# Andrei Suslin's works

Andrei Suslin, a well known mathematician, died 10 July 2018. (https://en.wikipedia.org/wiki/Deaths_in_2018) I believe it may be appropriate to give an overview of his work on this site. Personally, I know the Quillen–Suslin theorem but little of more recent works.

[EDIT] I will try to give a presentation of the Quillen–Suslin theorem as I understand it. (As was suggested in the comments. Experts, please feel free to set me straight if I get anything wrong.) Let $k$ be an arbitrary field and $A=k[x_1,\dots, x_n]$ a polynomial right over it. The theorem states that a finitely generated projective module over $A$ is free. In down to earth terms it means that if $$\sum_{i=1}^n a_ib_i=1,$$ then there exists an invertible $n\times n$ matrix with the first column $(a_1,\dots, a_n)^\top$. (Although the equivalence of these statements is not exactly obvious.) The problem was originally formulated in 1955 by Serre and became known as the Serre conjecture (apparently to his chagrin). It took twenty years to prove; this was done independently by Quillen and Suslin in 1976.

Why this result is interesting? (1) It is just interesting (polynomial rings are rigid'' very unlike, say, fields). (2) It has a hypothetical generalization with a field replaced by a regular Noetherian ring, called a Bass–Quillen conjecture. (3) It has natural analytic analogs with projective modules interpreted as vector bundles (e.g. Is analytic Quillen-Suslin simple? ) (4) It has connections with other K-theoretical problems.

(I better stop at this before I say some nonsense.) By the way, the Wikipedia article says that a simple and short proof of the theorem may be found in Lang's "Algebra". Unfortunately, the copy of "Algebra" I have is a 1965 edition. Can anyone give a hint what this short proof is like? Also, I would appreciate some comments on the Merkurjev–Suslin theorem which I do not really understand.

• May be you can add your view on Quillen-Suslin theorem as you said you are familiar with that... Adding some more information to just saying name might make this more useful to others... Jul 12, 2018 at 13:41
• There are better places than this forum for such recognition. I recommend that you edit the question to ask about a particular aspect of Suslin's work. Surely continuing his studies is an honor that Andrei Suslin would appreciate, and this forum is well suited to handle such specific inquiries and proper responses. Gerhard "MathOverflow Does Its Recognizing Differently" Paseman, 2018.07.12. Jul 12, 2018 at 14:36
• meta.mathoverflow.net/a/3800 Jul 12, 2018 at 16:40
• I'm voting to close this question because MO is not a place for memorializing the departed; if there is a question about a mathematical result, then that should be asked separately Jul 12, 2018 at 19:44
• @YemonChoi MO is the only place which I know where so many mathematicians working in all areas are active. If some of them explain the essence of important works of the departed mathematicians, it looks to be a good contribution to our community. Jan 16, 2019 at 4:44