# t-structures on the derived category of finitely generated abelian groups

is it possible to explicitly parametrise all the t-structures on the derived category of finitely generated abelian groups?

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I think that the answer to your question is 'yes'.:) Inside you category is the direct sum of derived categories of finitely generated $p$-groups; the localization is the derived category of finitely generated $\mathbb{Q}$-vector spaces. It seems that this information should yield a way to answer your question.:) – Mikhail Bondarko Feb 19 '11 at 3:08

I guess the answer is the following. Take arbitrary subset $S$ of the set of all prime numbers. Let $A_S = <Z,\{Z/pZ\}_{p \not\in S},\{Z/pZ[-1]\}_{p \in S}>$. Then $A_S$ is a heart of the t-structure which is obtained by a simple tilting from the standard one. The claim is that any bounded t-structure is a shift of one of those.

The main part of the proof is to show that all objects $Z/pZ$ and $Z$ are pure in any t-structure. For this use the fact that $Z$-modules have homological dimension 1 and that the objects from the heart of a t-structure don't have negative $Ext$'s between.

EDIT: Let us show that $Z/pZ$ is pure. By shifting the t-structure we can assume that $Z/pZ \in D^{\le 0}$, but $Z/\pZ \not\in D^{\le 1}$. Then consider the triangle $$Z/pZ \to H^0(Z/pZ) \to \tau^{\le -1}(Z/pZ)[1].$$ Recall that any object in $D^b(Z)$ is a direct sum of $Z[i]$ and $Z/q^r Z[j]$ with $q$ prime. If there is a summand $M$ in $H^0(Z/pZ)$ such that $Hom(Z/pZ,M) = 0$ then $M$ is also a summand of $\tau^{\le -1}(Z/pZ)[1]$, hence $M \in D^{\le -1}[1] = D^{\le -2}$ and simultaneously $M \in D^0$ (since both subcategories are closed w.r.t. taking direct summands), so there is a nontrivial $Hom$ from $D^{\le -2}$ to $D^0$ which is impossible.

So, all summands in $H^0(Z/pZ)$ should have a nontrivial $Hom$ from $Z/pZ$ into them. Thus we can only have $Z[1]$, $Z/p^rZ$ or $Z/p^rZ[1]$. Assume $H^0(Z/pZ) = Z^a[1] + \oplus_i Z/p^{r_i}Z + \oplus_j Z/p^{s_j}Z[1]$ and the above triangle has form $$Z/pZ \to Z^a[1] + \oplus_i Z/p^{r_i}Z + \oplus_j Z/p^{s_j}Z[1] \to \tau^{\le -1}(Z/pZ)[1].$$ Now if we look at it in the standard t-structure we shall see that the third term has a $p$-torsion cohomology in degree 0 and a cohomology in degree $-1$ which is a sum of $Z^a$ and a $p$-torsion. Since any complex in $D(Z)$ is a sum of its cohomology, we conclude that $\tau^{\le -1}(Z/pZ)$ is a sum of $Z^a$, $p$-torsion and $p$-torsion shifted by $-1$. Once again, we cannot have same summands in $D^0$ and $D^{\le -1}$, hence the only possibility is that $H^0(Z/pZ) = Z/pZ$, so it is pure.

A similar argument shows that $Z$ is also pure.

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Many thanks for your response. If you could add some more details to the second paragraph, I would be very grateful. Many thanks – SGP Feb 22 '11 at 22:27
@SGP: Detail added, hope now its ok. – Sasha Feb 23 '11 at 8:21
Thanks a lot for all the details! The classification result is very beautiful and satisfying! – SGP Feb 24 '11 at 23:45

The question and Sasha's answer can be generalized to a Noetherian ring $R$. The parametrization is in terms of functions from the integers to a specialization closed subset of $Spec R$. I do not have access to precise references at the moment, but see the paper 'Invariants of t-structures and classiﬁcation of nullity classes' by Don Stanley and further references therein.

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There is a difference between constructing t-structure in small categories (like the derived category of finitely generated modules) and big categories (like the derived category of all modules). The second is much simpler since you can always construct an adjoint functor as a certain limit. Are you sure that Don Stanley works with a small category? For $Z$ it is very helpful to use the fact that homological dimension is 1. – Sasha Feb 20 '11 at 17:47
I agree that constructing t-structures in big categories is easier than in small categories. I have not read the Stanley paper carefully, but having looked it over again, I see that he treats both big and small categories. The main result about $D^{b}_{fg}(R)$, the bounded derived category of finitely generated modules for $R$ Noetherian, commutative, and finite dimensional, is Theorem 5.6 in the published version. This gives a complete invariant of t-structures in terms of perversity functions. When $R$ has a dualizing complex, then Stanley gives references for a converse: – Chris Brav Feb 21 '11 at 9:46
A perversity function comes from such a t-structure if it is monotone and comonotone. The reference is to the paper of Bezrukavnikov on perverse coherent sheaves. It is possible I've misunderstood the point that Sasha is making, however, and I'd be grateful for clarification. – Chris Brav Feb 21 '11 at 9:48
Many thanks for the references! While Sasha's response addresses my particular question, your response is also very useful to me: I did intend to follow my question with another as to what happens for other varieties/schemes. Thanks to your references, I do not need to! – SGP Feb 22 '11 at 22:26
An alternate approach by Jeremías, Saorín at myself: dx.doi.org/10.1016/j.jalgebra.2010.04.023 (also arxiv.org/abs/0905.2063). – Leo Alonso Feb 23 '11 at 10:59