On the space $\operatorname{Ext}$ for the group~$SL(2,q)$

V. P. Burichenko

Matematičeskie trudy, Tome 16 (2013) no. 1, pp. 28-55

Résumé

We consider the space $\operatorname{Ext}^r(A,B)=\operatorname{Ext}^r_{KG}(A,B)$, where $G=SL(2,q)$, $q=p^n$, $K$ is an algebraically closed field of characteristic $p$, $A$ and $B$ are irreducible $KG$-modules, and $r\geq1$. Carlson [6] described a basis of $\operatorname{Ext}^r(A,B)$ in arithmetical terms. However, there are certain difficulties concerning the dimension of such a space. In the present article, we find the dimension of $\operatorname{Ext}^r(A,B)$ for $r=1,2$ (in the above-mentioned article, Carlson presents the corresponding assertions without proofs; moreover, there are errors in their formulations). As a corollary, we find the dimension of the space $H^r(G,A)$, where $A$ is an irreducible $KG$-module. This result can be used in studying nonsplit extensions of $L_2(q)$.