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CATEGORIES:Algebra and Representation Theory Seminar
SUMMARY:Flatness and Completion Revisited - Amnon Yekuteli
(Ben Gurion)
DTSTART;TZID=Europe/London:20180207T163000
DTEND;TZID=Europe/London:20180207T173000
UID:TALK97558AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/97558
DESCRIPTION:In this talk I consider an old and exhaustively st
udied situation: A is a commutative ring\, and \\a
is an ideal in it. ("\\a" stands for gothic "a".)
We are interested in the \\a-adic completion oper
ation for A-modules\, and in flatness of A-modules
.\nThe departure from the classical and familiar s
ituation is this: the A-modules we care about are
not finitely generated. The following was consider
ed an open problem by commutative algebraists: if
A is a noetherian ring and M is a flat A-module\,
is the \\a-adic completion of M also flat? Partial
positive answers were published in the literature
over the years. A few months ago I found a proof
of the general case. But then\, a series of emails
led me to prior proofs\, that are embedded in pre
tty recent texts\, and were unknown to the algebra
community.\nNonetheless\, my methods gave more de
tailed variants of the general result mentioned ab
ove\, that are actually new. One of them concerns
the case of a ring A that is not noetherian\, but
where the ideal \\a is weakly proregular. The latt
er is a condition discovered by Grothendieck a lon
g time ago\, but became prominent only very recent
ly (in the context of derived completion).\nIn the
talk I will give the background\, mention the new
and not-so-new results (with some proofs)\, and g
ive a few concrete examples. There will be no deri
ved categories in this talk (!). The talk should b
e totally understandable to anyone with knowledge
of \\a-adic completion and flatness.
LOCATION:MR12
CONTACT:Eugenio Giannelli
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