Generalized support varieties for finite group schemes. (Q612979)

Let \(k\) be a field of finite characteristic \(p\), and let \(G\) be a finite group scheme whose order is a multiple of \(p\). Let \(kG\) be the dual of the coordinate algebra \(k[G]\). In previous work the authors, along with others, have created \(\pi\)-points, flat \(K\)-algebra maps \(\alpha_K\colon K[t]/(t^p)\to KG\) for \(K\) an extension of \(k\), as an aide in studying support varieties of \(kG\)-modules. The scheme of equivalence classes of \(\pi\)-points is denoted \(\Pi(G)\). Each finite dimensional \(kG\)-module \(M\) gives rise to a closed subset \(\Pi(G)_M\) of \(\Pi(G)\), and every closed subset is of this form. However, this is not a one-to-one correspondence, i.e., it is possible for \(M_1\) and \(M_2\) to be nonisomorphic \(kG\)-modules which give the same closed subset of \(\Pi(G)\): examples can be constructed using the observation that \(\Pi(G)_M\) is empty whenever \(M\) is a projective \(kG\)-module. The non-maximal subvariety \(\Gamma(G)_M\) provides some refinement to the support variety \(\Pi(G)_M\). In the work under review, the authors introduce a new family of invariants which they call ``generalized support varieties''. For a finite dimensional \(kG\)-module \(M\) and \(1\leq j<p\) they define \(\Gamma^j(G)\) to be the subsets of \(\Pi(G)\) which satisfy a certain non-maximal rank condition which depends on both \(j\) and \(M\). The \(\Gamma^j(G)\) are called non-maximal rank varieties. The collection \(\{\Gamma^j(G)_M\}\) is finer than \(\Pi(G)\), and each \(\Gamma^j(G)_M\) is a proper closed subset of \(\Pi(G)\). Other properties are proved, such as \(\Gamma^j(G)_M\) is empty if and only if \(M\) have constant \(j\)-rank, these varieties do not differentiate between stably isomorphic \(kG\)-modules nor modules in the same component of the stable Auslander-Reiten quiver, and the union of the \(\Gamma^j(G)_M\) is equal to \(\Gamma(G)_M\). Another class of invariants is introduced when \(M\) is of constant rank, i.e., then the rank of the operator \(M_K\to M_K\) induced from a \(\pi\)-point is independent of the choice of \(\pi\)-point. A cohomology class \(\zeta\in H^1(G,M)\) gives rise to an extension \(E_\eta\) of \(k\) by \(M\). In an effort to generalize the zero locus in \(\text{Spec}(H^\bullet(G,k))\) a subset \(Z(\zeta)\) is constructed. Here \(Z(\zeta)\) is shown to be \(\Pi(G)\) if the extension with \(E_\zeta\) as above is locally split, otherwise \(Z(\zeta)=\Gamma^1(G)_{E_\zeta}\), establishing that \(Z(\zeta)\) is necessarily closed in \(\Pi(G)\). This construction is then generalized to extension classes \(\zeta\in\text{Ext}_G^n(M,N)\) where \(M\) and \(N\) are \(kG\) modules of constant Jordan type.