I am reading Dolbeault Isomorphism from Raghavan Narasimhan's "Compact Riemann Surfaces". I am facing some difficulties to understand some portions. If someone can explain these that will be very helpful.
$\bullet$ Let $X$ be the riemann surface and $\mathbf{E}$ is a holomorphic vector bundle over $X$. The author defines $\mathcal{A}_E ^{0,1}(W):=C_E ^{\infty}(W) \otimes_{C^{\infty}(W)} \mathcal{A}^{0,1}(W)$ where $W\subseteq X$ open, $\mathcal{A}^{0,1} (W)=$ space of $1$-forms on $W$ of type $(0,1)$ [i.e. locally of the form $b \hspace{0.3ex}d\bar{z}$] and $C_E ^{\infty}(W)=$ the space of $C^{\infty}$ sections of $E$ over $W$. I know what is a tensor product, but I can't understand what the space $\mathcal{A}_E ^{0,1}(W)$ really is. Is the tensor necessary just to have forms with coefficients in smooth functions? Please explain.
$\bullet$ I think this is kind of related to my first question. I need some explanations how $ 0\rightarrow \mathbf{E}\rightarrow \mathbf{E^\infty}\rightarrow \mathcal{A}_E ^{0,1}(W)\rightarrow 0$ forms a short exact sequence of sheaves. Here $\mathbf{E}^\infty$ is the sheaf with $\mathbf{E}^\infty (W)=C_E ^\infty (W)$ for $W\subseteq X$ open.
Thanks in advance.
The discussion above solves it. All credit goes to Nicolas Hemelsoet.