In my thesis, in the chapter on finite flat groupschemes, I made the mistake of thinking that a finite flat group scheme is the same thing as a finite locally free group scheme. In other words, I made the classic mistake of thinking that a finite flat module over a ring is finite locally free (or equivalently finitely presented). A counter example is given in the stacks project, see examples.pdf. Luckily I discovered this error (or maybe somebody else did and pointed it out to me) and the published version of my thesis does not have this mistake.

Why I made this mistake I am not sure, maybe because I read Matsumura’s Commutative Algebra, where you can find the result that a finite flat module over a local ring is finite free.

I have since learned that this is not as bad a mistake as one may think. Namely, it turns out that whether or not every finite flat R-module is finite locally free, is a property of R which depends only on the topology of X = Spec(R). The result is that every finite flat R-module is finite locally free if and only if every Z ⊂ X which is closed and closed under generalizations is also open. A similar result holds for schemes. (I found this in some paper a while back, but now I cannot remember which paper.)

I just added this to the stacks project this morning, see Algebra, Lemma Tag 052U and Morphisms, Lemma Tag 053N.

Pingback: Flat finite type ring extensions « Stacks Project Blog

This is Thm 5.7 in Lazards “Disconnexités des spectres d’anneaux et des préschémas” (Bull SMF 1967, p. 95-108).