TITLE: Etymology of Tor and Ext Functors
QUESTION [15 upvotes]: The names of the derived functors $\operatorname{Tor}$ and $\operatorname{Ext}$ seem quite cryptic to me. Does anyone know what these abbreviations stand for? I would be glad if someone could tell me where these names come from.

REPLY [17 votes]: Ext stands for extension, as the group $\operatorname{Ext}^1(X,Y)$ parameterises extensions $Z$ fitting into a short exact sequence:
$$0\to Y\to Z\to X\to 0$$
modulo the trivial extension $X\oplus Y$.
According to Wikipedia, Tor is short for torsion, as if $r\in R$ is not a zero divisor and $B$ is an $R$-module, then $\operatorname{Tor}_1(R/(r),B)$ can be identified with the $r$-torsion part of $B$, i.e. $b\in B$ such that $rb=0$.

REPLY [3 votes]: Torsion and Extension.............