The Cohen-Macaulay Property and F-Rationality in Certain Rings of Invariants

Glassbrenner, Donna

doi:10.1006/jabr.1995.1275

Cited by 7 publications

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“…Singh in [8]. Both papers make use of special identities involving the elementary symmetric polynomials, viz., [2,Section 11] and [8,Section 5]. We provide a more conceptual proof of these result using F.S.…”

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confidence: 79%

“…In this section we show how the viewpoint of Section 1 can be applied to the questions of F-rationality considered in [2]. In particular the examples given there are generic in the sense that we could have started with almost any regular sequence in F[z 1 , .…”

Section: Application To F-rationalitymentioning

confidence: 98%

“…For the sake of completeness we include a short summary the salient points of Macaulay's theory and the K ⊂ L-paradigm of [6]. 2 We use the notation m for the maximal ideal of a commutative graded connected algebra such as …”

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confidence: 99%

“…In addition they are integral, i.e., a E ∈ Z for all E ∈ N n 0 : in fact a E = ±1 for all E in the support of either ∆ n or ∇ n . The fact that they are integral is what makes the examples of[2] characteristic dependent.Specifically let f 1 , . .…”

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confidence: 99%

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On alternating invariants and Hilbert ideals

Smith¹

2004

Journal of Algebra

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In a paper on F-rationality [J. Algebra 176 (1995) ] Donna Glassbrenner showed that over a field of odd characteristic p the Hilbert ideals of the tautological representations of the symmetric group Σ n and alternating group A n coincide if n ≡ 0, 1 mod p. She asked if this was always the situation in the modular case. We answer this in the affirmative using Macaulay's theory of irreducible ideals in polynomial algebras: a somewhat forgotten bywater of commutative algebra. As a bonus, the method yields applications back to the original question of F-rationality studied in [J. Algebra 176 (1995) 824-860].  2004 Elsevier Inc. All rights reserved.Let ρ : G → GL(n, F) be a representation of a finite group G over the field F. Denote by V = F n the representation space on which G acts via ρ and F[V ] the algebra of polynomial functions on V . If z 1 , . . . , z n ∈ V * is a basis for the space of linear forms V * we use the alternate notation F[z 1 , . . . , z n ] for F [V ]. Associated with ρ is the subalgebra F[V ] G of invariant polynomial functions and the Hilbert ideal h(ρ) ⊂ F[V ]; the latter is the ideal in F[V ] generated by all the homogeneous invariant forms of strictly positive degree. If ρ is clear from context we simply write h(G) for the Hilbert ideal.In her study [2] of F-rationality of invariant rings Donna Glassbrenner proved that the Hilbert ideals of the symmetric group Σ n on n letters and its alternating subgroup A n coincide for the standard representation of Σ n on V = F n by permutation of the coordinates

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confidence: 79%

Section: Application To F-rationalitymentioning

confidence: 98%

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confidence: 99%

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confidence: 99%

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On alternating invariants and Hilbert ideals

Smith¹

2004

Journal of Algebra

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“…If the order |G| of G is invertible in F, the nonmodular case, then Steinberg's Theorem holds for finite groups over F (see [12] for the case of finite fields and [6] for the general case). However, over a field F of nonzero characteristic the ring of coinvariants F[V ] G of a representation ρ : G ↩→ GL(n, F) of a finite group may well be a Poincaré duality algebra though G contains no reflections at all (see e.g., [7,28]). …”

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Projective bundle ideals and Poincaré duality algebras

Smith¹,

Stong

2011

Journal of Pure and Applied Algebra

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a b s t r a c tThe study of maximal-primary irreducible ideals in a commutative graded connected Noetherian algebra over a field is in principle equivalent to the study of the corresponding quotient algebras. Such algebras are Poincaré duality algebras. A prototype for such an algebra is the cohomology with field coefficients of a closed oriented manifold. Topological constructions on closed manifolds often lead to algebraic constructions on Poincaré duality algebras and therefore also on maximal-primary irreducible ideals. It is the purpose of this note to examine several of these and develop some of their basic properties.

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F-rationality of the ring of modular invariants

Hashimoto

2017

Journal of Algebra

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Using the description of the Frobenius limit of modules over the ring of invariants under an action of a finite group on a polynomial ring over a field of characteristic p > 0 developed by Symonds and the author, we give a characterization of the ring of invariants with a positive dual F -signature. Combining this result and Kemper's result on depths of the ring of invariants under an action of a permutation group, we give an example of an F -rational, but non-F -regular ring of invariants under the action of a finite group.

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The Cohen-Macaulay Property and F-Rationality in Certain Rings of Invariants

Cited by 7 publications

References 0 publications

On alternating invariants and Hilbert ideals

On alternating invariants and Hilbert ideals

Projective bundle ideals and Poincaré duality algebras

F-rationality of the ring of modular invariants

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