Serre's conjecture on vector bundles

Tags: #open/conjectures

Statement: every finitely generated free;

For \(k\in \mathsf{Field}\), is every \(M\in {}_{k[x_1, \cdots, x_{n}]}{\mathsf{Mod}}^{\mathrm{fg}}\) actually free? - True by Quillen-Suslin 76

  • Open problem (Bass-Quillen):
    • Is the pullback-induced map \({ \mathsf{Vect} }_r(X) \to { \mathsf{Vect} }_r(X \times{\mathbf{A}}^n)\) a bijection?
    • Proved in special cases
#open/conjectures