Dear All Here is a story: Editor A sends a paper to B with a refereeing request. Referee B sends in his report with a negative recommendation, and editor A rejects the paper. The author complains: The referee does not know the A-B paper. Best Johannes HUEBSCHMANN Johannes Professeur émérite Université de Lille - Sciences et Technologies Département de Mathématiques CNRS-UMR 8524 Laboratoire Paul Painlevé Labex CEMPI (ANR-11-LABX-0007-01) 59 655 VILLENEUVE D'ASCQ Cedex/France Johannes.Huebschmann@univ-lille.fr ________________________________ De: "Dusko Pavlovic" <duskgoo@gmail.com> À: "Michael Barr, Prof." <barr.michael@mcgill.ca> Cc: "categories" <categories@mq.edu.au> Envoyé: Dimanche 14 Janvier 2024 22:58:19 Objet: Re: Two stories about refereeing hi mike. thank you so much for sharing memories! i note that the early ones were simply enjoyable stories, then you got to subtle detaila of history, and now you illustrate what i guess many people said many times: that the peer reviewing system is full of cracks, if not completely broken. ironically, peer reviewing was introduced to preempt the non-peer reviewing by pre-WW2 and cold war politicians, and ended up sawing politics among the peers. do you have any suggestions how to reform the system? some people (eg perlman) suggested that on the web the interested readers are the reviewers. that there is a natural selection of math results. but in the meantime, the web selection is hardly natural, as there is a whole industry of promoting and burying ideas. is there a better way? :) -- dusko On Sat, Jan 13, 2024 at 9:17 PM Michael Barr, Prof. <barr.michael@mcgill.ca<mailto:barr.michael@mcgill.ca>> wrote: Refereeing is supposed to be the gold standard of acceptability. But is it? I am going to tell two stories below. The second one does no credit to anybody, especially me, with one exception and so I will be vague on the details so no one can be identified. Curiously, both stories as well as the one I told a week ago involve the same technique: the equivalence between elements of the second Hochschild cohomology group and singular extensions. The first one involves a sequence of theorems I am about to describe. Let A be a ring with center C and M be a 2-sided A module such that cm = mc for all m\in M and c\in C. This is equivalent to M being a module over the enveloping algebra A^e = A \otimes_C A^{op}. If d: A \to M is a derivation, then it is trivial to show that the restriction d|C : C \to M^A = {m\in M | am = ma for all a\in A}. When is the converse true? The answer appeared in a series of theorems that took the form: If A is separable over C, then every derivation d: C \to M^A, extends to a derivation \bar d: A \to M. The first such theorem (proved by Hochschild, incidentally, but not using his cohomology) was for the case that C was a field. The second iteration, proved by Roy and Sridharan, was for the case that C was a semilocal ring (that is, had only finitely many maximal ideals). The third was by Max Knus, doing it for arbitrary commutative C. I spent the academic year 1970-71 at the University of Fribourg thanks to a research grant received by Heinrich Kleisli. Every Friday we got on a train and traveled to Zurich to participate in Eckmann's category theory seminar. One day, Max was lecturing on his proof of the theorem above. I didn't actually listen to the lecture because as soon as he stated the result I thought I saw a trivial proof. Well before he finished, I had the details. What I had was that assuming that A is C-projective and that the second Hochschild cohomology group H^2(A,M) = 0, then any derivation d: C \to M^A can be extended to a derivation A \to M. How does this apply? It was well known that if A is a separable C algebra, then it is C-projective. The definition of separability is that A is A^e projective which means that H^n(A,M) = Ext^n_{A^e}(A,M) = 0 for all n > 0, and in particular H^2(A,M) = 0. This gives a more general result since it would apply, for example, to any ring of polynomials in non-commutating variables over C and these are not separable. Here is the proof. Form B = A \oplus M with multiplication (a,m)(a',m') = (aa',am'+ma'). That is the split extension, but not as C-algebras, when I use d to twist the C-action: c(a,m) = (ca, cm + dc). But still H^2(A,M) = 0 and that implies there is a splitting s: A \to B and it is a simple computation that s(a) = (s,\bar d(s)) and that \bar d is a derivation. Since s must also be C-linear, another simple computation is that \bar d extends d. Max and I wrote it up in a paper that is about 1½ pages long and sent it to an editor of the AMS Proceedings who was also a friend. I expected it would be accepted quickly since it was so easy to read and tied up loose ends so neatly. Indeed the report came back quickly and said, in part, "The only possible reason to publish this is that the question has been so badly handled in the literature." I was astonished. Badly handled? Meaning the previous authors used much more complicated arguments and I thought that this ended the question. Anyway, the report was sufficiently equivocal that the editor in question left it up to me and I said certainly, publish it. And it was duly published. But if the editor hadn't been a friend? I don't know. Until I started writing this, I had never given a thought to who the editor might have been, but it has suddenly occurred to me who it likely was. I still think the report was fatuous. The second story is different. I was asked to referee a paper. It had sort of been provisionally accepted (odd story, but I won't elaborate). The author was a protegé of someone, call him K, who knew I was the referee. I won't discuss how or why he knew, but he did. As I read it, I really one part could be vastly simplified by using the connection between H^2 and singular extensions very similar to the above. I told the editor that and left it up to him. I guess he told K who came to see me and started a song and dance about how you have to encourage young researchers just getting started, etc., etc. Finally, I yielded. I won't defend my action, I just did it. But I did write to the author, not saying I was the referee, but saying that they should know this technique for the future. And they withdrew the paper! As I said there was one person who came out of this looking good. I hope he rewrote it and sent it off again. If I had ready access to Math Reviews, I would try to find out. But the two stories make me wonder about refereeing in general. 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