The Jouanolou-Thomason homotopy lemma

The goal of this note is to prove what is now known as the Jouanolou-Thomason homotopy lemma or simply “Jouanolou’s trick.” Our main reason for discussing this here is that i) most statements (that I have seen) assume unncessary quasi-projectivity hypotheses, and ii) most applications of the result that I know (e.g., in homotopy K-theory) appeal to the result as merely a “black box,” while the proof indicates that the construction is quite geometric and relatively explicit. For simplicity, throughout the word scheme means separated Noetherian scheme.