By Apostolos Beligiannis

During this paper the authors examine homological and homotopical features of an idea of torsion that's normal adequate to hide torsion and cotorsion pairs in abelian different types, $t$-structures and recollements in triangulated different types, and torsion pairs in reliable different types. the right kind conceptual framework for this research is the overall atmosphere of pretriangulated different types, an omnipresent category of additive different types inclusive of abelian, triangulated, good, and extra normally (homotopy different types of) closed version different types within the feel of Quillen, as certain situations. the main target in their research is at the research of the powerful connections and the interaction among (co)torsion pairs and tilting thought in abelian, triangulated and strong different types on one hand, and common cohomology theories caused by means of torsion pairs nonetheless. those new common cohomology theories offer a average generalization of the Tate-Vogel (co)homology concept. The authors additionally learn the connections among torsion theories and closed version constructions, which enable them to categorise all cotorsion pairs in an abelian classification and all torsion pairs in a sturdy type, in homotopical phrases. for example they receive a class of (co)tilting modules alongside those strains. eventually they provide torsion theoretic purposes to the constitution of Gorenstein and Cohen-Macaulay different types, which supply a ordinary generalization of Gorenstein and Cohen-Macaulay jewelry.

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FC (γ) Σ(X ) ⊆ X and if XC −−→ C → − B→ − Σ(XC ) is a triangle in ∇, then C(XC , Ω(B)) = 0. (iii) The pair (X , Y) is a torsion pair in C. Proof. We prove that (i) ⇔ (iii). The proof of (ii) ⇔ (iii) is similar. 3, it remains to show that (i) ⇒ (iii). For any C ∈ C, let fC : R(C) → − C be the counit of the adjoint pair (i, R), and fC gC hC β α let (∗): R(C) −−→ C −−→ Y C −−→ ΣR(C) be a triangle in ∇. If Ω(Y C ) − →A− → g C C −−→ Y C is a triangle in ∆, then since fC ◦ g C = 0, there exists a morphism κ : R(C) → − A such that: κ ◦ α = fC .

Define a functor F ∗ : X → − D by F ∗ = F i. Then F ∗ is obviously left exact and the above isomorphism shows that ∼ = → F . If G : X → − D is another exact functor endowed with a natural F (f ) : F ∗ R − ∼ ∼ = = → F i = F ∗ . Hence F ∗ is the isomorphism ξ : GR − → F , then ξi : G ∼ = GRi − 5. LIFTING TORSION PAIRS 42 unique up to isomorphism left exact functor which extends F . Then R : C → − X represents X as the stabilization of C. Conversely assume that R represents X as the stabilization of C with respect to its left triangulation.

Hence f is a weak kernel of g. The parenthetical case is dual. 3. TORSION PAIRS 37 The following characterizes when a pair of subcategories forms a torsion pair. 7. Let (X , Y) be a pair of subcategories of a pretriangulated category C. If C(X , Y) = 0, then the following are equivalent: (i) (α) The inclusion i : X → C has a right adjoint R and the counit fC : R(C) → − C is a weak kernel, ∀C ∈ C. gC fC hC (β) If R(C) −−→ C −−→ Y C −−→ ΣR(C) is a triangle in ∇, then Y C ∈ Y. gC (γ) Ω(Y) ⊆ Y and if Ω(Y C ) → − A→ − C −−→ Y C is a triangle in ∆, then C C(Σ(A), Y ) = 0.

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Homological and Homotopical Aspects of Torsion Theories by Apostolos Beligiannis

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