The category of F-modules has finite global dimension

Let R   be a regular ring of characteristic p>0p>0. In [4], Hochster showed that the category of Lyubeznikʼs FRFR-modules has enough injectives, so that every FRFR-module has an injective resolution in this category. We show in this paper that under mild conditions on R, for example when R is essentially of finite type over an F-finite regular local ring, the category of F  -modules has finite global dimension d+1d+1 where d=dimR. In [4], Hochster also showed that when M and N   are FRFR-finite FRFR-modules, HomFR(M,N)HomFR(M,N) is finite. We show that in general ExtFR1(M,N) is not necessarily finite.