A remark on singular cohomology and sheaf cohomology
DOI:
https://doi.org/10.7146/math.scand.a-132191Abstract
We prove a comparison isomorphism between singular cohomology and sheaf cohomology.
References
Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos. Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat, Lecture Notes in Mathematics, 269, Springer-Verlag, Berlin-New York, 1972
Barwick, C., On left and right model categories and left and right Bousfield localizations, Homology Homotopy Appl. 12 (2010), no. 2, 245–320. http://projecteuclid.org/euclid.hha/1296223884
Bredon, G. E., Sheaf theory, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1967.
Brown, K. S., Abstract homotopy theory and generalized sheaf cohomology, Trans. Amer. Math. Soc. 186 (1973), 419–458. https://doi.org/10.2307/1996573
Brown, K. S. and Gersten, S. M., Algebraic $K$-theory as generalized sheaf cohomology, in Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 266–292. Lecture Notes in Math., 341 Springer, Berlin, 1973.
Choudhury, U. and Gallauer Alves de Souza, M., Homotopy theory of dg sheaves, Comm. Algebra 47 (2019), no. 8, 3202–3228. https://doi.org/10.1080/00927872.2018.1554744
Dugger, D., Hollander, S., and Isaksen, D. C., Hypercovers and simplicial presheaves, Math. Proc. Cambridge Philos. Soc. 136 (2004), no. 1, 9–51. https://doi.org/10.1017/S0305004103007175
Dugger, D. and Isaksen, D. C., Topological hypercovers and $mathbb A^1$-realizations, Math. Z. 246 (2004), no. 4, 667–689. https://doi.org/10.1007/s00209-003-0607-y
Elmendorf, A. D., Kriz, I., Mandell, M. A., and May, J. P., Rings, modules, and algebras in stable homotopy theory. With an appendix by M. Cole, Mathematical Surveys and Monographs, 47, American Mathematical Society, Providence, RI, 1997,
https://doi.org/10.1090/surv/047
Hinich, V., Deformations of sheaves of algebras, Adv. Math. 195 (2005), no. 1, 102–164. https://doi.org/10.1016/j.aim.2004.07.007
Hinich, V. A. and Schechtman, V. V., On homotopy limit of homotopy algebras, in $K$-theory, arithmetic and geometry (Moscow, 1984–1986), pp. 240–264, Lecture Notes in Math., 1289, Springer, Berlin, 1987. https://doi.org/10.1007/BFb0078370
Jardine, J. F., Simplicial presheaves, J. Pure Appl. Algebra 47 (1987), no. 1, 35–87. https://doi.org/10.1016/0022-4049(87)90100-9
Jardine, J. F., Presheaves of chain complexes, $K$-Theory 30 (2003), no. 4, 365–420, https://doi.org/10.1023/B:KTHE.0000021707.27987.c9
Lurie, J., Higher topos theory, Annals of Mathematics Studies, 170, Princeton University Press, Princeton, NJ, 2009. https://doi.org/10.1515/9781400830558
Sella, Y., Comparison of sheaf cohomology and singular cohomology, arXiv:1602.06674, 2016.
Shipley, B., $Hmathbb Z$-algebra spectra are differential graded algebras, Amer. J. Math. 129 (2007), no. 2, 351–379. https://doi.org/10.1353/ajm.2007.0014
Spaltenstein, N., Resolutions of unbounded complexes, Compositio Math. 65 (1988), no. 2, 121–154.
Spanier, E. H., Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966.
Thomason, R. W., Algebraic $K$-theory and étale cohomology, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 3, 437–552.
