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| ON THE FIRST KIND OF RELATIVE k-JET COHOMOLOGY OF SINGULARITIES OF MAPGERMS |
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Citation： |
Xiao Erjian.ON THE FIRST KIND OF RELATIVE k-JET COHOMOLOGY OF SINGULARITIES OF MAPGERMS[J].Chinese Annals of Mathematics B,1986,7(1):24~33 |
| Page view： 786
Net amount： 704 |
Authors： |
Xiao Erjian; |
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| Abstract： |
In this paper the author generalizes the computations about the firs kind of k-jet cohomology in [5] to mapgerms. The main results are as follows:
$$\[{H^o}({\Omega _{\varphi ,k - ,x}})\]) = \underbrace {{C_{M,x}} \oplus \cdots \oplus }_{( + k)}{C_{M,x}},\]$$
$$\[{H^p}({\Omega _{\varphi ,k - ,x}}) = 0,0 < p < m - \dim {C_{M,x}}/I{(\varphi )_x} - 1\begin{array}{*{20}{c}}
{or}&{p = m.}\end{array}\]$$
There exists an integer s, such that
$$\[{(I{(\varphi )_x})^s}{H^p}({\Omega _{\varphi ,k - ,x}}),m - \dim {C_{M,x}}/I{(\varphi )_x} - 1 \le p \le m - 1.\]$$
Hence, $\[{H^p}({\Omega _{\varphi ,k - ,x}})\]$ are finitely generated $\[{C_{M,x}}/{(I{(\varphi )_x})^s}\]$-modules. If $\[{\dim _C}{C_{M,x}}/I{(\varphi )_x} < \infty \]$
then
$$\[{H^p}({\Omega _{\varphi ,k - ,x}}) = 0,0 < p < m - 1\begin{array}{*{20}{c}}{or}&{p = m}\end{array},\]$$
$$\[{\dim _C}{H^{m - 1}}({\Omega _{\varphi ,k - ,x}}) = \sum\limits_r^{k - m} {{{\dim }_C}\Omega _{\varphi ,k - r - m,x}^{m,r}} < \infty .\]$$ |
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