Stillman’s question for twisted commutative algebras. (English) Zbl 1503.13004

Ganapathy, Karthik

J. Commut. Algebra 14, No. 3, 315-318 (2022).

Summary: Let \(\mathbb{A}_{n, m}\) be the polynomial ring \(\mathrm{Sym}(\mathbb{C}^n \otimes \mathbb{C}^m)\) with the natural action of \(\mathrm{GL}_m (\mathbb{C})\). We consider a family of \(\mathrm{GL}_m (\mathbb{C})\)-stable ideals \(J_{n, m}\) in \(\mathbb{A}_{n, m}\), each equivariantly generated by one homogeneous polynomial of degree 2 and show that the regularity of this family is unbounded. Using this, we negatively answer a question raised by D. Erman et al. [Int. Math. Res. Not. 2021, No. 16, 12281–12304 (2021; Zbl 1495.13026)] on a generalization of Stillman’s conjecture.

MSC:

13A50	Actions of groups on commutative rings; invariant theory
13D02	Syzygies, resolutions, complexes and commutative rings

Keywords:

twisted commutative algebras; regularity; Stillman’s conjecture; polynomial ring

Citations:

Zbl 1495.13026

Software:

Macaulay2; SINGULAR

References:

[1]	T. Ananyan and M. Hochster, “Small subalgebras of polynomial rings and Stillman’s conjecture”, J. Amer. Math. Soc. 33:1 (2020), 291-309. · Zbl 1439.13037 · doi:10.1090/jams/932
[2]	W. Decker, G.-M. Greuel, G. Pfister, and H. Schönemann, “Singular 4-1-1: A computer algebra system for polynomial computations”, software, available at http://www.singular.uni-kl.de. · Zbl 1344.13002
[3]	J. Draisma, “Topological Noetherianity of polynomial functors”, J. Amer. Math. Soc. 32:3 (2019), 691-707. · Zbl 1439.13044 · doi:10.1090/jams/923
[4]	D. Erman, S. V. Sam, and A. Snowden, “Big polynomial rings and Stillman’s conjecture”, Invent. Math. 218:2 (2019), 413-439. · Zbl 1427.13018 · doi:10.1007/s00222-019-00889-y
[5]	D. Erman, S. V. Sam, and A. Snowden, “Generalizations of Stillman’s conjecture via twisted commutative algebra”, Int. Math. Res. Not. 2021:16 (2021), 12281-12304. · Zbl 1495.13026 · doi:10.1093/imrn/rnz123
[6]	D. R. Grayson and M. E. Stillman, “Macaulay2: A software system for research in algebraic geometry”, software, available at http://www.math.uiuc.edu/Macaulay2.
[7]	J. McCullough, “Stillman’s question for exterior algebras and Herzog’s conjecture on Betti numbers of syzygy modules”, J. Pure Appl. Algebra 223:2 (2019), 634-640. · Zbl 1410.13010 · doi:10.1016/j.jpaa.2018.04.012
[8]	J. McCullough and A. Seceleanu, “Bounding projective dimension”, pp. 551-576 in Commutative algebra, Springer, 2013. · Zbl 1282.13029 · doi:10.1007/978-1-4614-5292-8_17
[9]	S. V. Sam and A. Snowden, “Introduction to twisted commutative algebras”, 2012.
[10]	S. V. Sam and A. Snowden, “GL-equivariant modules over polynomial rings in infinitely many variables, II”, Forum Math. Sigma 7 (2019), art. id. e5. · Zbl 1441.16008 · doi:10.1017/fms.2018.27
[11]	S. V. Sam, A. Snowden, and J. Weyman, “Homology of Littlewood complexes”, Selecta Math. (N.S.) 19:3 (2013), 655-698 · Zbl 1269.05117 · doi:10.1007/s00029-013-0119-5