Higher Auslander-Reiten Sequences Revisited

Abstract

Let (

\mathcal{C}\mathcal{,}\mathbb{E}\mathcal{,}\mathfrak{s}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathscr{C},\mathbb{E},\mathfrak{s}$$\end{document}

) be an n-exangulated category with enough projectives and enough injectives, and

\mathcal{X}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathscr{X}$$\end{document}

be a cluster-tilting subcategory of

\mathcal{C}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathscr{C}$$\end{document}

. Liu and Zhou have shown that the quotient category

\mathcal{C}\mathcal{/}\mathcal{X}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathscr{C}/\mathscr{X}$$\end{document}

is an n-abelian category. In this paper, we prove that if

\mathcal{C}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathscr{C}$$\end{document}

has Auslander–Reiten n-exangles, then

\mathcal{C}\mathcal{/}\mathcal{X}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathscr{C}/\mathscr{X}$$\end{document}

has Auslander–Reiten n-exact sequences. Moreover, we also show that if a Frobenius n-exangulated category

\mathcal{C}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathscr{C}$$\end{document}

has Auslander–Reiten n-exangles, then the stable category

\overline{\mathcal{C}}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\overline{\mathscr{C}}$$\end{document}

of

\mathcal{C}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathscr{C}$$\end{document}

has Auslander–Reiten (n + 2)-angles.

Keywords

n-exangulated category / n-abelian category / (n + 2)-angulated category / Auslander–Reiten sequence / 18G80 / 18E10

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