Minimal Free Resolutions over Complete Intersections / by David Eisenbud, Irena Peeva.

Format:

Book

Author/Creator:

Eisenbud, David, Author.

Peeva, Irena, Author.

Series:

Lecture Notes in Mathematics, 1617-9692 ; 2152

Language:

English

Subjects (All):

Commutative algebra.

Commutative rings.

Geometry, Algebraic.

Algebra, Homological.

Mathematical physics.

Commutative Rings and Algebras.

Algebraic Geometry.

Category Theory, Homological Algebra.

Theoretical, Mathematical and Computational Physics.

Local Subjects:

Commutative Rings and Algebras.

Algebraic Geometry.

Category Theory, Homological Algebra.

Theoretical, Mathematical and Computational Physics.

Physical Description:

1 online resource (X, 107 p.)

Edition:

1st ed. 2016.

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2016.

Language Note:

English

Summary:

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.

Contents:

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.

Notes:

Bibliographic Level Mode of Issuance: Monograph

Includes bibliographical references and index.

ISBN:

3-319-26437-0

OCLC:

944445429