Codimension and projective dimension up to symmetry. Issue 2 (5th December 2019)

• Record Type: Journal Article
• Authors: Le, Dinh Van; Nagel, Uwe; Nguyen, Hop D.; Römer, Tim
• Is Part Of: Mathematische Nachrichten. Volume 293:Issue 2(2020)
• Page Start: 346
• Page End: 362
• Publication Date: 2019-12-05
• Subjects: invariant ideal -- monoid -- polynomial ring -- symmetric group -- 13A50 -- 13C15 -- 13D02 -- 13F20 -- 16P70 -- 16W22; Mathematics -- Periodicals; 510.5
• DOI: 10.1002/mana.201800413
• Languages: English
• ISSNs: 0025-584X

Abstract

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.