I didn't quite understand some definitions on some homology topics. Let $R$ be a commutative ring, $A, A', C$ $R$-modules, consider the map $f^*: Ext^1_R(C,A) \rightarrow Ext^1_R(C,A') $ induced by $f : A \rightarrow A'$ how is the map defined? In what sense is the short exact sequence $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow0$ an element of $Im f^*$?
Asked
Active
Viewed 81 times
0
-
It's probably easier to define using Hom and projective resolutions. – Quimey Apr 19 '21 at 08:38
-
Consider the pushout $B'=A'\coprod_A B$ and show that $0\to A'\to B'\to C\to 0$ is a short exact sequence. This give an extension in $\operatorname{Ext}^1_R(C,A')$ which is the image by $f^*$ of the extension given by the initial short exact sequence. – Roland Apr 19 '21 at 15:16