## Geometry & Topology

### Homology of FI-modules

#### Abstract

We prove an explicit and sharp upper bound for the Castelnuovo–Mumford regularity of an FI-module in terms of the degrees of its generators and relations. We use this to refine a result of Putman on the stability of homology of congruence subgroups, extending his theorem to previously excluded small characteristics and to integral homology while maintaining explicit bounds for the stable range.

An equivalent version of this paper can be found on arXiv.

#### Article information

Source
Geom. Topol., Volume 21, Number 4 (2017), 2373-2418.

Dates
Revised: 31 August 2016
Accepted: 3 September 2016
First available in Project Euclid: 19 October 2017

https://projecteuclid.org/euclid.gt/1508437644

Digital Object Identifier
doi:10.2140/gt.2017.21.2373

Mathematical Reviews number (MathSciNet)
MR3654111

Zentralblatt MATH identifier
1371.18012

#### Citation

Church, Thomas; Ellenberg, Jordan. Homology of FI-modules. Geom. Topol. 21 (2017), no. 4, 2373--2418. doi:10.2140/gt.2017.21.2373. https://projecteuclid.org/euclid.gt/1508437644

#### References

• A Borel, J-P Serre, Corners and arithmetic groups, Comment. Math. Helv. 48 (1973) 436–491
• F Calegari, M Emerton, Homological stability for completed homology, Math. Ann. 364 (2016) 1025–1041
• R Charney, On the problem of homology stability for congruence subgroups, Comm. Algebra 12 (1984) 2081–2123
• T Church, J S Ellenberg, B Farb, FI-modules and stability for representations of symmetric groups, Duke Math. J. 164 (2015) 1833–1910
• T Church, J S Ellenberg, B Farb, R Nagpal, FI-modules over Noetherian rings, Geom. Topol. 18 (2014) 2951–2984
• J S Ellenberg, A Venkatesh, C Westerland, Homological stability for Hurwitz spaces and the Cohen–Lenstra conjecture over function fields, Ann. of Math. 183 (2016) 729–786
• W L Gan, A long exact sequence for homology of FI-modules, preprint (2016)
• W L Gan, L Li, On central stability, preprint (2016)
• W L Gan, L Li, A remark on FI-module homology, Michigan Math. J. 65 (2016) 855–861
• L Li, Upper bounds of homological invariants of $F\!I\sb{G}$–modules, preprint (2016)
• L Li, N Yu, Filtrations and homological degrees of FI-modules, J. Algebra 472 (2017) 369–398
• A Putman, Stability in the homology of congruence subgroups, Invent. Math. 202 (2015) 987–1027
• S V Sam, A Snowden, Stability patterns in representation theory, Forum Math. Sigma 3 (2015) e11 (108 pages)
• S V Sam, A Snowden, GL-equivariant modules over polynomial rings in infinitely many variables, Trans. Amer. Math. Soc. 368 (2016) 1097–1158
• C A Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge Univ. Press (1994)
• J D Wiltshire-Gordon, Uniformly presented vector spaces, preprint (2014)