$$d\mathbb {Z}$$-Cluster tilting subcategories of singularity categories

Abstract

For an exact category $${{\mathcal {E}}}$$ E with enough projectives and with a $$d\mathbb {Z}$$ d Z -cluster tilting subcategory, we show that the singularity category of $${{\mathcal {E}}}$$ E admits a $$d\mathbb {Z}$$ d Z -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 $${{\mathcal {E}}}$$ E and $${\underline{{{\mathcal {E}}}}}$$ E ̲ . We also deduce that the Gorenstein projectives of $${{\mathcal {E}}}$$ E admit a $$d\mathbb {Z}$$ d Z -cluster tilting subcategory under some assumptions. Finally, we compute the $$d\mathbb {Z}$$ d Z -cluster tilting subcategory of the singularity category for a finite-dimensional algebra which is not Iwanaga–Gorenstein.

Higher Ideal Approximation Theory

“…As an example of the above discussion, set A = mod-Λ and let B = Gprj-Λ be the full subcategory of A consisting of all Gorenstein-projective Λ-modules. By [AHS,Theorem 3.16], if C is an n-cluster tilting subcategory of mod-Λ with enough injectives, then C ∩ Gprj-Λ is an n-cluster tilting subcategory with enough injectives of Gprj-Λ, see also [Kv,Theorem 7.3]. Hence in this case, Ext n C ∩Gprj-Λ is the same as the functor Ext n Gprj-Λ on C ∩ Gprj-Λ.…”

Higher Auslander correspondence for exact categories

“…Remark 2.7. Let E be an exact category and M be a full subcategory of E. By the proof of [24,Proposition 4…”