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Take the projective space $\mathbb{P}^n$ over a ring $W$. We call $\mathcal{O}(q)$ the usual twisted line bundle.

Now take the map $f: \mathbb{P}^n\to\mathbb{P}^n$ defined by $$[x_0,\ldots, x_n]\mapsto [x_0^g,\ldots, x_n^g]$$ for some fixed $g$.

Now take the inverse limit $X = \varprojlim \mathbb{P}^n$, where the transition maps are the map $f$, every time. Call $p : X\to\mathbb{P}^n$ the projection.

What is the cohomology $H^r(X,p^*\mathcal{O}(q))$.

Of course, it vanishes for all $r \neq 0,n$.

I am trying to do it, but I am confused by the notation, and all the indexes.

I have an educated guess for $H^0$

$$H^0(X, p^*\mathcal{O}(q)) = \bigoplus_{k\in\mathbb{Z}}\left( \bigoplus_{\{i_1,\ldots, i_j\}\subset\{0,\ldots, n\}, n_1 + \ldots, n_j = q}W \cdot x_{i_1}^{n_1/g^k}\cdot x_{i_j}^{n_j/g^k}\right)$$

Motivation My friend and I are reading a paper we just wrote. We don't want to make a mistake on this calcùlation. It's easy to make mistakes, with all those indices.

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    $\begingroup$ The basis should be indexed by points in the standard simplex with coordinates in $Z[1/g]$, which seems to agree with what you wrote. Similarly for any toric variety and the toric Frobenius. $\endgroup$
    – Piotr Achinger
    Commented May 19, 2018 at 20:30
  • $\begingroup$ Fantastic! What thank you so much :-) $\endgroup$
    – user124171
    Commented May 19, 2018 at 20:31
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    $\begingroup$ The formula has errors. How do you get confused about your own notation, if this is about your own work (given that the direct limits involved are not so complicated)? Or are you borrowing from work of others which has you confused? What does it mean to "read a paper we just wrote"? $\endgroup$
    – nfdc23
    Commented May 19, 2018 at 20:55
  • $\begingroup$ @nfdc23 Which error? We are borrowing from work of others $\endgroup$
    – user124171
    Commented May 19, 2018 at 20:56
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    $\begingroup$ You should be able to figure this out for yourself; this kind of computation is a student-exercise level, not at the level for Math Overflow. Maybe what you are reading from others has errors? Think about a special case (e.g., $n=1$). $\endgroup$
    – nfdc23
    Commented May 19, 2018 at 20:57

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