Thanks. Now that this understood, I got the rest (I recall the since the sequence is exact $g \circ f = 1$ the trivial map.

@stankewicz See my last comment, it may help.

@Wilberd van der Kallen I replied your answer in a different answer (see below) due to technical reasons. I still have questions as written there. Thanks,

I'm not sure. He first states that $$g( \beta_\sigma (\sigma \cdot \beta_\tau ) \beta_{\sigma \tau}^{-1}) = \gamma_\sigma \sigma \cdot \gamma_\tau \gamma_{\sigma \tau}^{-1} = 1 $$ and then deduces from that, that $$ \beta_\sigma (\sigma \cdot \beta_\tau ) \beta_{\sigma \tau}^{-1} = f( \alpha_{\sigma , \tau})$$ (I'm quoting: " = 1, so $\beta ... = f( \alpha_{\sigma , \tau})$ for some unique $\alpha_{\sigma , \tau} \in A$). However, what about the general case? Why it is not =1 regardless the $f,g$ homomorphisms?

I have to think about it.

@Wilberd van der Kallen We proved a Lemma that said that the ring of regular functions on the relevant open sets is stabilized by $H$. Let $U \subset Y$ be an open set, then one can show (it is a long proof, though), that $$ \mathcal{O}_Y(U) \circ \varphi = \left( \mathcal{O}_G(\varphi^{-1}(U)) \right)^H$$ so I guess this satisfies the demands regarding the scheme. Note that in class we havn't used the "schemes" and "sheaves" concepts and use more basic tools (Alg. group, Lie algebras, ringed spaces, differential etc).

We build $Y=G/H$ as a projective space in order to have a H-invariant vector $0\ne v \in k^n$ such that $\forall h \in H : h \cdot v = kv \in \mathrm{Span}_k(v)$. However, this is not the important thing in my question.