Homological and Homotopical Aspects of Torsion Theories by Apostolos Beligiannis

By Apostolos Beligiannis

During this paper the authors examine homological and homotopical elements of an idea of torsion that is normal sufficient to hide torsion and cotorsion pairs in abelian different types, $t$-structures and recollements in triangulated different types, and torsion pairs in strong different types. the correct conceptual framework for this learn is the final environment of pretriangulated different types, an omnipresent category of additive different types inclusive of abelian, triangulated, strong, and extra commonly (homotopy different types of) closed version different types within the feel of Quillen, as distinctive situations. the main target in their examine is at the research of the powerful connections and the interaction among (co)torsion pairs and tilting idea in abelian, triangulated and good different types on one hand, and common cohomology theories brought about via torsion pairs nevertheless. those new common cohomology theories offer a common generalization of the Tate-Vogel (co)homology concept. The authors additionally learn the connections among torsion theories and closed version constructions, which permit them to categorise all cotorsion pairs in an abelian type and all torsion pairs in a solid classification, in homotopical phrases. for example they receive a category of (co)tilting modules alongside those strains. ultimately they offer torsion theoretic purposes to the constitution of Gorenstein and Cohen-Macaulay different types, which offer a ordinary generalization of Gorenstein and Cohen-Macaulay jewelry.

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Example text

If C(P, Σt (P)) = 0, ∀t ≥ 1, then C(P, Σt (C)) = 0, ∀t ≥ 1, for any C ∈ Add(Q) n and n ≥ 1. Using this lemma we get a description of the torsion class XP under an additional assumption. 7 below. 6. If C(P, Σn (P)) = 0, ∀n ≥ 1, then XP = Loc+ (P). Proof. Since XP is closed under coproducts, extensions, Σ and contains P, it follows by construction that Loc+ (P) ⊆ XP . 3, where C = Y−1 . Setting χ0 = f0 and Q0 = T0 , we can construct inductively a tower of objects τ0 τ1 T0 −→ T1 −→ T2 → − · · · and the following tower of triangles: χ0 hC g0 χ1 −−−−→ Y0 −−−−→ Σ(T0 )     g1 Σ(τ ) χ2 g0 ◦g1 ◦g2 T0 −−−−→ C   τ0 T1 −−−−→ C   τ 1 0 g0 ◦g1 −−−−→ Y1 −−−−→ Σ(T1 )     g2 Σ(τ ) 1 T2 −−−−→ C −−−−−−→ Y2 −−−−→ Σ(T2 )       ..

The proof of (2) is dual and is left to the reader. Note. It is easy to see that the above proof works in any pretriangulated category with coproducts, resp. products. This more general result will be useful in the study of the torsion pair induced by Cohen-Macaulay objects in the stable category of a Nakayama category, see the last chapter. Example. Let (T , F) be a torsion pair in a Grothendieck category G and assume that (T , F) is of finite type, or more generally that F is closed under coproducts.

6, R commutes with Ω, it follows easily that Ωn R(C) ∼ = RΩn (C). Then the morphism Ωn (fC ) : Ωn R(C) → − Ωn (C) is invertible. From the triangle ∆(C) above we infer that Ωn+1 (YC ) = 0. Hence for any object C in C there exists n ≥ 0, such that Ωn (YC ) = 0. Let now Y ∈ Y be an arbitrary object. h gY fY Y Then in the standard triangle Ω(Y ) −−→ YY −−→ R(Y ) −−→ Y we have that fY = 0. This implies trivially that YY = R(Y ) ⊕ Ω(Y ). Since R(Y ) ∈ X and YY ∈ Y we have R(Y ) = 0. Hence YY = Ω(Y ).

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