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SUMMARY:Flatness and Completion Revisited - Amnon Yekuteli (Ben Gurion)
DTSTART:20180207T163000Z
DTEND:20180207T173000Z
UID:TALK97558@talks.cam.ac.uk
CONTACT:Eugenio Giannelli
DESCRIPTION:In this talk I consider an old and exhaustively studied situat
 ion: A is a commutative ring\, and \\a is an ideal in it. ("\\a" stands fo
 r gothic "a".) We are interested in the \\a-adic completion operation for 
 A-modules\, and in flatness of A-modules.\nThe departure from the classica
 l and familiar situation is this: the A-modules we care about are not fini
 tely generated. The following was considered an open problem by commutativ
 e 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 publish
 ed in the literature over the years. A few months ago I found a proof of t
 he general case. But then\, a series of emails led me to prior proofs\, th
 at are embedded in pretty recent texts\, and were unknown to the algebra c
 ommunity.\nNonetheless\, my methods gave more detailed variants of the gen
 eral result mentioned above\, that are actually new. One of them concerns 
 the case of a ring A that is not noetherian\, but where the ideal \\a is w
 eakly proregular. The latter is a condition discovered by Grothendieck a l
 ong time ago\, but became prominent only very recently (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 give a few concrete e
 xamples. There will be no derived categories in this talk (!). The talk sh
 ould be totally understandable to anyone with knowledge of \\a-adic comple
 tion and flatness.
LOCATION:MR12
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