On the timing of our common work BGRS (Bautista, Gabriel, Roiter, Salmeron)
Ringel’s repeated misstatements raise many questions. Obviously, he feels quite safe on a website of the University of Bielefeld. But things might really look bad for the mathematical community, if his procedure caught on. Ringel’s behaviour is therefore to blame even in the present case of a still moderate mathematical achievement.
It is not our intention to examine here Ringel’s simplistic hints at his sources. Particularly problematic are his allegations on the internal teamwork because the team members are sworn to secrecy there where Ringel claims a right to inspection.
After a first misstatement in the Math. Reviews, Ringel was corrected by Bautista in 1989. Thereupon he called me quite humbly and promised immediate rectification. But he also claimed very strongly a copy of my original proof, in disregard of the written request of secrecy of my co-authors. After 25 years I do not remember the conclusion of the discussion – my biggest worry then concerned 3 seriously ill members of my family. Be that as it may, I met Ringel several times until 1998. He even invited me to Bielefeld, but he never reminded me of any promise of mine.
The question now is whether I can refute his assertions, without mentioning individual contributions of team members. Let us try to do so by stressing the timing rather than the mathematical content of our work.
1) I was in Mexico in 1982 between July 15 and august 15. BS (Bautista+Salmeron) then showed me a booklet full of notes taken during discussions with Roiter in Kiev. All these notes concerned Roiter’s article Generalization of Bongartz‚ Theorem. The intention was to correct Roiter‘s wrong proof of the existence of multiplicative bases in representation-finite algebras. In this purpose BS asked me to join their team.
I had not read Roiter‘s article before and consulted it only marginally afterwards. I did neither get nor ask for any copy of the notes. Thus my answer was quite clear: Those who find a proof, have to write it down. I am not available for pure cosmetics. But I am ready to think about the proposal and to join the team if I can be of some help with a real mathematical contribution.
2) BS then explained me that the lacking proof consisted in two parts. One half concerned the existence of a semimultiplicative basis. The other part, as I try to remember now, concerned the reduction of a semimultiplicative basis to a multiplicative one. BS proposed to try and solve the second part. And they asked me to have a look at the first one.
3) We met again in Luminy between september 20 and 25. Being the managing editor of the Commentarii Mathematici Helvetici, I had been obliged to perform in meantime the work postponed during my visit to Mexico. I had found no time to think about semimultiplicative bases, whereas BS had completed their part of the proof. They entrusted me a manuscript and we decided to examine separately the yet non-solved half of the proof. On my side, I again expressed my intention to join the team only if I was able to offer a substantial mathematical contribution to the solution.
4) BS never informed me about what happened in Mexico between October 1982 and March 1983. In Zurich the school year began at the beginning of October. Until Christmas I was able to read thoroughly their manuscript and to convince myself that the existence of semimultiplicative bases really was the only lacking part of the whole proof. And I went through Fischbacher’s list of representation-finite algebras with 3 simple modules. But no time was left before Christmas for tough research.
So I decided to devote the Christmas holidays (1½ weeks!) to semimultiplicative bases. As is well known, finite representation-type is a sophisticated notion, whereas semimultiplicativity is a rather simple one. So I first tried to understand what the existence of a semimultiplicative basis really means for a given algebra. Using Fischbacher’s lists, I first looked at the algebras with at most 3 simple modules. The result was surprisingly simple and easily implied the existence of such bases in the representation-finite case (1983 January 3). In other words, my proof was a piece of what I like to call experimental mathematics.
5) What could I do? I had no information from BS. But, due to the conditions I had imposed on a collaboration, it was up to me to inform them about my results. I knew that Roiter should arrive in Mexico in January. Therefore I wrote quickly 11 pages summarizing the results of my research. Up to now, not a single word has been changed in this summary entitled Roiter’s theorem on multiplicative bases. I then learned from Roiter that his arrival in Mexico had been postponed to February 16 by the soviet administration. So I decided to wait 1 month before sending him my summary to Mexico. The reason was that I didn’t trust the soviet post office – for instance, letters which Roiter wrote in 1983 on November 16 and November 28 did not reach me in Trondheim, which I left on December 17 –. On the other hand, Roiter seemed to be the right man for discussing with BS and informing me, whether my results could still be of any use.
6) In meantime, I presented my results – as a possible contribution to a common work with BS – to a small group of 5, 6 students.
Finally, I sent my summary to Mexico on February 22. Roiter immediately acknowledged the receipt and sent me a more complete answer on March 14, simply saying that BS had worked very hard, have ‘some text’ and invite me to participate in a common team-work.
7) At the end of March, I received from Mexico 27 handwritten pages, promised to read them carefully and confirmed my intention to participate in a common work. As I remember, the manuscript was well written but far too special to reach the central points of their conceptual approach to semimultiplicativity. In my answer, I therefore added: I,t seems urgent to write my proof down. Only after this work it will be possible to have a fair comparison.
8) Until June 20, I then sent successively 4 chapters (34 pages) containing the proofs lacking in the summary. The last chapter consisted in the proof of a lemma used before. In the meantime BS sent me around June 10 the last part of their approach to semimultiplicativity, a quite impressive booklet of 68 further pages.
In fact, the 2 approaches started from opposite points. So, what about Ringel’s comment: And I know that Bautista sent his proof to Gabriel, who then proposed many changes?
9) The month of July was used by both sides to prepare the final version of the reduction of semimultiplicativity to multiplicativity. BS moreover found the time to get acquainted with my version.
Clearly, no more results were needed. But we still had to agree on the results to be published in common, the required prior knowledge and the approach. For this purpose, BS proposed on august 2 three possible devices, enumerated in the order of their preference. The 4 authors finally agreed on the first device and I presented the corresponding results as a common achievement at the Torun Conference on Representation Theory (1983 august 16-20).
10) Writing up the chosen results – in accordance with the chosen device – lasted about 5 months and was happily achieved by 4 authors then staying for quite some time in Sao Paulo, Trondheim, Kiev and Mexico-City, respectively. For those interested in the question who did what, I can whisper: Bautista wrote the sentences starting with the letters A, or E, I …, Gabriel, Roiter and Salmeron those starting with B, or F, J …, with C, or G, K … and with D, or H, L … , respectively.
It is trite to say that oral descriptions of events which happened 20 years ago, are mostly unreliable, all the more so in mathematics, all the more so if the storyteller did not take part in the events. I therefore have one question for Ringel, just one: What are your sources?
pg, Ennetbühl, 2015 June 16