Cohomology and projectivity of modules for finite group schemes (Q2758154)

Let \(G\) be a finite group scheme over a field \(k\) of positive characteristic \(p\). Let \(M\) be a rational \(G\)-module. Two questions are investigated here. From the paper: NEWLINENEWLINENEWLINE(a)``Does there exist a family of subgroup schemes of \(G\) which detects whether \(M\) is projective?'' NEWLINENEWLINENEWLINE(b)``Does there exist a family of subgroup schemes of \(G\) which detects whether a cohomology class \(z\in \)Ext\(_{G}^{n}(M,M)\) (for \(M\) finite dimensional) is nilpotent?'' NEWLINENEWLINENEWLINEIf the connected component of \(G\) is also unipotent, then the answer to both questions is ``yes''. This paper gives the subgroup schemes necessary to answer both of these questions. NEWLINENEWLINENEWLINEThe subgroup schemes in question are the elementary group schemes. Here the elementary group scheme \({\mathcal E}_{r,s}\) is the product \({\mathbb G}_{a(r)}\times E_{s},\) where \({\mathbb G}_{a(r)}\) is the \(r\)-th Frobenius kernel of the additive group scheme (referred to \(\alpha_{p^{r}}\) sometimes in the literature) and \(E_{s}\) is the constant group scheme for an elementary abelian \(p\)-group of rank \(s\). The cohomology of \({\mathcal E}_{r,s}\) is well-known (since \(H^{\ast }({\mathcal E}_{r,s},k)=H^{\ast }({\mathbb G}_{a(r)},k)\otimes H^{\ast }(E_{s},k)\)). NEWLINENEWLINENEWLINEThe second of the two questions is resolved first. Theorem 6.1 shows that if there is a \(z\in H^{n}(G,\Lambda)\) so that for any \(K/k\) finite and \(v:{\mathcal E}_{r,s}\otimes _{k}K\hookrightarrow G\otimes _{k}K\) any embedding we have \(v^{\ast }(z)\in H^{n}({\mathcal E}_{r,s}\otimes _{k}K,\Lambda \otimes _{k}K)\) nilpotent, then \(z\) is nilpotent. Detecting projectivity is handled in theorem 8.1, which states that \(M\) is projective if and only if for every \(K/k\) finite and every \(H\subset G\otimes _{k}K\) closed with \(H\cong {\mathcal E}_{r,s}\otimes _{k}K\) the restriction of \(M\otimes _{k}K\) to \(H\) is projective. Finally, the author asks if either of these results can be extended to any finite \(G\).