Janet Page

Symbolic vs. ordinary powers and the containment problem for Hibi rings

The relationship between symbolic powers and ordinary powers of prime ideals is of much interest in commutative algebra. It is easily verified that n-th ordinary power of a prime ideal is contained in its n-th symbolic power, and typically we strive for containments in the other direction. In characteristic 0, Ein, Lazarsfeld, and Smith showed that any regular ring R satisfies what is now called the "uniform symbolic topology property" (USTP). Namely, they showed that there is a d (in this case the dimension of R) such that for every prime ideal p in R, we have that the dn-th symbolic power of p is contained in its n-th ordinary power. Since then, this result has been extended to characteristic p by Hochster and Huneke and much more recently to mixed characteristic by Ma and Schwede, but there are still few results known in the non regular case. Recently, Smolkin and Carvajal-Rojas came up with a new way to get at this containment problem (in characteristic p) by introducing diagonal F-regularity, and using this method they proved that Segre products of polynomial rings also satisfy USTP. In this talk, I will review the background of this problem and discuss new results (joint with Daniel Smolkin and Kevin Tucker) which extend this work, by introducing a certain class of combinatorially defined rings called Hibi rings.