First and second cohomologies of grading-restricted vertex algebras

Let $V$ be a grading-restricted vertex algebra and $W$ a $V$-module. We show that for any $m\in {\mathbb{Z}}_{+}$, the first cohomology $H_m^1(V,W)$ of $V$ with coefficients in $W$ introduced by the author is linearly isomorphic to the space of derivations from $V$ to $W$. In particular, $H_m^1(V,W)$ for $m\in \mathbb{N}$ are equal (and can be denoted using the same notation $H^1(V,W)$). We also show that the second cohomology $H^{2}_{\frac{1}{2}}(V, W)$of$V$withcoefficientsin$W$ introduced by the author corresponds bijectively to the set of equivalence classes of square-zero extensions of $V$ by $W$. In the case that $W=V$, we show that the second cohomology $H_{\frac{1}{2}}^2(V,V)$ corresponds bijectively to the set of equivalence classes of first order deformations of $V$.