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

#### Janet Page, University of Bristol. Part of the algebra seminar series.

The relationship between symbolic powers and ordinary powers of prime ideals is an active area of research 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 there is a uniform containment which holds for every prime ideal in a regular ring R.  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.  In this talk, I will review the background of this problem and discuss new results (joint with Daniel Smolkin and Kevin Tucker) in this direction, by introducing a class of toric rings called Hibi rings, which can be combinatorially defined in terms of finite posets.

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