A classification of localizing subcategories by relative homological algebra

by George Nadareishvili
Doctoral thesis
Date of Examination:2015-10-16
Date of issue:2015-12-17
Advisor:Prof. Dr. Ralf Meyer
Referee:Prof. Dr. Ralf Meyer
Referee:Prof. Dr. Thomas Schick
Persistent Address: http://dx.doi.org/10.53846/goediss-5429

 

 

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Abstract

English

In this thesis, we use the tools of relative homological algebra in triangulated categories to define a sensible notion of support for objects in the bootstrap class of a Kasparov category of C*-algebras over a finite topological space with totally ordered lattice of open subsets. This category is equivalent to a bootstrap category of filtrations of C*-algebras. As a consequence, we provide a full classification of localizing subcategories of the bootstrap category in terms of a product of lattices of noncrossing partitions of a regular polygon. In addition, we consider the 2-periodic derived category of countable modules over the ring of upper triangular matrices. Since the homological algebra is the same, the lattices of localizing subcategories in this category and the bootstrap category are isomorphic. 
Keywords: C*-algebra; Kasparov category; homological algebra; triangulated category; non-commutative topology; filtrations of C*-algebras; C*-algebras over a topological space; localizing subcategory; localising subcategory; Bivariant K-theory; Filtrated K-theory; relative homological algebra; bootstrap class; KK-theory; ring of upper triangular matrices
 
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