Depth and detection in modular invariant theory

JP Elmer

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1 Citation (Scopus)


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 RG modules Hi(G,R), for i⩾0 with RG itself as a special case. There are lower bounds for depthRG(Hi(G,R)) and for depth(RG). 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(RG) attains its lower bound, a process begun in [J. Elmer, P. Fleischmann, On the depth of modular invariant rings for the groups Cp×Cp, 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 RG attains its lower bound if and only if the depth of RP does so.
Translated title of the contributionDepth and detection in modular invariant theory
Original languageEnglish
Pages (from-to)1653 - 1666
Number of pages14
JournalJournal of Algebra
Volume322, issue 5
Publication statusPublished - Sep 2009

Bibliographical note

Publisher: Springer

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