in Appendix A, Sections A.8 and A.9, the latter rings contain informations about arbitrary small Zariski neighbourhoods of 0 ∈ K n . Such neighbourhoods turn out to be still quite large, for instance, if n = 1 then they consist of K minus a finite number of points. If we are working over the field K = ℂ, respectively K = ℝ, we can use the convergent power series ring K{x} which contains information about arbitrary small Euclidean neighbourhoods of 0, and this is what we are usually interested in. For arbitrary fields, however, we have to consider the formal power series ring K[[x]] instead.
Gert-Martin Greuel, Gerhard Pfister
7. Homological Algebra
Abstract
In Section 2.7, we saw that the tensor product is right exact, but in general not exact. We shall establish criteria for the exactness in terms of homology.
Gert-Martin Greuel, Gerhard Pfister
Backmatter
Metadata
Title
A Singular Introduction to Commutative Algebra
Authors
Gert-Martin Greuel
Gerhard Pfister
Copyright Year
2002
Publisher
Springer Berlin Heidelberg
Electronic ISBN
978-3-662-04963-1
Print ISBN
978-3-540-42897-8
DOI
https://doi.org/10.1007/978-3-662-04963-1