Section 1. (0MV6) · ScalingStacks

1. Definitions, terminology and notation

In this section we shall introduce some of the basic definitions, terminology and notation which we shall use throughout this paper. We start by recalling the concept of a preenvelope and the notation of perpendicular categories; then we give the definition of a $t$ -structure on a triangulated category and introduce the definition of a co- $t$ -structure and some of its basic properties.

Throughout this paper $\mathcal{T}$ will be a triangulated category with set indexed coproducts. We shall denote the suspension functor on $\mathcal{T}$ by $\Sigma:\mathcal{T}\rightarrow\mathcal{T}$ , and we shall write $\textnormal{Hom}(X,Y)$ instead of $\textnormal{Hom}_{\mathcal{T}}(X,Y)$ for the Hom-sets of $\mathcal{T}$ .

1.1. Preenvelopes and perpendicular categories

We recall the following definition from [8].

0MU0

Definition 1.1. Let $\mathcal{F}$ be a full subcategory of a category $\mathcal{C}$ and suppose $X$ is an object of $\mathcal{C}$ . A morphism $\phi:X\rightarrow F$ with $F\in\mathcal{F}$ is called an $\mathcal{F}$ -preenvelope if for each morphism $X\rightarrow F^{\prime}$ with $F^{\prime}\in\mathcal{F}$ there exists a morphism $F\rightarrow F^{\prime}$ making the following triangle commute.

An $\mathcal{F}$ -preenvelope is sometimes called a left $\mathcal{F}$ -approximation. We obtain the notion of an $\mathcal{F}$ -precover by dualising the definition above.

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Definition 1.2. Let $S$ be an object of a triangulated category $\mathcal{T}$ . The subcategory right $n$ -perpendicular to $S$ , denoted by $S^{\perp_{n}}$ , is given by:

S^{\perp_{n}}:=\{X\in\mathcal{T}\ |\ \textnormal{Hom}(S,\Sigma^{i}X)=0% \textnormal{ for }i=1,\ldots,n\}.

The subcategory right $\infty$ -perpendicular to $S$ , denoted by $S^{\perp_{\infty}}$ , is given by:

S^{\perp_{\infty}}:=\{X\in\mathcal{T}\ |\ \textnormal{Hom}(S,\Sigma^{i}X)=0% \textnormal{ for }i>0\}.

Similarly, one can also define the subcategories left $n$ -perpendicular and left $\infty$ -perpendicular to $S$ .

For a subcategory $\mathcal{S}$ of $\mathcal{T}$ , we define:

	$\displaystyle\mathcal{S}^{\perp}$	$\displaystyle:=$	$\displaystyle\{X\in\mathcal{T}\ \|\ \textnormal{Hom}(S,X)=0\textnormal{ for all% }S\in\mathcal{S}\},$
	$\displaystyle{}^{\perp}\mathcal{S}$	$\displaystyle:=$	$\displaystyle\{X\in\mathcal{T}\ \|\ \textnormal{Hom}(X,S)=0\textnormal{ for all% }S\in\mathcal{S}\}.$

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].

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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.

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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.

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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.

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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.

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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.

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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. Definitions, terminology and notation

1.1. Preenvelopes and perpendicular categories

1.2. t-structures and co-t-structures

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