On the natural transformations of extension functors

Abstract

Assume that $n$ is a non-negative integer and $\C$ an $n$-Frobenius category. This paper is concerned with the natural transformations between extension functors on $\C$. The main result of the paper indicates that for any pair of objects $M, N\in \C$, 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 Yoneda extensions of length $n$ arising as a push-out along a morphism $P\rt\syz^nN$, with $P$ $n$-projective. This result is fruitful from the point of view that, $n$-Frobenius categories may not have projective objects.