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 \(\mathcal{S}\).

(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].

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 \(\mathcal{T}\).

(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].

Reviewer: Matthias Künzer (Aachen)

##### MSC:

18E30 | Derived categories, triangulated categories (MSC2010) |

18E35 | Localization of categories, calculus of fractions |