Geodesic
On the space $\operatorname{Ext}$ for the group~$SL(2,q)$
Matematičeskie trudy, Tome 16 (2013) no. 1, pp. 28-55.

Voir la notice de l'article provenant de la source Math-Net.Ru

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 $L_2(q)$. 
@article{MT_2013_16_1_a2,
     author = {V. P. Burichenko},
     title = {On the space $\operatorname{Ext}$ for the group~$SL(2,q)$},
     journal = {Matemati\v{c}eskie trudy},
     pages = {28--55},
     publisher = {mathdoc},
     volume = {16},
     number = {1},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2013_16_1_a2/}
}
TY  - JOUR
AU  - V. P. Burichenko
TI  - On the space $\operatorname{Ext}$ for the group~$SL(2,q)$
JO  - Matematičeskie trudy
PY  - 2013
SP  - 28
EP  - 55
VL  - 16
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MT_2013_16_1_a2/
LA  - ru
ID  - MT_2013_16_1_a2
ER  - 
%0 Journal Article
%A V. P. Burichenko
%T On the space $\operatorname{Ext}$ for the group~$SL(2,q)$
%J Matematičeskie trudy
%D 2013
%P 28-55
%V 16
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MT_2013_16_1_a2/
%G ru
%F MT_2013_16_1_a2
V. P. Burichenko. On the space $\operatorname{Ext}$ for the group~$SL(2,q)$. Matematičeskie trudy, Tome 16 (2013) no. 1, pp. 28-55. http://geodesic.mathdoc.fr/item/MT_2013_16_1_a2/

[1] Bakhturin Yu. A., Osnovnye struktury sovremennoi algebry, Nauka, M., 1990 | MR | Zbl

[2] Braun K. S., Kogomologii grupp, Nauka, M., 1987 | MR

[3] Burichenko V. P., “Rasshireniya abelevoi 2-gruppy s pomoschyu $L_2(q)$ s neprivodimym deistviem”, Algebra i logika, 39:3 (2000), 280–319 | MR | Zbl

[4] Burichenko V. P., “O nerasschepimykh rasshireniyakh abelevykh $p$-grupp s pomoschyu $L_2(p^n)$ c neprivodimym deistviem”, Matem. tr., 16:2 (2013) (to appear)

[5] Godeman R., Algebraicheskaya topologiya i teoriya puchkov, Izd-vo inostr. lit., M., 1961 | MR

[6] Kartan A., Eilenberg S., Gomologicheskaya algebra, Izd-vo inostr. lit., M., 1960 | MR

[7] V. D. Mazurov (red.), Kourovskaya tetrad, Nereshennye voprosy teorii grupp, 12-e izd., Institut matematiki SO RAN, Novosibirsk, 1992 http://math.nsc.ru/~alglog/17kt.pdf | MR | Zbl

[8] Kuzmin Yu. V., Gomologicheskaya teoriya grupp, Faktorial press, M., 2006

[9] Maklein S., Gomologiya, Mir, M., 1966

[10] Brauer R., Nesbitt C., “On the modular characters of groups”, Ann. of Math. (2), 42 (1941), 556–590 | DOI | MR | Zbl

[11] Carlson J. F., “The cohomology of irreducible modules over $SL(2,p^n)$”, Proc. London Math. Soc (3), 47:3 (1983), 480–492 | DOI | MR | Zbl