Subsection 2.2. (0MVB) · ScalingStacks

2.2. Chain and cochain DGAs

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 $i<0$ . A cochain DGA $R$ is called simply connected if, in addition, $H^{0}(R)$ is a division ring and $H^{1}(R)=0$ .

0MU8

Example 2.1. Let $R$ be a chain DGA. Let $\mathcal{D}(R)$ be the derived category of DG $R$ -modules, see [6], and define a pair of subcategories of $\mathcal{D}(R)$ as follows:

	$\displaystyle\mathcal{X}$	$\displaystyle=$	$\displaystyle\{M\in\mathcal{D}(R)\ \|\ H^{i}(M)=0\textnormal{ for }i>0\},$
	$\displaystyle\mathcal{Y}$	$\displaystyle=$	$\displaystyle\{M\in\mathcal{D}(R)\ \|\ H^{i}(M)=0\textnormal{ for }i<0\}.$

It is easy to show that the pair $(\mathcal{X},\mathcal{Y})$ forms a $t$ -structure on $\mathcal{D}(R)$ . Again, this $t$ -structure is non-degenerate and its heart consists of DG $R$ -modules whose cohomology is concentrated in degree zero.

0MU9

Example 2.2. Let $R$ be a simply connected cochain DGA. Let $\mathcal{D}(R)$ be the derived category of DG $R$ -modules and define a pair of subcategories of $\mathcal{D}(R)$ as follows:

	$\displaystyle\mathcal{A}$	$\displaystyle=$	$\displaystyle\{M\in\mathcal{D}(R)\ \|\ H^{i}(M)=0\textnormal{ for }i<0\},$
	$\displaystyle\mathcal{B}$	$\displaystyle=$	$\displaystyle\{M\in\mathcal{D}(R)\ \|\ H^{i}(M)=0\textnormal{ for }i>0\}.$

It is easy to show that the pair $(\mathcal{A},\mathcal{B})$ forms a co- $t$ -structure on $\mathcal{D}(R)$ . As in Example 2.1, this co- $t$ -structure is non-degenerate and its coheart consists of DG $R$ -modules whose cohomology sits in degree zero.

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

David Pauksztello

Original source: arXiv:0705.0102v2