Section 4. (0MVF) · ScalingStacks

4. A co- $t$ -structure induced by a compact simply connected corigid object

The aim of this section is to give a proof of the following theorem, which is the first half of Theorem 2.4.

Theorem 4.1. Let $\mathcal{T}$ be a triangulated category with set indexed coproducts. Suppose $S$ is a compact simply connected corigid object of $\mathcal{T}$ . Further assume that $\{\Sigma^{i}S\ |\ i\in\mathbb{Z}\}$ is a generating set in $\mathcal{T}$ . Then the following forms a non-degenerate co- $t$ -structure on $\mathcal{T}$ :

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

Note that $\mathcal{B}=S^{\perp_{\infty}}$ .

In order to prove this we need a lemma analogous to Lemma 3.3. This is an immediate consequence of the next lemma.

0MUQ

Lemma 4.2. Let $S$ be a compact object of a triangulated category $\mathcal{T}$ and suppose we have a sequence of objects and morphisms

X_{0}\stackrel{{\scriptstyle\alpha_{0}}}{{\longrightarrow}}X_{1}\stackrel{{% \scriptstyle\alpha_{1}}}{{\longrightarrow}}X_{2}\stackrel{{\scriptstyle\alpha_% {2}}}{{\longrightarrow}}X_{3}\stackrel{{\scriptstyle\alpha_{3}}}{{% \longrightarrow}}\cdots

such that $\textnormal{Hom}(S,\alpha_{n}):\textnormal{Hom}(S,X_{n})\rightarrow\textnormal% {Hom}(S,X_{n+1})$ is an isomorphism for each $n\geqslant 0$ . Then $\textnormal{Hom}(S,X_{0})\cong\textnormal{Hom}(S,\textnormal{hocolim}(X_{n}))$ .

0MUR

Proof: It is well-known that the filtered colimit, $\textnormal{colim}\,\textnormal{Hom}(S,X_{n})$ , is isomorphic to $\textnormal{Hom}(S,X_{0})$ . The assertion now follows by Lemma 3.5. $\Box$

0MUS

Lemma 4.3. Under the assumptions of Proposition 3.6 we have that

\textnormal{Hom}(S,\Sigma^{i}\mu):\textnormal{Hom}(S,\Sigma^{i}M)\rightarrow% \textnormal{Hom}(S,\Sigma^{i}\bar{M})

is an isomorphism for $i<1$ .

0MUT

Proof: In Lemma 3.3, $\textnormal{Hom}(S,\Sigma^{i}\mu):\textnormal{Hom}(S,\Sigma^{i}M_{k})% \rightarrow\textnormal{Hom}(S,\Sigma^{i}M_{k+1})$ is an isomorphism for $i<1$ . Now apply Lemma 4.2. $\Box$

Proof of Theorem 4.1: Conditions $(0)$ and $(1)$ of the definition of a co- $t$ -structure are clear.

In order to show $(2)$ assume $X\in\Sigma^{-1}A$ and $Y\in\mathcal{B}$ . Recall that $\mathcal{B}=S^{\perp_{\infty}}$ . By Proposition 3.6 there exists an $S^{\perp_{\infty}}$ -preenvelope $\mu:X\rightarrow\bar{X}$ , that is, we have a surjection of Hom spaces

\textnormal{Hom}(\bar{X},Y)\twoheadrightarrow\textnormal{Hom}(X,Y).

It is therefore sufficient to show $\textnormal{Hom}(\bar{X},Y)=0$ .

We have the following isomorphism of Hom-spaces and trivial Hom-spaces:

$\displaystyle\textnormal{Hom}(S,\Sigma^{i}X)$	$\displaystyle\cong$	$\displaystyle\textnormal{Hom}(S,\Sigma^{i}\bar{X})\textnormal{ for all }i<1% \textnormal{\ \ \ (Lemma \ref{corollary1})}$
$\displaystyle\textnormal{Hom}(S,\Sigma^{i}X)$	$\displaystyle=$	$\displaystyle 0\textnormal{ for all }i<1\textnormal{\ \ \ (since $X\in\Sigma^{% -1}\mathcal{A}$)}$
$\displaystyle\textnormal{Hom}(S,\Sigma^{i}\bar{X})$	$\displaystyle=$	$\displaystyle 0\textnormal{ for all }i>0\textnormal{\ \ \ (since $\bar{X}\in% \mathcal{B}$)}.$

It follows that $\textnormal{Hom}(S,\Sigma^{i}\bar{X})=0$ for all $i\in\mathbb{Z}$ . The assumption that $\{\Sigma^{i}S\ |\ i\in\mathbb{Z}\}$ is a generating set in $\mathcal{T}$ implies that $\bar{X}=0$ . Thus $\textnormal{Hom}(\bar{X},Y)=0$ and we see that $\textnormal{Hom}(X,Y)=0$ . Hence $\textnormal{Hom}(\Sigma^{-1}\mathcal{A},\mathcal{B})=0$ .

We next show condition $(3)$ . Suppose $X$ is an object of $\mathcal{T}$ . By Proposition 3.6 there is an $S^{\perp_{\infty}}$ -preenvelope $\mu:X\rightarrow\bar{X}$ . Write $B=\bar{X}$ and extend the morphism $\mu:X\rightarrow B$ to a distinguished triangle:

(4.1)

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

We claim that $\textnormal{Hom}(S,\Sigma^{i}A)=0$ for $i<0$ . Consider the following long exact sequence obtained from (4.1):

\textnormal{Hom}(S,\Sigma^{i-1}A)\rightarrow\textnormal{Hom}(S,\Sigma^{i}X)% \rightarrow\textnormal{Hom}(S,\Sigma^{i}B)\rightarrow\textnormal{Hom}(S,\Sigma% ^{i}A).

Now, by Lemma 4.3, we see that $\textnormal{Hom}(S,\Sigma^{i}X)\rightarrow\textnormal{Hom}(S,\Sigma^{i}B)$ is an isomorphism for all $i<1$ . Hence $\textnormal{Hom}(S,\Sigma^{i}A)=0$ for all $i<0$ and $A\in\mathcal{A}$ . Hence the distinguished triangle in (4.1) above gives us the required distinguished triangle.

It is clear that $\cap_{n\in\mathbb{Z}}\Sigma^{n}\mathcal{A}=\cap_{n\in\mathbb{Z}}\Sigma^{n}% \mathcal{B}=\{0\}$ because $\{\Sigma^{i}S\ |\ i\in\mathbb{Z}\}$ is a generating set for $\mathcal{T}$ . Hence $(\mathcal{A},\mathcal{B})$ is a non-degenerate co- $t$ -structure on $\mathcal{T}$ . $\Box$

0MUU

Remark 4.4. In [7], a co- $t$ -structure $(\mathcal{A},\mathcal{B})$ is called right adjacent to a $t$ -structure $(\mathcal{X},\mathcal{Y})$ if $\mathcal{A}=\mathcal{Y}$ . By [4], the full subcategory $\mathcal{A}$ of $\mathcal{T}$ occurs in a $t$ -structure $(\mathcal{X},\mathcal{A})$ on $\mathcal{T}$ , where

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

Therefore, the co- $t$ -structure on $\mathcal{T}$ obtained in Theorem 4.1 is right adjacent to the $t$ -structure $(\mathcal{X},\mathcal{A})$ .

David Pauksztello

Original source: arXiv:0705.0102v2

4. A co-t-structure induced by a compact simply connected corigid object

4. A co- $t$ -structure induced by a compact simply connected corigid object