= Solution
Let $X$ be a separated <Noetherian scheme> over a field $k$ and $\mathcal F$ a <coherent sheaf>. <Coherent cohomology> studies the groups $H^q(X,\mathcal F)$: <global sections> in degree zero and the obstructions to gluing local sections in higher degrees. Coherence means finite generation locally, and on a Noetherian scheme it is preserved by kernels, cokernels and extensions. This makes <coherent sheaves> suitable for exact-sequence calculations.
For an ordered open cover $\mathcal U=(U_i)$, the <Čech cochain groups> are
$$
\check C^q(\mathcal U,\mathcal F)=\prod_{i_0<\cdots<i_q}\mathcal F(U_{i_0}\cap\cdots\cap U_{i_q}),
$$
with differential
$$
(\delta c)_{i_0\ldots i_{q+1}}
=\sum_{j=0}^{q+1}(-1)^j\left.c_{i_0\ldots\widehat{i_j}\ldots i_{q+1}}\right|_{U_{i_0}\cap\cdots\cap U_{i_{q+1}}}.
$$
Each double omission occurs twice with opposite signs, so $\delta^2=0$. The <Čech cohomology> of the cover is $\check H^q=\ker\delta/\operatorname{im}\delta$, and taking the direct limit over refinements defines cover-independent Čech cohomology. In degree zero the <sheaf gluing axiom> identifies it with $\Gamma(X,\mathcal F)$.
The comparison with derived-functor <sheaf cohomology> is particularly effective algebraically. Positive-degree cohomology of a <quasi-coherent sheaf> on an <affine scheme> vanishes. A separated scheme has affine intersections of affine opens, so a finite affine cover is acyclic for $\mathcal F$. Its <Čech cochain complex> consequently computes $H^q(X,\mathcal F)$ directly. For a <short exact sequence of sheaves>, sections over each affine intersection are exact on quasi-<coherent sheaves>. The resulting short exact sequence of Čech complexes gives the <long exact sequence in sheaf cohomology>. In particular, the connecting map measures the failure of a global section of the quotient to lift globally.
Several central projective results make the theory finite and computable. On a <projective variety> over $k$, every coherent cohomology group is finite-dimensional. The <Grothendieck vanishing> theorem gives $H^q(X,\mathcal F)=0$ for $q>\dim X$. If $\mathcal O_X(1)$ is ample, <Serre vanishing> gives $H^{q>0}(X,\mathcal F(m))=0$ for sufficiently large $m$, and these large twists are <globally generated sheaves>. On <projective space> a <coherent sheaf> has a <finite twisting resolution of a coherent sheaf on projective space>: finite sums of $\mathcal O(a)$ resolve it. The explicit line-bundle cohomology below and the long exact sequences prove finiteness and positive-twist vanishing. For a closed embedding $i:X\hookrightarrow\mathbb P^n$, $i_*\mathcal F$ is coherent and $H^q(X,\mathcal F)=H^q(\mathbb P^n,i_*\mathcal F)$, reducing these results to <projective space>. A sufficiently positive twist of a finite presentation by sums of $\mathcal O(-a)$ also proves global generation.
The <Euler characteristic of a coherent sheaf>
$$
\chi(X,\mathcal F)=\sum_q(-1)^q\dim_kH^q(X,\mathcal F)
$$
is therefore defined, and the long exact sequence makes it additive. With a fixed projective embedding, $\chi(X,\mathcal F(m))$ is its <Hilbert polynomial>, and for sufficiently large $m$ it equals $h^0(X,\mathcal F(m))$. On a smooth projective variety of dimension $r$, <Serre duality> gives, for a <vector bundle> $E$,
$$
H^q(X,E)^*\cong H^{r-q}(X,E^*\otimes\omega_X).
$$
For an arbitrary <coherent sheaf> the correct counterpart is $H^q(X,\mathcal F)^*\cong\operatorname{Ext}^{r-q}_X(\mathcal F,\omega_X)$; replacing this Ext group indiscriminately by the cohomology of the ordinary sheaf dual would lose local extension information.
We now calculate the basic examples. For $n\ge1$, use the standard cover $U_i=D_+(x_i)$ of $\mathbb P^n$. Sections of $\mathcal O(m)$ on an intersection are degree-$m$ <Laurent monomials> in which negative exponents are allowed only for coordinates inverted on that intersection. Decompose the <Čech cochain complex> by exponent vectors $(a_0,\ldots,a_n)$ with sum $m$. If no exponent is negative, the corresponding simplex complex contributes only in degree zero. If all exponents are negative, the monomial occurs only on the full intersection and contributes only in degree $n$. When the negative-index set is nonempty and proper, inserting any index outside it gives a contracting homotopy, so the monomial complex is acyclic. Hence the <cohomology of twisting sheaves on projective space> is
$$
H^q(\mathbb P^n,\mathcal O(m))\cong
\begin{cases}
k[x_0,\ldots,x_n]_m,&q=0,\ m\ge0,\\
k^{\binom{-m-1}{n}},&q=n,\ m\le-n-1,\\
0,&\text{otherwise}.
\end{cases}
$$
In the top group a basis consists of the <Laurent monomials> with every exponent at most $-1$. Writing $a_i=-1-b_i$ reduces the count to nonnegative solutions of $\sum b_i=-m-n-1$, giving the displayed binomial coefficient. In particular, all intermediate cohomology vanishes, and $\omega_{\mathbb P^n}=\mathcal O(-n-1)$ agrees with <Serre duality>. For $\mathbb P^0$ separately, every twist is trivial and $H^0=k$.
On the <projective line> this gives
$$
h^0(\mathcal O(m))=\max(m+1,0),\qquad h^1(\mathcal O(m))=\max(-m-1,0),\qquad\chi(\mathcal O(m))=m+1.
$$
Thus positive and negative degrees produce quite different groups, while their alternating difference is a single polynomial. The <twisted cubic> is isomorphic to $\mathbb P^1$ and its hyperplane bundle corresponds to $\mathcal O_{\mathbb P^1}(3)$. Consequently $h^0(X,\mathcal O_X(m))=3m+1$ for $m\ge0$, agreeing with Q2, and $h^1(X,\mathcal O_X(m))=\max(-3m-1,0)$.
For a smooth plane curve $C$ of degree $d\ge1$, the <ideal-sheaf sequence of a projective hypersurface> is
$$
0\longrightarrow\mathcal O_{\mathbb P^2}(-d)\longrightarrow\mathcal O_{\mathbb P^2}\longrightarrow i_*\mathcal O_C\longrightarrow0.
$$
The <long exact sequence in sheaf cohomology> and the preceding formula give $H^0(C,\mathcal O_C)=k$ and $H^1(C,\mathcal O_C)\cong H^2(\mathbb P^2,\mathcal O(-d))$. Therefore
$$
\boxed{g(C)=\frac{(d-1)(d-2)}2.}
$$
For $d=1,2$ this is zero; for a plane cubic it is one. Twisting the same exact sequence gives $\chi(C,\mathcal O_C(m))=dm+1-g$. These examples illustrate how <Čech cohomology>, vanishing, <Serre duality> and exact sequences convert local equations into global numerical invariants.
Back to article page
