Partial silting objects and smashing subcategories
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Mathematische Zeitschrift
Partial silting objects and smashing subcategories Lidia Angeleri Hügel1 · Frederik Marks2 · Jorge Vitória3 Received: 15 February 2019 / Accepted: 16 November 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract We study smashing subcategories of a triangulated category with coproducts via silting theory. Our main result states that for derived categories of dg modules over a non-positive differential graded ring, every compactly generated localising subcategory is generated by a partial silting object. In particular, every such smashing subcategory admits a silting t-structure. Keywords Smashing subcategory · Silting object · Partial silting object · t-Structure Mathematics Subject Classification 18E30 · 18E35 · 18E40
1 Introduction Smashing subcategories of a triangulated category T occur naturally as kernels of localisation functors which preserve coproducts. These subcategories have proved to be useful as they often induce a decomposition of T into smaller triangulated categories, yielding a so-called recollement of triangulated categories in the sense of [9]. A typical example of a smashing subcategory is given by a localising subcategory of T , i.e. a triangulated subcategory closed under coproducts, which is generated by a set of compact objects from T . The claim that all smashing subcategories of a compactly generated triangulated category are of this form is sometimes referred to as the Telescope conjecture, first stated for the stable homotopy category in Algebraic Topology (see [11,38]). Affirmative answers to the conjecture were provided, for example, in [29] and [25] for derived module categories of
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Lidia Angeleri Hügel [email protected] Frederik Marks [email protected] Jorge Vitória [email protected]
1
Dipartimento di Informatica - Settore di Matematica, Università degli Studi di Verona, Strada le Grazie 15 - Ca’ Vignal, 37134 Verona, Italy
2
Frederik Marks, Institut für Algebra und Zahlentheorie, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
3
Dipartimento di Matematica e Informatica, Università degli Studi di Cagliari, Palazzo delle Scienze, via Ospedale, 72, 09124 Cagliari, Italy
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commutative noetherian rings and of hereditary rings, respectively. On the other hand, Keller provided an example of a smashing subcategory in the derived category of modules over a (non-noetherian) commutative ring that does not contain a single non-trivial compact object from the ambient derived category (see [19]). So, although the Telescope conjecture does not hold in general, it is difficult to determine whether a given triangulated category satisfies it. In this article, we study smashing subcategories via silting theory. This approach is motivated by some recent work indicating that localisations of categories and rings are intrinsically related to silting objects (see [6,27,36,37]). In particular, it was shown in [27] that universal localisations of rings in the sense of [3
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