Detecting nilpotence and projectivity over finite unipotent supergroup schemes
Let \(k\) be a field, \(\mathrm{char}(K)=p>2\), and let \(G\) be a finite unipotent supergroup scheme over \(k\). The cohomology of \(G\) will be denoted \(H^{*,*}(G,k)\), which is isomorphic to \(\mathrm{Ext}_{kG}^{*,*}(k,k)\): the latter index in the superscript arising from the \(\mathbb Z/2\mathbb Z\)-grading. Of interest is the nilpotent elements of this cohomology group, and the authors reduce this question to one involving elementary supergroup schemes. The main result is that \(x\in H^{*,*}(G,k)\) is nilpotent if and only if \(x_K\in H^{*,*}(G\times_k K,K)\), restricted to \(H^{*,*}(E,K)\), is nilpotent, where \(K\) is an extension of \(k\) and \(E\le G\times_k K\) is elementary. Additionally, if \(M\) is a \(kG\)-module, then \(M\) is projective if and only if the restriction of \(M\times_k K\) to \(E\) is projective. These results are then applied to finite dimensional sub-Hopf algebras of the Steenrod algebra over \(\mathbb F_p\).
scientific article