On the problem of spectra of solvability of varieties of modules (Q1907427)

Let \({\mathfrak M}\) be a variety of left modules (over a fixed associative \(R\) with 1) and \(A\) an \(R\)-module. Then \({\mathfrak M}(A)\) is the smallest submodule of \(A\) in the set of all submodules for which the quotient of \(A\) by the submodules belongs to \({\mathfrak M}\). For any ordinal \(\alpha\) the author defines a submodule \({\mathfrak M}^\alpha (A)\) as follows: \({\mathfrak M}^0(A)=A\), \({\mathfrak M}^{\alpha+1}(A)={\mathfrak M}({\mathfrak M}^\alpha(A))\) and \({\mathfrak M}^\alpha(A)=\bigcap_{\beta<\alpha}{\mathfrak M}^\beta(A)\), if \(\alpha\) is a limit ordinal. The smallest ordinal \(\gamma\) such that \({\mathfrak M}^\gamma(A)=0\) is called the \({\mathfrak M}\)-solvability level of \(A\) and is denoted by \(\text{step}({\mathfrak M},A)\). The \(S\)-spectrum of \({\mathfrak M}\) in a class \({\mathfrak N}\) of modules is the class \(\text{Spec}({\mathfrak M},{\mathfrak N})\) of all ordinals \(\text{step}({\mathfrak M},A)\) where \(A\in{\mathfrak N}\). The author answers the following question in the negative: Is it true that for any variety \({\mathfrak M}\) and any subvariety \({\mathfrak N}\sqsubseteq{\mathfrak M}\), \(\text{spec}({\mathfrak M},{\mathfrak N})\) is either finite or unbounded? Namely, he proves that for any variety \({\mathfrak M}\) of modules over a local left upper-perfect ring and for any subvariety \({\mathfrak N}\sqsubseteq{\mathfrak M}\) the \(S\)-spectrum \(\text{Spec}({\mathfrak M},{\mathfrak N})\) is one of the segments \(W(n)\) (\(0<n<\omega\)), \(W(\omega+2)\) and each of these sets is the \(S\)-spectrum for some \({\mathfrak M}\), \({\mathfrak N}\) and \(R\).