Models for singularity categories

By Hanno Becker. 2012

104 citations recorded by the source corpus.

In this article we construct various models for singularity categories of modules over differential graded rings. The main technique is the connection between abelian model structures, cotorsion pairs and deconstructible classes, and our constructions are based on more general results about localization and transfer of abelian model structures. We indicate how recollements of triangulated categories can be obtained model categorically, discussing in detail Krause's recollement for the stable derived category. In the special case of curved mixed Z-graded complexes, we show that one of our singular models is Quillen equivalent to Positselski's contraderived model for the homotopy category of matrix factorizations.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cdbddd11437807be1294c3b5becf140200a3c53c

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In Algebra and Number Theory

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Citations in the corpus

Hereditary abelian model categories (2015, methodology): “Hanno Becker made this precise in an appendix of his PhD thesis [6].”
How to construct a Hovey triple from two cotorsion pairs (2014, background): “ry. The author again wishes to thank Hanno Becker and the referee of his paper [Gil13]. The construction of Hovey triples that we give here is a direct generalization of a construction of Becker from [Bec14]. In particular, when the two given cotorsion pairs are injective, then our construction is e”
The stable module category of a general ring (2014, methodology): “The other half can also be interpreted in terms of model structures; this has been done very nicely in [Bec12].”
On purity and applications to coderived and singularity categories (2014, methodology): “ C(G) is called coacyclic, if Ext1 C(G)(X,Y ) = 0 for each Y ∈ C(InjG). We denote the class of all coacyclic complexes by C coac(G). Remark 6.8. Thisdeﬁnition perfectly agrees with the terminology of [Bec14]. For a locally noetherianG it matches also theoriginal andsomewhat diﬀerent deﬁnition of Pos”
Exact model categories, approximation theory, and cohomology of quasi-coherent sheaves (2013, background): “Interestingly, although model structures in connection with singularity categories are first explicitly mentioned in [6] by Becker, Murfet [56] and Neeman [57, 58, 59] still implicitly use parts of the theory which we are going to present.”