Projectivity of modules for infinitesimal unipotent group schemes

Abstract

In this paper, it is shown that the projectivity of a rational module for an infinitesimal unipotent group scheme over an algebraically closed field of positive characteristic can be detected on a family of closed subgroups. Let k be an algebraically closed field of characteristic p > 0 and G be an infinitesimal group scheme over k, that is, an affine group scheme G over k whose coordinate (Hopf) algebra k[G] is a finite-dimensional local k-algebra. A rational G-module is equivalent to a k[G]-comodule and further equivalent to a module for the finitedimensional cocommutative Hopf algebra k[G]∗ ≡ Homk(k[G], k). Since k[G]∗ is a Frobenius algebra (cf. [Jan]), a rational G-module (even infinite-dimensional) is in fact projective if and only if it is injective (cf. [FW]). Further, for any rational G-module M and any closed subgroup scheme H ⊂ G, if M is projective over G, then it remains projective upon restriction to H (cf. [Jan]). We consider the question of whether there is a “nice” collection of closed subgroups of G upon which projectivity (over G) can be detected. For an example of what we mean by a “nice” collection, consider the situation of modules over a finite group. Over a field of characteristic p > 0, a module over a finite group is projective if and only if it is projective upon restriction to a p-Sylow subgroup (cf. [Rim]). For a p-group (and hence for any finite group), L. Chouinard [Ch] showed that a module is projective if and only if it is projective upon restriction to every elementary abelian subgroup. If the module is assumed to be finite-dimensional, this result follows from the theory of varieties for finite groups (cf. [Ca] or [Ben]). Indeed, elementary abelian subgroups play an essential role in this theory. In work of A. Suslin, E. Friedlander, and the author [SFB1], [SFB2], a theory of varieties for infinitesimal group schemes was developed. In this setting, subgroups of the form Ga(r) (the rth Frobenius kernel of the additive group scheme Ga) play the role analogous to that of elementary abelian subgroups in the case of finite groups. Not surprisingly then, for finite-dimensional modules, one obtains the following analogue of Chouinard’s Theorem. Proposition 1 ([SFB2, Proposition 7.6]). Let k be an algebraically closed field of characteristic p > 0, r > 0 be an integer, G be an infinitesimal group scheme over k of height ≤ r, and M be a finite-dimensional rational G-module. Then M is Received by the editors January 27, 1998 and, in revised form, March 24, 1998 and May 19, 1999. 2000 Mathematics Subject Classification. Primary 14L15, 20G05; Secondary 17B50. c ©2000 American Mathematical Society