Consider a degree $6$ weighted projective hypersurface $Y$ in $w\mathbb{P} := \mathbb{P}(1,1,1,2,3)$ (it's an index $2$ Fano threefold of degree $1$ in regular projective space). Let $H$ be the ample line bundle on $Y$ which gives an embedding of $Y$ into regular projective space.
I'm interested in computing the cohomology groups $H^i(Y, \mathcal{O}_Y(H))$. If this were, say, a cubic threefold (degree $3$ hypersurface in $\mathbb{P}^4$), then I'd use the short exact sequence $ 0 \to \mathcal{O}_{\mathbb{P}^4}(-1) \to \mathcal{O}_{\mathbb{P}^4}(1) \to \mathcal{O}_Y(H) \to 0 $ to compute the analogous cohomology groups.
Since my current case is a weighted projective hypersurface though, I'm not entirely sure of a way to proceed. The short exact sequence I mention above is a twist of the ideal sheaf sequence $0 \to I_Y = \mathcal{O}_{\mathbb{P}^4}(-Y) \to \mathcal{O}_{\mathbb{P}^4} \to \mathcal{O}_{Y} \to 0$ so I was wondering if there is an analogous ideal sheaf sequence for weighted projective spaces?
Another thought was to use the weighted projective Euler exact sequence: it would look something like $$ 0 \to \Omega_{w\mathbb{P}}^1 \to \bigoplus_{i=0}^4 \mathcal{O}_{w \mathbb{P}}(-D_i) \to \mathcal{O}_{w\mathbb{P}} \to 0 $$ where we would replace $D_i$ by the weights of $w \mathbb{P}$ when doing the computation. I suppose the next step would be to twist this and restrict somehow (?), so that the last term becomes $\mathcal{O}_Y(H)$. I'm not sure if this, as well as computing the resulting cohomologies from the first two terms is feasible though?
Many thanks.
It is more convenient to think here of $w\mathbb{P}$ as of a quotient stack; anyway, if $Y$ i smooth it does not pass through the stacky points $$ (0,0,0,1,0), (0,0,0,0,1) \in w\mathbb{P} $$ and therefore the stacky structure of $w\mathbb{P}$ plays no role for $Y$.
The advantage of $w\mathbb{P} = \mathbb{P}(w_0,w_1,\dots,w_n)$ as of a stack is that it comes with the line bundle sequence $\mathcal{O}(i)$ such that $$ H^p(w\mathbb{P}, \mathcal{O}(i)) = \begin{cases} A_i, & \text{if $i \ge 0$ and $p = 0$},\\ A_{-w-i}, & \text{if $i \le -w$ and $p = n$},\\ 0, & \text{otherwise}, \end{cases} $$ where $A = \Bbbk[x_0,x_1,\dots,x_n]$ with $\deg(x_k) = w_k$, and $w = w_0 + w_1 + \dots + w_n$. Now you can compute cohomology of restrictions of these line bundles to arbitrary hypersurfaces in the same way as in the usual projective space using the Koszul resolution $$ 0 \to \mathcal{O}(-d) \to \mathcal{O} \to \mathcal{O}_Y \to 0 $$ for a hypersurface $Y$ of degree $d$.