Abstract
Symmetric ideals in increasingly larger polynomial rings that form an ascending chain are investigated. We focus on the asymptotic behavior of codimensions and projective dimensions of ideals in such a chain. If the ideals are graded it is known that the codimensions grow eventually linearly. Here this result is extended to chains of arbitrary symmetric ideals. Moreover, the slope of the linear function is explicitly determined. We conjecture that the projective dimensions also grow eventually linearly. As part of the evidence we establish two non-trivial lower linear bounds of the projective dimensions for chains of monomial ideals. As an application, this yields Cohen–Macaulayness obstructions.
| Original language | English |
|---|---|
| Pages (from-to) | 346-362 |
| Number of pages | 17 |
| Journal | Mathematische Nachrichten |
| Volume | 293 |
| Issue number | 2 |
| DOIs | |
| State | Published - Feb 1 2020 |
Bibliographical note
Publisher Copyright:© 2019 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Funding
We are grateful to the anonymous referees for insightful comments and suggestions that helped a lot to improve the clarity of the paper. The first author wishes to thank Lorenzo Venturello for his help with computations using Macaulay2. The second named author was partially supported by Simons Foundation grant #317096. The third named author was partially supported by a postdoctoral fellowship from the Vietnam Institute for Advanced Study in Mathematics (VIASM). Nguyen was also partially supported by Project ICRTM01_2019.01 of the International Centre for Research and Postgraduate Training in Mathematics (ICRTM), Institute of Mathematics, VAST. We are grateful to the anonymous referees for insightful comments and suggestions that helped a lot to improve the clarity of the paper. The first author wishes to thank Lorenzo Venturello for his help with computations using Macaulay2. The second named author was partially supported by Simons Foundation grant #317096. The third named author was partially supported by a postdoctoral fellowship from the Vietnam Institute for Advanced Study in Mathematics (VIASM). Nguyen was also partially supported by Project ICRTM01 _ 2019.01 of the International Centre for Research and Postgraduate Training in Mathematics (ICRTM), Institute of Mathematics, VAST.
| Funders | Funder number |
|---|---|
| NASU - Institute of Mathematics | |
| Simons Foundation | 317096 |
| American Institute of Mathematics Structured Quartet Research Ensembles | |
| Vietnam Academy of Science and Technology | |
| Vietnam Institute for Advanced Study in Mathematics | ICRTM01 _ 2019.01 |
Keywords
- 13A50
- 13C15
- 13D02
- 13F20
- 16P70
- 16W22
- invariant ideal
- monoid
- polynomial ring
- symmetric group
ASJC Scopus subject areas
- General Mathematics