Subsection 1.2. (0MV8) · ScalingStacks

1.2. $t$ -structures and co- $t$ -structures

The concept of a $t$ -structure on a triangulated category $\mathcal{T}$ was first introduced by Beilinson, Bernstein and Deligne in [3]. The basic theory of $t$ -structures can be found in [3] and [13].

0MU2

Definition 1.3. Let $\mathcal{T}$ be a triangulated category. A pair of full subcategories of $\mathcal{T}$ , $(\mathcal{X},\mathcal{Y})$ , is called a $t$ -structure on $\mathcal{T}$ if it satisfies the following properties:

$(1)$ $\mathcal{X}\subseteq\Sigma^{-1}\mathcal{X}$ and $\Sigma^{-1}\mathcal{Y}\subseteq\mathcal{Y}$ ;

$(2)$ $\textnormal{Hom}(\mathcal{X},\Sigma^{-1}\mathcal{Y})=0$ ;

$(3)$ For any object $Z$ of $\mathcal{T}$ there exists a distinguished triangle

X\rightarrow Z\rightarrow\Sigma^{-1}Y\rightarrow\Sigma X

with $X\in\mathcal{X}$ and $Y\in\mathcal{Y}$ .

The full subcategories $\mathcal{X}$ and $\mathcal{Y}$ are often denoted by $\mathcal{T}^{t\leqslant 0}$ and $\mathcal{T}^{t\geqslant 0}$ , or simply by $\mathcal{T}^{\leqslant 0}$ and $\mathcal{T}^{\geqslant 0}$ , respectively; see [7] and [11].

The notion of a $t$ -structure has become widespread in the study of triangulated categories and lends itself particularly well to induction arguments in this setting, see for example [5].

We next introduce the almost dual notion of a co- $t$ -structure. We note that co- $t$ -structures have also recently been introduced by Bondarko in [7] where they are called weight structures.

0MU3

Definition 1.4. Let $\mathcal{T}$ be a triangulated category. A pair of full subcategories of $\mathcal{T}$ , $(\mathcal{A},\mathcal{B})$ , will be called a co- $t$ -structure on $\mathcal{T}$ if it satisfies the following properties:

$(0)$ $\mathcal{A}$ and $\mathcal{B}$ are closed under direct summands;

$(1)$ $\Sigma^{-1}\mathcal{A}\subseteq\mathcal{A}$ and $\mathcal{B}\subseteq\Sigma^{-1}\mathcal{B}$ ;

$(2)$ $\textnormal{Hom}(\Sigma^{-1}\mathcal{A},\mathcal{B})=0$ ;

$(3)$ For any object $X$ of $\mathcal{T}$ there exists a distinguished triangle

\Sigma^{-1}A\rightarrow X\rightarrow B\rightarrow A

with $A\in\mathcal{A}$ and $B\in\mathcal{B}$ .

In [7] the full subcategories $\mathcal{A}$ and $\mathcal{B}$ are denoted by $\mathcal{T}^{w\geqslant 0}$ and $\mathcal{T}^{w\leqslant 0}$ , respectively.

It is easy to see that $\mathcal{A}$ is closed under direct summands if and only if $\Sigma^{-1}\mathcal{A}$ is closed under direct summands; similarly for $\mathcal{B}$ .

One can see that by interchanging the roles of $\mathcal{X}$ and $\mathcal{Y}$ in the definition of a $t$ -structure, properties $(1)$ , $(2)$ and $(3)$ in Definition 1.3 become the corresponding properties in the definition of a co- $t$ -structure. The inclusion of condition $(0)$ in Definition 1.4 is the reason why a co- $t$ -structure is almost dual to a $t$ -structure rather than simply being its dual. We also note that properties $(2)$ and $(3)$ in Definitions 1.3 and 1.4 make $(\mathcal{X},\Sigma^{-1}\mathcal{Y})$ and $(\Sigma^{-1}\mathcal{A},\mathcal{B})$ into examples of torsion theories in the sense of [12].

The notion of a non-degenerate co- $t$ -structure can be defined in a manner analogous to that of a non-degenerate $t$ -structure.

0MU4

Definition 1.5. A co- $t$ -structure $(\mathcal{A},\mathcal{B})$ on a triangulated category $\mathcal{T}$ will be called non-degenerate if we have

\bigcap_{n\in\mathbb{Z}}\Sigma^{n}\mathcal{A}=\bigcap_{n\in\mathbb{Z}}\Sigma^{% n}\mathcal{B}=\{0\}.

Recall that a full subcategory $\mathcal{X}$ of a triangulated category $\mathcal{T}$ is said to be closed under extensions if, whenever we have a distinguished triangle

X^{\prime}\rightarrow X\rightarrow X^{\prime\prime}\rightarrow\Sigma X^{\prime}

with $X^{\prime}$ and $X^{\prime\prime}$ objects of $\mathcal{X}$ , then $X$ is also an object of $\mathcal{X}$ . We next give some elementary properties of co- $t$ -structures.

0MU5

Proposition 1.6. Let $\mathcal{T}$ be a triangulated category and suppose $(\mathcal{A},\mathcal{B})$ is a co- $t$ -structure on $\mathcal{T}$ . We have:

(i) For all objects $X$ of $\mathcal{T}$ there exists an $\Sigma^{-1}\mathcal{A}$ -precover $\alpha:\Sigma^{-1}A\rightarrow X$ .

(ii) For all objects $X$ of $\mathcal{T}$ there exists a $\mathcal{B}$ -preenvelope $\beta:X\rightarrow B$ .

(iii) We have $\Sigma^{-1}\mathcal{A}={}^{\perp}\mathcal{B}$ and $\mathcal{B}=(\Sigma^{-1}\mathcal{A})^{\perp}$ .

(iv) $\mathcal{A}$ is closed under extensions.

(v) $\mathcal{B}$ is closed under extensions.

0MU6

Proof: Properties (i) and (ii) are immediate consequences of the definition of a co- $t$ -structure: the $\Sigma^{-1}\mathcal{A}$ -precover $\alpha:\Sigma^{-1}A\rightarrow X$ and the $\mathcal{B}$ -preenvelope $\beta:X\rightarrow B$ are just the first and second morphisms in the distinguished triangle given by property $(3)$ of Definition 1.4. Property (iii) is a consequence of condition $(0)$ and the orthogonality condition $(2)$ of Definition 1.4, and properties (iv) and (v) are easy consequences of property (iii), see [12]. $\Box$

One sees in Proposition 1.6 that preenvelopes and precovers replace the truncation functors associated with $t$ -structures. In order to obtain the equalities of property (iii), and thus the fact that both halves of the co- $t$ -structure are closed under extensions, we need to assume condition $(0)$ of Definition 1.4 which says that both halves of a co- $t$ -structure are closed under direct summands.

Let $(\mathcal{X},\mathcal{Y})$ be a $t$ -structure on a triangulated category $\mathcal{T}$ . The intersection $\mathcal{H}=\mathcal{X}\cap\mathcal{Y}$ , of both halves of the $t$ -structure is called the heart of the $t$ -structure. It has the nice property that it is an abelian subcategory of $\mathcal{T}$ . In particular, the hearts of $t$ -structures provide a means of obtaining abelian categories from triangulated categories. We define an analogous notion for co- $t$ -structures.

0MU7

Definition 1.7. Let $(\mathcal{A},\mathcal{B})$ be a co- $t$ -structure for a triangulated category $\mathcal{T}$ . The intersection $\mathcal{C}=\mathcal{A}\cap\mathcal{B}$ will be called the coheart of the co- $t$ -structure.

It would be hoped that the coheart of a co- $t$ -structure on $\mathcal{T}$ would be an abelian subcategory of $\mathcal{T}$ . Unfortunately, this is not the case, see [7]. However, in this paper we present an example in which the coheart does turn out to be an abelian category.

In the next section we give some examples of co- $t$ -structures.

David Pauksztello

Original source: arXiv:0705.0102v2

1.2. t-structures and co-t-structures

1.2. $t$ -structures and co- $t$ -structures