Weight structures and simple dg modules for positive dg algebras

By B. Keller, Pedro Nicolás. 2010

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Using techniques due to Dwyer-Greenlees-Iyengar we construct weight structures in triangulated categories generated by compact objects. We apply our result to show that, for a dg category whose homology vanishes in negative degrees and is semi-simple in degree 0, each simple module over the homology lifts to a dg module which is unique up to isomorphism in the derived category. This allows us, in certain situations, to deduce the existence of a canonical t-structure on the perfect derived category of a dg algebra. From this, we can obtain a bijection between hearts of t-structures and sets of so-called simple-minded objects for some dg algebras (including Ginzburg algebras associated to quivers with potentials). In three appendices, we elucidate the relation between Milnor colimits and homotopy colimits and clarify the construction of t-structures from sets of compact objects in triangulated categories as well as the construction of a canonical weight structure on the unbonded derived category of a non positive dg category.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/e4cc522612a456ac1558072952a5b24a001e67dd

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Citations in the corpus

Exchange graphs and Ext quivers (2011, background): “morphism S: Hom• D(L,M) −→∼ Hom• D(M,L) ∨[N]. (7.1) 28 ALASTAIR KING AND YU QIU An object Sin a CY-Ncategory is spherical if Hom•(S,S) = k⊕k[−N]. Note that, as N≥ 2, any spherical object is rigid. By [19] (see also [22],[26]), we know that D(ΓN Q) is a CY-N category, which admits a canonical heart H”
Silting reduction and Calabi--Yau reduction of triangulated categories (2014, methodology): “at any silting subcategory gives a co-t-structure on T. The following proposition is well-known, which was proved as [35, Theorem 5.5], see also [2, Proposition 2.22], [9, proof of Theorem 4.3.2] and [28]. Proposition 2.8. Let Pbe a silting subcategory of Twith P= addP. (a) Then (T≥0,T≤0) is a bound”
Realisation functors in tilting theory (2015, background): “This description covers the previous definitions of silting objects occurring in the literature ([1,8,48,50,51,78]).”
On compactly generated torsion pairs and the classification of co--structures for commutative noetherian rings (2012, background): “As in [43], we need to assume that the triangulated category in question is at the base of a stable derivator (see §2.”
Discreteness of silting objects and t ‐structures in triangulated categories (2017, unknown): “The following result appeared in an earlier version of [47].”
Direct limits in the heart of a t-structure: The case of a torsion pair☆ (2013, background): “urcia (04555/GERM/06), with a part of FEDER funds. The authors thank these institutions for their help. 1 approach has led to classiﬁcation results in many cases (see, e.g., [Br], [GKR], [AJS], [SS], [KN]...). A second line of research consists in starting with a well-behaved class of t-structures a”
Silting Objects (2018, background): “This interplay with t-structures led Psaroudakis and Vitória [90], and independently, Nicolás, Saoŕın, and Zvonareva [84], to develop the following concept where the requirement of compactness is dropped (cf. [90, Definition 4.1] and [84, Remark 3]).”