Small subalgebras of polynomial rings and Stillman’s Conjecture
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- by Tigran Ananyan and Melvin Hochster
- J. Amer. Math. Soc. 33 (2020), 291-309
- DOI: https://doi.org/10.1090/jams/932
- Published electronically: October 1, 2019
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Abstract:
Let $n, d, \eta$ be positive integers. We show that in a polynomial ring $R$ in $N$ variables over an algebraically closed field $K$ of arbitrary characteristic, any $K$-subalgebra of $R$ generated over $K$ by at most $n$ forms of degree at most $d$ is contained in a $K$-subalgebra of $R$ generated by $B \leq {}^\eta \mathcal {B}(n,d)$ forms ${G}_1, \ldots , {G}_{B}$ of degree $\leq d$, where ${}^\eta \mathcal {B}(n,d)$ does not depend on $N$ or $K$, such that these forms are a regular sequence and such that for any ideal $J$ generated by forms that are in the $K$-span of ${G}_1, \ldots , {G}_{B}$, the ring $R/J$ satisfies the Serre condition $\mathrm {R}_\eta$. These results imply a conjecture of M. Stillman asserting that the projective dimension of an $n$-generator ideal $I$ of $R$ whose generators are forms of degree $\leq d$ is bounded independent of $N$. We also show that there is a primary decomposition of $I$ such that all numerical invariants of the decomposition (e.g., the number of primary components and the degrees and numbers of generators of all of the prime and primary ideals occurring) are bounded independent of $N$.References
- Tigran Ananyan and Melvin Hochster, Ideals generated by quadratic polynomials, Math. Res. Lett. 19 (2012), no. 1, 233–244. MR 2923188, DOI 10.4310/MRL.2012.v19.n1.a18
- T. Ananyan and M. Hochster, Strength conditions, small subalgebras, and Stillman bounds in degree $\leq 4$, preprint, arXiv:1810.00413 [math.AC], September 30, 2018.
- Jesse Beder, Jason McCullough, Luis Núñez-Betancourt, Alexandra Seceleanu, Bart Snapp, and Branden Stone, Ideals with larger projective dimension and regularity, J. Symbolic Comput. 46 (2011), no. 10, 1105–1113. MR 2831475, DOI 10.1016/j.jsc.2011.05.011
- Winfried Bruns, “Jede” endliche freie Auflösung ist freie Auflösung eines von drei Elementen erzeugten Ideals, J. Algebra 39 (1976), no. 2, 429–439. MR 399074, DOI 10.1016/0021-8693(76)90047-8
- Lindsay Burch, A note on the homology of ideals generated by three elements in local rings, Proc. Cambridge Philos. Soc. 64 (1968), 949–952. MR 230718, DOI 10.1017/s0305004100043632
- Frederick W. Call, A theorem of Grothendieck using Picard groups for the algebraist, Math. Scand. 74 (1994), no. 2, 161–183. MR 1298359, DOI 10.7146/math.scand.a-12487
- Giulio Caviglia and Manoj Kummini, Some ideals with large projective dimension, Proc. Amer. Math. Soc. 136 (2008), no. 2, 505–509. MR 2358490, DOI 10.1090/S0002-9939-07-09159-9
- L. van den Dries and K. Schmidt, Bounds in the theory of polynomial rings over fields. A nonstandard approach, Invent. Math. 76 (1984), no. 1, 77–91. MR 739626, DOI 10.1007/BF01388493
- J. A. Eagon and D. G. Northcott, Ideals defined by matrices and a certain complex associated with them, Proc. Roy. Soc. London Ser. A 269 (1962), 188–204. MR 142592, DOI 10.1098/rspa.1962.0170
- Bahman Engheta, Bounds on projective dimension, ProQuest LLC, Ann Arbor, MI, 2005. Thesis (Ph.D.)–University of Kansas. MR 2708123
- Bahman Engheta, A bound on the projective dimension of three cubics, J. Symbolic Comput. 45 (2010), no. 1, 60–73. MR 2568899, DOI 10.1016/j.jsc.2009.06.005
- Bahman Engheta, On the projective dimension and the unmixed part of three cubics, J. Algebra 316 (2007), no. 2, 715–734. MR 2358611, DOI 10.1016/j.jalgebra.2006.11.018
- A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Inst. Hautes Études Sci. Publ. Math. 28 (1966), 255. MR 217086
- M. Hochster and John A. Eagon, Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci, Amer. J. Math. 93 (1971), 1020–1058. MR 302643, DOI 10.2307/2373744
- Melvin Hochster and Craig Huneke, $F$-regularity, test elements, and smooth base change, Trans. Amer. Math. Soc. 346 (1994), no. 1, 1–62. MR 1273534, DOI 10.1090/S0002-9947-1994-1273534-X
- Melvin Hochster and Joel L. Roberts, Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay, Advances in Math. 13 (1974), 115–175. MR 347810, DOI 10.1016/0001-8708(74)90067-X
- Craig Huneke, Paolo Mantero, Jason McCullough, and Alexandra Seceleanu, The projective dimension of codimension two algebras presented by quadrics, J. Algebra 393 (2013), 170–186. MR 3090065, DOI 10.1016/j.jalgebra.2013.06.038
- Craig Huneke, Paolo Mantero, Jason McCullough, and Alexandra Seceleanu, A tight bound on the projective dimension of four quadrics, J. Pure Appl. Algebra 222 (2018), no. 9, 2524–2551. MR 3783004, DOI 10.1016/j.jpaa.2017.10.005
- Peter Kohn, Ideals generated by three elements, Proc. Amer. Math. Soc. 35 (1972), 55–58. MR 296064, DOI 10.1090/S0002-9939-1972-0296064-1
- Jason McCullough, A family of ideals with few generators in low degree and large projective dimension, Proc. Amer. Math. Soc. 139 (2011), no. 6, 2017–2023. MR 2775379, DOI 10.1090/S0002-9939-2010-10792-X
- J. McCullough and A. Seceleanu, Bounding projective dimension, Commutative Algebra. Expository Papers Dedicated to David Eisenbud on the Occasion of His 65th Birthday (I. Peeva, Ed.), Sparinger-Verlag London Ltd., London, 2013.
- Irena Peeva and Mike Stillman, Open problems on syzygies and Hilbert functions, J. Commut. Algebra 1 (2009), no. 1, 159–195. MR 2462384, DOI 10.1216/JCA-2009-1-1-159
- A. Seidenberg, Constructions in algebra, Trans. Amer. Math. Soc. 197 (1974), 273–313. MR 349648, DOI 10.1090/S0002-9947-1974-0349648-2
- Jean-Pierre Serre, Algèbre locale. Multiplicités, Lecture Notes in Mathematics, vol. 11, Springer-Verlag, Berlin-New York, 1965 (French). Cours au Collège de France, 1957–1958, rédigé par Pierre Gabriel; Seconde édition, 1965. MR 0201468, DOI 10.1007/978-3-662-21576-0
Bibliographic Information
- Tigran Ananyan
- Affiliation: Altair Engineering, 1820 E. Big Beaver Road, Troy, Michigan 48083
- MR Author ID: 902622
- Email: antigran@umich.edu
- Melvin Hochster
- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109–1043
- MR Author ID: 86705
- ORCID: 0000-0002-9158-6486
- Email: hochster@umich.edu
- Received by editor(s): August 9, 2018
- Received by editor(s) in revised form: July 14, 2019
- Published electronically: October 1, 2019
- Additional Notes: The second author was partially supported by grants from the National Science Foundation (DMS–0901145 and DMS–1401384)
- © Copyright 2019 American Mathematical Society
- Journal: J. Amer. Math. Soc. 33 (2020), 291-309
- MSC (2010): Primary 13D05, 13F20
- DOI: https://doi.org/10.1090/jams/932
- MathSciNet review: 4066476