For a field $\mathbb{ F}$, let $L_k(\mathbb{ F})$ be the Lie algebra of derivations $f(t)\frac{ d}{ dt}$ of the polynomial ring $\mathbb{ F}[t]$, where $f(t)$ is a polynomial of degree $\geqslant k$. For any $k\geqslant -1$, we present a basis of the space of the cohomology with finite-dimensional support of the Lie algebra $L_k(\mathbb{ F})$ with coefficients in the trivial module $\mathbb{ F}$ for the case where ${ \rm char}(\mathbb{ F})=2$. The main result obtained is an analog of the famous Goncharova's Theorem for the case ${ \rm char}(\mathbb{ F})=0$ and $k\geqslant 1$.
Categories: math.RT