Periodic dimensions and some homological properties of eventually periodic algebras (Q6588183)

In the paper under review there are studied the so called eventually periodic modules, by introducing a new invariant called the periodic dimension. In order to be more precise, let \(R\) be a left noetherian ring. If \(R\) is supposed in addition to be left (semi)perfect, then any left (finitely generated) \(R\)-module \(M\) has a projective cover, henceforth we can consider its minimal projective resolution \(P_\bullet\to M\), which is unique up to isomorphism in the category of complexes. The \(n\)-th syzygy of \(M\), denoted \(\Omega_R^n(M)\), is defined as usual to be the kernel of the \((n-1)\)-th differential \(d_{n-1}: P_{n-1}\to P_{n-2}\) of \(P_\bullet\) (which is equal, by the exactness of \(P_\bullet\), with the image of \(d_n:P_n\to P_{n-1}\)). The module \(M\) is called periodic if there is \(p>0\) such that \(M\cong\Omega_R^p(M)\); the smallest positive integer \(p>0\) with this property is called the period of \(M\). The module \(M\) is called eventually periodic, if there is \(n\geq0\) such that \(\Omega_R^n(M)\) is periodic, e.g., a module of finite projective dimension is eventually periodic with the period \(p=1\). The periodic dimension of \(M\), denoted \(\mathrm{per.dim}_RM\), is the degree \(n\) of the first periodic syzygy of \(M\), or \(\infty\) if such a periodic syzygy does not exist. The first main result in the paper under review says that if \(R\) is a noetherian (semi)perfect ring and \(M\) is a (finitely generated) \(R\)-module of Gorenstein projective dimension \(r<\infty\), then \(r\leq\mathrm{per.dim}_RM\leq r+1\). This result specializes further to the case of an eventually periodic \(d\)-Gorenstein algebra, whose periodic dimension is either \(d\) or \(d+1\). The paper ends with some applications, where it is shown that some important homological conjectures, e.g., the finitistic dimension conjecture or the Gorenstein symmetric conjecture, hold for eventually periodic algebras.