The homological dimension of commutative group schemes over a perfect field

Abstract

In [lo] and [7], Serre and Oort have shown that the category of commutative group schemes of finite type over an algebraically closed field has homological dimension one if the field has characteristic zero and two otherwise. We extend their result by relating the homological dimension of this category of group schemes over any perfect field to the cohomological dimension of the Galois group of the field. In particular, if ;Z and B are abelian varieties over a finite field then our result implies that Exti(-4, B) = 0 for i > 2, and this completes the computation of these groups (see [4]). Also, Oort and Oda [8] have shown that Exts(A, B) = 0 if ;1 and B are abelian varieties over an algebraically closed field. We give a short proof of