Homological degrees of representations of categories with shift functors

Li, Liping

doi:10.1090/tran/7041

Homological degrees of representations of categories with shift functors
HTML articles powered by AMS MathViewer

by Liping Li PDF

Trans. Amer. Math. Soc. 370 (2018), 2563-2587 Request permission

Abstract:

Let $\mathbb {k}$ be a commutative Noetherian ring and let $\underline {\mathscr {C}}$ be a locally finite $\mathbb {k}$-linear category equipped with a self-embedding functor of degree 1. We show under a moderate condition that finitely generated torsion representations of $\underline {\mathscr {C}}$ are super finitely presented (that is, they have projective resolutions, each term of which is finitely generated). In the situation that these self-embedding functors are genetic functors, we give upper bounds for homological degrees of finitely generated torsion modules. These results apply to quite a few categories recently appearing in representation stability theory. In particular, when $\mathbb {k}$ is a field of characteristic 0, using the result of Church and Ellenberg [arXiv:1506.01022], we obtain another upper bound for homological degrees of finitely generated $\mathscr {FI}$-modules.

References

Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), no. 2, 473–527. MR 1322847, DOI 10.1090/S0894-0347-96-00192-0
Thomas Church and Jordan S. Ellenberg, Homology of FI-modules, Geom. Topol. 21 (2017), no. 4, 2373–2418. MR 3654111, DOI 10.2140/gt.2017.21.2373
Thomas Church and Benson Farb, Representation theory and homological stability, Adv. Math. 245 (2013), 250–314. MR 3084430, DOI 10.1016/j.aim.2013.06.016
Thomas Church, Jordan S. Ellenberg, and Benson Farb, FI-modules and stability for representations of symmetric groups, Duke Math. J. 164 (2015), no. 9, 1833–1910. MR 3357185, DOI 10.1215/00127094-3120274
Thomas Church, Jordan S. Ellenberg, Benson Farb, and Rohit Nagpal, FI-modules over Noetherian rings, Geom. Topol. 18 (2014), no. 5, 2951–2984. MR 3285226, DOI 10.2140/gt.2014.18.2951
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
B. Farb, Representation stability, Proceedings of the International Congress of Mathematicians, Seoul 2014, Vol. II, pp. 1173–1196, arXiv:1404.4065.
Vincent Franjou, Jean Lannes, and Lionel Schwartz, Autour de la cohomologie de Mac Lane des corps finis, Invent. Math. 115 (1994), no. 3, 513–538 (French, with English and French summaries). MR 1262942, DOI 10.1007/BF01231771
Wee Liang Gan, A long exact sequence for homology of FI-modules, New York J. Math. 22 (2016), 1487–1502. MR 3603074
Wee Liang Gan and Liping Li, Noetherian property of infinite EI categories, New York J. Math. 21 (2015), 369–382. MR 3358549
W. L. Gan and L. Li, Koszulity of directed categories in representation stability theory, preprint, arXiv:1411.5308.
Wee Liang Gan and Liping Li, Coinduction functor in representation stability theory, J. Lond. Math. Soc. (2) 92 (2015), no. 3, 689–711. MR 3431657, DOI 10.1112/jlms/jdv043
Wee Liang Gan and Liping Li, A remark on FI-module homology, Michigan Math. J. 65 (2016), no. 4, 855–861. MR 3579189, DOI 10.1307/mmj/1480734023
Liping Li, A generalized Koszul theory and its application, Trans. Amer. Math. Soc. 366 (2014), no. 2, 931–977. MR 3130322, DOI 10.1090/S0002-9947-2013-05891-6
Liping Li, Upper bounds of homological invariants of $FI_G$-modules, Arch. Math. (Basel) 107 (2016), no. 3, 201–211. MR 3538516, DOI 10.1007/s00013-016-0921-3
L. Li, Two homological proofs of the Noetherianity of $FI_G$, arXiv:1603.04552.
L. Li and E. Ramos, Depth and the local cohomology of FIG-modules, arXiv:1602.04405.
Liping Li and Nina Yu, Filtrations and homological degrees of FI-modules, J. Algebra 472 (2017), 369–398. MR 3584882, DOI 10.1016/j.jalgebra.2016.11.019
Volodymyr Mazorchuk, Serge Ovsienko, and Catharina Stroppel, Quadratic duals, Koszul dual functors, and applications, Trans. Amer. Math. Soc. 361 (2009), no. 3, 1129–1172. MR 2457393, DOI 10.1090/S0002-9947-08-04539-X
Rohit Nagpal, FI-modules and the cohomology of modular representations of symmetric groups, ProQuest LLC, Ann Arbor, MI, 2015. Thesis (Ph.D.)–The University of Wisconsin - Madison. MR 3358218
Andrew Putman, Stability in the homology of congruence subgroups, Invent. Math. 202 (2015), no. 3, 987–1027. MR 3425385, DOI 10.1007/s00222-015-0581-0
A. Putman and S. Sam, Representation stability and finite linear groups, Duke Math. J. 166 (2017), no. 12, 2521-2598.
E. Ramos, Homological invariants of FI-modules and $FI_G$-modules, preprint, arXiv:1511.03964.
E. Ramos, On the degreewise coherence of $FI_G$-modules, New York J. Math. 23 (2017) 873-895.
Steven V. Sam and Andrew Snowden, GL-equivariant modules over polynomial rings in infinitely many variables, Trans. Amer. Math. Soc. 368 (2016), no. 2, 1097–1158. MR 3430359, DOI 10.1090/tran/6355
Steven V. Sam and Andrew Snowden, Gröbner methods for representations of combinatorial categories, J. Amer. Math. Soc. 30 (2017), no. 1, 159–203. MR 3556290, DOI 10.1090/jams/859
Jennifer C. H. Wilson, $\rm FI_\scr W$-modules and stability criteria for representations of classical Weyl groups, J. Algebra 420 (2014), 269–332. MR 3261463, DOI 10.1016/j.jalgebra.2014.08.010