We consider the ideal
in
and in finite graded extensions
(e.g.
,
where
is a homogeneous integral equation for
over
)
and describe an algorithm to compute the tight closure
. The graded resolution of the ideal is
-
where the map on the left is given by sending the generators to
-
On the projective line
this corresponds to
-
We may pull back this sequence along the finite morphism
-
to obtain the corresponding exact sequence over the curve
which can be used to compute the tight closure of the ideal in
. A homogeneous element
of degree
yields a cohomology class in
-
![{\displaystyle {}H^{1}(C,\operatorname {Syz} {\left(x^{4},y^{4},xy^{3}\right)}(m))\cong H^{1}(C,{\mathcal {O}}_{C}(m-5))\oplus H^{1}(C,{\mathcal {O}}_{C}(m-7))\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e84aa54001a0341ea2746aaf6a06b8dab0e8dab5)
which can be easily computed using Čech cohomology. On
,
comes from
and on
it comes from
. Their difference, the syzygy
-
equals
-
Hence the components of this cohomology class are
-
Therefore the issue whether
belongs to the tight closure of
depends on these two components, which both correspond to a parameter situation.
First of all, if
,
then both degrees are non-negative and therefore these classes are tightly
by
(the proof of)
fact.
If
,
we only have to look at the second component inside
. For the monomial
the second component is
(independent of
),
hence it belongs to the tight closure, though the first component need not be
. The monomial
yields
, which is not
unless the equation has low degree. This class is
(with some exceptions in small characteristics)
not tightly
. Hence
does not belong to the tight closure. For
,
still only the second component is interesting, therefore
belongs to the tight closure, but
not
(under the same restrictions).
For
both components lie in negative degree, so an element will belong to the tight closure only if it belongs to the ideal itself. For the element
the second component is
, but not the first component.