I am reading the book "The Homology of Banach and Topological Algebras" by A.Y. Helemskii and couldn't understand one lemma on page 204 about complex splitting. enter image description here I understood how to prove that complex

$$ 0 \leftarrow O \left( V \right) \stackrel{\pi}{\leftarrow} O \left( V \times V \right) \stackrel{d}{\leftarrow} O \left( V \times V \right) \leftarrow 0 \, \, \, \left( \mathcal{V} \right) $$ splits, where $O \left( V \right)$ is a space of holomorphic functions, $\pi: g (x, y) \mapsto g(x, x)$ and $d: g (x, y) \mapsto (y-x) g(x, y)$. I understood the isomorphism of complexes $ Y \hat{\bigoplus}_{O \left( V \right) } \left( \mathcal{V} \right) $ and $\left( \mathcal{U} \right) $, but I couldn't understand why this complex $ Y \hat{\bigoplus}_{O \left( V \right) } \left( \mathcal{V} \right) $ splits (preceding lemma was about splitting of complex $\left( \mathcal{V} \right)$). It seems that the additivity of the functor is not enough to preserve the splitting.


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