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

Algebra and Geometry Seminar

17th October 2018, 2:30 pm – 3:30 pm

Howard House, 4th Floor Seminar Room

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.

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