$$d\mathbb {Z}$$ d Z -Cluster tilting subcategories of singularity categories
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Mathematische Zeitschrift
dZ-Cluster tilting subcategories of singularity categories Sondre Kvamme1 Received: 4 September 2018 / Accepted: 26 March 2020 © The Author(s) 2020
Abstract For an exact category E with enough projectives and with a dZ-cluster tilting subcategory, we show that the singularity category of E admits a dZ-cluster tilting subcategory. To do this we introduce cluster tilting subcategories of left triangulated categories, and we show that there is a correspondence between cluster tilting subcategories of E and E . We also deduce that the Gorenstein projectives of E admit a dZ-cluster tilting subcategory under some assumptions. Finally, we compute the dZ-cluster tilting subcategory of the singularity category for a finite-dimensional algebra which is not Iwanaga–Gorenstein. Keywords Cluster tilting · Exact category · Homological algebra · Singularity category · Triangulated category Mathematics Subject Classification 18E10 · 18E30 · 16E65
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Exact categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Left triangulated categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Cluster tilting subcategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Cluster tilting subcategories of left triangulated categories . . . . . . . . . . . . . . . . . . . . . . . . 6 d-cluster tilting in stable categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Gorenstein projectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The author was supported by a public grant as part of the FMJH. He would like to thank the anonymous referee for several useful comments and corrections.
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Sondre Kvamme [email protected] Department of Mathematics, Uppsala University, 75106 Uppsala, Sweden
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S. Kvamme
1 Introduction Auslander–Reiten theory is a fundamental tool to describe the module category of finitedimensional algebras, see [7] and [4,44,45]. A generalization of this theory, called higher Auslander–Reiten theory, was introduced by Iyama in [25] and further developed in [24,27]. In this case, the objects of study are module categories equipped with a d-cluster tilting subcategory. We refer to [2,16,21–23,29–33,35,37] for some other important papers. Also, see [26] for a survey of the theory and [36] for an introduction. Let be a finite-dimensional algebra and let mod be the category of finitely generated right -modules. Assume has global dimension d. If M is a d-cluster tilting subcategory in mod then the subcategory U = add{M[di] ∈ D b (mod ) | M ∈ M and i ∈ Z}
is d-c
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