Tight closure in non-equidimensional rings

Abstract

An equidimensional local ring is F-rational if and only if one ideal generated by a system of parameters is tightly closed. The question of whether a non-equidimensional local ring can have a tightly closed ideal generated by a system of parameters has been a long-standing open problem, and for certain classes of non-equidimensional rings we prove that this is not possible. A key point is that tight closure has a colon capturing property in equidimensional rings that it does not have in non-equidimensional rings. We define a new closure operation, one that rectifies the absence of the colon capturing property of tight closure in non-equidimensional rings. This closure operation agrees with tight closure when the ring is equidimensional, and we prove that the F-rationality of a local ring is equivalent to a single system of parameters being closed with respect to this new closure operation.