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Localization theory for triangulated categories. (English) Zbl 1232.18012
Holm, Thorsten (ed.) et al., Triangulated categories. Based on a workshop, Leeds, UK, August 2006. Cambridge: Cambridge University Press (ISBN 978-0-521-74431-7/pbk). London Mathematical Society Lecture Note Series 375, 161-235 (2010).
This is an introductory article with several original contributions by its author. Its main objective is to give conditions on a full subcategory $$\mathcal{S}$$ of a Verdier triangulated category $$\mathcal{T}$$ to be the kernel of some Bousfield localisation functor $$L : \mathcal{T}\to\mathcal{T}$$.
Fix a Grothendieck universe and call its members small sets. A coproduct indexed by a small set is called a small coproduct. Suppose $$\mathcal{T}$$ to be a triangulated category with small coproducts.
A cardinal is called regular if it cannot be written as a sum, indexed by a strictly smaller cardinal, of strictly smaller cardinals. Let $$\alpha$$ be a regular cardinal. An object $$X$$ of $$\mathcal{T}$$ is called $$\alpha$$-small if for any small set $$I$$ and any tuple of objects $$(Y_i)_{i\in I}$$, any morphism $$X\to\coprod_{i\in I} Y_i$$ factors over $$\coprod_{i\in J} Y_i$$ for some $$J\subseteq I$$ with $$\text{card}J < \alpha$$.
Now $$\mathcal{T}$$ is called $$\alpha$$-well-generated if there exists a small set $$T_0\subseteq\text{Ob}\mathcal{T}$$ consisting of $$\alpha$$-small objects such that the only full triangulated subcategory of $$\mathcal{T}$$ that contains $$T_0$$ and that is closed under summands and small coproducts is $$\mathcal{T}$$ itself, and such that for all small sets $$I$$ and all tuples $$(f_i)_{i\in I}$$ of morphisms in $$I$$, surjectivity of $$\mathcal{T}(X,f_i)$$ for all $$X\in T_0$$ and all $$i\in I$$ implies surjectivity of $$\mathcal{T}(X,\coprod_{i\in I} f_i)$$ for all $$X\in T_0$$. Finally, $$\mathcal{T}$$ is called well-generated if it is $$\beta$$-well-generated for some regular cardinal $$\beta$$ (§§5.1, 6.1, 6.3). This notion is due to Neeman. Such a $$T_0$$ should be seen as a “well-behaved generating subset” of $$\text{Ob}\mathcal{T}$$.
Let $$\mathcal{T}$$ be a well-generated triangulated category. Let $$\mathcal{S}\subseteq\mathcal{T}$$ be a full triangulated category that is closed under small coproducts.
An exact functor $$L : \mathcal{T}\to\mathcal{T}$$ together with a transformation $$\eta : \text{id}_\mathcal{T}\to L$$ such that $$L\eta = \eta L$$ is an isotransformation, is called a Bousfield localisation functor.
Consider the following conditions on the subcategory $$\mathcal{S}$$.
(1)
There exists a small-coproducts-preserving Bousfield localisation functor $$(L,\eta)$$ with $$\text{Ob}\mathcal{S} = \{ X\in\text{Ob}\mathcal{T} : LX\simeq 0 \}$$.
(2)
The subcategory $$\mathcal{S}$$ is an aisle of a constant t-structure on $$\mathcal{T}$$ whose co-aisle $$\mathcal{S}^\perp$$ is closed under small coproducts.
(3)
The subcategory $$\mathcal{S}$$ is well-generated.
(4)
The Verdier quotient $$\mathcal{T}/\mathcal{S}$$ is well-generated.
(5)
There exists a locally presentable abelian category $$\mathcal{A}$$ and a small-coproducts-preserving cohomological functor $$H :\mathcal{T}\to\mathcal{A}$$ such that $$\text{Ob}\mathcal{S} = \{ X\in\text{Ob}\mathcal{T} : H(X[n])\simeq 0\text{ for }n\in\mathbb{Z} \}$$.
(6)
There is a small subset $$S_0\subseteq\text{Ob}\mathcal{S}$$ such that the only full triangulated subcategory of $$\mathcal{S}$$ that contains $$S_0$$ and that is closed under small coproducts is $$\mathcal{S}$$ itself.
(7)
There exists a Bousfield localisation functor $$(L,\eta)$$ with $$\text{Ob}\mathcal{S} = \{ X\in\text{Ob}\mathcal{T} : LX\simeq 0 \}$$.
(8)
The subcategory $$\mathcal{S}$$ is an aisle of a constant t-structure on $$\mathcal{T}$$.
We have (1) $$\Leftrightarrow$$ (2) $$\Rightarrow$$ (3) $$\Leftrightarrow$$ (4) $$\Leftrightarrow$$ (5) $$\Leftrightarrow$$ (6) $$\Rightarrow$$ (7) $$\Leftrightarrow$$ (8) (p. 162, Cor. 7.4.3, Prop. 5.5.1 and Prop. 4.9.1.(3)).
(There is a misprint on p. 177, l. -3, where “$$\phi\circ\phi'$$ and $$\phi''\circ\phi$$” should read “$$\phi'\circ\phi$$ and $$\phi''\circ\phi'$$”; cf. H. Schubert [Categories. Translated from the German by Eva Gray. Berlin-Heidelberg-New York: Springer-Verlag. XI, 385 p. (1972; Zbl 0253.18002)], 19.3.3.(a).)
For the entire collection see [Zbl 1195.18001].

MSC:
 1.8e+31 Derived categories, triangulated categories (MSC2010) 1.8e+36 Localization of categories, calculus of fractions
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