Hopf algebra structures and tensor products for group algebras (Q526162)

Let \(E\) be an elementary abelian \(p\)-group of order \(p^r\) for a prime \(p\) and \(k\) be a field of characteristic \(p\). Set \(A := kE\), the group algebra of \(E\) over \(k\). As a group algebra, \(A\) has a natural Hopf algebra structure. As an algebra, \(A\) may also be identified with the restricted enveloping algebra of an \(r\)-dimensional \(p\)-restricted Lie algebra having trivial bracket and trivial \([p]\)-mapping. From this perspective, \(A\) may also be given the structure of a Hopf algebra, with a different co-algebra structure. These two different co-algebra structures, give rise two (potentially) different module structures on the tensor product of modules, which leads to a difference at the level of cohomology; a difference that is the focus of this paper. The structure of the cohomology ring \(H^*(E,k) \cong \text{Ext}_{A}^*(k,k)\) is well known. In characteristic 2, it is a polynomial algebra on \(r\) generators (in degree 1), and, in odd characteristics, it is a polynomial algebra on \(r\) generators (in degree 2) tensored with an exterior algebra on \(r\) generators (in degree 1). Note that the structure of the cohomology ring depends only on the algebra structure of \(A\), not on the co-algebra structure. However, given an \(A\)-module \(M\), tensoring with \(M\) gives rise to a ring homomorphism \(\text{Ext}_{A}^*(k,k) \to \text{Ext}_{A}^*(M,M)\), which does depend on the chosen co-algebra structure. Let \(S \subset \text{Ext}_{A}^*(k,k)\) denote the polynomial subring generated by the Bocksteins of degree 1 cohomology classes. The main result of this paper is that, when this ring homomorphism is restricted to \(S\), it is the same for both co-algebra structures. A key element of the proof, which is reviewed in this paper, is a result of \textit{J. Pevtsova} and \textit{S. Witherspoon} [Algebr. Represent. Theory 12, No. 6, 567--595 (2009; Zbl 1223.16003)], which showed that this ring homomorphism factors through the Hochshild cohomology ring \(H^*(A/k;A) = \text{Ext}_{A^e}^*(A,A)\), where \(A^e := A\otimes A^{\text{op}}\). That is, the map factors as \[ \text{Ext}_{A}^*(k,k) \to H^*(A/k;A) \to \text{Ext}_{A}^*(M,M). \] Here, the authors explicitly compute the first map, which does depend on the co-algebra structure. The result follows from this computation and the fact that the second map depends only on \(M\) and not on the co-algebra structure. The authors apply this result to tensoring with a Carlson module. Given a homogeneous cohomology class \(\zeta \in \text{Ext}_{A}^*(k,k)\), there is an associated Carlson module \(L_{\zeta}\), obtained as a submodule of an associated syzygy module, that (in general) has interesting properties with respect to the cohomology map \(\text{Ext}_{A}^*(k,k) \to \text{Ext}_{A}^*(L_{\zeta},L_{\zeta})\). From the aforementioned main result, if \(\zeta \in S\) and \(M\) is any \(A\)-module, then the module structures on \(L_{\zeta}\otimes M\) given by the two co-algebra structures on \(A\) are isomorphic.