Some adjoints in homotopy categories. (English) Zbl 1205.18008

Neeman, Amnon

Ann. Math. (2) 171, No. 3, 2143-2155 (2010).

Let \(R\) be a ring. Let \({\mathcal K}(R\text{-Flat})\) (\({\mathcal K}(R\text{-Proj})\)) denote the homotopy category of cochain complexes of flat, (respectively, projective) \(R\)-modules.
In a previous paper [see Invent. Math. 174, No. 2, 255–308 (2008; Zbl 1184.18008)], the author proved that \({\mathcal K}(R\text{-Proj})\) is a Verdier quotient of \({\mathcal K}(R\text{-Flat})\). Let \(j^*:{\mathcal K}(R\text{-Flat})\to {\mathcal K}(R\text{-Proj})\) denote the quotient functor.
The main theorem in the paper proves that \(j^*\) has a right adjoint \(j_*\). The author mentions that this theorem has a generalization to (non-affine) schemes.

Reviewer: Adrian Langer (Warszawa)

MSC:

18E30	Derived categories, triangulated categories (MSC2010)
16B50	Category-theoretic methods and results in associative algebras (except as in 16D90)
14A22	Noncommutative algebraic geometry
14F05	Sheaves, derived categories of sheaves, etc. (MSC2010)
18G55	Nonabelian homotopical algebra (MSC2010)
18F20	Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)

Keywords:

homotopy category; projective modules; flat modules

Citations:

Zbl 1184.18008

References:

[1]	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. · Zbl 1029.16002
[2]	P. C. Eklof and J. Trlifaj, ”How to make Ext vanish,” Bull. London Math. Soc., vol. 33, iss. 1, pp. 41-51, 2001. · Zbl 1030.16004
[3]	E. E. Enochs, ”Injective and flat covers, envelopes and resolvents,” Israel J. Math., vol. 39, iss. 3, pp. 189-209, 1981. · Zbl 0464.16019
[4]	E. E. Enochs and J. R. Garc’ia Rozas, ”Flat covers of complexes,” J. Algebra, vol. 210, iss. 1, pp. 86-102, 1998. · Zbl 0931.13009
[5]	J. Gillespie, ”The flat model structure on \({ Ch}(R)\),” Trans. Amer. Math. Soc., vol. 356, iss. 8, pp. 3369-3390, 2004. · Zbl 1056.55011
[6]	J. Gillespie, ”The flat model structure on complexes of sheaves,” Trans. Amer. Math. Soc., vol. 358, iss. 7, pp. 2855-2874, 2006. · Zbl 1094.55016
[7]	M. Hovey, ”Cotorsion pairs, model category structures, and representation theory,” Math. Z., vol. 241, iss. 3, pp. 553-592, 2002. · Zbl 1016.55010
[8]	S. Iyengar and H. Krause, ”Acyclicity versus total acyclicity for complexes over Noetherian rings,” Documenta Math., vol. 11, pp. 207-240, 2006. · Zbl 1119.13014
[9]	D. S. Murfet, ”The mock homotopy category of projectives and Grothendieck duality,” PhD Thesis , Aust. National U., 2007.
[10]	A. Neeman, Triangulated Categories, Princeton, NJ: Princeton Univ. Press, 2001. · Zbl 0996.19003
[11]	A. Neeman, ”The homotopy category of flat modules, and Grothendieck duality,” Invent. Math., vol. 174, iss. 2, pp. 255-308, 2008. · Zbl 1184.18008