**Contemporary Mathematics**

Volume: 375;
2005;
276 pp;
Softcover

MSC: Primary 14;
Secondary 18; 32

Print ISBN: 978-0-8218-3705-4

Product Code: CONM/375

List Price: $92.00

Individual Member Price: $73.60

**Electronic ISBN: 978-0-8218-7965-8
Product Code: CONM/375.E**

List Price: $92.00

Individual Member Price: $73.60

# Variance and Duality for Cousin Complexes on Formal Schemes

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*Joseph Lipman; Suresh Nayak; Pramathanath Sastry*

Robin Hartshorne's classical 1966 book "Residues and Duality" [RD] developed
Alexandre Grothendieck's ideas for a pseudofunctorial variance theory of
residual complexes and duality for maps of noetherian schemes.

The three articles in this volume rework the main parts of the last two
chapters in [RD], in greater generality—for Cousin complexes on formal
schemes, not just residual complexes on ordinary schemes—and by more concrete
local methods which clarify the relation between local properties of residues
and global properties of dualizing pseudofunctors. A new approach to pasting
pseudofunctors is applied in using residual complexes to construct a dualizing
pseudofunctor over a fairly general category of formal schemes, where
compactifications of maps may not be available.

A theory of traces and duality with respect to pseudo-proper maps is then
developed for Cousin complexes. For composites of compactifiable maps of formal
schemes, this, together with the above pasting technique, enables integration
of the variance theory for Cousin complexes with the very different approach to
duality initiated by Deligne in the appendix to [RD].

The book is suitable for advanced graduate students and researchers in
algebraic geometry.

#### Readership

Graduate students and researchers interested in algebraic geometry.

# Table of Contents

## Variance and Duality for Cousin Complexes on Formal Schemes

- Contents v6 free
- Preface vii8 free
- Part 1. Pseudofunctorial behavior of Cousin complexes on formal schemes 114 free
- 1. Introduction and main results 417
- 2. Preliminaries on formal schemes 1326
- 3. Local cohomology and Cousin complexes 3144
- 4. Generalized fractions and pseudofunctors 4558
- 5. Pseudofunctorial behavior for smooth maps 6376
- 6. Closed immersions and base change 7285
- 7. The retract case 7992
- 8. The main theorem 91104
- 9. Residual and dualizing complexes 104117
- 10. Some explicit descriptions 112125
- References 132145

- Part 2. Duality for Cousin complexes 135148
- 1. Introduction 139152
- 2. Traces 144157
- 3. The twisted inverse image pseudofunctor 157170
- 4. The comparison map 161174
- 5. Smooth maps 168181
- 6. The Cousin of the comparison map 175188
- 7. The Comparison map for flat morphisms 179192
- 8. The universal property of the trace 182195
- 9. Variants 186199
- References 192205

- Part 3. Pasting pseudofunctors 193206
- 1. Introduction 195208
- 2. The abstract pasting results 199212
- 3. Proofs I (generalized isomorphisms in the labeled setup) 207220
- 4. Proofs II (the cocycle condition) 225238
- 5. Proofs III (old isomorphisms and linearity) 241254
- 6. Proofs IV (the output) 251264
- 7. Applications 259272
- References 271284

- Index 273286 free