Bounds on homological invariants of VI-modules.

*(English)*Zbl 1452.18013The 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.

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.

Reviewer: Tiago Cruz (Stuttgart)

##### MSC:

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, Künneth formula (category-theoretic aspects) |

##### References:

[1] | T. Church, Bounding the homology of \(\text{FI} \)-modules, preprint, arXiv:1612.07803v1. |

[2] | T. Church and J. S. Ellenberg, Homology of \(\text{FI} \)-modules, Geom. Topol. 21 (2017), no. 4, 2373-2418, arXiv:1506.01022v2. · Zbl 1371.18012 |

[3] | T. Church, J. Miller, R. Nagpal, and J. Reinhold, Linear and quadratic ranges in representation stability, Adv. Math. 333 (2018), 1-40, arXiv:1706.03845v2. · Zbl 1392.15030 |

[4] | W. L. Gan, A long exact sequence for homology of \(\text{FI} \)-modules, New York J. Math. 22 (2016), 1487-1502, arXiv:1602.08873v3. · Zbl 1358.18006 |

[5] | W. L. Gan and L. Li, Coinduction functor in representation stability theory, J. Lond. Math. Soc. (2) 92 (2015), no. 3, 689-711, arXiv:1502.06989v3. · Zbl 1358.18001 |

[6] | W. L. Gan, L. Li, and C. Xi, An application of Nakayama functor in representation stability theory, preprint, arXiv:1710.05493v1. |

[7] | W. L. Gan and J. Watterlond, A representation stability theorem for \(\text{VI} \)-modules, Algebr. Represent. Theory 21 (2018), no. 1, 47-60, arXiv:1602.00654v3. · Zbl 06839337 |

[8] | L. Li, Upper bounds of homological invariants of \(\text{FI}_G \)-modules, Arch. Math. (Basel) 107 (2016), no. 3, 201-211, arXiv:1512.05879v3. · Zbl 1395.16003 |

[9] | L. Li, Homological degrees of representations of categories with shift functors, Trans. Amer. Math. Soc. 370 (2018), no. 4, 2563-2587, arXiv:1507.08023v3. · Zbl 1439.16006 |

[10] | L. Li and E. Ramos, Depth and the local cohomology of \(\text{FI}_G \)-modules, Adv. Math. 329 (2018), 704-741, arXiv:1602.04405v3. · Zbl 1398.13016 |

[11] | L. Li and N. Yu, Filtrations and homological degrees of \(\text{FI} \)-modules, J. Algebra 472 (2017), 369-398, arXiv:1511.02977v3. · Zbl 1371.13015 |

[12] | J. Miller and J. C. H. Wilson, Quantitative representation stability over linear groups, preprint, arXiv:1709.03638. |

[13] | R. Nagpal, \( \text{VI} \)-modules in non-describing characteristic, Part I, preprint, arXiv:1709.07591v1. |

[14] | A. Putman and S. V. Sam, Representation stability and finite linear groups, Duke Math. J. 166 (2017), no. 13, 2521-2598, arXiv:1408.3694v3. · Zbl 1408.18003 |

[15] | S. · Zbl 1347.05010 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.