Depth of modular invariant rings

Campbell, H. E. A.; Hughes, Ian; Kemper, Gregor; Shank, R. James; Wehlau, David L.

doi:10.1007/bf01237176

Cited by 18 publications

(25 citation statements)

References 14 publications

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“…Since a p-group always fixes nonzero vectors, it follows from (0.2) that depth R G ≥ min 3 n , so in particular, R G is Cohen-Macaulay if n ≤ 3 (see Smith [51]). The result of Ellingsrud and Skjelbred on cyclic p-groups was recently generalized by Campbell et al [14], who proved that if G acts as a shallow group on V (see Remark 2.8(c) in this paper), then G V is flat. Typical examples of shallow groups are abelian groups with a cyclic Sylow p-subgroup P. Unlike Ellingsrud and Skjelbred's proof, the proof in [14] uses elementary methods.…”

Section: Previous Workmentioning

confidence: 83%

“…The result of Ellingsrud and Skjelbred on cyclic p-groups was recently generalized by Campbell et al [14], who proved that if G acts as a shallow group on V (see Remark 2.8(c) in this paper), then G V is flat. Typical examples of shallow groups are abelian groups with a cyclic Sylow p-subgroup P. Unlike Ellingsrud and Skjelbred's proof, the proof in [14] uses elementary methods. The most recent contribution comes from Shank and Wehlau [49], who studied the depth of invariants of SL 2 p acting on symmetric powers of the natural two-dimensional representation.…”

Section: Previous Workmentioning

confidence: 83%

“…The question of the depth of modular invariant rings has enjoyed considerable interest recently (see Bourguiba and Zarati [8], Smith [52], Campbell et al [14], Shank and Wehlau [49]). However, the story of results (known to the author) which study the depth of modular invariant rings for certain classes of groups is rather short.…”

Section: Previous Workmentioning

confidence: 99%

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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: Previous Workmentioning

confidence: 83%

Section: Previous Workmentioning

confidence: 83%

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The Depth of Invariant Rings and Cohomology

Kemper

2001

Journal of Algebra

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“…The first such instance is the result of Ellingsrud and Skjelbred [4], who showed that (7) holds if G is a cyclic p-group. This was extended by Campbell et al [3] to the case where G is abelian and has a cyclic Sylow p-subgroup (see Theorems 1 and 9 in [3]). Corollary 5.3 clearly generalizes this latter result.…”

Section: Proof By Theorem 52 (A) Implies That There Is M G With G/mentioning

confidence: 97%

“…One might hope that the inequality is actually an equality. In fact, this was proved to be true if p 2 does not divide |G| (see [10,Theorem 3.1]) or if G is cyclic (see [4] or [3]). However, the following example shows that equality does not hold in general.…”

Section: Introductionmentioning

confidence: 99%

Depth and cohomological connectivity in modular invariant theory

Fleischmann

Kemper²,

Shank

2004

Trans. Amer. Math. Soc.

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Abstract. Let G be a finite group acting linearly on a finite-dimensional vector space V over a field K of characteristic p. Assume that p divides the order of G so that V is a modular representation and let P be a Sylow p-subgroup for G. Define the cohomological connectivity of the symmetric algebra S(V * ) to be the smallest positive integer m such that H m (G, S(V * )) = 0. We show that min dim K (V P ) + m + 1, dim K (V ) is a lower bound for the depth of S(V * ) G . We characterize those representations for which the lower bound is sharp and give several examples of representations satisfying the criterion. In particular, we show that if G is p-nilpotent and P is cyclic, then, for any modular representation, the depth of S(V * ) G is min dim K (V P ) + 2, dim K (V ) .

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