The homotopy category of flat modules, and Grothendieck duality
Tóm tắt
Let R be a ring. We prove that the homotopy category K(R-Proj) is always
$\aleph_1$
-compactly generated, and, depending on the ring R, it may or may not be compactly generated. We use this to give a description of K(R-Proj) as a quotient of K(R-Flat). The remarkable fact is that this new description of K(R-Proj) generalizes to non-affine schemes; this will appear in Murfet’s thesis.
Tài liệu tham khảo
Balmer, P.: Presheaves of triangulated categories and reconstruction of schemes. Math. Ann. 324(3), 557–580 (2002)
Balmer, P.: The spectrum of prime ideals in tensor triangulated categories. J. Reine Angew. Math. 588, 149–168 (2005)
Bökstedt, M., Neeman, A.: Homotopy limits in triangulated categories. Compos. Math. 86, 209–234 (1993)
Bousfield, A.K.: The localization of spectra with respect to homology. Topology 18, 257–281 (1979)
Enochs, E.E., García Rozas, J.R.: Flat covers of complexes. J. Algebra 210(1), 86–102 (1998)
Gabriel, P., Zisman, M.: Calculus of fractions and homotopy theory. Ergeb. Math. Grenzgeb., vol. 35. Springer, New York (1967)
Iyengar, S., Krause, H.: Acyclicity versus total acyclicity for complexes over Noetherian rings. Doc. Math. 11, 207–240 (2006)
Jørgensen, P.: The homotopy category of complexes of projective modules. Adv. Math. 193(1), 223–232 (2005)
Krause, H.: The stable derived category of a Noetherian scheme. Compos. Math. 141(5), 1128–1162 (2005)
Neeman, A.: The Grothendieck duality theorem via Bousfield’s techniques and Brown representability. J. Am. Math. Soc. 9, 205–236 (1996)
Neeman, A.: Triangulated Categories. Ann. Math. Stud., vol. 148. Princeton University Press, Princeton, NJ (2001)
Verdier, J.-L.: Des catégories dérivées des catégories abeliennes. Astérisque, vol. 239. Société Mathématique de France, Paris (1996)