Minimal Free Resolutions over Complete Intersections
Published on 8. March 2016
X, 107 pages
978-3-319-26437-0 (ISBN)
Description
This book introduces a theory of higher matrix factorizations for regular sequences and uses it to describe the minimal free resolutions of high syzygy modules over complete intersections. Such resolutions have attracted attention ever since the elegant construction of the minimal free resolution of the residue field by Tate in 1957.
The theory extends the theory of matrix factorizations of a non-zero divisor, initiated by Eisenbud in 1980, which yields a description of the eventual structure of minimal free resolutions over a hypersurface ring. Matrix factorizations have had many other uses in a wide range of mathematical fields, from singularity theory to mathematical physics.
Reviews / Votes
"The text provides a wonderful introduction describing the background which led to the development of higher matrix factorizations and includes (with proofs and examples) all the theory required to understand the new material and put it in context." (Benjamin P. Richert, Mathematical Reviews, May, 2017)More details
Series
Edition
1st ed. 2016
Language
English
Place of publication
Cham
Switzerland
Publishing group
Springer International Publishing
Target group
Professional and scholarly
Product notice
Reflowable
Illustrations
X, 107 p.
ISBN-13
978-3-319-26437-0 (9783319264370)
DOI
10.1007/978-3-319-26437-0
Schweitzer Classification
Other editions
Additional editions
David Eisenbud | Irena Peeva
Minimal Free Resolutions over Complete Intersections
Book
03/2016
Springer
€40.65
Shipment within 10-15 days
Content
- Intro
- Preface
- Acknowledgments
- Contents
- 1 Introduction and Survey
- 1.1 How We Got Here
- Revealing the Pattern
- 1.2 What is a Higher Matrix Factorization?
- Matrix Factorizations of a Sequence of Elements
- 1.3 What's in This Book?
- High Syzygies are Higher Matrix Factorization Modules
- Minimal R-Free and S-Free Resolutions
- Syzygies over Intermediate Quotient Rings
- 1.4 Notation and Conventions
- 2 Matrix Factorizations of One Element
- 2.1 Matrix Factorizations and Resolutions over a Hypersurface
- 3 Finite Resolutions of HMF Modules
- 3.1 The Minimal S-Free Resolution of a Higher Matrix Factorization Module
- 3.2 Consequences
- 3.3 Building a Koszul Extension
- 3.4 Higher Homotopies
- 4 CI Operators
- 4.1 CI Operators
- 4.2 The Action of the CI Operators on Ext
- 4.3 Resolutions with a Surjective CI Operator
- 5 Infinite Resolutions of HMF Modules
- 5.1 The Minimal R-Free Resolution of a Higher Matrix Factorization Module
- 5.2 Betti Numbers
- 5.3 Strong Matrix Factorizations
- 5.4 Resolutions over Intermediate Rings
- 6 Far-Out Syzygies
- 6.1 Pre-stable Syzygies and Generic CI Operators
- 6.2 The Graded Case
- 6.3 The Box Complex
- 6.4 From Syzygies to Higher Matrix Factorizations
- 6.5 Betti Numbers of Pre-stable Matrix Factorizations
- 7 The Gorenstein Case
- 7.1 Syzygies and Maximal Cohen-Macaulay Modules
- 7.2 Stable Syzygies in the Gorenstein Case
- 7.3 Maximal Cohen-Macaulay Approximations
- 7.4 Stable Matrix Factorizations over a Gorenstein Ring
- 8 Functoriality
- 8.1 HMF Morphisms
- References
- Index
