Stable Homotopy theory
Let be a triangulated category. A t-structure on is a pair of full subcategories
for all and the hom object is the zero object: ;
the subcategories are closed under suspension/desuspension: and .
For all objects there is a fiber sequence with and .
Given a t-structure, its heart is the intersection
(Higher Algebra, def. 18.104.22.168).
The heart of a stable -category is an abelian category.
(BBD 82, Higher Algebra, remark 22.214.171.124)
Application to spectral sequence
If a the heart of a t-structure on a stable (∞,1)-category with sequential limits is an abelian category, then the spectral sequence of a filtered stable homotopy type converges (see there).
Related Lab entries include Bridgeland stability?.
For triangulated categories
S. I. Gelfand, Yuri Manin, Methods of homological algebra, Nauka 1988, Springer 1998, 2003
Donu Arapura, Triangulated categories and -structures (pdf)
Alexander Beilinson, Joseph Bernstein, Pierre Deligne, Faisceaux pervers, Asterisque 100, Volume 1, 1982
- D. Abramovich, A. Polishchuk, Sheaves of t-structures and valuative criteria for stable complexes, J. reine angew. Math. 590 (2006), 89–130
- A. L. Gorodentsev, S. A. Kuleshov, A. N. Rudakov, t-stabilities and t-structures on triangulated categories, Izv. Ross. Akad. Nauk Ser. Mat. 68 (2004), no. 4, 117–150
- A. Polishchuk, Constant families of t-structures on derived categories of coherent sheaves, Moscow Math. J. 7 (2007), 109–134
- John Collins, Alexander Polishchuk, Gluing stability conditions, arxiv/0902.0323
For stable (∞,1)-categories
Revised on November 19, 2013 03:38:32
by Zoran Škoda