Innnitely Many Associated Primes of Frobenius Powers and Local Cohomology

A modiication of Katzman's example is given to produce a two-generated ideal in a two-dimensional Noetherian integral domain for which the set of associated primes of all the Frobenius powers is innnite. A further modiication yields a four-dimensional Noetherian integral domain and a ve-dimensional Noetherian local integral domain for which an explicit second local cohomology module has innnitely many associated primes. Katzman gave an example in K1] of an ideal I in a two-dimensional ring for which the set of associated primes of all the Frobenius powers of I is innnite. The ring in Katzman's example was not an integral domain. In this paper it is shown that the innnite cardinality of the set of associated primes of all the Frobenius powers of an ideal can happen even in a two-dimensional integral domain. An application is another example of a local cohomology module with innnitely many associated primes. Singh in Si] found the rst example of such a module. His example was a non-local six-dimensional integral domain R for which H 3 I (R) has innnitely many associated primes for some ideal I. Katzman in K2] revisited his own example from K1] to construct a ve-dimensional local integral domain R for which H 2 I (R) has innnitely many associated primes for some ideal I. Similarly also the ideal in this paper yields a ve-dimensional local integral domain R for which H 2 I (R) has innnitely many associated primes for some ideal I. Theorem 8 gives a general method for constructing local cohomology modules with innnitely many associated primes from certain families of matrices. Both Katzman's ideal and the ideal in this paper yield such families of matrices. The author thanks the NSF for partial support on grants DMS-0073140 and DMS-9970566. She also thanks Kamran Divaani-Aazar and the Institute for Studies in Theoretical Physics and Mathematics (IPM) in Tehran, Iran, for their interest and hospitality.