Cohomological finite generation for the group scheme \(SL2\) (Q2437949)

In [\textit{A. Touzé}, Duke Math. J. 151, No. 2, 219--249 (2010; Zbl 1196.20052)] and [\textit{A. Touzé} and \textit{W. van der Kallen}, Duke Math. J. 151, No. 2, 251--278 (2010; Zbl 1196.20053)] the author and Touzé proved that any linear algebraic group \(G\) over a field \(k\) is reductive if and only if it satisfies the CFG-property, namely, for all finitely generated \(k\)-algebra \(A\) acted on by \(G\), the cohomology algebra \(H^*(G,A)\) is finite generated. In the paper under review, the condition that \(k\) is a field is relaxed to a noetherian ring but the paper only consider the group \(\mathrm{SL}_2\). The main result is that \(\mathrm{SL}_2\) over a noetherian ring \(k\) has the CFG-property. More specifically, the paper first explicitly constructs a \(2\)-cochain \(c^{\mathbb{Z}}_r\) in the Hochschild complex \(C^{\bullet}(\mathbb{G}_a,\mathbb{Z})\). Using it, the paper then constructs a universal class \(c_r[m]^{(j)}\) in \(H^{2mp^{r-1}}(G,\Gamma^m\Gamma^{p^{r+j}}(\mathfrak{gl}_2))\) where \(\Gamma^mV\) denotes the \(m\)-th module of divided powers of any \(\mathbb{Z}\)-module \(V\). This allows one to show that \(H^{\mathrm{even}}(G,A) \to H^0(G,H^*(\bar{G}_r,\bar{A}))\) is noetherian where \(\bar{G}_r\) is the \(r\)-th Frobenius kernel of \(G_{k/pk}\) and \(A\) is a finite generated commutative \(k\)-algebra with \(G\) action and there is a power surjective equivariant homomorphism from \(A\) to \(\bar{A}\), where \(\bar{A}\) is a finitely generated commutative \(k/pk\)-algebra with \(G_{k/pk}\)-action such that \(\bar{G}_r\) acts trivially. This is what is needed to argue in the same way as one of author's previous paper [CRM Proceedings \& Lecture Notes 35, 127--138 (2004; Zbl 1080.20039)] to conclude that \(H^*(G,A)\) is finitely generated with some extra technical details involving filtrations and Graossahans graded ring of \(A\).