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## Stratifying triangulated categories (2009)

Citations: | 18 - 5 self |

### Citations

216 |
The Grothendieck duality theorem via BousfieldÕs techniques and Brown representability,
- Neeman
- 1996
(Show Context)
Citation Context ...morphisms between each pair of objects in the Verdier quotient T/S form a set. The quotient functor Q : T → T/S preserves coproducts, and a standard argument based on Brown’s representability theorem =-=[23, 27]-=- yields an exact right adjoint Qρ. Note that Qρ is fully faithful; see [17, Proposition I.1.3]. It follows that the composite L = QρQ is a localization functor satisfying KerL = S; see [7, Lemma 3.1].... |

189 |
Axiomatic stable homotopy theory,
- Hovey, Palmieri, et al.
- 1997
(Show Context)
Citation Context ...and could be tracked easily under changes of categories. This is the notion of stratification introduced in [8] for tensor triangulated categories, inspired by work of Hovey, Palmieri, and Strickland =-=[19]-=-. For the stable module category of G, this condition yields a parameterization of localizing subcategories reminiscent of, and enhancing, Quillen stratification [30] of the cohomology algebra of G, w... |

188 | Calculus of fractions and homotopy theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, - Gabriel, Zisman |

186 |
Triangulated categories,
- Neeman
- 2001
(Show Context)
Citation Context ...X2 f2 −→ X3 f3 −→ · · · be a sequence of morphisms in T. Its homotopy colimit, denoted hocolimXn, is defined by an exact triangle⊕ n>1 Xn θ −→ ⊕ n>1 Xn −→ hocolimXn −→ where θ is the map (id−fn); see =-=[11]-=-. Now fix a homogeneous element r ∈ R of degree d. For each X in T and each integer n set Xn = Σ ndX and consider the commuting diagram X r X r2 X r3 · · · X1 r // X2 r // X3 r // · ... |

181 | Representations and cohomology - Benson - 1998 |

178 |
The spectrum of an equivariant cohomology ring
- Quillen
- 1971
(Show Context)
Citation Context ...f Hovey, Palmieri, and Strickland [19]. For the stable module category of G, this condition yields a parameterization of localizing subcategories reminiscent of, and enhancing, Quillen stratification =-=[30]-=- of the cohomology algebra of G, whence the name. In this work we present a notion of stratification for any compactly generated triangulated category T, and establish a number of consequences which f... |

113 | The connection between the K-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of - Neeman - 1992 |

86 | Deriving DG categories, Ann - Keller - 1994 |

85 |
Deriving DG categories”, Ann. Sci. École Norm
- Keller
- 1994
(Show Context)
Citation Context ...morphisms between each pair of objects in the Verdier quotient T/S form a set. The quotient functor Q : T → T/S preserves coproducts, and a standard argument based on Brown’s representability theorem =-=[23, 27]-=- yields an exact right adjoint Qρ. Note that Qρ is fully faithful; see [17, Proposition I.1.3]. It follows that the composite L = QρQ is a localization functor satisfying KerL = S; see [7, Lemma 3.1].... |

77 |
The Varieties and the Cohomology Ring of a Module
- Carlson
- 1983
(Show Context)
Citation Context ...port varieties has played an increasingly important role in various aspects of representation theory. The original context was Carlson’s support varieties for modular representations of finite groups =-=[12]-=-, but the method soon spread to restricted Lie algebras [14], complete intersections in commutative algebra [1, 2], Hochschild cohomological support for certain finite dimensional algebras [13], and f... |

76 | Support varieties and cohomology over complete intersections,
- Avramov, Buchweitz
- 2000
(Show Context)
Citation Context ...al context was Carlson’s support varieties for modular representations of finite groups [12], but the method soon spread to restricted Lie algebras [14], complete intersections in commutative algebra =-=[1, 2]-=-, Hochschild cohomological support for certain finite dimensional algebras [13], and finite group schemes [15, 16]. One of the themes in this development has been the classification of thick or locali... |

64 |
Support varieties for restricted Lie algebras
- Friedlander, Parshall
- 1986
(Show Context)
Citation Context ...various aspects of representation theory. The original context was Carlson’s support varieties for modular representations of finite groups [12], but the method soon spread to restricted Lie algebras =-=[14]-=-, complete intersections in commutative algebra [1, 2], Hochschild cohomological support for certain finite dimensional algebras [13], and finite group schemes [15, 16]. One of the themes in this deve... |

58 | Local cohomology and support for triangulated categories
- Benson, Iyengar, et al.
(Show Context)
Citation Context ...s for T. The context is that we are given an action of a graded commutative ring R on T, namely a map from R to the graded center of T. We write SpecR for the set of homogeneous prime ideals of R. In =-=[7]-=- we developed a theory of support for objects in T, based on a construction of exact functors Γp : T→ T for each p ∈ SpecR, which are analogous to local cohomology functors from commutative algebra. T... |

55 |
Modules of finite virtual projective dimension,
- Avramov
- 1989
(Show Context)
Citation Context ...al context was Carlson’s support varieties for modular representations of finite groups [12], but the method soon spread to restricted Lie algebras [14], complete intersections in commutative algebra =-=[1, 2]-=-, Hochschild cohomological support for certain finite dimensional algebras [13], and finite group schemes [15, 16]. One of the themes in this development has been the classification of thick or locali... |

52 | Les foncteurs dérivés de lim ←− et leurs applications en théorie des modules - Jensen - 1972 |

51 |
Thick subcategories of the stable module category
- Benson, Carlson, et al.
- 1997
(Show Context)
Citation Context ...2 DAVE BENSON, SRIKANTH B. IYENGAR, AND HENNING KRAUSE a notion of support for complexes living in the prime ideal spectrum of R. Somewhat later came the classification by Benson, Carlson and Rickard =-=[6]-=- of the thick subcategories of the stable module category of finite dimensional representation of a finite group G in terms of the spectrum of its cohomology ring. In [8] we established an analogous c... |

42 |
The chromatic tower for D(R)’, Topology 31
- Neeman
- 1992
(Show Context)
Citation Context ...ries of representations. This story started with Hopkins’ classification [18] of thick subcategories of the perfect complexes over a commutative Noetherian ring R, followed by Neeman’s classification =-=[25]-=- of localizing subcategories of the full derived category of R; both involved 2010 Mathematics Subject Classification. 18G99 (primary); 13D45, 18E30, 20J06, 55P42. Key words and phrases. localizing su... |

40 |
Complexity and varieties for infinitely generated modules
- Benson, Carlson, et al.
(Show Context)
Citation Context ...ne has X ∈ LocT({ΓpX | p ∈ SpecR}). (3) For any X ∈ T and any specialization closed subset V of SpecR, one has ΓVX ∈ LocT({ΓpX | p ∈ V}) . (4) For any X ∈ T, one has LocT(X) = LocT({Xp | p ∈ SpecR}). =-=(5)-=- For any X ∈ T and any localizing subcategory S of T, one has X ∈ S ⇐⇒ Xp ∈ S for each p ∈ SpecR . (6) For any X ∈ T, one has X ∈ LocT({Xp | p ∈ SpecR}). The proof uses some results which may also be ... |

40 |
Global methods in homotopy theory. Homotopy theory
- Hopkins
- 1985
(Show Context)
Citation Context ... the themes in this development has been the classification of thick or localizing subcategories of various triangulated categories of representations. This story started with Hopkins’ classification =-=[18]-=- of thick subcategories of the perfect complexes over a commutative Noetherian ring R, followed by Neeman’s classification [25] of localizing subcategories of the full derived category of R; both invo... |

38 | Support varieties for selfinjective algebras
- Erdmann, Holloway, et al.
(Show Context)
Citation Context ...groups [12], but the method soon spread to restricted Lie algebras [14], complete intersections in commutative algebra [1, 2], Hochschild cohomological support for certain finite dimensional algebras =-=[13]-=-, and finite group schemes [15, 16]. One of the themes in this development has been the classification of thick or localizing subcategories of various triangulated categories of representations. This ... |

36 |
Representations and Cohomology I, Cambridge Studies in Advanced Mathematics 30, Cambridge Univ.
- Benson
- 1991
(Show Context)
Citation Context ...as X ∈ S ⇐⇒ ΓpX ∈ S for each p ∈ SpecR . (2) For any X ∈ T, one has X ∈ LocT({ΓpX | p ∈ SpecR}). (3) For any X ∈ T and any specialization closed subset V of SpecR, one has ΓVX ∈ LocT({ΓpX | p ∈ V}) . =-=(4)-=- For any X ∈ T, one has LocT(X) = LocT({Xp | p ∈ SpecR}). (5) For any X ∈ T and any localizing subcategory S of T, one has X ∈ S ⇐⇒ Xp ∈ S for each p ∈ SpecR . (6) For any X ∈ T, one has X ∈ LocT({Xp ... |

32 | Noncrossing partitions and representations of quivers,
- Ingalls, Thomas
- 2009
(Show Context)
Citation Context ...)(X) ∼= k for each indecomposable object X . The localizing subcategories of D(kQ) are parameterized by the noncrossing partitions associated to Q; this can be deduced from work of Ingalls and Thomas =-=[20]-=-. Thus the triangulated category D(kQ) is stratified by some ring acting on it if and only if the quiver consists of one vertex and has no arrows. 5. Orthogonality Let X and Y be objects in T. The dis... |

31 | Homology of perfect complexes
- Avramov, Buchweitz, et al.
(Show Context)
Citation Context ...as desired. Proposition 2.11. Let a be an ideal in R. For each object X in T the following statements hold: (1) X//a is in ThickT(ΓV(a)X) and ΓV(a)X is in LocT(X//a); (2) LocT(X//a) = LocT(ΓV(a)X); =-=(3)-=- ΓV(a)X and LV(a)X are in LocT(X). Proof. (1) By construction X//a is in ThickT(X). As ΓV(a) is an exact functor, one obtains that ΓV(a)(X//a) is in ThickT(ΓV(a)X). This justifies the first claim in (... |

30 | Stratifying modular representations of finite groups
- Benson, Iyengar, et al.
(Show Context)
Citation Context ...opy category of complexes of injective kG-modules, K(Inj kG), is a compactly generated tensor triangulated category with a canonical action of the cohomology ring H∗(G, k). One of the main results of =-=[8]-=-, Theorem 9.7, is that K(Inj kG) is stratified by this action. The same is true also of the stable module category StMod kG; see [8, Theorem 10.3]. 2For these results to hold, the R action should be c... |

26 | Π-supports for modules for finite group schemes
- Friedlander, Pevtsova
(Show Context)
Citation Context ...n spread to restricted Lie algebras [14], complete intersections in commutative algebra [1, 2], Hochschild cohomological support for certain finite dimensional algebras [13], and finite group schemes =-=[15, 16]-=-. One of the themes in this development has been the classification of thick or localizing subcategories of various triangulated categories of representations. This story started with Hopkins’ classif... |

20 | Dimensions of triangulated categories via Koszul objects - Bergh, Iyengar, et al. |

9 |
Representation theoretic support spaces for finite group schemes
- Friedlander, Pevtsova
(Show Context)
Citation Context ...n spread to restricted Lie algebras [14], complete intersections in commutative algebra [1, 2], Hochschild cohomological support for certain finite dimensional algebras [13], and finite group schemes =-=[15, 16]-=-. One of the themes in this development has been the classification of thick or localizing subcategories of various triangulated categories of representations. This story started with Hopkins’ classif... |

6 | Quillen stratification for the Steenrod algebra
- Palmieri
- 1999
(Show Context)
Citation Context ...tion, notably, modules over cocommutative Hopf algebras and modules over the Steenrod algebra, where analogues of Quillen stratification have been proved by Friedlander and Pevtsova [15] and Palmieri =-=[29]-=- respectively. One goal of [7] and the present work is to pave the way to such results. Acknowledgments. It is our pleasure to thank Zhi-Wei Li for a critical reading of an earlier version of this man... |

2 | Noncrossing partitions and representations of quivers, arXiv: math/0612219 - Ingalls, Thomas |

2 |
Thick subcategories and localizing subcategories of derived categories, lectures at workshop “Algebraic triangulated categories and related topics
- Iyengar
- 2009
(Show Context)
Citation Context ... is inspired by an analogous statement for modules over complete intersection local rings, due to Avramov and Buchweitz [2]. A stratification theorem is not yet available in this context; see however =-=[21]-=-. The stratification condition also implies that Ravenel’s ‘telescope conjecture’ [31], sometimes called the ‘smashing conjecture’, holds for T. Theorem 1.4. If T is noetherian and stratified by R and... |

1 | Cosupport and colocalising subcategories of modules over finite groups, in preparation - Benson, Iyengar, et al. |

1 |
Localization and periodicity in homotopy theory, Homotopy Theory
- Ravenel
- 1985
(Show Context)
Citation Context ...ngs, due to Avramov and Buchweitz [2]. A stratification theorem is not yet available in this context; see however [21]. The stratification condition also implies that Ravenel’s ‘telescope conjecture’ =-=[31]-=-, sometimes called the ‘smashing conjecture’, holds for T. Theorem 1.4. If T is noetherian and stratified by R and L : T→ T is a localization functor that preserves arbitrary coproducts, then the loca... |

1 | Colocalising subcategories and cosupport - Benson, Iyengar, et al. |