# Filtrations on the Homology of Algebraic Varieties

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*Eric M. Friedlander; Barry Mazur*

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 analyzed. 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.

#### Table of Contents

# Table of Contents

## Filtrations on the Homology of Algebraic Varieties

- Contents v6 free
- Preface ix10 free
- Introduction 112 free
- Chapter 1. Questions and Speculations 516 free
- Chapter 2. Abelian monoid varieties 920
- 2.1. Monoids 920
- 2.2. Limits 920
- 2.3. Directed systems attached to an abelian monoid 1021
- 2.4. Limits of covariant functors 1122
- 2.5. Bi-algebras 1324
- 2.6. Abelian group completions 1425
- 2.7. Constructing limH*(m) from H*(M) 1627
- 2.8. Base points 1930
- 2.9. Primitive elements 2031
- 2.10. Mixed Hodge Structure 2233

- Chapter 3. Chow varieties and Lawson homology 2536
- Chapter 4. Correspondences and Lawson homology 3950
- Chapter 5. "Multiplication" of algebraic cycles 4758
- Chapter 6. Operations in Lawson homology 5566
- 6.1. The structure of the algebra A 5566
- 6.2. A "geometric" description of s 5869
- 6.3. A homological description of iterates of s 5869
- 6.4. The connection between s and the correspondence homomorphism 6071
- 6.5. The operator σ[sub(j)]and the Chow correspondence homomorphism 6273
- 6.6. The operator h 6374

- Chapter 7. Filtrations 6778
- Appendix A. Mixed Hodge Structures, Homology, and Cycle classes 7384
- A.1. Mixed Hodge Structure on homology and cohomology 7384
- A.2. Homology and cohomology of smooth varieties 7485
- A.3. Cycle classes in homology 7586
- A.4. Change of cycle class under l.c.i. morphisms 7687
- A.5. Relation to birational change of correspondence 7788
- A.6. Correspondences and suspensions 7889

- Appendix B. Trace maps and the Dold-Thom Theorem 8192
- Appendix Q. On the group completion of a simplicial monoid 89100
- Q.1. Rings of fractions 90101
- Q.2. Grading of RS-[sup(1)] 91102
- Q.3. The Eilenberg-Moore spectral sequence 92103
- Q.4. A comparison lemma 94105
- Q.5. Good simplicial monoids 95106
- Q.6. Homology of the group completion 97108
- Q.7. Applications to K-theory 98109
- Q.8. A theorem of Mather 102113
- Q.9. The group completion theorem in Segal's setup 103114
- References for Appendix Q 105116

- Bibliography 107118