Abstract
Let $R$ be a ring. In a previous paper [11] we found a new description for the category $\mathbf{K}(R\text{-Proj})$; it is equivalent to the Verdier quotient $\mathbf{K}(R\text{-Flat})/{\mathscr S}$, for some suitable $\mathscr{S}\subset\mathbf{K}(R\text{-Flat})$. In this article we show that the quotient map from $\mathbf{K}(R\text{-Flat})$ to $\mathbf{K}(R\text{-Flat})/\mathscr{S}$ always has a right adjoint. This gives a new, fully faithful embedding of $\mathbf{K}(R\text{-Proj})$ into $\mathbf{K}(R\text{-Flat})$. Its virtue is that it generalizes to nonaffine schemes.
-
[Bican-ElBashir-Enochs01]
L. Bican, R. El Bashir, and E. Enochs, "All modules have flat covers," Bull. London Math. Soc., vol. 33, iss. 4, pp. 385-390, 2001.@article {Bican-ElBashir-Enochs01, MRKEY = {1832549},
AUTHOR = {Bican, L. and El Bashir, R. and Enochs, E.},
TITLE = {All modules have flat covers},
JOURNAL = {Bull. London Math. Soc.},
FJOURNAL = {The Bulletin of the London Mathematical Society},
VOLUME = {33},
YEAR = {2001},
NUMBER = {4},
PAGES = {385--390},
ISSN = {0024-6093},
CODEN = {LMSBBT},
MRCLASS = {16D40 (16D90 16E30)},
MRNUMBER = {2002e:16002},
MRREVIEWER = {Lidia Angeleri H{ü}gel},
DOI = {10.1017/S0024609301008104},
ZBLNUMBER = {1029.16002},
} -
[Eklof-Trlifaj01] P. C. Eklof and J. Trlifaj, "How to make Ext vanish," Bull. London Math. Soc., vol. 33, iss. 1, pp. 41-51, 2001.
@article {Eklof-Trlifaj01, MRKEY = {1798574},
AUTHOR = {Eklof, Paul C. and Trlifaj, Jan},
TITLE = {How to make {E}xt vanish},
JOURNAL = {Bull. London Math. Soc.},
FJOURNAL = {The Bulletin of the London Mathematical Society},
VOLUME = {33},
YEAR = {2001},
NUMBER = {1},
PAGES = {41--51},
ISSN = {0024-6093},
CODEN = {LMSBBT},
MRCLASS = {16E30 (18G15 20K40)},
MRNUMBER = {2001i:16015},
MRREVIEWER = {Enrico Gregorio},
DOI = {10.1112/blms/33.1.41},
ZBLNUMBER = {1030.16004},
} -
[Enochs81] E. E. Enochs, "Injective and flat covers, envelopes and resolvents," Israel J. Math., vol. 39, iss. 3, pp. 189-209, 1981.
@article {Enochs81, MRKEY = {636889},
AUTHOR = {Enochs, Edgar E.},
TITLE = {Injective and flat covers, envelopes and resolvents},
JOURNAL = {Israel J. Math.},
FJOURNAL = {Israel Journal of Mathematics},
VOLUME = {39},
YEAR = {1981},
NUMBER = {3},
PAGES = {189--209},
ISSN = {0021-2172},
CODEN = {ISJMAP},
MRCLASS = {16A53 (13H05 16A33 16A49 18B30)},
MRNUMBER = {83a:16031},
MRREVIEWER = {M. Teply},
DOI = {10.1007/BF02760849},
ZBLNUMBER = {0464.16019},
} -
[Enochs-Garcia98] E. E. Enochs and J. R. Garc’ia Rozas, "Flat covers of complexes," J. Algebra, vol. 210, iss. 1, pp. 86-102, 1998.
@article {Enochs-Garcia98, MRKEY = {1656416},
AUTHOR = {Enochs, Edgar E. and Garc{\'ı}a Rozas, J. R.},
TITLE = {Flat covers of complexes},
JOURNAL = {J. Algebra},
FJOURNAL = {Journal of Algebra},
VOLUME = {210},
YEAR = {1998},
NUMBER = {1},
PAGES = {86--102},
ISSN = {0021-8693},
CODEN = {JALGA4},
MRCLASS = {13D25 (16E99)},
MRNUMBER = {99m:13028},
MRREVIEWER = {Hans-Bjørn Foxby},
DOI = {10.1006/jabr.1998.7582},
ZBLNUMBER = {0931.13009},
} -
[Gillespie04] J. Gillespie, "The flat model structure on ${ Ch}(R)$," Trans. Amer. Math. Soc., vol. 356, iss. 8, pp. 3369-3390, 2004.
@article {Gillespie04, MRKEY = {2052954},
AUTHOR = {Gillespie, James},
TITLE = {The flat model structure on {${\rm Ch}(R)$}},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {356},
YEAR = {2004},
NUMBER = {8},
PAGES = {3369--3390},
ISSN = {0002-9947},
CODEN = {TAMTAM},
MRCLASS = {18G50 (18G15 18G35 55U35)},
MRNUMBER = {2005b:18020},
MRREVIEWER = {David A. Blanc},
DOI = {10.1090/S0002-9947-04-03416-6},
ZBLNUMBER = {1056.55011},
} -
[Gillespie06] J. Gillespie, "The flat model structure on complexes of sheaves," Trans. Amer. Math. Soc., vol. 358, iss. 7, pp. 2855-2874, 2006.
@article {Gillespie06, MRKEY = {2216249},
AUTHOR = {Gillespie, James},
TITLE = {The flat model structure on complexes of sheaves},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {358},
YEAR = {2006},
NUMBER = {7},
PAGES = {2855--2874},
ISSN = {0002-9947},
CODEN = {TAMTAM},
MRCLASS = {55U35 (18G15)},
MRNUMBER = {2007a:55024},
MRREVIEWER = {Timothy Porter},
DOI = {10.1090/S0002-9947-06-04157-2},
ZBLNUMBER = {1094.55016},
} -
[Hovey02] M. Hovey, "Cotorsion pairs, model category structures, and representation theory," Math. Z., vol. 241, iss. 3, pp. 553-592, 2002.
@article {Hovey02, MRKEY = {1938704},
AUTHOR = {Hovey, Mark},
TITLE = {Cotorsion pairs, model category structures, and representation theory},
JOURNAL = {Math. Z.},
FJOURNAL = {Mathematische Zeitschrift},
VOLUME = {241},
YEAR = {2002},
NUMBER = {3},
PAGES = {553--592},
ISSN = {0025-5874},
CODEN = {MAZEAX},
MRCLASS = {55U35 (18E30 18G55)},
MRNUMBER = {2003m:55027},
DOI = {10.1007/s00209-002-0431-9},
ZBLNUMBER = {1016.55010},
} -
[Iyengar-Krause06] S. Iyengar and H. Krause, "Acyclicity versus total acyclicity for complexes over Noetherian rings," Documenta Math., vol. 11, pp. 207-240, 2006.
@article {Iyengar-Krause06, MRKEY = {2262932},
AUTHOR = {Iyengar, Srikanth and Krause, Henning},
TITLE = {Acyclicity versus total acyclicity for complexes over {N}oetherian rings},
JOURNAL = {Documenta Math.},
FJOURNAL = {Documenta Mathematica},
VOLUME = {11},
YEAR = {2006},
PAGES = {207--240},
ISSN = {1431-0635},
MRCLASS = {16E05 (13D25 18E30)},
MRNUMBER = {2007h:16013},
MRREVIEWER = {Zhaoyong Huang},
ZBLNUMBER = {1119.13014},
} -
[Murfet07] D. S. Murfet, "The mock homotopy category of projectives and Grothendieck duality," PhD Thesis , Aust. National U., 2007.
@phdthesis{Murfet07,
author={Murfet, D. S.},
TITLE={The mock homotopy category of projectives and {Grothendieck} duality},
SCHOOL={Aust. National U.},
YEAR={2007},
} -
[Neeman99] A. Neeman, Triangulated Categories, Princeton, NJ: Princeton Univ. Press, 2001.
@book {Neeman99, MRKEY = {1812507},
AUTHOR = {Neeman, Amnon},
TITLE = {Triangulated Categories},
SERIES = {Annals of Math. Studies},
NUMBER = {148},
PUBLISHER = {Princeton Univ. Press},
ADDRESS = {Princeton, NJ},
YEAR = {2001},
PAGES = {viii+449},
ISBN = {0-691-08685-0; 0-691-08686-9},
MRCLASS = {18E30 (55-02 55N20 55U35)},
MRNUMBER = {2001k:18010},
MRREVIEWER = {Stanis{\l}aw Betley},
ZBLNUMBER = {0996.19003},
ZBLNUMBER = {0974.18008},
} -
[Neeman08] A. Neeman, "The homotopy category of flat modules, and Grothendieck duality," Invent. Math., vol. 174, iss. 2, pp. 255-308, 2008.
@article {Neeman08, MRKEY = {2439608},
AUTHOR = {Neeman, Amnon},
TITLE = {The homotopy category of flat modules, and {G}rothendieck duality},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {174},
YEAR = {2008},
NUMBER = {2},
PAGES = {255--308},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {16E05 (16B50 18E30)},
MRNUMBER = {2009h:16008},
MRREVIEWER = {Wolfgang Rump},
DOI = {10.1007/s00222-008-0131-0},
ZBLNUMBER = {05377113},
}
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