On the structure of modules indexed by small categories

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Crichton Ogle
March 9, 2021
4:00PM - 5:00PM
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Zoom

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Add to Calendar 2021-03-09 16:00:00 2021-03-09 17:00:00 On the structure of modules indexed by small categories Speaker:  Crichton Ogle (OSU) Title:  On the structure of modules indexed by small categories Speaker's URL:   https://math.osu.edu/people/ogle.1 Abstract:  (joint work with S. Sultan) Given a small category C, a C-module M is a functor from C to the category of finite-dimensional vector spaces over a field k. Associated to M is its local structure, given as a functor from C to the category of bi-closed multi-flags over k. When the local structure of M is stable (a condition satisfied whenever both the category C and the field k are finite), it determines a quasi-tame cover QTC(M) (a finite direct sum of quasi-blocks), indexed by the same category, for which the associated graded local structure is canonically isomorphic to that of M. If C is a poset category, this cover is tame (a finite direct sum of blocks). QTC(M) represents the closest approximation to M by a quasi-tame module, and recovers M precisely when M itself is tame. In the case M has stable local structure and is equipped with an inner product compatible with that structure, there exists a C-module surjection QTC(M) -> M inducing the above-mentioned isomorphism on associated graded local structures. This map is an isomorphism iff the excess of M vanishes (where the excess numerically measures the failure of the local structure of M to be in general position). URL associated with Seminar https://tgda.osu.edu/activities/tdga-seminar/ Zoom Department of Mathematics math@osu.edu America/New_York public
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Speaker:  Crichton Ogle (OSU)

Title:  On the structure of modules indexed by small categories

Speaker's URL:   https://math.osu.edu/people/ogle.1

Abstract:  (joint work with S. Sultan)

Given a small category C, a C-module M is a functor from C to the category of finite-dimensional vector spaces over a field k. Associated to M is its local structure, given as a functor from C to the category of bi-closed multi-flags over k. When the local structure of M is stable (a condition satisfied whenever both the category C and the field k are finite), it determines a quasi-tame cover QTC(M) (a finite direct sum of quasi-blocks), indexed by the same category, for which the associated graded local structure is canonically isomorphic to that of M. If C is a poset category, this cover is tame (a finite direct sum of blocks). QTC(M) represents the closest approximation to M by a quasi-tame module, and recovers M precisely when M itself is tame. In the case M has stable local structure and is equipped with an inner product compatible with that structure, there exists a C-module surjection QTC(M) -> M inducing the above-mentioned isomorphism on associated graded local structures. This map is an isomorphism iff the excess of M vanishes (where the excess numerically measures the failure of the local structure of M to be in general position).

URL associated with Seminar
https://tgda.osu.edu/activities/tdga-seminar/

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