Huge progress in the study of the homological properties of commutative rings has been made through the use of tight closure, an idea developed by Mel Hochster and Craig Huneke. Unfortunately, tight closure is only defined for equicharacteristic local rings. There appears to be no way to extend it to mixed characteristic local rings. Thus there is a natural desire to find a suitable closure operation for this setting.
One interesting alternate closure operation, which sometimes has similar properties, is the plus closure. The plus closure appears to be harder to work with but may in fact coincide with the tight closure in characteristic p rings. It also does have a certain degree of utility in the mixed characteristic case. Unfortunately, it is essentially worthless for equicharacteristic zero rings. More importantly, because much of the internal structure of mixed characteristic rings resembles that of equicharacteristic zero rings, it simply is not a satisfactory analogue for tight closure.
In the mixed characteristic case, the plus closure is most effective for ideals which contain a power of the prime integer p. Of course, in the characteristic p case, where the plus closure is at its best, all ideals have this property. The idea behind the research undertaken here is that we can use the plus closure of I q p n R for large n to define a new closure for I.
In this article, we define and explore four new closely related closure operations. Two of the closures, the full extended plus closure and the full rank one closure, are defined first; then the remaining two potentially smaller closures, the extended plus closure and the rank one closure, are defined by applying the full closures to ideals in finitely generated Z-subalgebras of the original ring. The full closures seem to provide the basis for a ลฝ solid theory for characteristic p rings where the tight closure already . performs this role . It is our hope that the full closures will actually 801