On the natural transformations of extension functors

Abstract

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 $M, N\in \C$ and any non-negative integer $n$, the group of all natural transformations from $\Ext^{n+1}_{\C}(N, -)$ to $\Ext^{n+1}_{\C}(M, -)$ is isomorphic to the quotient group $\Ext^n_{\C}(M, \syz^nN)/{\p}$, where $\p$ is the subgroup consisting of those extensions of length $n$ arising as a push-out along a morphism $P\rt\syz^nN$, 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 $\Tor_{n+1}^R(-, M)$ to $\Tor_{n+1}^R(-, N)$. 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.