关于扩张函子的自然变换
On the natural transformations of extension functors
浏览论文内容
中文总结 AI 辅助
本文研究正合范畴$\C$上扩张函子的自然变换，给出两个主要结果。一是在$\C$有足够多投射对象时相关自然变换群的同构结论及在特定范畴下的情况，二是在$n$-Frobenius范畴中类似结论成立，推广了Hilton - Rees定理。
中文摘要 AI 辅助
假设$\C$是一个正合范畴。本文关注$\C$上扩张函子之间的自然变换。第一个主要结果表明，若$\C$有足够多投射对象，对于$\C$中任意对象对$M,N$及任意非负整数$n$，从$\Ext^{n + 1}_{\C}(N, -)$到$\Ext^{n + 1}_{\C}(M, -)$的所有自然变换群同构于商群$\Ext^n_{\C}(M, \syz^nN)/{\p}$，其中$\p$是由沿态射$P\to\syz^nN$（$P$为投射对象）的推出产生的长度为$n$的扩张组成的子群。结合Auslander - Gruson - Jensen对偶性可知当$\C$是结合环$R$上所有有限表示左模的范畴时，该商群同构于从$\Tor_{n + 1}^R(-, M)$到$\Tor_{n + 1}^R(-, N)$的自然变换。第二个主要结果证明若$\C$是$n$-Frobenius范畴，将投射对象换为$n$-投射对象时第一个结果仍成立。这些结果是Hilton - Rees定理的深远推广，$n = 0$时可恢复该定理。
英文摘要
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.
