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TZID:America/Chicago
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DTSTART:20211107T020000
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UID:calendar.25285.field_event_date_2.0@math.wustl.edu
CREATED:20211122T153343Z
DESCRIPTION:Abstract: In 1921, Emmy Noether proved that ideals in commutat
ive Noetherian rings decompose into primary components. Noether's theorem
is an analogue of the fundamental theorems of arithmetic and algebra, whi
ch say that integers and complex polynomials in a single variable can be d
ecomposed into powers of prime factors. While factorizations of integers a
nd 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 pr
imary 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 c
omparison between ordinary and symbolic powers of arbitrary ideals in arbi
trary 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 t
hat 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 res
ult was shown by Ein-Lazarsfeld-Smith and Hochster-Huneke using complex an
alytic and positive characteristic techniques, respectively. In mixed cha
racteristic, Ma and Schwede proved the result for radical ideals in excel
lent regular rings using perfectoid spaces. My proof also uses perfectoid
spaces to extend the results of Ma and Schwede to arbitrary ideals in arbi
trary regular rings. \n\nHost: Roya Beheshti
DTSTART;TZID=America/Chicago:20211203T160000
DTEND;TZID=America/Chicago:20211203T170000
LAST-MODIFIED:20211201T145611Z
SUMMARY:Colloquium: 'Symbolic powers of ideals in regular rings'
URL;TYPE=URI:https://math.wustl.edu/events/colloquium-symbolic-powers-ideal
s-regular-rings
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