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@SebastianGoette I presume that you mean $\widetilde{\omega} \in \Lambda^2TM$ defined by $\widetilde{\omega}(X) = \omega(X,-)$ and $\Lambda_\omega F_\nabla$ is just abusively using $\Lambda_\omega $ instead of $\Lambda_{\widetilde{\omega}}$?
I found and old paper by Donaldson "Anti Self-Dual Yang-Mills Connections Over Complex Algebraic Surfaces and Stable Vector Bundles." where the author states that $\Lambda_\omega\left(a_{\alpha\beta}dz^\alpha \wedge d\bar{z}^\beta\right) = -2i\sum a_{\alpha\alpha}$, though at the moment this just expands my confusion. @QuartoBendir
Thank you, I need to go back and read about these notions on delta functors before I understand your answer thoroughly. I was kinda hoping that I could have done this using injective resolution for $\mathcal{F}$. @ben-c
Thank you! I guess there is also a minor mistake in the last sentence of the proof. It should be probably the product/Leibniz rule instead of the chain rule? @RichardLärkäng
One more question, in $\xi \mapsto\psi_i^{-1} ((d + A_i) (\psi_{i} \xi))$ do you identify $\psi^{-1}_i : V \times \Bbb C^r \to E|_{V}$ with a map $$\psi^{-1}_i : \Omega^1 \otimes (V \times \Bbb C^r) \to \Omega^1 \otimes E|_{V}$$ that sends $\omega \otimes s \mapsto \omega \otimes \psi^{-1}_i(s)$, since $(d + A_i) (\psi_{i} \xi)$ is a section of $\Omega^1 \otimes (V \times \Bbb C^r)$? @richard-lärkäng
Thanks for updating this. It clears things up for me. About regarding $\psi_i$'s as matrices, equation 4.4 states $$\psi^{-1}_i \circ \partial \circ \psi_i -\psi^{-1}_j \circ \partial \circ \psi_j = \psi^{-1}_jA_j\psi_j -\psi^{-1}_iA_i\psi_i,$$ where the compositions on the left-hand side are compositions of linear maps, but I thought that these products on the rhs are matrix products which would indeed mean that $\psi_i$'s are identified as matrices. Is this just being lazy and dropping the composition symbol? @richard-lärkäng
