There should be an (anti-)equivalence between finite commutative group schemes over perfect fields, and certain modules over the Dieudonné ring of that field. Despite being universally known, the proof is rarely given in the literature. I am looking for a reference that satisfies the following criteria.
- It's written in English.
- It's written in modern language (so no barely-readable typewriter stuff with archaic language).
- It's no-nonsense (so no crystalline cohomology, no prisms, no Breuil-Kisin modules, just a simple proof for a simple beginner).
Does such a reference exist?
I recommend Pink's notes on finite group schemes as main source and as an additional reference Stix's notes on finite flat group schemes & $p$-div. groups.
Both sources are self-contained, if you are familiar with the basics of algebraic geometry. The classification focuses on $p$-group schemes, but I assume this is also the case you are interested in.
I will warn you though that, as far as I know, most further theory revolving around this topic will likely include French/old/"nonsense approach" sources.