Isomorphism of the cubical and categorical cohomology groups of a higher-rank graph

Gillaspy, Elizabeth; Wu, Jianchao

doi:10.1090/btran/38

Isomorphism of the cubical and categorical cohomology groups of a higher-rank graph
HTML articles powered by AMS MathViewer

by Elizabeth Gillaspy and Jianchao Wu HTML | PDF

Trans. Amer. Math. Soc. Ser. B 8 (2021), 442-480

Abstract:

We use category-theoretic techniques to provide two proofs showing that for a higher-rank graph $\Lambda$, its cubical (co-)homology and categorical (co-)homology groups are isomorphic in all degrees, thus answering a question of Kumjian, Pask and Sims in the positive. Our first proof uses the topological realization of a higher-rank graph, which was introduced by Kaliszewski, Kumjian, Quigg, and Sims. In our more combinatorial second proof, we construct, explicitly and in both directions, maps on the level of (co-)chain complexes that implement said isomorphism. Along the way, we extend the definition of cubical (co-)homology to allow arbitrary coefficient modules.

References

Teresa Bates, Jeong Hee Hong, Iain Raeburn, and Wojciech Szymański, The ideal structure of the $C^*$-algebras of infinite graphs, Illinois J. Math. 46 (2002), no. 4, 1159–1176. MR 1988256
Jonathan Brown, Lisa Orloff Clark, Cynthia Farthing, and Aidan Sims, Simplicity of algebras associated to étale groupoids, Semigroup Forum 88 (2014), no. 2, 433–452. MR 3189105, DOI 10.1007/s00233-013-9546-z
Jonathan H. Brown, Gabriel Nagy, Sarah Reznikoff, Aidan Sims, and Dana P. Williams, Cartan subalgebras in $C^*$-algebras of Hausdorff étale groupoids, Integral Equations Operator Theory 85 (2016), no. 1, 109–126. MR 3503181, DOI 10.1007/s00020-016-2285-2
Lisa Orloff Clark, Astrid an Huef, and Aidan Sims, AF-embeddability of 2-graph algebras and quasidiagonality of $k$-graph algebras, J. Funct. Anal. 271 (2016), no. 4, 958–991. MR 3507995, DOI 10.1016/j.jfa.2016.04.024
J. Cuntz, A class of $C^{\ast }$-algebras and topological Markov chains. II. Reducible chains and the Ext-functor for $C^{\ast }$-algebras, Invent. Math. 63 (1981), no. 1, 25–40. MR 608527, DOI 10.1007/BF01389192
Joachim Cuntz and Wolfgang Krieger, A class of $C^{\ast }$-algebras and topological Markov chains, Invent. Math. 56 (1980), no. 3, 251–268. MR 561974, DOI 10.1007/BF01390048
Kenneth R. Davidson and Dilian Yang, Periodicity in rank 2 graph algebras, Canad. J. Math. 61 (2009), no. 6, 1239–1261. MR 2588421, DOI 10.4153/CJM-2009-058-0
D. Drinen and M. Tomforde, Computing $K$-theory and $\textrm {Ext}$ for graph $C^*$-algebras, Illinois J. Math. 46 (2002), no. 1, 81–91. MR 1936076, DOI 10.1215/ijm/1258136141
Masatoshi Enomoto and Yasuo Watatani, A graph theory for $C^{\ast }$-algebras, Math. Japon. 25 (1980), no. 4, 435–442. MR 594544
D. Gwion Evans and Aidan Sims, When is the Cuntz-Krieger algebra of a higher-rank graph approximately finite-dimensional?, J. Funct. Anal. 263 (2012), no. 1, 183–215. MR 2920846, DOI 10.1016/j.jfa.2012.03.024
Elizabeth Gillaspy and Alexander Kumjian, Cohomology for small categories: $k$-graphs and groupoids, Banach J. Math. Anal. 12 (2018), no. 3, 572–599. MR 3824741, DOI 10.1215/17358787-2017-0041
Jeong Hee Hong and Wojciech Szymański, The primitive ideal space of the $C^\ast$-algebras of infinite graphs, J. Math. Soc. Japan 56 (2004), no. 1, 45–64. MR 2023453, DOI 10.2969/jmsj/1191418695
Astrid an Huef, Marcelo Laca, Iain Raeburn, and Aidan Sims, KMS states on $C^*$-algebras associated to higher-rank graphs, J. Funct. Anal. 266 (2014), no. 1, 265–283. MR 3121730, DOI 10.1016/j.jfa.2013.09.016
Astrid an Huef, Marcelo Laca, Iain Raeburn, and Aidan Sims, KMS states on the $C^*$-algebra of a higher-rank graph and periodicity in the path space, J. Funct. Anal. 268 (2015), no. 7, 1840–1875. MR 3315580, DOI 10.1016/j.jfa.2014.12.006
S. Kaliszewski, Alex Kumjian, John Quigg, and Aidan Sims, Topological realizations and fundamental groups of higher-rank graphs, Proc. Edinb. Math. Soc. (2) 59 (2016), no. 1, 143–168. MR 3439127, DOI 10.1017/S0013091515000061
Sooran Kang and David Pask, Aperiodicity and primitive ideals of row-finite $k$-graphs, Internat. J. Math. 25 (2014), no. 3, 1450022, 25. MR 3189779, DOI 10.1142/S0129167X14500220
Alex Kumjian and David Pask, Higher rank graph $C^\ast$-algebras, New York J. Math. 6 (2000), 1–20. MR 1745529
Alex Kumjian, David Pask, and Iain Raeburn, Cuntz-Krieger algebras of directed graphs, Pacific J. Math. 184 (1998), no. 1, 161–174. MR 1626528, DOI 10.2140/pjm.1998.184.161
Alex Kumjian, David Pask, Iain Raeburn, and Jean Renault, Graphs, groupoids, and Cuntz-Krieger algebras, J. Funct. Anal. 144 (1997), no. 2, 505–541. MR 1432596, DOI 10.1006/jfan.1996.3001
Alex Kumjian, David Pask, and Aidan Sims, Homology for higher-rank graphs and twisted $C^\ast$-algebras, J. Funct. Anal. 263 (2012), no. 6, 1539–1574. MR 2948223, DOI 10.1016/j.jfa.2012.05.023
Alex Kumjian, David Pask, and Aidan Sims, On twisted higher-rank graph $C^*$-algebras, Trans. Amer. Math. Soc. 367 (2015), no. 7, 5177–5216. MR 3335414, DOI 10.1090/S0002-9947-2015-06209-6
Wolfgang Lück, Transformation groups and algebraic $K$-theory, Lecture Notes in Mathematics, vol. 1408, Springer-Verlag, Berlin, 1989. Mathematica Gottingensis. MR 1027600, DOI 10.1007/BFb0083681
Wolfgang Lück, Chern characters for proper equivariant homology theories and applications to $K$- and $L$-theory, J. Reine Angew. Math. 543 (2002), 193–234. MR 1887884, DOI 10.1515/crll.2002.015
Saunders Mac Lane, Homology, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1975 edition. MR 1344215
David Pask, Iain Raeburn, Mikael Rørdam, and Aidan Sims, Rank-two graphs whose $C^*$-algebras are direct limits of circle algebras, J. Funct. Anal. 239 (2006), no. 1, 137–178. MR 2258220, DOI 10.1016/j.jfa.2006.04.003
Daniel Quillen, Higher algebraic $K$-theory. I, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973, pp. 85–147. MR 0338129
Iain Raeburn, Aidan Sims, and Trent Yeend, The $C^*$-algebras of finitely aligned higher-rank graphs, J. Funct. Anal. 213 (2004), no. 1, 206–240. MR 2069786, DOI 10.1016/j.jfa.2003.10.014
Iain Raeburn and Wojciech Szymański, Cuntz-Krieger algebras of infinite graphs and matrices, Trans. Amer. Math. Soc. 356 (2004), no. 1, 39–59. MR 2020023, DOI 10.1090/S0002-9947-03-03341-5
David I. Robertson and Aidan Sims, Simplicity of $C^\ast$-algebras associated to higher-rank graphs, Bull. Lond. Math. Soc. 39 (2007), no. 2, 337–344. MR 2323468, DOI 10.1112/blms/bdm006
Guyan Robertson and Tim Steger, Affine buildings, tiling systems and higher rank Cuntz-Krieger algebras, J. Reine Angew. Math. 513 (1999), 115–144. MR 1713322, DOI 10.1515/crll.1999.057
Efren Ruiz, Aidan Sims, and Adam P. W. Sørensen, UCT-Kirchberg algebras have nuclear dimension one, Adv. Math. 279 (2015), 1–28. MR 3345177, DOI 10.1016/j.aim.2014.12.042
Adam Skalski and Joachim Zacharias, Entropy of shifts on higher-rank graph $C^*$-algebras, Houston J. Math. 34 (2008), no. 1, 269–282. MR 2383707
Jack Spielberg, Graph-based models for Kirchberg algebras, J. Operator Theory 57 (2007), no. 2, 347–374. MR 2329002
Fei Xu, On the cohomology rings of small categories, J. Pure Appl. Algebra 212 (2008), no. 11, 2555–2569. MR 2440268, DOI 10.1016/j.jpaa.2008.04.004