# Origin of exact sequences

I have seen exact sequences appearing a lot in algebraic texts with different purposes. But I've never seen names of the people associated with it. Also I don't understand what's so good about showing a certain sequence is exact.

1. Does anyone know who first invented the exact sequences or how they came to be?
2. What were they invented for originally and what's so useful about them?
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–  archipelago Mar 12 '14 at 21:42
Thanks. I've already seen that. I'm interested in their history and origin. –  Spock Mar 12 '14 at 21:43
I wonder if the Poincaré lemma and syzygy modules (à la Hilbert) should count as progenitors of the notion of an exact sequence. –  Igor Khavkine Mar 13 '14 at 14:46
What exactly was Hilbert's formulation of the syzygy theorem? –  Mariano Suárez-Alvarez Mar 13 '14 at 22:11
My apologies, @MarianoSuárez-Alvarez, but my knowledge does not really extend beyond en.wikipedia.org/wiki/Hilbert%27s_syzygy_theorem. –  Igor Khavkine Mar 14 '14 at 9:24

From Lefschetz's obituary of Hurewicz: "At a later date (1941) and in a very short abstract of this Bulletin Hurewicz introduced the concept of exact sequence whose mushroom like expansion in recent topology is well known."

Here is the abstract in its entirety:

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This abstract not only answers the OP's first question; it's also a great answer for the second. –  Steven Landsburg Mar 12 '14 at 22:52

A short exact sequence $$0\to A\to B\to C\to 0$$ tells you that to understand $B$ you often can split the work in three parts: understand $A$ first, then understand $C$, then finally try to glue your understanding of $A$ and $C$ into understanding of $B$ itself.

They show up all over the place because this strategy is very often successful.

P.S. Sometimes one proceeds differently. For example, you may be interested inknowing something about $A$, but it is difficult, so you find a larger $B$, which is hopefully easier to study, but then the information you got is not about $A$ but about $B$, so $C$ measures the difference. Similarly, one is often interested in $C$, and it is convenient to describe it as the quotient of a simpler object $B$: but then $B$ and $C$ are not the same thing, of course, and you need to study their difference, which is encoded in the object $A$.

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There is a long exact sequence in a paper of Hurewicz in the Bulletin of the AMS, 1941. The paper is called "On Duality Theorems". According to Weibel's History of Homological Algebra, this is the first appearance of an exact sequence.

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Right, and Weibel goes on to say that after "the application of this notion in the 1945 axiomatization of homology theory [Eilenberg-Steenrod]", "The next step was taken in 1947 by Kelley and Pitcher, who coined the term “exact sequence” [Exact homomorphism sequences in homology theory, Annals of Math. 48 (1947), 682–709]". –  Francois Ziegler Mar 12 '14 at 21:52
I couldn't fine the 1941 paper on the AMS Bulletin website. –  Spock Mar 12 '14 at 22:28
@Spock: Neither could I. But it's in Hurewicz's collected works. –  Steven Landsburg Mar 12 '14 at 22:48

In the book "A History of Algebraic and Differential Topology, 1900 - 1960" by Jean Dieudonne http://www.amazon.com/History-Algebraic-Differential-Topology-1900/dp/0817649069 there is a section "Exact Sequences" on pages 85-89 where a somewhat detailed history of this concept is described. According to Dieudonne, the first appearence of the exact sequence occured in a short paper on dimension theory without proofs by Hurewicz (mentioned in the answers above). Hurewicz's essential remark was that in a given sequence of homomorphisms the image of each homomorphism was the kernel of the next one, but he didn't use the name "exact sequence" to characterize this property. At the same time, independently, Eckmann and Ehresmann and Feldbau each described what later was called the homotopy exact sequence of a fiber space. At that time nobody noted any relation between these results and that of Hurewicz.

The next two appearances of exact sequences in homology theory occured in 1945. The first, under the name "natural system of groups and homomorphisms" was in the axiomatic theory of homology by Eilenberg and Steenrod. The second, independently, was in the first topology paper of H. Cartan, although he didn't used the term "exact sequence". This term first appeared in 1947 paper by Kelley and Pitcher (also mentioned in the answers above). They observed that the notion of "exact sequence" is meaningful for arbitrary commutative groups and homomorphisms of groups and that all previously considered exact sequences in homology are just special cases of a purely algebraic result applying to the chain complexes of commutative groups.

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From Peter Freyd's book, "Abelian Categories", page 156: "And then one day at Princeton, my advisor, Norman Steenrod , calmly told me how he and Eilenberg --- just a few years before --- had chosen the word "exact"."

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