Book contents
- Frontmatter
- Contents
- Foreword
- Introduction
- Chapter 1 Krull Dimension
- Chapter 2 Syzygetic Sequences
- Chapter 3 Approximation Complexes
- Chapter 4 Linkage and Koszul Homology
- Chapter 5 Arithmetic of Rees Algebras
- Chapter 6 Factoriality
- Chapter 7 Ideal Transforms
- Chapter 8 The Equations of Rees Algebras
- Chapter 9 Commuting Varieties of Algebras
- Chapter 10 Computational Methods in Commutative Algebra
- Bibliography
- Index
Chapter 4 - Linkage and Koszul Homology
Published online by Cambridge University Press: 19 August 2009
- Frontmatter
- Contents
- Foreword
- Introduction
- Chapter 1 Krull Dimension
- Chapter 2 Syzygetic Sequences
- Chapter 3 Approximation Complexes
- Chapter 4 Linkage and Koszul Homology
- Chapter 5 Arithmetic of Rees Algebras
- Chapter 6 Factoriality
- Chapter 7 Ideal Transforms
- Chapter 8 The Equations of Rees Algebras
- Chapter 9 Commuting Varieties of Algebras
- Chapter 10 Computational Methods in Commutative Algebra
- Bibliography
- Index
Summary
Linkage theory deals, grosso modo, with the following general problem. Let X be an algebraic variety and Y a closed subvariety of X contained in the closed subvariety Z. What are the properties, with respect to Z and Y, of the closed subvarieties W of Z such that Z = Y ∪ W?
The aspects of this notion which will be discussed in this chapter have a more specialized character. The first of those has a vivid directness, which we recall. Let R be a Cohen–Macaulay Noetherian ring, and let I and J be two ideals of R. I and J are said to be directly linked if there exists a regular sequence z = z1, …, zn I ∩ J such that I = (z): J and J = (z): I. Furthermore, they are said to be geometrically linked ideals if they are unmixed ideals of the same grade n, without common components and I ∩ J = (z) for some regular sequence of grade n. In both cases, the schemes V(I) and V(J) together define a complete intersection: V(I) ∪ V(J) = V(z). Another notion of linkage, residual intersection, replaces the regular sequence z by more general ideals.
These relationships shall be noted by: I ∼ J. We will then also say that J is a direct link of I.
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- Chapter
- Information
- Arithmetic of Blowup Algebras , pp. 73 - 98Publisher: Cambridge University PressPrint publication year: 1994