Non Cohen-Macaulay Vector Invariants and a Noether Bound for a Gorenstein Ring of Invariants

Campbell, H. E. A.; Geramita, Antony V.; Hughes, Ian; Shank, R. James; Wehlau, David L.

doi:10.4153/cmb-1999-018-4

Cited by 16 publications

(10 citation statements)

References 6 publications

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“…In other words, given a permutation action on indeterminates x 1 x n , we take new indeterminates x i j , with 1 ≤ i ≤ r and l ≤ j ≤ n, on which G acts by the second index. Vector invariants (usually of linear representations) have been a classical area of interest in invariant theory and have recently enjoyed some interest in modular invariant theory (Richman [48], Campbell and Hughes [11], Campbell et al [13]). Vector invariants of permutation groups have been studied by Fleischmann [23] and Kemper [34].…”

Section: Theorem 33 In the Above Situation We Havementioning

confidence: 99%

The Depth of Invariant Rings and Cohomology

Kemper

2001

Journal of Algebra

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Let G be a finite group acting linearly on a vector space V over a field K of positive characteristic p and let P ≤ G be a Sylow p-subgroup. Ellingsrud and Skjelbred [Compositio Math. 41 (1980), 233-244] proved the lower boundfor the depth of the invariant ring, with equality if G is a cyclic p-group. Let us call the pair G V flat if equality holds in the above. In this paper we use cohomological methods to obtain information about the depth of invariant rings and in particular to study the question of flatness. For G of order not divisible by p 2 it ensues that G V is flat if and only if H 1 G K V = 0 or dim V P + 2 ≥ dim V . We obtain a formula for the depth of the invariant ring in the case that G permutes a basis of V and has order not divisible by p 2 . In this situation G V is usually not flat. Moreover, we introduce the notion of visible flatness of pairs G V and prove that this implies flatness. For example, the groups SL 2 q SO 3 q SU 3 q Sz q , and R q with many interesting representations in defining characteristic are visibly flat. In particular, if G = SL 2 q and V is the space of binary forms of degree n or a direct sum of such spaces, then G V is flat for all q = p r with the exception of a finite number of primes p. Along the way, we obtain results about the Buchsbaum property of invariant rings and about the depth of the cohomology modules H i G K V . We also determine the support of the positive cohomology H + G K V as a module over K V G . In the Appendix, the visibly flat pairs G V of a group with BN-pair and an irreducible representation are classified.

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Section: Theorem 33 In the Above Situation We Havementioning

confidence: 99%

The Depth of Invariant Rings and Cohomology

Kemper

2001

Journal of Algebra

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“…The first part has been shown. For the second part, note that β(R G ) does not change if we extend the ground field F (see [2,Lemma 2.4] The following example shows that the first bound can be better than the second bound in the above corollary. …”

Section: It Follows That R H As An A-module Is Generated By Homogeneomentioning

confidence: 95%

“…The ring F[mV ] G is called the (m-dimensional) vector invariant ring of G [17]. More discussion of vector invariant rings can be found in [14], [2], [3] and [4].…”

Section: It Follows That R H As An A-module Is Generated By Homogeneomentioning

confidence: 99%

A new degree bound for invariant rings

Chuai

2004

Proc. Amer. Math. Soc.

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“…finite groups G has been done, see [3][4][5]7,[9][10][11]13,16]. These papers contain many explicit examples of non-Cohen-Macaulay invariant rings of finite groups.…”

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confidence: 99%

Non-Cohen–Macaulay invariant rings of infinite groups

Kohls

2006

Journal of Algebra

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We give explicit examples of invariant rings that are not Cohen-Macaulay for all classical groups SL n (K), GL n (K), Sp 2n (K), SO n (K) and O n (K), where K is an algebraically closed field of positive characteristic. We prove that every non-trivial unipotent group over K has representations such that the invariant ring is not Cohen-Macaulay.One of the main topics in invariant theory is the study of structural properties of invariant rings. In particular, it is a much studied question whether for a linear algebraic group G over a field K and a G-module V (i.e., a finite-dimensional vector space over K with a morphism G → GL(V ) of algebraic groups) the invariant ring K[V ] G is Cohen-Macaulay or not. Any account of the research done so far on this question should start with the theorem of Hochster and Roberts [6], which says that if G is linearly reductive, then K[V ] G is always Cohen-Macaulay. Much later, Kemper [9] obtained some sort of a converse, which states that for every group G which is reductive but not linearly reductive, there exists a G-module V such that K[V ] G is not Cohen-Macaulay. Notice that (at least an important step in) the proof of [9] is non-constructive. Apart from this, a fair amount of research about the Cohen-Macaulay property of K[V ] G for

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Non Cohen-Macaulay Vector Invariants and a Noether Bound for a Gorenstein Ring of Invariants

Cited by 16 publications

References 6 publications

The Depth of Invariant Rings and Cohomology

The Depth of Invariant Rings and Cohomology

A new degree bound for invariant rings

Non-Cohen–Macaulay invariant rings of infinite groups

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