Vector bundles without intermediate cohomology and the trichotomy result

Horrocks (Proc London Math Soc 14:689-713, 1964) that vector bundles on Pⁿ P n without intermediate cohomology split as direct sum of line bundles. This result has been the starting point of a great research activity on other varieties, showing interesting connections with derived categories and other areas. We follow some paths into this fascinating story, which has classical roots. The story has a culmination with the trichotomy result (finite/tame/wild) for arithmetically Cohen Macaulay (ACM) varieties obtained by Faenzi and Pons-Llopis in 2021. This is an expanded version of the talk given at the conference“Homemade Algebraic Geometry”in July 2023 at Alcalá de Henares celebrating Enrique Arrondo’s 60th birthday.