Compactly generated t-structures in the derived category of a commutative ring
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Mathematische Zeitschrift
Compactly generated t-structures in the derived category of a commutative ring Michal Hrbek1 Received: 7 August 2018 / Accepted: 7 May 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract We classify all compactly generated t-structures in the unbounded derived category of an arbitrary commutative ring, generalizing the result of Alonso Tarrío et al. (J Algebra 324(3):313–346, 2010) for noetherian rings. More specifically, we establish a bijective correspondence between the compactly generated t-structures and infinite filtrations of the Zariski spectrum by Thomason subsets. Moreover, we show that in the case of a commutative noetherian ring, any bounded below homotopically smashing t-structure is compactly generated. As a consequence, all cosilting complexes are classified up to equivalence. Keywords Commutative ring · Derived category · t-Structure · Cosilting complex Mathematics Subject Classification 13C05 · 13D09 · 13D30 · 18G35
1 Introduction There is a large supply of classification results for various subcategories of the unbounded derived category D(R) of a commutative noetherian ring R. Since the work of Hopkins [16] and Neeman and Bökstedt [30], we know that the localizing subcategories of D(R) are parametrized by data of a geometrical nature—the subsets of the Zariski spectrum Spec(R). As a consequence, the famous telescope conjecture, stating that any smashing localizing subcategory is generated by a set of compact objects, holds for the category D(R). These compactly generated localizations then correspond to those subsets of Spec(R), which are specialization closed, that is, upper subsets in the poset (Spec(R), ⊆). A more recent work [1] provides a “semistable” version of the latter classification result. Specifically, it establishes a bijection between compactly generated t-structures and infinite decreasing sequences of specialization closed subsets of Spec(R) indexed by the integers. The concept of a t-structure in a triangulated category was introduced by Be˘ılinson et al. [9], and can be seen as a way of constructing abelian categories inside triangulated categories, together with cohomological functors. In the setting of the derived category of a ring, t-
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Michal Hrbek [email protected] Institute of Mathematics CAS, Žitná 25, 115 67 Prague, Czech Republic
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structures proved to be an indispensable tool in various general instances of tilting theory, replacing the role played by the ordinary torsion pairs in more traditional tilting frameworks (see e.g. [6,26,31,34]). Not all the mentioned results carry well to the generality of an arbitrary commutative ring R (that is, without the noetherian assumption). Namely, the classification of all localizing subcategories via geometrical invariants is hopeless (see [29]), and the telescope conjecture does not hold in general, the first counterexample of this is due to Keller [19]. However, when restricting to the subcategories induced by compact objects, the situation is far more opti
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