2026/07/26 by Abdolnaser Bahlekeh, Shokrollah Salarian
#math.RT
Assume that \C is an exact category. This paper is concerned with the natural transformations between extension functors on \C. The first main result indicates that if \C has enough projective objects, then for any pair of objects M, N∈ \C and any non-negative integer n, the group of all natural transformations from \Extn+1\C(N, -) to \Extn+1\C(M, -) is isomorphic to the quotient group \Extn\C(M, \syznN)/\p, where \p is the subgroup consisting of those extensions of length n arising as a push-out along a morphism P\rt\syznN, with P projective. This, together with the Auslander-Gruson-Jensen duality yields that if \C is the category of all finitely presented left modules over an associative ring R, then the quotient group is isomorphic to the natural transformations from \Torn+1R(-, M) to \Torn+1R(-, N). The second main result proves that if \C is an n-Frobenuis category, then the statement of the first result remains true, whenever projectives are replaced by n-projectives. This result is fruitful from the point of view that, n-Frobenius categories may not have projective objects. These results provide a far-reaching generalization of the Hilton-Rees theorem, in the sense that the case n=0, recover the Hilton-Rees theorem.