# Colloquium: "Symbolic powers of ideals in regular rings"

*Abstract: In 1921, Emmy Noether proved that ideals in commutative Noetherian rings decompose into primary components. Noether's theorem is an analogue of the fundamental theorems of arithmetic and algebra, which say that integers and complex polynomials in a single variable can be decomposed into powers of prime factors. While factorizations of integers and complex polynomials behave well with respect to taking powers, primary decompositions of ideals are not always well-behaved. In 1928, Wolfgang Krull introduced symbolic powers of ideals, which do have well-behaved primary decompositions. Since then, comparing ordinary and symbolic powers of ideals has become an important open problem in commutative algebra and algebraic geometry. In this talk, I will discuss my proof of a uniform comparison between ordinary and symbolic powers of arbitrary ideals in arbitrary regular rings. In particular, I resolve a question of Hochster and Huneke that has been open for almost two decades, which asks whether for every finite-dimensional regular ring R, there exists an integer h such that for every ideal I, the hn-th symbolic power of I is contained in the n-th ordinary power of I for all n > 0. In equal characteristic, this result was shown by Ein-Lazarsfeld-Smith and Hochster-Huneke using complex analytic and positive characteristic techniques, respectively. In mixed characteristic, Ma and Schwede proved the result for radical ideals in excellent regular rings using perfectoid spaces. My proof also uses perfectoid spaces to extend the results of Ma and Schwede to arbitrary ideals in arbitrary regular rings. *

*Host: Roya Beheshti*