Bounds on homological invariants of VI-modules (Q2195009)

The representation theory of the category FI, as in [\textit{T. Church} et al., Duke Math. J. 164, No. 9, 1833--1910 (2015; Zbl 1339.55004)], can be regarded as a framework to address problems on representation stability of the symmetric group. A linear analogue is the representation theory of the category VI, as studied in [\textit{W. L. Gan} and \textit{J. Watterlond}, Algebr. Represent. Theory 21, No. 1, 47--60 (2018; Zbl 1485.20036)]. The latter is related with problems on representation stability of the general linear group over a finite field. Formally, the category VI has finite dimensional vector spaces over a finite field with order \(q\) as its objects and injective linear maps as its morphisms. VI-modules over a commutative Noetherian ring \(k\) (containing \(q\) as an invertible element) are functors from the category VI to the module category over \(k\). In the paper under review, the authors compute upper bounds for: the Castelnuovo-Mumford regularity, the injective dimension, and degrees of local cohomology of finitely generated VI-modules. Moreover, these bounds are very similar to their analogues in the context of FI-modules.