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On the number of generators of ideals in polynomial rings
- Source :
- Annals of Mathematics, Annals of Mathematics, Princeton University, Department of Mathematics, 2016, 184 (1), pp.315-331. ⟨10.4007/annals.2016.184.1.3⟩
- Publication Year :
- 2016
- Publisher :
- HAL CCSD, 2016.
-
Abstract
- Let $R$ be a smooth affine algebra over an infinite perfect field $k$. Let $I\subset R$ be an ideal, $\omega_I:(R/I)^n\to I/I^2$ a surjective homomorphism and $Q_{2n}\subset \mathbb{A}^{2n+1}$ be the smooth quadric defined by the equation $\sum x_iy_i=z(1-z)$. We associate with the pair $(I,\omega_I)$ an obstruction in the set of homomorphisms $\mathrm{Hom}_{\mathbb{A}^1}(\mathrm{Spec}(R),Q_{2n})$ up to naive homotopy whose vanishing is sufficient for $\omega_I$ to lift to a surjection $R^n\to I$. Subsequently, we prove that the obstruction vanishes in case $R=k[T_1,\ldots,T_m]$ for $m\in \mathbb{N}$ where $k$ is an infinite perfect field having characteristic different from $2$ thus resolving an old conjecture of M. P. Murthy.<br />Comment: Paper withdrawn. Due to an error in Lemma 3.2.3, the proof of the main result collapses
- Subjects :
- Sequence
Noetherian ring
Conjecture
Mathematics::Commutative Algebra
Polynomial ring
010102 general mathematics
13C05, 14C17, 14M10
Base field
16. Peace & justice
Commutative Algebra (math.AC)
Mathematics - Commutative Algebra
01 natural sciences
MSC 13E15 13C10 13A15 13C40 14M10 14R10 19M05
Combinatorics
Mathematics - Algebraic Geometry
Mathematics (miscellaneous)
0103 physical sciences
FOS: Mathematics
010307 mathematical physics
Ideal (ring theory)
0101 mathematics
Statistics, Probability and Uncertainty
[MATH]Mathematics [math]
Algebraic Geometry (math.AG)
Mathematics
Subjects
Details
- Language :
- English
- ISSN :
- 0003486X
- Database :
- OpenAIRE
- Journal :
- Annals of Mathematics, Annals of Mathematics, Princeton University, Department of Mathematics, 2016, 184 (1), pp.315-331. ⟨10.4007/annals.2016.184.1.3⟩
- Accession number :
- edsair.doi.dedup.....a9a0b72f017518b453ec49106d13f6f8