When is the derived category of representations of a finite poset equivalent to its opposite?

2012-09-26T19:32:50Z

If I have a finite partially ordered set $K$, I can look at its derived category of finite dimensional representations $D(K)$. Note that $D(K^{op}) \simeq D(K)^{op}$ by linear duality.

But when do we have an equivalence $D(K) \simeq D(K^{op})$?

The kind of thing I have in mind is in this paper of Justin Curry: http://www.math.upenn.edu/~jucurry/papers/co_sheaf_dereq.pdf

When the poset comes from a finite cell complex, there is such a duality which interchanges the standard injectives with skyscrapers.

Does this depend only on the poset, and can this always be done?

Answer by David Treumann for When is the derived category of representations of a finite poset equivalent to its opposite?

2012-09-26T23:50:41Z

I don't know a very general answer. Your duality on cell complexes resembles Verdier duality and has a local nature, but some of these equivalences aren't like that. E.g. $K = (0 < 1 < 2)$ and $K^{op} = (2 < 1 < 0)$ are isomorphic as posets and so we get $D(K) = D(K^{op})$ from that, but I am pretty sure $K$ doesn't have a Verdier dualizing sheaf.

But here is an answer to a narrower question: when is there an equivalence $D(K) = D(K^{op})$ that is similar to the one on regular cell complexes? I mean an equivalence that is Verdier-like in the sense that it takes the standard injectives on $K$ to the skyscrapers on $K^{op}$, up to a shift.

For each $x \in K$, define $J_x:K \to \mathrm{Mod}_k$ by $$ J_x(w) = \begin{array}{cc} k & \text{if $w \leq x$}\\ 0 & \text{otherwise} \end{array} $$ These are the indecomposable injective objects in the abelian category of functors $K \to \mathrm{Mod}_k$.

The $J_x$ have a simple Hom pattern: $\mathrm{Hom}(J_x,J_y) = k$ if $x \geq y$. All other Homs (and Exts) vanish. (They form an "exceptional collection" in $D(K)$.)

The simple objects are the skyscrapers $\delta_x$. You can compute the Homs and Exts between $\delta_x$ and $\delta_y$ by writing down an injective resolution of $\delta_y$ whose $p$th term is $$ \bigoplus_{y = y_0 \geq y_1 \geq y_2 \cdots \geq y_p} J_{y_p} $$ The differentials have degree $+1$. Then $\mathrm{Hom}(\delta_x,\text{that injective resolution})$ is a complex whose $p$th term is $$ \bigoplus_{y = y_0 \geq y_1 \geq y_2 \cdots \geq y_p = x} k $$ This is the cochain complex that computes something like the relative cohomology $H^*(N,\partial N)$ where $N$ is the nerve of the interval $ \{p \in K \mid x \leq p \leq y\} $ and $\partial N$ is the subcomplex of simplices that don't contain the edge $[x < y]$. So a necessary condition is for this $\partial N$ to be a homology sphere.

I think this condition is sufficient too, except you have to worry a little bit about the dimensions of those spheres. You have to be able to choose integers $d(x)$ for each $x \in K$ so that the dimension of that homology sphere is $d(y) - d(x)$ or something.

Somebody once told me that this condition on the intervals in posets has a standard name, maybe "Gorenstein star posets" but I am not sure I am remembering that right.

When is the derived category of representations of a finite poset equivalent to its opposite? - MathOverflow

When is the derived category of representations of a finite poset equivalent to its opposite?

Answer by David Treumann for When is the derived category of representations of a finite poset equivalent to its opposite?