A cotorsion theory in the homotopy category of flat quasicoherent sheaves
Authors:
E. Hosseini and Sh. Salarian
Journal:
Proc. Amer. Math. Soc. 141 (2013), 753762
MSC (2010):
Primary 18E30, 16E40, 16E05, 13D05, 14F05
Published electronically:
August 7, 2012
MathSciNet review:
3003669
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Additional Information
Abstract: Let be a Noetherian scheme, be the homotopy category of flat quasicoherent modules and be the homotopy category of all flat complexes. It is shown that the pair ,  is a complete cotorsion theory in , where  is the essential image of the homotopy category of dgcotorsion complexes of flat modules. Then we study the homotopy category (  ). We show that in the affine case, this homotopy category is equal with the essential image of the embedding functor which has been studied by Neeman in his recent papers. Moreover, we present a condition for the inclusion (  ) to be an equality, where is the essential image of the homotopy category of complexes of cotorsion flat sheaves.
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 [AEGO]
 S. T. Aldrich, E. Enochs, J. R. García Rozas, L. Oyonarte, Covers and envelopes in Grothendieck categories: flat covers of complexes with applications, J. Algebra, 243 (2001), 615630. MR 1850650 (2002i:18010)
 [AJPV]
 L. Alonso Tarrío, A. Jeremías López, M. Pérez Rodríguez, and M. Vale Gonsalves, The derived category of quasicoherent sheaves and axiomatic stable homotopy, Adv. Math. 218 (2008), no. 4, 12241252. MR 2419383 (2010a:14024)
 [AS]
 J. Asadollahi, Sh. Salarian, Cohomology theories based on flats, J. Algebra, 353 (2012), 93120. MR 2872438
 [B]
 A. I. Bondal, Representations of associative algebras and coherent sheaves, Math. USSRIzv. 34(1) (1990) 9 (2006), 2342. MR 992977 (90i:14017)
 [BEE]
 L. Bican, R. El Bashir, E. Enochs, All modules have flat cover, Bull. Lond. Math. Soc. 33 (4)(2001), 385390. MR 1832549 (2002e:16002)
 [BEIJR]
 D. Bravo, E. Enochs, A. Iacob, O. Jenda, J. Rada, Cotorsion pairs in C(RMod), to appear in Rocky Mountain Journal.
 [E]
 R. El Bashir, Covers and directed colimits, Alg. Repr. Theory, 9 (2006), 423430. MR 2252654 (2007k:16003)
 [EG]
 E. Enochs, J. García Rozas, Flat covers of complexes, J. Algebra, 210 (1998), 86102. MR 1656416 (99m:13028)
 [EE]
 E. Enochs, S. Estrada, Relative homological algebra in the category of quasicoherent sheaves, Adv. Math. 194 (2005), 284295. MR 2139915 (2006a:16012)
 [ET]
 P. Eklof, J. Trlifaj, How to make Ext vanish, Bull. Lond. Math. Soc. 33 (2001), 4151. MR 1798574 (2001i:16015)
 [IK]
 S. Iyengar, H. Krause, Acyclicity versus total acyclicity for complexes over noetherian rings, Doc. Math. 11 (2006), 207240. MR 2262932 (2007h:16013)
 [M]
 D. Murfet, The mock homotopy category of projectives and Grothendieck duality, Ph.D. thesis, Canberra (2007).
 [N1]
 A. Neeman, The homotopy category of flat modules, and Grothendieck duality, Invent. Math. 174 (2008), 255308. MR 2439608 (2009h:16008)
 [N2]
 A. Neeman, Some adjoints in homotopy categories, Annals of Math. (2) 171 (3) (2010), 21432155. MR 2680406 (2011i:18026)
 [S]
 L. Salce, Cotorsion theories for abelian groups, Symposia Mathematica, Vol. XXIII (Conf. Abelian Groups and Their Relationship to the Theory of Modules, INDAM,
Rome, 1977), Academic Press, London, 1979, pp. 1132. MR 565595 (81j:20078)
 [TT]
 R. W. Thomason, T. Trobaugh, Higher algebraic theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247435. Adv. Math. 218 (2008), 12241252. MR 1106918 (92f:19001)
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Additional Information
E. Hosseini
Affiliation:
Department of Mathematics, University of Isfahan, Isfahan, Iran
Email:
e.hosseini@sci.ui.ac.ir
Sh. Salarian
Affiliation:
Department of Mathematics, University of Isfahan, P.O. Box 8174673441, Isfahan, Iran – and – School of Mathematics, Institute for Research in Fundamental Science (IPM), P.O. Box 193955746, Tehran, Iran
Email:
salarian@ipm.ir
DOI:
http://dx.doi.org/10.1090/S000299392012113644
Keywords:
Cotorsion theory,
quasicoherent sheaves,
homotopy category,
adjoint functors,
dualizing complex
Received by editor(s):
May 8, 2011
Received by editor(s) in revised form:
July 11, 2011, and July 12, 2011
Published electronically:
August 7, 2012
Additional Notes:
This research was in part supported by a grant from IPM, No. 90130218
Communicated by:
Lev Borisov
Article copyright:
© Copyright 2012
American Mathematical Society
