The rate of a standard graded K-algebra A is a measure of the growth of the shifts in a minimal free resolution of K as an A-module. In particular A has rate one if and only if it is Koszul. It is known that a generic Artinian Gorenstein algebra of embedding dimension n >= 3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \ge 3$$\end{document} and socle degree s=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=3$$\end{document} is Koszul. We prove that a generic Artinian Gorenstein algebra with n >= 4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 4$$\end{document} and s >= 3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \ge 3 $$\end{document} has rate s2.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\lfloor \frac{s}{2} \right\rfloor .$$\end{document} In the process we show that such an algebra is generated in degree s2+1.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\lfloor \frac{s}{2} \right\rfloor +1.$$\end{document} This gives a partial positive answer to a longstanding conjecture stated by the first author on the minimal free resolution of a generic Artinian Gorenstein ring of odd socle degree.
On the rate of generic Gorenstein K-algebras
De Negri E.;De Stefani A.;Rossi M. E.
2025-01-01
Abstract
The rate of a standard graded K-algebra A is a measure of the growth of the shifts in a minimal free resolution of K as an A-module. In particular A has rate one if and only if it is Koszul. It is known that a generic Artinian Gorenstein algebra of embedding dimension n >= 3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \ge 3$$\end{document} and socle degree s=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=3$$\end{document} is Koszul. We prove that a generic Artinian Gorenstein algebra with n >= 4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 4$$\end{document} and s >= 3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \ge 3 $$\end{document} has rate s2.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\lfloor \frac{s}{2} \right\rfloor .$$\end{document} In the process we show that such an algebra is generated in degree s2+1.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\lfloor \frac{s}{2} \right\rfloor +1.$$\end{document} This gives a partial positive answer to a longstanding conjecture stated by the first author on the minimal free resolution of a generic Artinian Gorenstein ring of odd socle degree.
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