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On \(t\)-structures and torsion theories induced by compact objects. (English) Zbl 1006.18011
The authors present in this paper a new approach and a generalisation of J. Rickard’s theory of tilting complexes. Rickard’s theory describes equivalences between derived module categories. The authors study in the present paper the relation with \(t\)-structures of triangulated categories, torsion theories of abelian categories, and the consequences for tilting theory given by two-term tilting complexes. The restriction to two-term complexes seems to be motivated by Okuyama’s method for proving Broué’s conjecture. Okuyama’s method uses two-term tilting complexes, and is at the moment the most successful way to prove the conjecture in examples.
The authors first show that for a compact object \(C\) in a triangulated category so that for positive \(n\), \(\text{Hom}(C,C[n])=0\), \(C\) generates a \(t\)-structure so that \(\text{Hom}(C,-)\) induces an equivalence between its core and the endomorphism ring of \(C\). In case of the derived category of an abelian category \(A\) and \(C\) being a two-term complex, the authors show when \(C\) defines a torsion theory of \(A\). The condition is related to the condition of being a tilting complex, but is weaker. As a consequence, if this happens the authors deduce an equivalence between the endomorphism ring of \(C\) and the core of the \(t\)-structure, as well as a torsion theory for the core. Finally, the authors make the case of a module category of an algebra, in particular of an Artin algebra, more precise. They provide an explicit interesting example for these phenomenons.

18E30 Derived categories, triangulated categories (MSC2010)
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
18E40 Torsion theories, radicals
16G10 Representations of associative Artinian rings
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[1] M. Auslander, Coherent functors, in: Proceedings of the Conference on Categorical Algebra, La Jolla 1965, Springer, Berlin, 1966, pp. 189-231.
[2] Beilinson, A.A.; Bernstein, J.; Deligne, P., Faisceaux pervers, Astérisque, 100, (1982) · Zbl 0536.14011
[3] Bökstedt, M.; Neeman, A., Homotopy limits in triangulated categories, Compositio math., 86, 209-234, (1993) · Zbl 0802.18008
[4] Dickson, S.E., A torsion theory for abelian categories, Trans. amer. math. soc., 121, 233-235, (1966) · Zbl 0138.01801
[5] Happel, D.; Ringel, C.M., Tilted algebras, Trans. amer. math. soc., 274, 399-443, (1982) · Zbl 0503.16024
[6] Holm, T., Derived equivalence classification of algebras of dihedral, semidihedral, and quaternion type, J. algebra, 211, 159-205, (1999) · Zbl 0932.16009
[7] Hoshino, M., Tilting modules and torsion theories, Bull. London math. soc., 14, 334-336, (1982) · Zbl 0486.16019
[8] Hoshino, M., On splitting torsion theories induced by tilting modules, Commun. algebra, 11, 4, 427-439, (1983) · Zbl 0506.16018
[9] M. Hoshino, Y. Kato, Tilting complexes defined by idempotents, preprint. · Zbl 1002.16004
[10] Neeman, A., The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. amer. math. soc., 9, 205-236, (1996) · Zbl 0864.14008
[11] Hartshorne, R., Residues and duality, Lecture notes in mathematics, vol. 20, (1966), Springer Berlin
[12] Rickard, J., Morita theory for derived categories, J. London math. soc., 39, 2, 436-456, (1989) · Zbl 0642.16034
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