On the length of generalized fractions of modules having polynomial type \(\leq 2\) (Q1277222)

This is a short presentation of the author's results on the structure of a finitely generated module \(M\) over a local ring which satisfies the condition \[ n_1\cdots n_de(x_1,\ldots,x_d;M)- \ell\big(M(1/(x_1^{n_1},\ldots,x_d^{n_d},1))\big) < \infty, \] for every system of parameters \(x_1,\ldots,x_d\) of \(M\) and all positive integers \(n_1,\ldots,n_d\), where \(M(1/(x_1^{n_1},\ldots,x_d^{n_d},1))\) is the module of generalized fractions introduced by \textit{R. Y. Sharp} and \textit{H. Zakeri} [Mathematika 29, 32-41 (1982; Zbl 0497.13006)]. The proofs of the presented results will appear in ``On the length of generalized fractions of modules having small polynomial type'' (Math. Proc. Camb. Philos. Soc.).