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Small subalgebras of polynomial rings and Stillman’s Conjecture

Authors: Tigran Ananyan and Melvin Hochster
Journal: J. Amer. Math. Soc. 33 (2020), 291-309
MSC (2010): Primary 13D05, 13F20
Published electronically: October 1, 2019
MathSciNet review: 4066476
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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 $N$.

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Additional Information

Tigran Ananyan
Affiliation: Altair Engineering, 1820 E. Big Beaver Road, Troy, Michigan 48083
MR Author ID: 902622

Melvin Hochster
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109–1043
MR Author ID: 86705
ORCID: 0000-0002-9158-6486

Keywords: Polynomial ring, ideal, form, projective dimension, regular sequence, \text{R}$_\eta$
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)
Article copyright: © Copyright 2019 American Mathematical Society