On projective resolutions of simple modules over the Borel subalgebra S+(n,r) of the Schur algebra S(n,r) for n⩽3

We construct projective resolutions for simple modules over the Borel subalgebras S+(2,r) and S+(3,r) of the Schur algebras S(2,r) and S(3,r) over a field of positive characteristic. In the case S+(2,r) we get a minimal projective resolution. We find also projective resolutions of minimal possible length for Weyl modules over the Schur algebra S(2,r), corresponding to the regular weights.