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.

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