By Apostolos Beligiannis
During this paper the authors examine homological and homotopical points of an idea of torsion that's basic sufficient to hide torsion and cotorsion pairs in abelian different types, $t$-structures and recollements in triangulated different types, and torsion pairs in solid different types. the right kind conceptual framework for this learn is the final atmosphere of pretriangulated different types, an omnipresent category of additive different types such as abelian, triangulated, good, and extra commonly (homotopy different types of) closed version different types within the experience of Quillen, as designated instances. the main target in their examine is at the research of the robust 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 triggered via torsion pairs however. those new common cohomology theories supply a traditional generalization of the Tate-Vogel (co)homology conception. The authors additionally examine the connections among torsion theories and closed version constructions, which permit them to categorise all cotorsion pairs in an abelian classification and all torsion pairs in a strong type, in homotopical phrases. for example they receive a class of (co)tilting modules alongside those strains. eventually they offer torsion theoretic functions to the constitution of Gorenstein and Cohen-Macaulay different types, which offer a usual generalization of Gorenstein and Cohen-Macaulay jewelry.