Refining thick subcategory theorems

Sunil K. Chebolu¹

¹ Department of Mathematics University of Western Ontario London, Ontario Canada, N6A 5B7

Fundamenta Mathematicae, Tome 189 (2006) no. 1, pp. 61-97

Résumé

We use a $K$-theory recipe of Thomason to obtain classifications of triangulated subcategories via refining some standard thick subcategory theorems. We apply this recipe to the full subcategories of finite objects in the derived categories of rings and the stable homotopy category of spectra. This gives, in the derived categories, a complete classification of the triangulated subcategories of perfect complexes over some commutative rings. In the stable homotopy category of spectra we obtain only a partial classification of the triangulated subcategories of the finite $p$-local spectra. We use this partial classification to study the lattice of triangulated subcategories. This study gives some new evidence for a conjecture of Adams that the thick subcategory $\mathbb C_2$ can be generated by iterated cofiberings of the Smith–Toda complex. We also discuss several consequences of these classification theorems.

Détail

DOI : 10.4064/fm189-1-5

Keywords: k theory recipe thomason obtain classifications triangulated subcategories via refining standard thick subcategory theorems apply recipe full subcategories finite objects derived categories rings stable homotopy category spectra gives derived categories complete classification triangulated subcategories perfect complexes commutative rings stable homotopy category spectra obtain only partial classification triangulated subcategories finite p local spectra partial classification study lattice triangulated subcategories study gives evidence conjecture adams thick subcategory mathbb generated iterated cofiberings smith toda complex discuss several consequences these classification theorems