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* A curious history
@ 2024-01-07  0:21 Michael Barr, Prof.
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From: Michael Barr, Prof. @ 2024-01-07  0:21 UTC (permalink / raw)
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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.


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