On the local cohomology of L-shaped integral FI-modules (Q2079228)

In representation stability, representations of the category of finite sets and injections, called FI-modules, were introduced by \textit{T. Church} et al. in [Duke Math. J. 164, No. 9, 1833--1910 (2015; Zbl 1339.55004)]. For an FI-module \(M\), its torsion submodule \(TM\) is defined to consist of those elements of \(M\) which are sent to \(0\) by inclusions into large enough finite sets. An important topic in representation stability is the calculation of local cohomology, i.e., the derived functors of the FI-torsion of a given FI-module. Rationally, this was solved by \textit{S. V. Sam} and \textit{A. Snowden} (see [Trans. Am. Math. Soc. 368, No. 2, 1097--1158 (2016; Zbl 1436.13012); Forum Math. Sigma 7, Paper No. e5, 71 p. (2019; Zbl 1441.16008)]), but remain unknown integrally. In the paper under review the author compute the local cohomology of the integral Spechtral FI-module corresponding to the \(L\)-shaped Young diagrams i.e. the partitions \((2, 1, \ldots , 1)\). Furthermore the presence of torsion is discussed.