Let \(A\) be a ring, let \(n\ge1\), put \(S=A[x_0,\ldots,x_n]\), and write \(\PP^n=\PP_A^n\). Then
\(H^0(\PP^n, \OO(d)) = S_d\), a free \(A\)-module of rank \(\binom{n+d}{n}\) for \(d \geq 0\), and \(0\) for \(d < 0\);
\(H^n(\PP^n,\OO(d))\) is the free \(A\)-module on the monomials \(x_0^{a_0}\cdots x_n^{a_n}\) with every \(a_i\le-1\) and \(\sum_i a_i=d\). It is zero for \(d>-n-1\) and has rank \(\binom{-d-1}{n}\) for \(d\le-n-1\). It is \(A\)-linearly dual to \(S_{-d-n-1}\);
\(H^p(\PP^n, \OO(d)) = 0\) for \(0 < p < n\) and for \(p > n\), for every \(d\).
[@Har10a, Theorem III.5.1] The projective-space cohomology calculation gives these formulas over every ring \(A\).
(Projective dimension zero)
For \(n=0\), the scheme \(\PP_A^0\) is \(\Spec A\), and every \(\OO(d)\) is trivial. Thus \(H^0(\PP_A^0,\OO(d))\cong A\) for every integer \(d\), and all positive-degree cohomology groups vanish. The negative-degree sections are represented on its single standard chart by \(Ax_0^d\), as in P-AGH2510SATIDEAL, step <2>1.
(Consequences over a field)
For \(n\ge2\), \(H^1(\PP_k^n,\OO(d))=0\) for every integer \(d\). On \(\PP_k^1\), one has \[ h^0(\OO(d))=\max\{d+1,0\},\qquad h^1(\OO(d))=\max\{-d-1,0\}=h^0(\OO(-d-2)). \] In particular \(h^1(\OO(-2))=1\). The Euler sequence gives \(\Omega_{\PP_k^1/k}^1\cong\OO(-2)\), so the projective line has no nonzero global regular differentials. The duality between the top and bottom groups is Serre duality, with canonical sheaf \(\omega_{\PP_k^n}=\OO(-n-1)\).
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P-AGXMISCCALCULATIONSOral calculations on projective space, sections and divisorsrelated-toT-COHFINFiniteness of cohomology for coherent sheaves on a projective schemeusesT-MWDVLRiemann--Roch and Serre duality on a curveusesT-COHSDSerre duality in dimension $n$usesT-CRVHASSEThe Hasse invariant criterion and the Hasse polynomial $h_p(\lambda)$usesPR-COHLESThe long exact sequence, and the ideal sequence as a computational deviceuses