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Structure theorems over polynomial rings

Advances in Mathematics
Publication Date
  • Mathematics


doi:10.1016/j.aim.2006.02.012 Advances in Mathematics 208 (2007) 408–421 Structure theorems over polynomial rings Peter Symonds 1 School of Mathematics, University of Manchester, PO Box 88, Manchester M60 1QD, UK Received 28 June 2005; accepted 27 February 2006 Available online 18 April 2006 Communicated by David J. Benson Abstract Given a polynomial ring R over a field k and a finite group G, we consider a finitely generated graded RG-module S. We regard S as a kG-module and show that various conditions on S are equivalent, such as only containing finitely many isomorphism classes of indecomposable summands or satisfying a struc- ture theorem in the sense of [D. Karagueuzian, P. Symonds, The module structure of a group action on a polynomial ring: A finiteness theorem, preprint,]. © 2006 Elsevier Inc. All rights reserved. Keywords: Polynomial ring; Structure theorem; Group action 1. Introduction Consider a polynomial ring R = k[d1, . . . , dn], finitely generated over a commutative ring k and graded in such a way that the di are homogeneous of positive degree. We are most interested in the case when k is a field of finite characteristic, but we allow k to be any complete local noetherian commutative ring, for example the p-adic integers. Let G be a finite group and let S =⊕∞i=N Si be a finitely generated graded left RG-module, where G preserves the grading. By a structure theorem for S over RG we mean a set of finitely generated graded kG-sub- modules X¯I ⊆ S, one for each I ⊆ {1, . . . , n}, such that S ∼=⊕I⊆{1,...,n} k[di | i ∈ I ] ⊗k X¯I as a graded kG-module, where the map from right to left is induced by the action of R on S. E-mail address: [email protected] 1 Partially supported by a grant from the Leverhulme Trust. 0001-8708/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.aim.2006.02.012 P. Symonds / Advances in Mathematics 208 (2007) 408–421 409

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