We give a sufficient condition for an Ext-finite triangulated category to be saturated. Saturatedness means that every contravariant cohomological functor of finite type to vector spaces is representable. The condition consists in the existence of a strong generator. We prove that the bounded derived categories of coherent sheaves on smooth proper commutative and noncommutative varieties have strong generators, and are hence saturated. In contrast, the similar category for a smooth compact analytic surface with no curves is not saturated. 2000 Math. Subj. Class. Primary 18E30.

Generators and representability of functors in commutative and noncommutative geometry

We construct many examples of nonslice knots in 3-space that cannot be distinguished from slice knots by previously known invariants. Using Whitney towers in place of embedded disks, we define a geometric filtration of the 3-dimensional topological knot concordance group. The bottom part of the filtration exhibits all classical concordance invariants, including the CassonGordon invariants. As a first step, we construct an infinite sequence of new obstructions that vanish on slice knots. These take values in the L-theory of skew fields associated to certain universal groups. Finally, we use the dimension theory of von Neumann algebras to define an L 2 -signature and use this to detect the first unknown step in our obstruction theory.

Knot concordance, Whitney towers and L^2 signatures

We define a dimension for a triangulated category. We prove a representabilityTheorem for a certain class of functors on finite dimensional triangulatedcategories. We study the dimension of the boundedderived category of an algebra or a scheme and we show in particularthat the bounded derived category of coherent sheaves over avariety has a finite dimension. For a self-injective algebra, a lowerbound for Auslander's representation dimension is given by the dimensionof the stable category. We use this to compute the representationdimension of exterior algebras. This provides the first known examplesof representation dimension >3. We deduce that theLoewy length of the group algebra over F_2 of a finite group is strictly bounded below by2-rank of the group (a conjecture of Benson).

Dimensions of triangulated categories

These notes provide an introduction to the theory of localization for triangulated categories Localization is a machinery to formally invert morphisms in a category We explain this formalism in some detail and we show how it is applied to triangulated categories

pdf/localization-theory-for-triangulated-categories-2v9tta234q.pdf

Localization theory for triangulated categories

We prove a modified version of Ravenel's telescope conjecture. It is shown that every smashing subcategory of the stable homotopy category is generated by a set of maps between finite spectra. This result is based on a new characterization of smashing subcategories, which leads in addition to a classification of these subcategories in terms of the category of finite spectra. The approach presented here is purely algebraic; it is based on an analysis of pure-injective objects in a compactly generated triangulated category, and covers therefore also situations arising in algebraic geometry and representation theory.

Smashing subcategories and the telescope conjecture - an algebraic approach

Preface. Symbols. 1. Some Basic Facts of Ring Theory. 2. Regular and Strongly Regular Rings. 3. Rings of Bounded Index and Io-rings. 4. Semiregular and Weakly Regular Rings. 5. Max Rings and pi-regular Rings. 6. Exchange Rings and Modules. 7. Separative Exchange Rings. Bibliography. Index.

https://www.springer.com/productFlyer_978-1-4020-0851-1.pdf?SGWID=0-0-1297-33635801-0

Rings Close to Regular

Symbols radicals, local and semisimple modules projective and injective modules Bezout and regular modules continuous and finite-dimensional modules rings of quotients flat modules and semiperfect rings semihereditary and invariant rings endomorphism rings distributive rings with maximum conditions self-injective and skew-injective rings semidistributive and serial rings monoid rings and related topics.

Semidistributive Modules and Rings

Preface 1. Preliminaries 2. Rings characterized by their proper factor rings 3. Rings each of whose proper cyclic modules has a chain condition 4. Rings each of whose cyclic modules is injective (or CS) 5. Rings each of whose proper cyclic modules is injective 6. Rings each of whose simple modules is injective (or -injective) 7. Rings each of whose (proper) cyclic modules is quasi-injective 8. Rings each of whose (proper) cyclic modules is continuous 9. Rings each of whose (proper) cyclic modules is pi-injective 10. Rings with cyclics @0-injective, weakly injective or quasi-projective 11. Hypercyclic, q-hypercyclic and pi-hypercyclic rings 12. Cyclic modules essentially embeddable in free modules 13. Serial and distributive modules 14. Rings characterized by decompositions of their cyclic modules 15. Rings each of whose modules is a direct sum of cyclic modules 16. Rings each of whose modules is an I0-module 17. Completely integrally closed modules and rings 18. Rings each of whose cyclic modules is completely integrally closed 19. Rings characterized by their one-sided ideals References Index

Cyclic Modules and the Structure of Rings

This work contains some new and known results on modules over formal matrix rings. The main results are presented with proofs.

Modules over formal matrix rings

1. Projective and Injective Modules 2. Bezout and Regular Modules 3. Continuous and Finite-Dimensional Modules 4. Classical Localizations 5. Flat Modules and Semiperfect Rings 6. Semihereditary Rings 7. Endomorphism Rings and Distributivity 8. Q max (A) and Nonsingular Rings 9. Polynomials, Series and Quaternions 10. Semidistributive Rings and Modules 11. Distributive Group and Monoid Rings 11. Rings Related to Commutative Rings 12. Self-injective and Skew-injective Rings 13. Radicals, Local and Semisimple Modules 14. Rings with Maximun Conditions

Distributive Modules and Related Topics

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