Some homological properties of GL(m|n) in arbitrary characteristic

We show that Penkov's approach to a superanalog of Borel-Bott-Weil theorem for $G=GL(m|n)$ over a field of zero characteristic can be extended for a perfect field of arbitrary odd characteristic. We also prove some partial version of Kempf's vanishing theorem and characteristic free formula for Euler characteristic $\chi(B, \lambda^{\epsilon})$, where $B$ is a Borel subgroup of $G$.