The recent kerfuffle about plagiarism puts reminded me, I'm not sure why, about a paper published with my name as sole author to which I never contributed a single word.
The paper is called, A note on commutative algebra cohomology, Bull. Amer. Math. Soc.
74 (1968), 31--313. So who wrote it, what Bulletin editor did he send it to, who refereed it, and who accepted it for publication? The answer to all four questions is the same: Murray Gerstenhaber.
I had sent him an example of a commutative ring R and an injective R-module M, for which
the commutative (Harrison) second cohomology group does not vanish. The example is ultra-simple and I will describe it below. The significance of this is that it demonstrated that the Harrison cohomology groups could not be fit into the Cartan-Eilenberg
frame in which these groups were Ext(E,M) where E is an enveloping module and M is the coefficient module. Murray asked me to write the example as a note and he, as editor-in-chief of the Bulletin would publish it. I told to include there example, with credit
to me, in his paper. His response was to write the paper, submit it to himself, referee it and accept it.
I was quite surprised, after looking at the paper preparatory to writing this note,
to discover that even section 1, called An example, is not the example I had originally sent him. In fact, he gave an example of the non-vanishing of the third cohomology group using an explicit cocycle. My example was the non-vanishing of the second group
and didn't mention cocycles at all. There were two more sections of the paper that I had nothing at all to do with.
Here is the example. Let k be a (commutative) field and R = k[x]/(x^2), sometimes called
the ring of dual numbers over k. R is well known to be self-injective. The proof is a consequence of a theorem in Cartan-Eilenberg. Let S = R[x]/(x^4). The obvious exact sequence
0 ---> <x^2,x^3> ---> S ---> R ---> 0
whose kernel is the linear span of x^2 and x^3 quite obviously cannot split. As an
R-module, the kernel is R itself under the map taking 1 to x^2 and x to x^3.
But this example didn't even make it into the paper. I should mention that Murray was
the mathematician who is responsible for my going into mathematics.
Michael