Relative cohomology of polynomial mappings
| dc.creator | Bonnet, Philippe | |
| dc.date | 2006-02-13 | |
| dc.date.accessioned | 2026-07-07T07:03:22Z | |
| dc.date.available | 2026-07-07T07:03:22Z | |
| dc.description | Let $F$ be a polynomial mappping from $\mathbb{C}^n$ to $\mathbb{C}^q$ with $n>q$. We study the De Rham cohomology of its fibres and its relative cohomology groups, by introducing a special fibre $F^{-1}(\infty)$ "at infinity" and its cohomology. Let us fix a weighted homogeneous degree on $\mathbb{C}[x_1,...,x_n]$ with strictly positive weights. The fibre at infinity is the zero set of the leading terms of the coordinate functions of $F$. We introduce the cohomology groups $H^k(F^{-1}(\infty))$ of $F$ at infinity. These groups enable us to compute all the other cohomology groups of $F$. For instance, if the fibre at infinity has an isolated singularity at the origin, we prove that every weighted homogeneous basis of $H^{n-q}(F^{-1}(\infty))$ is a basis of all the groups $H^{n-q}(F^{-1}(y))$ and also a basis a the $(n-q)^{th}$ relative cohomology group of $F$. Moreover the dimension of $H^{n-q}(F^{-1}(\infty))$ is given by a global Milnor number of $F$, which only depends on the leading terms of the coordinate functions of $F$. | |
| dc.description | 20 pages | |
| dc.identifier | https://arxiv.org/abs/math/0602273 | |
| dc.identifier | http://arxiv.org/abs/math/0602273 | |
| dc.identifier | Manuscripta mathematica 110, 413-432 (2003) | |
| dc.identifier | doi:10.1007/s00229-002-0313-9 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/108948 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Commutative Algebra | |
| dc.subject | 14F10; 14F40; 14R99 | |
| dc.title | Relative cohomology of polynomial mappings | |
| dc.type | text |