Relative Homological Algebra and Purity in Triangulated Categories

By Apostolos Beligiannis. 2000

213 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e30cca35512544a63e60259385d0d8c9d337c40e

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Citations in the corpus

Cohomological quotients and smashing localizations (2003, unknown): “The next definition is taken from [3].”
Subcategories of singularity categories via tensor actions (2011, unknown): “category R-Mod. Since the subcategory of compacts in R-Mod is precisely R-mod every object is thus a coproduct of compact objects (so in particular R-Mod is a pure-semisimple triangulated categorycf. [8] Corollary 12.26). It follows that every localizing subcategory of R-Mod is generated by objects ”
Cohen-Macaulay modules, (co)torsion pairs and virtually Gorenstein algebras (2005, background): “We refer to [16,42] for more details concerning purity in triangulated categories.”
On algebras of finite Cohen-Macaulay type (2011, methodology): “lim ÝÑ projΛ is compactly generated and phantomless if and only if the module category Mod-X is Frobenius [22].”
The phantom cover of a module (2007, unknown): “For a compactly generated triangulated category T , a theory of purity has been developed independently by A. Beligiannis [ 4 ,5] and H. Krause [21].”
Homological and Homotopical Aspects of Torsion Theories (2007, methodology): “Another class of categories in which the conjecture is true is the class of compactly generated triangulated categories C which admit filtered colimits, that is: the pure-semisimple categories of [21].”
Silting Objects (2018, background): “The theory of purity in a compactly generated triangulated category T has been developed around 2000 in [69, 27].”
Failure of Brown representability in derived categories (2000, background): “The pure global dimension of the derived category D(R) = T is by definition [5] the supremum of the projective dimensions of the functors yX, X ∈ T.”
On compactly generated torsion pairs and the classification of co--structures for commutative noetherian rings (2012, background): “8(2) in [10] used for F = T (−, Z) provides us with a short exact sequence of abelian groups”
On the Freyd categories of an additive category (2000, background): “For further results in this direction we refer to [12].”
Ideal approximation theory (2013, background): “In the stable category of a finite group ring, the theory of phantom morphisms was developed in a series of papers [16,6,7,5] by Benson and Gnacadja; the triangulated version of this notion of a phantom morphism was studied in [3,24].”
Homological algebra in bivariant K-theory and other triangulated categories. II (2008, unknown): “More specifically, the Baum-Connes assembly map for coactions of certain compact Lie groups, which is studied in [17], is always an isomorphism and it is closely related to a universal coefficient theorem for equivariant Kasparov theory by Jonathan Rosenberg and Claude Schochet ([22]).”
Proper classes and Gorensteinness in extriangulated categories (2019, background): “One of interesting examples, due to Krause [13] and Beligiannis [5], is as follows: • Assume that (T ,Σ,∆) is a compactly generated triangulated category, where Σ is the suspension functor and ∆ is the triangulation.”
An Auslander-Type result for Gorenstein-projective modules (2007, background): “These modules play a central role in the theory of singularity [10–12, 14] and of relative homological algebra [9,17].”
Torsion pairs in silting theory (2016, unknown): “T c is developed in [Krause 2000, §2.3] and [Beligiannis 2000, §8.1].”
Generalized Brown representability in homotopy categories (2005, unknown): “(ii) Whenever a compactly generated triangulated category is ω-Brown on morphisms then it is ω-Brown on objects (see [4], 11.”