An explicit double-exponential upper bound on the regularity of OI-modules presented in finite degrees, together with a new inductive method for proving their structural properties.
A long exact sequence for homology of FI-modules
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abstract
We construct a long exact sequence involving the homology of an FI-module. Using the long exact sequence, we give two methods to bound the Castelnuovo-Mumford regularity of an FI-module which is generated and related in finite degree. We also prove that for an FI-module which is generated and related in finite degree, if it has a nonzero higher homology, then its homological degrees are strictly increasing (starting from the first homological degree).
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An inductive method for $\mathrm{OI}$-modules
An explicit double-exponential upper bound on the regularity of OI-modules presented in finite degrees, together with a new inductive method for proving their structural properties.