Subsection 2.4. (0MVD) · ScalingStacks

2.4. A co- $t$ -structure obtained from a corigid object

In this paper we shall look at the structure which is induced by an object behaving like a cochain DGA. The subsequent sections of this paper are devoted to proving the following theorem, which is the cochain analogue, or dual, of Theorem 2.3.

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Theorem 2.4. Let $\mathcal{T}$ be a triangulated category with set indexed coproducts. Suppose $S$ is a compact simply connected corigid 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 co- $t$ -structure on $\mathcal{T}$ :

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

Moreover, its coheart $\mathcal{C}=\mathcal{A}\cap\mathcal{B}$ is an abelian subcategory of $\mathcal{T}$ , and the functor

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

is an equivalence of categories.

Theorem 2.4 is the natural generalisation of Example 2.2 in the same way that Theorem 2.3 is the natural generalisation of Example 2.1.

In Section 3 we shall show that given any object $M$ of $\mathcal{T}$ , there exists a morphism $\mu:M\rightarrow\bar{M}$ with $\bar{M}\in S^{\perp_{\infty}}$ such that, given any other object $N\in S^{\perp_{\infty}}$ and a morphism $M\rightarrow N$ , then this morphism factors through $\mu:M\rightarrow\bar{M}$ ,

However, the factorisation is not necessarily unique. Thus we obtain an $S^{\perp_{\infty}}$ -preenvelope. Note that in Theorem 2.4 above, $\mathcal{B}=S^{\perp_{\infty}}$ . In Section 4 we show that this $S^{\perp_{\infty}}$ -preenvelope induces a non-degenerate co- $t$ -structure on $\mathcal{T}$ , while Section 5 is dedicated to proving that the coheart of this non-degenerate co- $t$ -structure is equivalent to the module category $\textnormal{Mod}(\textnormal{End}(S)^{\textnormal{op}})$ , and hence abelian.

We now make precise what we mean by an object of a triangulated category behaving like a cochain DGA. Following the definition of an $n$ -rigid object in a triangulated category of Iyama and Yoshino in [12], we make the following definitions of an $n$ -corigid object and a corigid object.

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Definition 2.5. An object $S$ of $\mathcal{T}$ will be called $n$ -corigid if we have

\textnormal{Hom}(\Sigma^{i}S,S)=0\textnormal{ for }0<i<n.

An object $S$ of $\mathcal{T}$ will be called corigid if we have

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

Note that a DGA $R$ is a cochain DGA if and only if it is a corigid object in its derived category $\mathcal{D}(R)$ .

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Definition 2.6. Let $S$ be an object of $\mathcal{T}$ . We shall call $S$ a simply connected corigid object of $\mathcal{T}$ if it satisfies the following assumptions:

$(1)$ $S$ is corigid, that is, $\textnormal{Hom}(\Sigma^{i}S,S)=0$ for $i>0$ ;

$(2)$ $\textnormal{Hom}(S,\Sigma S)=0$ ;

$(3)$ $\textnormal{End}(S)$ is a division ring.

Note that a DGA $R$ is a simply connected cochain DGA if and only if it is a simply connected corigid object in its derived category $\mathcal{D}(R)$ .

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Remark 2.7. For technical reasons, in the cochain analogue of Theorem 2.3 we must also insist that $S$ is simply connected in the sense of Definition 2.6. This is due to the lack of symmetry in the theory of chain and cochain DGAs mentioned in the introduction: one is able to construct a theory for chain DGAs, but in order to construct a viable dual theory for cochain DGAs one has to introduce the assumption of simply connectedness; see, for example, [2].

2.4. A co-t-structure obtained from a corigid object

2.4. A co- $t$ -structure obtained from a corigid object