Codimension and projective dimension up to symmetry

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. K E Y W O R D S invariant ideal, monoid, polynomial ring, symmetric group M S C ( 2 0 1 0 ) 13A50, 13C15, 13D02, 13F20, 16P70, 16W22 Let Sym( ) denote the symmetric group on {1, … , }. Considering it as stabilizer of + 1 in Sym( + 1), similarly one gets an ascending chain of symmetric groups. Define an action of Sym( ) on induced by ⋅ , = , ( ) for every ∈ Sym( ), 1 ≤ ≤ , 1 ≤ ≤ . 346

Betti tables of monomial ideals fixed by permutations of the variables

“…About such a chain, a natural interesting algebraic problem is to understand asymptotic behavior of I n . Indeed, Le,Nagel,Nguyen and Römer [LNNR1,LNNR2] recently studied asymptotic behavior of projective dimension and regularity of I n . They give certain linear bounds for these invariants and conjectured that they become linear functions on n for n ≫ 0.…”

Theorems of Carathéodory, Minkowski-Weyl, and Gordan up to symmetry

“…Examples of successful extensions include equivariant Hilbert's basis theorem [2,4,5,9,18], equivariant Hilbert-Serre theorem [11,16,17], equivariant Buchberger algorithm [8], equivariant Hochster's formula [15]. See, e.g., also [7,12,13,14,19,20,21] for related results.…”

Asymptotic behavior of symmetric ideals: A brief survey

“…This result leads to the following general problem (see [22, The aim of this note is to briefly summarize some recent results and open problems arising from the study of the previous problem. Apart from the aforementioned results of Nagel and Römer, we will discuss further results on the asymptotic behaviors of the codimension [22], the Castelnuovo-Mumford regularity [23,25,30], the projective dimension [22,25], and the Betti table [25,27] along Sym-invariant chains of ideals. For the sake of simplicity, some results will not be stated in their most general form.…”

“…In Section 2 we recall some basic notions and facts on invariant chains of ideals. Section 3 contains Nagel-Römer's result on rationality of Hilbert series and its consequences on the asymptotic behaviors of codimension and multiplicity, together with an improvement on the codimension obtained in [22]. The asymptotic behaviors of the Castelnuovo-Mumford regularity, the projective dimension, and the Betti table are discussed in Sections 4, 5, and 6, respectively.…”