A note on the Stanley-Reisner ring of a join and of a suspension
Let \(\Delta_ 1\) and \(\Delta_ 2\) be simplicial complexes, \(k[\Delta_ 1]\) and \(k[\Delta_ 2]\) their Stanley-Reisner rings over a field k, and \(\Delta_ 1*\Delta_ 2\) their join. The author proves the following theorem, which generalizes theorems of \textit{I. Pop} [Ann. Stiint. Univ. Al. I. Cuza Iaşi, N. Ser., Secţ. Ia 31, Suppl., 40-42 (1985; Zbl 0603.13018)] and \textit{C. Sava} [ibid. 31, 145-148 (1985; Zbl 0611.55007)]: \(k[\Delta_ 1\cdot \Delta_ 2]\) is Cohen-Macaulay [Gorenstein] if and only if \(k[\Delta_ 1]\) and \(k[\Delta_ 2]\) are both Cohen-Macaulay [Gorenstein]. The proof is purely algebraic and does not use the characterization of Cohen-Macaulay or Gorenstein complexes.
scientific article; zbMATH DE number 4042048