Let V be a grading-restricted vertex algebra and W a V-module. We show that for any m∈Z+\documentclass[12pt]{minimal}
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\begin{document}$${m\in \mathbb{Z}_{+}}$$\end{document}, the first cohomology Hm1(V,W)\documentclass[12pt]{minimal}
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\begin{document}$${H^{1}_{m}(V, W)}$$\end{document} of V with coefficients in W introduced by the author is linearly isomorphic to the space of derivations from V to W. In particular, Hm1(V,W)\documentclass[12pt]{minimal}
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\begin{document}$${H^{1}_{m}(V, W)}$$\end{document} for m∈N\documentclass[12pt]{minimal}
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\begin{document}$${m\in \mathbb{N}}$$\end{document} are equal (and can be denoted using the same notation H1(V, W)). We also show that the second cohomology H122(V,W)\documentclass[12pt]{minimal}
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\begin{document}$${H^{2}_{\frac{1}{2}}(V, W)}$$\end{document} of V with coefficients in 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 H122(V,V)\documentclass[12pt]{minimal}
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\begin{document}$${H^{2}_{\frac{1}{2}}(V, V)}$$\end{document} corresponds bijectively to the set of equivalence classes of first order deformations of V.
