Filtrations on the homology of algebraic varieties / Eric M. Friedlander, Barry Mazur.

Format:

Book

Author/Creator:

Friedlander, E. M. (Eric M.), 1944- author.

Mazur, Barry, author.

Series:

Memoirs of the American Mathematical Society ; Volume 110, Number 529.

Memoirs of the American Mathematical Society, 0065-9266 ; Volume 110, Number 529

Language:

English

Subjects (All):

Algebraic cycles.

Filters (Mathematics).

Homology theory.

Physical Description:

1 online resource (126 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island, United States : American Mathematical Society, 1994.

Language Note:

English

Summary:

This work provides a detailed exposition of a classical topic from a very recent viewpoint. Friedlander and Mazur describe some foundational aspects of ``Lawson homology'' for complex projective algebraic varieties, a homology theory defined in terms of homotopy groups of spaces of algebraic cycles. Attention is paid to methods of group completing abelian topological monoids. The authors study properties of Chow varieties, especially in connection with algebraic correspondences relating algebraic varieties. Operations on Lawson homology are introduced and analysed. These operations lead to a filtration on the singular homology of algebraic varieties, which is identified in terms of correspondences and related to classical filtrations of Hodge and Grothendieck.

Contents:

""Contents""; ""Preface""; ""Introduction""; ""Chapter 1. Questions and Speculations""; ""1.1. The nitrations""; ""1.2. Dependence on the projective imbedding""; ""1.3. The ""Strong Lefschetz"" mapping in Lawson Homology""; ""1.4. Equivalence relations on algebraic cycles""; ""1.5. Stabilized homotopy of moduli spaces""; ""1.6. Joins and Resultants""; ""Chapter 2. Abelian monoid varieties""; ""2.1. Monoids""; ""2.2. Limits""; ""2.3. Directed systems attached to an abelian monoid""; ""2.4. Limits of covariant functors""; ""2.5. Bi-algebras""; ""2.6. Abelian group completions""

""2.7. Constructing limH*(m) from H*(M)""""2.8. Base points""; ""2.9. Primitive elements""; ""2.10. Mixed Hodge Structure""; ""Chapter 3. Chow varieties and Lawson homology""; ""3.1. The Chow variety""; ""3.2. Functoriality and Chow varieties""; ""3.3. Functoriality: algebraic context""; ""3.4. The additive monoid""; ""3.5. Lawson homology""; ""Chapter 4. Correspondences and Lawson homology""; ""4.1. Correspondence homomorphisms""; ""4.2. The Chow correspondence homomorphism""; ""4.3. The Chow correspondence homomorphism and Lawson homology""

""Chapter 5. ""Multiplication"" of algebraic cycles""""5.1. The multiplicative structure on Chow varieties""; ""5.2. Bilinear pairings on group completions""; ""5.3. The multiplicative structure on Lawson homology""; ""5.4. The ring structure on Lawson homology of P[sup(0)]""; ""Chapter 6. Operations in Lawson homology""; ""6.1. The structure of the algebra A""; ""6.2. A ""geometric"" description of s""; ""6.3. A homological description of iterates of s""; ""6.4. The connection between s and the correspondence homomorphism""

""6.5. The operator Ï?[sub(j)]and the Chow correspondence homomorphism""""6.6. The operator h""; ""Chapter 7. Filtrations""; ""7.1. The Hodge Filtration""; ""7.2. The Geometric filtration""; ""7.3. The Topological filtration""; ""7.4. The Correspondence subspace""; ""7.5. Equality of correspondence and topological filtrations""; ""Appendix A. Mixed Hodge Structures, Homology, and Cycle classes""; ""A.1. Mixed Hodge Structure on homology and cohomology""; ""A.2. Homology and cohomology of smooth varieties""; ""A.3. Cycle classes in homology""

""A.4. Change of cycle class under l.c.i. morphisms""""A.5. Relation to birational change of correspondence""; ""A.6. Correspondences and suspensions""; ""Appendix B. Trace maps and the Dold-Thom Theorem""; ""B.1. The inverse image mapping on homology attached to a ""weighted map""""; ""B.2. The Dold-Thom theorem""; ""Appendix Q. On the group completion of a simplicial monoid""; ""Q.1. Rings of fractions""; ""Q.2. Grading of RS-[sup(1)]""; ""Q.3. The Eilenberg-Moore spectral sequence""; ""Q.4. A comparison lemma""; ""Q.5. Good simplicial monoids""; ""Q.6. Homology of the group completion""

""Q.7. Applications to K-theory""

Notes:

"July 1994, Volume 110, Number 529 (fourth of 6 numbers)"--Cover.

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-0108-8