Geometry of infinitesimal group schemes (Q1126705)

We consider affine group schemes \(G\) over a field \(k\) of characteristic \(p>0\). Equivalently, we consider finitely generated commutative \(k\)-algebras \(k[G]\) (the coordinate algebra of \(G\)) endowed with the structure of a Hopf algebra. The group scheme \(G\) is said to be finite if \(k[G]\) is finite dimensional (over \(k\)) and a finite group scheme is said to be infinitesimal if the (finite-dimensional) algebra \(k[G]\) is local. A rational \({\mathbf G}\)-module is a \(k\)-vector space endowed with the structure of a comodule for the Hopf algebra \(k[G]\). The abelian category of rational \(G\)-modules has enough injectives, so that \(\text{Ext}_G^i (M,N)\) is well defined for any pair of rational \(G\)-modules \(M,N\) and any nonnegative integer \(i\). Unlike the situation in characteristic 0, this category has many nontrivial extensions reflected by the cohomology groups we study. We sketch recent results concerning the cohomology algebras \(H^*(G,k)\) and the \(H^*(G,k)\)-modules \(\text{Ext}_G^* (M,M)\) for infinitesimal group schemes \(G\) and finite-dimensional rational \(G\)-modules \(M\). These results, obtained with \textit{A. Suslin} [Invent. Math. 127, 209-270 (1997; Zbl 0945.14028)] and with \textit{A. Suslin} and \textit{C. Bendel} [J. Am Math. Soc. 10, 729-759 (1997; Zbl 0960.14024)], are inspired by analogous results for finite groups. Indeed, we anticipate but have yet to realize a common generalization to the context of finite group schemes of our results and those for finite groups established by D. Quillen, J. Carlson, G. Avrunin and L. Scott, and others. Although there is considerable parallelism between the contexts of finite groups and infinitesimal group schemes, new techniques have been required to work with infinitesimal group schemes. Since the geometry first occuring in the context of finite groups occurs more naturally and with more structure in these recent developments, we expect these developments to offer new insights into the representation theory of finite groups. The most natural examples of infinitesimal group schemes arise as Frobenius kernels of affine algebraic groups \(G\) over \(k\) (i.e., affine group schemes whose coordinate algebras are reduced). Recall that the Frobenius map \(F: G\to G^{(1)}\) of an affine group scheme is associated to the natural map \(k[G]^{(1)}\to k[G]\) of \(k\)-algebras. (For any \(k\)-vector space \(V\) and any positive integer \(r\), the \(r\)th Frobenius twist \(V^{(r)}\) is the \(k\)-vector space obtained by base change by the \(p^r\)th power map \(k\to k\).) The \(r\)th Frobenius kernel of \(G\), denoted \(G_{(r)}\), is defined to be the kernel of the \(r\)th iterate of the Frobenius map, \(\ker\{F^r: G\to G^{(r)}\}\); thus, \[ k[G_{(r)}]= k[G]/ (X^{p^r};\;X\in{\mathcal M}_e), \] where \({\mathcal M}_e\subset k[G]\) is the maximal ideal at the identity of \(G\). If \(M\) is an irreducible rational \(G\)-module for an algebraic group \(G\), then \(M^{(r)}\) is again irreducible; moreover, for \(r\neq s\), \(M^{(r)}\) is not isomorphic to \(M^{(s)}\). It is easy to see that a rational \(G\)-module \(N\) is the \(r\)th twist of some rational \(G\)-module \(M\) if and only if \(G_{(r)}\) acts trivially on \(N\). Thus, much of the representation theory of an algebraic group \(G\) is lost when rational \(G\)-modules are viewed by restriction as \(G_{(r)}\)-modules. On the other hand, in favorable cases the category of rational \(G\)-modules is equivalent to the category locally finite modules for the hyperalgebra of \(G\), the ind-object \(\{G_{(r)}\), \(r\geq 0\}\). The special case of the first infinitesimal kernel \(G_{(1)}\) of an algebraic group \(G\) is a familiar object. The (\(k\)-linear) dual \(k[G_{(1)}]^\#\) of the coordinate algebra of \(G_{(1)}\) is naturally isomorphic to the restricted enveloping algebra of the \(p\)-restricted Lie algebra \(g= \text{Lie}(G)\). Thus, the category of rational \(G_{(1)}\)-modules is naturally isomorphic to the category of restricted \(g\)-modules. The results we describe in this paper are natural generalizations and refinements of results earlier obtained by the author and Brian Parshall for \(p\)-restricted Lie algebras.