2015/07/21 by Fasel, Jean · 1 citation
#13C05 #14C17 #14M10 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.1507.05734
Let R be a smooth affine algebra over an infinite perfect field k. Let I⊂ R be an ideal, ωI:(R/I)n→ I/I2 a surjective homomorphism and Q2n⊂ \mathbbA2n+1 be the smooth quadric defined by the equation ∑ xiyi=z(1-z). We associate with the pair (I,ωI) an obstruction in the set of homomorphisms Hom_\mathbbA1(Spec(R),Q2n) up to naive homotopy whose vanishing is sufficient for ωI to lift to a surjection Rn→ I. Subsequently, we prove that the obstruction vanishes in case R=k[T1,…,Tm] for m∈ ℕ where k is an infinite perfect field having characteristic different from 2 thus resolving an old conjecture of M. P. Murthy.