  We prove that the specialization to q=1 of a Kirillov-Reshetikhin module for
an untwisted quantum affine algebra of classical type is projective in a
suitable category. This yields a uniform character formula for the
Kirillov-Reshetikhin modules. We conjecture that these results holds for
specializations of minimal affinization with some restriction on the
corresponding highest weight. We discuss the connection with the conjecture of
Nakai and Nakanishi on q-characters of minimal affinizations. We establish this
conjecture in some special cases. This also leads us to conjecture an
alternating sum formula for Jacobi-Trudi determinants.
