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I am away from Torsion theory in abelian category for some while. So it might be a stupid question.

The definition of a torsion pair in the category of modules is as follows:

Definition: A pair $(\mathcal T,\mathcal F)$ of full subcategories of $A-\mathrm{mod}$ is called a torsion pair if following conditions hold:

  1. $\mathrm{Hom}_{A}(M,N)=0$ for all $M \in \mathcal T, N \in \mathcal F$.
  2. $\mathrm{Hom}_{A}(M,-)|_{\mathcal F}=0 \Rightarrow M \in \mathcal T$.
  3. $\mathrm{Hom}_{A}(-,N)|_{\mathcal T}=0 \Rightarrow N \in \mathcal F$.

Condition 2) and 3) means that the pair $(\mathcal T,\mathcal F)$ is maximal for $\mathrm{Hom}_{A}(M,N)=0$.

This definition is from the book elements of representation theory of associative algebras

I found this definition is similar to the definition of t-structures in derived category. I just quote the definition from Dimca, Sheaves in topology as follows:

A t-structure on a triangulated category $\mathcal D$ consists in two strictly full subcategories: $\mathcal D^{\leq 0}, \mathcal D^{\geq 0}$ such that the following conditions hold:

  1. $\mathrm{Hom}(X,Y)=0$ if $X \in \mathcal D^{\leq 0}, Y \in \mathcal D^{\geq 1}$.
  2. $\mathcal D^{\leq 0} \subseteq \mathcal D^{\leq 1}, \mathcal D^{\geq 1} \subseteq \mathcal D^{\geq 0}$.
  3. For any $X \in \mathcal D$, there is a distinguished triangle $$A\rightarrow X\rightarrow B\rightarrow A[+1], \qquad A \in \mathcal D^{\leq 0}, B \in \mathcal D^{\geq 1}.$$

Although the axiom 3) for t-structures looks different to the axioms of torsion pairs. However, there is a proposition of torsion pairs establishing the similar formula:

Let a pair $(\mathcal T,\mathcal F)$ be a torsion pair in $A-\mathrm{mod}$, and let $M$ be an $A$-module. Then there exists a short exact sequence $$0 \rightarrow tM \rightarrow M \rightarrow M/tM \rightarrow 0, \qquad tM \in \mathcal T, M/tM \in \mathcal F,$$ where $t$ is the idempotent radical (it behaves like radical of module).

My questions

  1. Is there any relationship between these two constructions?
  2. Is there a definition of torsion theory in triangulated categories? If there exists, does it coincide with t-structures in triangulated categories?
  3. t-structures played important roles in reconstruction schemes (or go back to abelian category) from derived category. So, is torsion theory in abelian category playing similar roles? (I suspected very much, so it might be stupid.)

Thank you in advance!

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up vote 16 down vote accepted

The two notions are related in the sense that they share a common generalization, namely the notion of torsion pair on a pre-triangulated category (this term has at least two meanings, here we mean a category which has compatible left and right triangulations - it covers several cases including triangulated categories and quasi-abelian categories). The reference for this material is

A. Beligiannis and I. Reiten: ''Homological and Homotopical Aspects of Torsion Theories''

which is available from Beligiannis' homepage. In fact one can take the analogy further and consider the analogy between TTF-triples on an abelian category and recollement of triangulated categories.

There is also another connection given by tilting theory. Suppose that $(\mathcal{T},\mathcal{F})$ is a torsion pair on an abelian category $\mathbf{A}$. Then we can obtain a t-structure on $D= D^b(\mathbf{A})$ by setting $D^{\leq 0} = \{ X\in D \; \vert \; H^i(X)=0 \; \text{for} \; i>0, H^0(X)\in \mathcal{T} \}$ and $D^{\geq 0} = \{ X\in D \; \vert \; H^i(X)=0 \; \text{for} \; i<-1, H^{-1}(X)\in \mathcal{F} \}$
For more information on this (in particular for some characterizations of when taking the derived category of the heart obtain from this t-structure is equivalent to $D$) one can see "Tilting in Abelian categories and quasitilted algebras" By Dieter Happel, Idun Reiten, Sverre O. Smalø.

I hope that at least goes some of the way toward answering (1) and (2).

As far as (3) is concerned I am not completely sure what to say. Certainly one can reconstruct a quasi-compact quasi-separated scheme from its derived category using the tensor structure, and if the scheme is particularly nice one can use the Serre functor. I am not aware of (or have forgotten if I knew) a way of reconstructing a scheme via t-structures (I guess one can use strictly localizaing subcategories which are particularly nice t-structures or take the heart of the standard one). One certainly can't just look at all t-structures - even for $D(\mathbb{Z})$ there is a proper class of t-structures.
In the abelian case the closest thing I can think of is taking the spectrum of indecomposable injectives. This is not directly torsion theoretic but it is true that injectives control hereditary torsion theories in the sense that every hereditary torsion theory in a Grothendieck abelian category has as its torsion class the left orthogonal to some injective object.

A particularly nice special case when one can really make the connection precise is the following (due to Krause). Suppose that $\mathbf{A}$ is a locally coherent Grothendieck abelian category i.e., it is a Grothendieck abelian category with a generating set of finitely presented objects and the finitely presented objects form an abelian subcategory. Then one can topologize the spectrum of indecomposable injectives in such a way that there is a bijection between hereditary torsion theories of finite type (those for which the right adjoint to the inclusion also commutes with filtered colimits) and closed subsets of the spectrum.

One last thought for the moment - although one can think of t-structures and torsion theories on abelian categories as common specializations of one more general definition the analogy can be misleading. However, there is a reasonably good analogy between hereditary torsion theories of finite type and smashing subcategories which can be made precise (again this is due to Krause). The heart of this is that every smashing subcategory of a compactly generated triangulated category is generated by an ideal of maps between compact objects. Corresponding to such an ideal there is a hereditary torsion theory of finite type in the category of additive presheaves of abelian groups on the compact objects. Something you may find particularly interesting about this (I certainly do) is that it links the theory of smashing subcategories (and the telescope conjecture) to the spectrum of indecomposable injectives in a nice abelian category.

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Thank you very much. What you mentioned about the injective spectrum can control hereditary torsion theories is interesting! Maybe I will ask a new question about this process – Shizhuo Zhang Feb 8 '10 at 13:33
I will wait for more answers to this question, but your answer is satisfactory enough – Shizhuo Zhang Feb 8 '10 at 13:34
What you mentioned about the hereditary torsion theories are corresponding to the closed subset of injective spectrum is also mentioned in several preprints of Rosenberg but in another language(coreflective categories). He has a spectrum which in general more general than Gabriel spectrum but coincide with it when the category has Gabriel-Krull dimension(in particular, locally noetherian abelian category). – Shizhuo Zhang Feb 9 '10 at 0:57
I asked this question to Rosenberg today, he told me that reconstruction theorem for derived category (Bondal-Orlov) is to produce t-structures. I will post this later in this page. – Shizhuo Zhang Feb 9 '10 at 0:58
I'll have to look at some of Rosenberg's preprints. I'd be interested in the connection between t-structures and Bondal-Orlov reconstruction. It has been a while since I looked at the paper but if I understood it and remember it correctly the t-structures were used for studying the autoequivalences. The actual reconstruction only uses the suspended structure and the existence of a Serre functor. – Greg Stevenson Feb 9 '10 at 1:11

As I understand it, all torsion classes correspond to t-structures in the way described by Greg, but there is almost always more t-structures than torsion classes (even taking into account the shifts). I think t-structures are closer to some kind of filtration on the abelian category. There's a paper on stability conditions (Gorodentsev, A. L.; Kuleshov, S. A.; Rudakov, A. N. $t$-stabilities and $t$-structures on triangulated categories. (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 68 (2004), no. 4, 117--150; translation in Izv. Math. 68 (2004), no. 4, 749--781) that talks about this a bit.

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I think that this is a helpful point of view and that the Gorodentsev, Kuleshov, Rudakov paper is definitely worth looking at (from memory they don't mention connections to abelian categories but I think I remember figuring out that one could get the finest t-stablities for the projective line by lifting from similar things on the category of coherent sheaves which are also minimal - but here everything is bounded and the category is hereditary so one expects things to work nicely!) Also I should point out that it was Don who showed there are a proper class of t-structures on $D(\mathbb{Z})$ – Greg Stevenson Feb 10 '10 at 20:33
Thank you for pointing out the paper t-stabilities and t-structures on triangulated categories – Shizhuo Zhang Feb 15 '10 at 12:11

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