Associated primes for cohomology modules

Elmer, Jonathan

doi:10.1007/s00013-008-2902-7

Cited by 4 publications

(3 citation statements)

References 10 publications

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“…The equivalence of (1), (2) and (3) was noted in [10], and the fact that (4) implies any of these is shown using only the results of [10]. The implication (3) ⇒ (4) is new, and requires the following explicit description of the associated primes of H m (G, R) found in [6]: …”

Section: Proposition 23 Let G Be a Finite Group Acting Linearly On mentioning

confidence: 82%

Depth and detection in modular invariant theory

Elmer

2009

Journal of Algebra

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Let G be a finite group acting linearly on a vector space V over a field of characteristic p dividing the group order, and let R := S(V * ). We study the R G modules H i (G, R), for i 0 with R G itself as a special case. There are lower bounds for depth R G (H i (G, R)) and for depth(R G ). We show that a certain sufficient condition for their attainment (due to Fleischmann, Kemper and Shank [P. Fleischmann, G. Kemper, R.J. Shank, Depth and cohomological connectivity in modular invariant theory, Trans. Amer. Math. Soc. 357 (2005) 3605-3621]) may be modified to give a condition which is both necessary and sufficient. We apply our main result to classify the representations of the Klein four-group for which depth(R G ) attains its lower bound, a process begun in [J. Elmer, P. Fleischmann, On the depth of modular invariant rings for the groups C p × C p , in: Proc. Symmetry and Space, Fields Institute, 2006, preprint, 2007]. We also use our new condition to show that if G = P × Q , with P a p-group and Q an abelian p -group, then the depth of R G attains its lower bound if and only if the depth of R P does so.

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Section: Proposition 23 Let G Be a Finite Group Acting Linearly On mentioning

confidence: 82%

Depth and detection in modular invariant theory

Elmer

2009

Journal of Algebra

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“…In the modular case, on the other hand, K[V ] G almost always fails to be Cohen-Macaulay, see [12]. The depth of K[V ] G has attracted much attention and has been determined for various families of representations, see for example [3,8,10,13,17]. In this paper we consider ideals of K[V ] G as modules over K[V ] G .…”

Section: Introductionmentioning

confidence: 99%

On Cohen–Macaulayness and depth of ideals in invariant rings

Kohls

Sezer

2016

Journal of Pure and Applied Algebra

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Abstract. We investigate the presence of Cohen-Macaulay ideals in invariant rings and show that an ideal of an invariant ring corresponding to a modular representation of a p-group is not Cohen-Macaulay unless the invariant ring itself is. As an intermediate result, we obtain that non-Cohen-Macaulay factorial rings cannot contain Cohen-Macaulay ideals. For modular cyclic groups of prime order, we show that the quotient of the invariant ring modulo the transfer ideal is always Cohen-Macaulay, extending a result of Fleischmann.

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“…[16, Lemma 2.4, Proposition 2.5], Ann k[V ] G (g) = I(V G ) ∩ k[V ] G , where I(V G ) denotes the ideal of polynomials f ∈ k[V ] vanishing on V G . Thus, the height of Ann k[V ] G (g) is codim(V G ) = 3 while, by[22, Corollary 1.6], its depth is only two.Using MAGMA[4] and the methods of[24, Section 2], one can verify that {a 1 := x 3 , a 2 := x 4 , a 3 := x 5 , a }…”

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The Cohen–Macaulay property of separating invariants of finite groups

Dufresne

Elmer

Kohls

2009

Transformation Groups

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Abstract. In the case of finite groups, a separating algebra is a subalgebra of the ring of invariants which separates the orbits. Although separating algebras are often better behaved than the ring of invariants, we show that many of the criteria which imply the ring of invariants is non Cohen-Macaulay actually imply that no graded separating algebra is Cohen-Macaulay. For example, we show that, over a field of positive characteristic p, given sufficiently many copies of a faithful modular representation, no graded separating algebra is Cohen-Macaulay. Furthermore, we show that, for a p-group, the existence of a Cohen-Macaulay graded separating algebra implies the group is generated by bireflections. Additionally, we give an example which shows that Cohen-Macaulay separating algebras can occur when the ring of invariants is not Cohen-Macaulay.

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Associated primes for cohomology modules

Cited by 4 publications

References 10 publications

Depth and detection in modular invariant theory

Depth and detection in modular invariant theory

On Cohen–Macaulayness and depth of ideals in invariant rings

The Cohen–Macaulay property of separating invariants of finite groups

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