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Mat. Sb., 2014, Volume 205, Number 3, Pages 41–82 (Mi msb8164)  

On the third cohomology of algebraic groups of rank two in positive characteristic

A. S. Dzhumadil'daeva, Sh. Sh. Ibraevb

a Institute of Mathematics, Ministry of Education and Science, Republic of Kazakhstan
b Bolashak University, Kyzylorda

Abstract: We evaluate the third cohomology of simple simply connected algebraic groups of rank 2 over an algebraically closed field of positive characteristic with coefficients in simple modules. It is assumed that the characteristic $p$ of the field is greater than $3$ for $\operatorname{SL}_3$, greater than $5$ for $\operatorname{Sp}_4$, and greater than $11$ for $G_2$. It follows from the main result that the dimensions of the cohomology spaces do not exceed the rank of the algebraic group in question. To prove the main results we study the properties of the first-quadrant Lyndon-Hochschild-Serre spectral sequence with respect to an infinitesimal subgroup, namely, the Frobenius kernel of the given algebraic group.
Bibliography: 49 titles.

Keywords: algebraic group, cohomology, simple module, Frobenius kernel.

Funding Agency Grant Number
Ministry of Education and Science of the Republic of Kazakhstan 055

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm8164

Full text: PDF file (801 kB)
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English version:
Sbornik: Mathematics, 2014, 205:3, 343–386

Bibliographic databases:

UDC: 512.815.6
MSC: Primary 20G15; Secondary 14L40, 17B45, 20G10, 20G15, 22E47
Received: 16.08.2012 and 04.01.2014

Citation: A. S. Dzhumadil'daev, Sh. Sh. Ibraev, “On the third cohomology of algebraic groups of rank two in positive characteristic”, Mat. Sb., 205:3 (2014), 41–82; Sb. Math., 205:3 (2014), 343–386

Citation in format AMSBIB
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