# Etymology of Tor and Ext

The names of the important functors Tor and 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.

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Tor comes from torsion, I think. The beginning of the second chapter in Weibel's book discusses this. – Potato Jul 1 '13 at 16:56
they come from their first versions. tor for torsion, ext for extensions – citedcorpse Jul 1 '13 at 16:57
I think this question would be more interesting if you modified it to include asking about the history and origin of the functors. – Potato Jul 1 '13 at 16:59
For the history and origins of these functors, see here: math.uiuc.edu/K-theory/0245/survey.pdf [History of Homological Algebra by Weibel] – Potato Jul 1 '13 at 17:03
I find it amazing that one can get to know what Ext and Tor are and not the origin of their names! – Mariano Suárez-Alvarez Dec 3 '14 at 8:17

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 $rb=0$.

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as for why torsion is called torsion, my google fu is too weak to bring up the relevant mathoverflow thread, but if i recall correctly it was to do with the fact that torsion in the algebraic sense leads to a nonorientability and "twisting" of a space onto itself – citedcorpse Jul 1 '13 at 17:03
@exitingcorpse This .pdf might answer your question: math.uiuc.edu/K-theory/0245/survey.pdf – Potato Jul 1 '13 at 17:04

Torsion and Extension.............

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Oh obviously. The dot dot dot's come off as arrogant. – Samuel Reid Jul 1 '13 at 17:58
@ Samuel Reid: "Body must be at least 30 characters" - I didn't have them :-) – Boris Novikov Jul 1 '13 at 18:36