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Proceedings of the American Mathematical Society

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Categories of dimension zero

Author: John D. Wiltshire-Gordon
Journal: Proc. Amer. Math. Soc. 147 (2019), 35-50
MSC (2010): Primary 18A25; Secondary 16G10
Published electronically: October 18, 2018
MathSciNet review: 3876729
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Abstract: If $ \mathcal {D}$ is a category and $ k$ is a commutative ring, the functors from $ \mathcal {D}$ to $ \mathbf {Mod}_{k}$ can be thought of as representations of $ \mathcal {D}$. By definition, $ \mathcal {D}$ is dimension zero over $ k$ if its finitely generated representations have finite length. We characterize categories of dimension zero in terms of the existence of a ``homological modulus'' (Definition 1.4) which is combinatorial and linear-algebraic in nature.

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John D. Wiltshire-Gordon
Affiliation: Department of Mathematics, University of Wisconsin-Madison, Van Vleck Hall, 480 Lincoln Drive, Madison, Wisconsin 53706

Keywords: Finite length category, representations of categories, homological modulus
Received by editor(s): June 28, 2017
Published electronically: October 18, 2018
Additional Notes: The author was supported by an NSF Graduate Research Fellowship (ID 2011127608). This work contains results that later appeared in his 2016 PhD thesis at the University of Michigan. The author acknowledges support from the algebra RTG at the University of Wisconsin, NSF grant DMS-1502553.
Communicated by: Jerzy Weyman
Article copyright: © Copyright 2018 American Mathematical Society