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.

中文翻译：

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$$d\mathbb {Z}$$-奇点类别的聚类倾斜子类别

对于具有足够投影和 $$d\mathbb {Z}$$ d Z -cluster 倾斜子类别的精确类别 $${{\mathcal {E}}}$$ E，我们证明了 $$ 的奇点类别{{\mathcal {E}}}$$ E 承认 $$d\mathbb {Z}$$ d Z -cluster 倾斜子类别。为此，我们引入了左三角化类别的聚类倾斜子类别，并且我们表明 $${{\mathcal {E}}}$$ E 和 $${\underline{{{\数学 {E}}}}}$$ E ̲ . 我们还推导出 $${{\mathcal {E}}}$$ E 的 Gorenstein 射影在某些假设下承认 $$d\mathbb {Z}$$ d Z -cluster 倾斜子范畴。最后，我们计算了非 Iwanaga-Gorenstein 的有限维代数的奇点范畴的 $$d\mathbb {Z}$$ d Z -cluster 倾斜子范畴。