Tight closure/x^4,y^4,xy^3/On P^1 and other curves/Example

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

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.