Duality in G-algebras
Let G be a finite group, and let R be a complete discrete valuation ring of characteristic zero with field of fractions K and residue field k of prime characteristic p. The author introduces the notion of a symmetric G-algebra which is a G-algebra A (free of finite rank over R) together with an R-linear map \(\tau\) : \(A\to R\) satisfying \(\tau (ab)=\tau (ba)\) and \(\tau (^ ga)=\tau (a)\) for \(a,b\in A\), \(g\in G\) such that the symmetric bilinear form \(\phi\) : \(A\times A\to R\), (a,b)\(\to \tau (ab)\), is non-degenerate. Then \(\phi\) extends to a non-degenerate symmetric bilinear form (again denoted by \(\phi)\) on \(K\otimes_ RA\), and the restriction \(\phi_ G\) of \(| G|^{-1}\phi\) is a non-degenerate symmetric bilinear form on \(K\otimes_ RA^ G\). Any full R-lattice L in \(K\otimes_ RA^ G\) has a dual \(L^*:=\{a:\) \(\phi_ G(a,L)\subseteq R\}\). If \(L=L^*\) then L is called unimodular. Suppose now that A itself is unimodular. Then one has \((A^ G)^*=A^ G_ 1\), the projective ideal of \(A^ G\). This implies that \(A^ G/A^ G_ 1\) is a symmetric algebra over the artinian ring R/\(| G| R\). The major part of the paper is devoted to properties of the almost projective ideal \(L_ G:=(J_ G)^*\) of \(A^ G\) where \(J_ G=J(A^ G)+A^ G_ 1\). We have \(A^ G_ 1\subseteq L_ G\subseteq A^ G\), and \(L_ G/A^ G_ 1\) is the socle of \(A^ G/A^ G_ 1.\) If M is an RG-lattice then \(A:=End_ R(M)\) is a unimodular symmetric G- algebra, and the almost projective ideal is closely related to almost split sequences. For example, it can be used to give an easy proof of the existence and properties of almost split sequences. Similarly, any block A of RG is a unimodular symmetric G-algebra, and in this case the almost projective ideal can be used to prove properties of defect groups and the central character of A. There are also interesting connections to virtually irreducible lattices (as introduced by R. Knörr) and splitting trace lattices (as defined by M. Auslander and J. F. Carlson).
scientific article; zbMATH DE number 4091728
scientific article