A book on Homological Algebra

A book on Homological Algebra

[Marco Grandis]

The following book has been published at World Scientific;
below there is a copy of its presentation in the WS web page.

Best regards to colleagues and friends

Marco Grandis

_____________________________________________________________

HOMOLOGICAL ALGEBRA
The Interplay of Homology with Distributive Lattices and Orthodox  
Semigroups
by Marco Grandis
World Scientific Publishing Co., 384pp

General information:     http://www.worldscibooks.com/mathematics/ 
8483.html
Table of Contents (45k):     http://www.worldscibooks.com/etextbook/ 
8483/8483_toc.pdf
Preface (37k):     http://www.worldscibooks.com/etextbook/ 
8483/8483_preface.pdf
Introduction (123k):   http://www.worldscibooks.com/etextbook/ 
8483/8483_intro.pdf
Chapter 1: Coherence and models in homological algebra (333k):      
http://www.worldscibooks.com/etextbook/8483/8483_chap01.pdf


In this book we want to explore aspects of coherence in homological  
algebra, that already appear in the classical situation of abelian  
groups or abelian categories.

Lattices of subobjects are shown to play an important role in the  
study of homological systems, from simple chain complexes to all the  
structures that give rise to spectral sequences. A parallel role is  
played by semigroups of endorelations.

These links rest on the fact that many such systems, but not all of  
them, live in distributive sublattices of the modular lattices of  
subobjects of the system.

The property of distributivity allows one to work with induced  
morphisms in an automatically consistent way, as we prove in a  
‘Coherence Theorem for homological algebra’. (On the contrary, a ‘non- 
distributive’ homological structure like the bifiltered chain complex  
can easily lead to inconsistency, if one explores the interaction of  
its two spectral sequences farther than it is normally done.)

The same property of distributivity also permits representations of  
homological structures by means of sets and lattices of subsets,  
yielding a precise foundation for the heuristic tool of Zeeman  
diagrams as universal models of spectral sequences.

We thus establish an effective method of working with spectral  
sequences, called ‘crossword chasing’, that can often replace the  
usual complicated algebraic tools and be of much help to readers that  
want to apply spectral sequences in any field.


Contents:
Introduction
Coherence and Models in Homological Algebra
Puppe-Exact Categories
Involutive Categories
Categories of Relations as RE-Categories
Theories and Models
Homological Theories and Their Universal Models
Appendix A: Some Points of Category Theory
Appendix B: A Proof for the Universal Exact System



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