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Bounds on homological invariants of VI-modules. (English) Zbl 1452.18013
The representation theory of the category FI, as in [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 [W. L. Gan and J. Watterlond, Algebr. Represent. Theory 21, No. 1, 47–60 (2018; Zbl 06839337)]. 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.

18G20 Homological dimension (category-theoretic aspects)
16E05 Syzygies, resolutions, complexes in associative algebras
16E10 Homological dimension in associative algebras
16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
16G99 Representation theory of associative rings and algebras
18G15 Ext and Tor, generalizations, K√ľnneth formula (category-theoretic aspects)
Full Text: DOI Euclid
[1] T. Church, Bounding the homology of \(\text{FI} \)-modules, preprint, arXiv:1612.07803v1.
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[15] S. · Zbl 1347.05010
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