Subsection 2.3. (0MVC) · ScalingStacks

2.3. A $t$ -structure obtained from a rigid object

Example 2.1 can be abstracted to an arbitrary triangulated category by looking at objects behaving like chain DGAs. Let $R$ be a chain DGA, and recall from the introduction that, given a DG $R$ -module $M$ we have

H^{i}(M)=\textnormal{Hom}_{\mathcal{D}(R)}(R,\Sigma^{i}M)\textnormal{ for }i% \in\mathbb{Z}.

Thus, $R$ being a chain DGA means that $\textnormal{Hom}_{\mathcal{D}(R)}(R,\Sigma^{i}R)=0$ for $i>0$ .

Now let $\mathcal{T}$ be an arbitrary triangulated category with set indexed coproducts. In the introduction, an object $S$ of $\mathcal{T}$ was called rigid if we had

\textnormal{Hom}_{\mathcal{T}}(S,\Sigma^{i}S)=0\textnormal{ for }i>0.

Hence, a DGA $R$ is a chain DGA if an only if it is a rigid object in its derived category $\mathcal{D}(R)$ . If we replace the chain DGA $R$ with some suitably nice rigid object $S$ of $\mathcal{T}$ , the following is a candidate for a $t$ -structure on $\mathcal{T}$ :

	$\displaystyle\mathcal{X}$	$\displaystyle=$	$\displaystyle\{X\in\mathcal{T}\ \|\ \textnormal{Hom}_{\mathcal{T}}(S,\Sigma^{i}% X)=0\textnormal{ for }i>0\},$
	$\displaystyle\mathcal{Y}$	$\displaystyle=$	$\displaystyle\{X\in\mathcal{T}\ \|\ \textnormal{Hom}_{\mathcal{T}}(S,\Sigma^{i}% X)=0\textnormal{ for }i<0\}.$

The suitably nice conditions we must place on $S$ to obtain this $t$ -structure are that $S$ must be a compact object of $\mathcal{T}$ and the set $\{\Sigma^{i}S\ |\ i\in\mathbb{Z}\}$ must be a generating set for $\mathcal{T}$ . Before stating the theorem in full, we recall the notions of a compact object and a generating set.

An object $S$ in a triangulated category $\mathcal{T}$ with set indexed coproducts is compact if the functor $\textnormal{Hom}(S,-)$ commutes with set indexed coproducts, that is the canonical map is an isomorphism

\textnormal{Hom}(S,\coprod_{i\in I}X_{i})\cong\coprod_{i\in I}\textnormal{Hom}% (S,X_{i})

for all families of objects $\{X_{i}\}_{i\in I}$ of $\mathcal{T}$ indexed by a set $I$ ; see [15] and [16]. A DGA $R$ is trivially a compact object of $\mathcal{D}(R)$ .

A set of objects $\mathcal{G}$ in a triangulated category $\mathcal{T}$ is called a generating set for $\mathcal{T}$ if given any object $X$ of $\mathcal{T}$ with $\textnormal{Hom}(G,X)=0$ for all objects $G$ of $\mathcal{G}$ , we have $X=0$ .

Example 2.1 is a special case of the following theorem of Hoshino, Kato and Miyachi, which appears in [11].

0MUA

Theorem 2.3 ([11], Theorem 1.3). Let $\mathcal{T}$ be a triangulated category with set indexed coproducts. Suppose $S$ is a compact rigid object of $\mathcal{T}$ and assume that $\{\Sigma^{i}S\ |\ i\in\mathbb{Z}\}$ is a generating set for $\mathcal{T}$ . Then the following forms a non-degenerate $t$ -structure on $\mathcal{T}$ :

	$\displaystyle\mathcal{X}$	$\displaystyle=$	$\displaystyle\{X\in\mathcal{T}\ \|\ \textnormal{Hom}_{\mathcal{T}}(S,\Sigma^{i}% X)=0\textnormal{ for }i>0\},$
	$\displaystyle\mathcal{Y}$	$\displaystyle=$	$\displaystyle\{X\in\mathcal{T}\ \|\ \textnormal{Hom}_{\mathcal{T}}(S,\Sigma^{i}% X)=0\textnormal{ for }i<0\}.$

Moreover, its heart $\mathcal{H}=\mathcal{X}\cap\mathcal{Y}$ is an admissible abelian subcategory of $\mathcal{T}$ in the sense of [3], and the functor

\textnormal{Hom}(S,-):\mathcal{H}\rightarrow\textnormal{Mod}(\textnormal{End}(% S)^{\textnormal{op}})

is an equivalence of categories.

The $t$ -structure is induced as follows: suppose $X$ is an object of $\mathcal{T}$ , a morphism $\alpha:X\rightarrow Y$ with $Y\in\mathcal{Y}$ is constructed such that, given any other object $Y^{\prime}\in\mathcal{Y}$ and a morphism $X\rightarrow Y^{\prime}$ , then this morphism factors uniquely through $\alpha:X\rightarrow Y$ ,

David Pauksztello

Original source: arXiv:0705.0102v2

2.3. A t-structure obtained from a rigid object

2.3. A $t$ -structure obtained from a rigid object