Categories of dimension zero
HTML articles powered by AMS MathViewer
- by John D. Wiltshire-Gordon PDF
- Proc. Amer. Math. Soc. 147 (2019), 35-50 Request permission
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.References
- Serge Bouc and Jacques Thévenaz, The representation theory of finite sets and correspondences, 2015, arXiv:1510.03034.
- Thomas Church, Jordan S. Ellenberg, and Benson Farb, FI-modules and stability for representations of symmetric groups, Duke Math. J. 164 (2015), no. 9, 1833–1910. MR 3357185, DOI 10.1215/00127094-3120274
- Thomas Church, Jordan S. Ellenberg, Benson Farb, and Rohit Nagpal, FI-modules over Noetherian rings, Geom. Topol. 18 (2014), no. 5, 2951–2984. MR 3285226, DOI 10.2140/gt.2014.18.2951
- Thomas Church and Benson Farb, Representation theory and homological stability, Adv. Math. 245 (2013), 250–314. MR 3084430, DOI 10.1016/j.aim.2013.06.016
- Ian G. Connell, On the group ring, Canadian J. Math. 15 (1963), 650–685. MR 153705, DOI 10.4153/CJM-1963-067-0
- Albrecht Dold, Homology of symmetric products and other functors of complexes, Ann. of Math. (2) 68 (1958), 54–80. MR 97057, DOI 10.2307/1970043
- Jordan S. Ellenberg and John D. Wiltshire-Gordon, Algebraic structures on cohomology of configuration spaces of manifolds with flows, 2015. arXiv:1508.02430.
- Vincent Franjou and Antoine Touzé (eds.), Lectures on functor homology, Progress in Mathematics, vol. 311, Birkhäuser/Springer, Cham, 2015. Lectures from the conference held in Nantes, April 2012. MR 3410176, DOI 10.1007/978-3-319-21305-7
- Andrew Gitlin, New examples of dimension zero categories, J. Algebra 505 (2018), 271–278. MR 3789913, DOI 10.1016/j.jalgebra.2018.03.009
- L. G. Kovács, Semigroup algebras of the full matrix semigroup over a finite field, Proc. Amer. Math. Soc. 116 (1992), no. 4, 911–919. MR 1123658, DOI 10.1090/S0002-9939-1992-1123658-6
- Henning Krause, Krull-Schmidt categories and projective covers, Expo. Math. 33 (2015), no. 4, 535–549. MR 3431480, DOI 10.1016/j.exmath.2015.10.001
- Nicholas J. Kuhn, Generic representation theory of finite fields in nondescribing characteristic, Adv. Math. 272 (2015), 598–610. MR 3303242, DOI 10.1016/j.aim.2014.12.012
- Jan Okniński, Semigroup algebras, Monographs and Textbooks in Pure and Applied Mathematics, vol. 138, Marcel Dekker, Inc., New York, 1991. MR 1083356
- Teimuraz Pirashvili, Dold-Kan type theorem for $\Gamma$-groups, Math. Ann. 318 (2000), no. 2, 277–298. MR 1795563, DOI 10.1007/s002080000120
- T. Pirashvili and B. Richter, Hochschild and cyclic homology via functor homology, $K$-Theory 25 (2002), no. 1, 39–49. MR 1899698, DOI 10.1023/A:1015064621329
- Eric M. Rains, The action of $S_n$ on the cohomology of $\overline M_{0,n}(\Bbb R)$, Selecta Math. (N.S.) 15 (2009), no. 1, 171–188. MR 2511203, DOI 10.1007/s00029-008-0467-8
- J. T. Schwartz, Fast probabilistic algorithms for verification of polynomial identities, J. Assoc. Comput. Mach. 27 (1980), no. 4, 701–717. MR 594695, DOI 10.1145/322217.322225
- Steven V. Sam and Andrew Snowden, GL-equivariant modules over polynomial rings in infinitely many variables, Trans. Amer. Math. Soc. 368 (2016), no. 2, 1097–1158. MR 3430359, DOI 10.1090/tran/6355
- Steven V. Sam and Andrew Snowden, Gröbner methods for representations of combinatorial categories, J. Amer. Math. Soc. 30 (2017), no. 1, 159–203. MR 3556290, DOI 10.1090/jams/859
- Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324, DOI 10.1017/CBO9781139644136
- John D. Wiltshire-Gordon, Uniformly presented vector spaces, 2014. arXiv:1406.0786.
- E. I. Zel′manov, Semigroup algebras with identities, Sibirsk. Mat. Ž. 18 (1977), no. 4, 787–798, 956 (Russian). MR 0486254
Additional Information
- John D. Wiltshire-Gordon
- Affiliation: Department of Mathematics, University of Wisconsin-Madison, Van Vleck Hall, 480 Lincoln Drive, Madison, Wisconsin 53706
- MR Author ID: 1017529
- Email: jwiltshiregordon@gmail.com
- 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
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 35-50
- MSC (2010): Primary 18A25; Secondary 16G10
- DOI: https://doi.org/10.1090/proc/14040
- MathSciNet review: 3876729