Infinitely many associated primes of Frobenius powers and local cohomology

A modification 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 infinite. A further modification yields a four-dimensional Noetherian integral domain and a five-dimensional Noetherian local integral domain for which an explicit second local cohomology module has infinitely 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 infinite. The ring in Katzman’s example was not an integral domain. In this paper it is shown that the infinite 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 infinitely many associated primes. Singh in [Si] found the first example of such a module. His example was a non-local six-dimensional integral domain R for which H I (R) has infinitely many associated primes for some ideal I. Katzman in [K2] revisited his own example from [K1] to construct a five-dimensional local integral domain R for which H I (R) has infinitely many associated primes for some ideal I. Similarly also the ideal in this paper yields a fivedimensional local integral domain R for which H I (R) has infinitely many associated primes for some ideal I. Theorem 8 gives a general method for constructing local cohomology modules with infinitely 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 DMS9970566. 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. 1991 Mathematics Subject Classification. 13C13, 13P05