Homology of sheaves via Brown representability

de Salas, Fernando

doi:10.1090/tran/8602

Homology of sheaves via Brown representability
HTML articles powered by AMS MathViewer

by Fernando Sancho de Salas HTML | PDF

Trans. Amer. Math. Soc. 376 (2023), 123-151 Request permission

Abstract:

We give an elementary construction of homology of sheaves from Brown representability for the dual and see how its main properties are derived easily from the construction. Comparison with Poincaré-Verdier duality and with homology of groups are also developed.

References

Glen E. Bredon, Sheaf theory, McGraw-Hill Book Co., New York-Toronto-London, 1967. MR 0221500
Justin Michael Curry, Dualities between cellular sheaves and cosheaves, J. Pure Appl. Algebra 222 (2018), no. 4, 966–993. MR 3720863, DOI 10.1016/j.jpaa.2017.06.001
René Deheuvels, Homologie des ensembles ordonnés et des espaces topologiques, Bull. Soc. Math. France 90 (1962), 261–321 (French). MR 168614, DOI 10.24033/bsmf.1582
Birger Iversen, Cohomology of sheaves, Universitext, Springer-Verlag, Berlin, 1986. MR 842190, DOI 10.1007/978-3-642-82783-9
Henning Krause, A Brown representability theorem via coherent functors, Topology 41 (2002), no. 4, 853–861. MR 1905842, DOI 10.1016/S0040-9383(01)00010-6
Saunders Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR 1712872
Juan Antonio Navarro González, Duality and finite spaces, Order 6 (1990), no. 4, 401–408. MR 1063821, DOI 10.1007/BF00346134
Amnon Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), no. 1, 205–236. MR 1308405, DOI 10.1090/S0894-0347-96-00174-9
Amnon Neeman, Triangulated categories, Annals of Mathematics Studies, vol. 148, Princeton University Press, Princeton, NJ, 2001. MR 1812507, DOI 10.1515/9781400837212
Amnon Neeman, Brown representability for the dual, Invent. Math. 133 (1998), no. 1, 97–105. MR 1626473, DOI 10.1007/s002220050240
Andrei V. Prasolov, Cosheafification, Theory Appl. Categ. 31 (2016), Paper No. 38, 1134–1175. MR 3660525
Andrei V. Prasolov, Precosheaves of pro-sets and abelian pro-groups are smooth, Topology Appl. 159 (2012), no. 5, 1339–1356. MR 2879363, DOI 10.1016/j.topol.2011.12.015
F. Sancho de Salas and J. F. Torres Sancho, Asphericity and Bökstedt-Neeman theorem, Proc. Amer. Math. Soc. 149 (2021), no. 11, 4583–4594. MR 4310087, DOI 10.1090/proc/15557
V. Carmona Sánchez, C. Maestro Pérez, F. Sancho de Salas, and J. F. Torres Sancho, Homology and cohomology of finite spaces, J. Pure Appl. Algebra 224 (2020), no. 4, 106200, 38. MR 4021914, DOI 10.1016/j.jpaa.2019.106200
N. Spaltenstein, Resolutions of unbounded complexes, Compositio Math. 65 (1988), no. 2, 121–154. MR 932640