The term extension closure appears in some papers constructing t-structures on triangulated categories, for example in Section 1.2 of
Bayer, Arend; Macrì, Emanuele; Toda, Yukinobu, Bridgeland stability conditions on threefolds. I Bogomolov-Gieseker type inequalities, J. Algebr. Geom. 23, No. 1, 117-163 (2014). ZBL1306.14005.
The text is roughly as follows. Let $X$ be a smooth projective $\mathbb{C}$-variety. Let $D^b(X)$ denote the derived category of bounded complexes of coherent sheaves. Let $\mathrm{Coh}(X)\subseteq D^b(X)$ denote the heart of the natural t-structure. Given a torsion pair $\mathcal{T},\mathcal{F}\subseteq\mathrm{Coh}(X)$, we want to consider a tilting heart
$$\langle \mathcal{F}[1],\mathcal{T}\rangle\subseteq D^b(X)$$
which is called the extension closure. Here are my questions
- What is $\langle \mathcal{F}[1],\mathcal{T}\rangle$? What are the objects?
- What are extensions in $D^b(X)$? I know short exact sequences in $D^b(X)$ split and thus I do not believe extensions here mean short exact sequences as for abelian categories.
- Does the order matter? Does $\langle \mathcal{T},\mathcal{F}[1]\rangle=\langle \mathcal{F}[1],\mathcal{T}\rangle$?
Best Answer
Let $\mathcal{A}$ denote an Abelian category and consider its bounded derived category $D^b(\mathcal{A})$. In this case, there is a "standard" $t$-structure coming from truncation of complexes; its heart is $\mathcal{A}$ viewed as complexes with cohomology only in degree $0$.
Given a pair of full subcategories of $\mathcal{A}$ denoted $\mathcal{T}$ and $\mathcal{F}$, $(\mathcal{T},\mathcal{F})$ is a torsion pair if every object $X\in \mathcal{A}$ fits into an exact sequence $0 \to T\to X\to F\to 0$ and $\mathrm{Hom}(\mathcal{T},\mathcal{F}) = 0$.
With this information, we can perform a (right) tilt to get a new heart of a bounded $t$-structure on $D^b(\mathcal{A})$ defined to be the extension closure of $\mathcal{F}[1]$ and $\mathcal{T}$. An extension in a triangulated category of an object $X$ by an object $Y$ is a distinguished triangle $Y\to Z\to X$, and we also say that the object $Z$ is an extension of $X$ by $Y$.
There is another characterisation of $\langle \mathcal{F}[1],\mathcal{T}\rangle$ as those $X\in D^b(\mathcal{A})$ with $H^{-1}_{\mathcal{A}}(X) \in \mathcal{F}$ and $H^0_{\mathcal{A}}(X) \in \mathcal{T}$. It is important to note that $\langle \mathcal{F}[1],\mathcal{T}\rangle$ is not closed under application of the shift functor, and consequently is not a triangulated subcategory of $D^b(\mathcal{A})$.
I believe the role of the ordering $\langle \mathcal{F}[1],\mathcal{T}\rangle$ is to emphasize that $\mathrm{Hom}(\mathcal{F}[1],\mathcal{T}) = 0$. Indeed, given $F\in \mathcal{F}$ and $T\in \mathcal{T}$, $\mathrm{Hom}(F[1],T) = \mathrm{Ext}^{-1}_{\mathcal{A}}(F,T) = 0$.
A short synopsis of some of these things can be found here also in the context of Bridgeland Stability.