Introduction to Serre’s Conjecture: 1955–1976

Abstract

On p. 243 of his famous article „Faisceaux algébriques cohérents“ (FAC, ca. 1955), Serre wrote: “On ignore s’il existe des A-modules projectifs de type fini qui ne soient pas libres” (A = k[t ₁, ..., t _n ], k a field). Shortly thereafter, the freeness of finitely generated projective modules over k[t ₁, ..., t _n ] became known to the mathematical world as “Serre’s Conjecture”. Serre had in no uncertain terms objected to the fact that what he raised as an open problem was turned into his “conjecture” by world acclaim.^(*) However, the fine distinction between “Serre’s Problem” and “Serre’s Conjecture” may now be safely left to the deliberations of the mathematical historian. Culminating almost twenty years of effort by algebraists, D. Quillen and A. Suslin proved independently, in January of 1976, that finitely generated projective modules over k[t ₁,..., t _n ] are, indeed, free.

In a letter to me on April 24, 1978 (right after the appearance of my Springer Lecture Notes), Serre reminisced: “No doubt about that: “Problem” is right. I never wrote anything indicating that I believed or disbelieved in a positive solution. Of course, I could not help that some people called the whole thing a “conjecture”, but I objected as often as I could.”