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Congruence modules and criteria for detecting free summands

Recent Developments in Commutative Algebra April 15, 2024 - April 19, 2024

April 15, 2024 (10:00 AM PDT - 11:00 AM PDT)
Speaker(s): Srikanth Iyengar (University of Utah)
Location: SLMath: Eisenbud Auditorium, Online/Virtual
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC
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Congruence modules and criteria for detecting free summands

Abstract

Wiles, in his work on modularity lifting, discovered a numerical criterion for a map R-->T of noetherian complete commutative local rings over a fixed discrete valuation ring O, and of relative dimension zero, to be an isomorphism of complete intersections. The criterion is in terms of the "congruence module" of T attached to an augmentation T-->O and the cotangent module of the composite map R-->O. Diamond generalized this result to a numerical criteria for a module over R to be free, again involving suitable congruence modules and cotangent modules. In my talk, I will present extensions of these results to higher relative dimension. These have applications in number theory, but I will focus mostly on the commutative algebra aspects. This is based on joint work with Khare, Manning, and Urban.

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Congruence modules and criteria for detecting free summands

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