# First Use of the Short Exact Sequence

A question I've been curious about for a long period of time and tried to find the answer to myself a number of times (but apparently never been able to figure out just the right thing to type into Google to make the search engine give me the right result) is, what mathematician was the first to use a short exact sequence to describe a set of objects (be they groups or rings or modules or something else), and what was the context?

Anyone happens to know? Would greatly satisfy me a lot to know the answer.

Although exact sequences appear in Hurewicz's short note from 1941, neither he nor other early authors (Eilenberg-Steenrod in 1945, Kelley-Pitcher in 1947) used short exact sequences, see Math SE, History behind Exact Sequences and MathOverflow, Origin of exact sequences for references. One can see one short exact sequence in Kelley-Pitcher, Exact Homomorphism Sequences in Homology Theory, but it is buried in Corollary 11.11b and no special note is made of it.

They appear to be Cartan's innovation motivated by sheaves. He introduced them at his seminar on algebraic topology in 1948/49. Here is from Weibel's History of Homological Algebra:

"The next year (1948-49), Henri Cartan ran a Seminar [C48] on algebraic topology, with 17 exposes published as unbound mimeographed notes. Exposes XII–XVII were devoted to an exposition of Leray’s theory of sheaves, but were withdrawn when Cartan’s viewpoint on sheaves changed later that year... In Expose 16 Cartan gave axioms for sheaf cohomology theory on a paracompact space $$X$$ (with or without supports in a family $$Φ$$ of closed subspaces of $$X$$, which we shall omit from our notation here). His axioms were: $$H^0(X,F)$$ is the group $$Γ(F)$$ of global sections of $$F$$ (with support in $$Φ$$); $$H^q(X, F)$$ depends functorially on $$F$$ and vanishes for negative $$q$$; a natural long exact cohomology sequence exists for each short exact sequence of sheaves; and if $$F$$ is a “fine” sheaf then $$H^q(X, F)=0$$ for all $$q\neq 0$$."

Judging by MathSciNet, it took a while for the lore to make its way into publications. The earliest published appearances of short exact sequences I found are from 1958, in Heller, Homological algebra in Abelian categories, and in Borel-Serre, Le théorème de Riemann-Roch.

1. The lower bound is (probably) 1941. Dieudonne in his book "A history of Algebraic and Differential Topology" writes when discussing Gysin's exact sequence (which appeared in Gysin's 1941 paper):

"Exact sequences were not yet used in 1941 but Gysin's main result can be expressed as exactness of the following homology sequence..."

Assuming that Dieudonne can be trusted on this matter, 1941 provides a lower bound.

1. The upper bound is 1945, the foundational paper

S. Eilenberg, N. Steenrod, Axiomatic approach to homology theory. Proceedings of the National Academy of Sciences of the United States of America. 31 (1945) 117–120.

were they define the long exact sequence of pairs as one of their axioms. They do not yet use the terminology "exact" and simply call it "a sequence" and define its key property which we now call "exactness." They do not refer to anybody else in the paper, so this might have been the earliest use of exact sequences as we know them. The terminology "exact sequence" is used by Eilenberg and Steenrod in their book "Foundations of Algebraic Topology", 1952. I do not know if this was the earliest use, but at least this provides an upper bound.

One should check, of course, earlier (than 1945) papers, especially ones by by Eilenberg. I checked only one, Eilenberg's 1944 Annals paper on singular homology theory. There are no exact sequences anywhere in the paper.

• Neither Eilenberg-Steenrod nor Kelley-Pitcher, who introduced the term "exact sequence" in 1947, single out short exact sequences or use them as such (one short sequence does occur in Kelley-Pitcher, buried in a statement). So 1945 is not the upper bound. The term "short exact sequence" starts to appear in print only in 1958 (Heller and Borel-Serre), but as already established. Commented Nov 24, 2021 at 1:09
• @Conifold: In the answer I was primarily addressing the question of the use of exact sequences rather than the name "exact sequence", which was my reading of the question. I was also taking the liberty of considering short exact sequences as a special case of general exact sequences. But maybe I was misinterpreting the OP by doing so. Commented Nov 24, 2021 at 18:27