{wiki=Leray's_theorem}

If every nonempty finite intersection of members of an open cover has vanishing higher <sheaf cohomology> for $\mathcal F$, then the cover's <Čech cohomology> computes $H^p(X,\mathcal F)$. An affine open cover of a separated scheme satisfies this condition for a quasi-coherent sheaf because its finite intersections are affine.

Acyclic cover theorem

For an indexed open cover $\mathcal U=(U_i)_{i\in I}$ and a sheaf $\mathcal F$, the <Čech cohomology> cochain groups are
$$
\check C^p(\mathcal U,\mathcal F)
=\prod_{i_0<\cdots<i_p}\mathcal F(U_{i_0}\cap\cdots\cap U_{i_p}).
$$
The differential is the alternating sum of restrictions:
$$
(dc)_{i_0\ldots i_{p+1}}
=\sum_{j=0}^{p+1}(-1)^j
c_{i_0\ldots\widehat{i_j}\ldots i_{p+1}}
\big|_{U_{i_0}\cap\cdots\cap U_{i_{p+1}}}.
$$
Since $d^2=0$, the cohomology
$$
\check H^p(\mathcal U,\mathcal F)
=\ker(d:\check C^p\to\check C^{p+1})/operatorname{im}(d:\check C^{p-1}\to\check C^p)
$$
is well defined.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 113 / 3 / a / Solution

Write $A=k[x,y]$ and $\mathfrak m=(x,y)$. Since $f(0,0)\ne0$, the <principal open subscheme> $D(f)=\operatorname{Spec}A_f$ contains $\mathfrak m$, and
$$
V=D(f)\setminus\{\mathfrak m\}.
$$
Cover $V$ by the two affine opens $D(xf)$ and $D(yf)$, whose intersection is $D(xyf)$. The degree-zero part of the resulting <Čech cohomology> complex gives
$$
H^0(V,\mathcal O_V)=A_{xf}\cap A_{yf}=A_f
$$
inside the <fraction field> of $A$. The last equality follows because $A_f$ is a <unique factorization domain> and a rational function regular after localizing at both $x$ and $y$ has no possible prime factor left in its denominator.

The same affine cover is acyclic, so its degree-one Čech group computes <sheaf cohomology> and gives
$$
H^1(V,\mathcal O_V)
\cong A_{xyf}/(A_{xf}+A_{yf}).
$$
Before localizing at $f$, the quotient
$$
Q=A_{xy}/(A_x+A_y)
$$
has the $k$-basis
$$
\{x^{-a}y^{-b}:a,b\geq1\}.
$$
Writing $f=c+h$ with $c=f(0,0)\in k^\times$ and $h\in(x,y)$, multiplication by $h$ is locally nilpotent on $Q$: for each negative monomial, a sufficiently high power of $(x,y)$ moves every term into $A_x+A_y$. Hence $c+h$ acts invertibly on $Q$ by a finite geometric series on each element. Localizing at $f$ therefore leaves $Q$ unchanged, and
$$
H^1(V,\mathcal O_V)\cong Q_f\cong Q.
$$
The displayed infinite basis proves that this vector space is infinite-dimensional.

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 113 / 3 / b / Solution

{c}
{title2=$\check H^p(\mathcal U,\mathcal F)$}
{wiki}

For an open cover $\mathcal U=(U_i)$, the Čech cochain group is
$$
\check C^p(\mathcal U,\mathcal F)=\prod_{i_0<\cdots<i_p}\mathcal F(U_{i_0}\cap\cdots\cap U_{i_p}),
$$
with the alternating sum of restrictions as differential. Its cohomology is the Čech cohomology of $\mathcal F$ with respect to $\mathcal U$.

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 Incoming links: Čech cohomology