ScalingStacks

The following example is taken from [7]. Let $𝒞$ be an additive category and let $𝒦(𝒞)$ be its homotopy category. We claim that the following pair of full subcategories of $𝒦(𝒞)$ forms a co-$t$-structure on $𝒯=𝒦(𝒞)$. Let

	$𝒜$	$=$
	$ℬ$	$=$	$\{\text{complexes in }𝒦(𝒞)\text{ isomorphic to complexes }C|C^i=0\text{ for }i>0\}.$

It is clear that $𝒜$ and $ℬ$ are closed under direct summands, and that ${\mathrm{\Sigma }}^{-1}𝒜\subseteq 𝒜$ and $ℬ\subseteq {\mathrm{\Sigma }}^{-1}ℬ$. It is also clear that $\text{Hom}({\mathrm{\Sigma }}^{-1}𝒜,ℬ)=0$. We need to show that property $(3)$ of Definition 1.4 holds. Suppose $X$ is an object of $𝒦(𝒞)$:

We obtain the following semi-split short exact sequence of complexes:

which gives us a distinguished triangle

$${\mathrm{\Sigma }}^{-1}A\to X\to B\to A$$

in $𝒦(𝒞)$. Hence $(𝒜,ℬ)$ is a co-$t$-structure on $𝒯=𝒦(𝒞)$. Moreover, it is non-degenerate and its coheart is just the class of complexes sitting in degree zero.

Recall from the introduction that a DGA $R$ is called a chain DGA if $H^i(R)=0$ for all $i>0$; similarly, a DGA $R$ is called a cochain DGA if $H^i(R)=0$ for all . A cochain DGA $R$ is called simply connected if, in addition, $H^0(R)$ is a division ring and $H^1(R)=0$.

Simply connected cochain DGAs arise naturally in algebraic topology as the cochain algebras of simply connected CW-complexes, see [9] and [17], for example.

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

$$H^i(M)={\text{Hom}}_{𝒟(R)}(R,{\mathrm{\Sigma }}^{i}M)\text{ for }i\in \mathbb{Z}.$$

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

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

$${\text{Hom}}_{𝒯}(S,{\mathrm{\Sigma }}^{i}S)=0\text{ for }i>0.$$

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

	$𝒳$	$=$	$\{X\in 𝒯|{\text{Hom}}_{𝒯}(S,{\mathrm{\Sigma }}^{i}X)=0\text{ for }i>0\},$
	$𝒴$	$=$

The suitably nice conditions we must place on $S$ to obtain this $t$-structure are that $S$ must be a compact object of $𝒯$ and the set $\{{\mathrm{\Sigma }}^{i}S|i\in \mathbb{Z}\}$ must be a generating set for $𝒯$. 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 $𝒯$ with set indexed coproducts is compact if the functor $\text{Hom}(S,-)$ commutes with set indexed coproducts, that is the canonical map is an isomorphism

$$\text{Hom}(S,\coprod _{i\in I}{X}_i)\cong \coprod _{i\in I}\text{Hom}(S,X_i)$$

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

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

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

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

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.

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 $𝒯$, there exists a morphism $\mu :M\to \overline{M}$ with $\overline{M}\in S^{{⟂}_{\mathrm{\infty }}}$ such that, given any other object $N\in S^{{⟂}_{\mathrm{\infty }}}$ and a morphism $M\to N$, then this morphism factors through $\mu :M\to \overline{M}$,

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

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

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 $𝒟(R)$.