Tight closure and strong test ideals Original Research Article

For certain classes of rings we give an affirmative answer to whether there exists a uniform bound on the least power of the tight closure of an arbitrary ideal which lies in the ideal. The rings need to have the property that modulo each minimal prime there exists a resolution of singularities obtained by blowing up an ideal. In this case we prove the existence of a ‘strong’ test ideal, and then apply this existence to give an affirmative answer to the uniform bound question.