ON THE EXISTENCE OF MOCK INJECTIVE MODULES FOR ALGEBRAIC GROUPS
WILLIAM D. HARDESTY, DANIEL K. NAKANO, AND PAUL SOBAJE
Abstract. Let G be an affine algebraic group scheme over an algebraically closed field k 
of characteristic p > 0, and let Gr denote the r-th Frobenius kernel of G. Motivated by 
recent work of Friedlander, the authors investigate the class of mock injective G-modules, 
which are defined to be those rational G-modules that are injective on restriction to Gr 
for all r # 1. In this paper the authors provide necessary and sufficient conditions for the 
existence of non-injective mock injective G-modules, thereby answering a question raised 
by Friedlander. Furthermore, the authors investigate the existence of non-injective mock 
injectives with simple socles. Interesting cases are discovered that show that this can occur 
for reductive groups, but will not occur for their Borel subgroups.