Some adjoints in homotopy categories. (English) Zbl 1205.18008

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.


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)


Zbl 1184.18008
Full Text: DOI Link


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