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Toën derived Hall numbers have the higher-Ext orbit formula

Statement

Let $x, y, z$ be perfect right dg $T$ -modules for a locally finite dg-enhancement $T$ . Define

[x, z]_{y} : = {f \in {Hom}_{D (T)} (x, z) : Cone (f) ≅ y}

and let $Aut (x)$ act on this finite set by precomposition. For $f \in [x, z]_{y}$ , write

Aut (f / z) : = {a \in Aut (x) : f \circ a = f} .

Then the coefficient in Toën derived Hall algebra by homotopy-cofiber convolution is

g_{x, y}^{z} = \sum_{[f] \in [x, z]_{y} / Aut (x)} \frac{1}{∣ Aut (f / z) ∣} \prod_{i > 0} ∣ {Ext}^{- i} (x, z) ∣^{(- 1)^{i}} ∣ {Ext}^{- i} (x, x) ∣^{(- 1)^{i + 1}}

and equivalently

g_{x, y}^{z} = \frac{∣ [x, z]_{y} ∣ \prod_{i > 0} ∣ {Ext}^{- i} (x, z) ∣^{(- 1)^{i}}}{∣ Aut (x) ∣ \prod_{i > 0} ∣ {Ext}^{- i} (x, x) ∣^{(- 1)^{i}}} .

All products are finite because perfect modules have finite total Ext.

Facts & Assumptions

Given: A finite field $k = F_{q}$ , a locally finite dg-enhancement $T$ , perfect modules $x, y, z$ , and the coefficient $g_{x, y}^{z}$ from Toën derived Hall algebra by homotopy-cofiber convolution.

[F1]

The coefficient $g_{x, y}^{z}$ is the weighted homotopy cardinality of the fiber of the target map over $z$ , restricted to arrows with source isomorphic to $x$ and cofiber isomorphic to $y$ (Toën derived Hall algebra by homotopy-cofiber convolution).

[F2]

The homotopy type of this restricted fiber is locally finite; its connected components are represented by arrows into $z$ (Derived Hall moduli types are locally finite and the input span is proper).

[F3]

For perfect modules, every total Ext space is finite-dimensional over $k$ and vanishes outside a finite range (Perfect dg modules have finite total Ext).

[F4]

For a component represented by $f : x \to z$ in the target fiber, its homotopy weight is $∣ Aut (f / z) ∣^{- 1} \prod_{i > 0} ∣ {Ext}^{- i} (x, z) ∣^{(- 1)^{i}} ∣ {Ext}^{- i} (x, x) ∣^{(- 1)^{i + 1}}$ (Mapping-space homotopy groups give the derived Hall stabilizer weight).

Proof

1.1F1F2chooseconstruct

Fix a representative $x$ . The fiber in [F1] fixes the target $z$ and restricts the source and cofiber to the isomorphism classes of $x$ and $y$ ; it retains no chosen equivalence of the source to $x$ or of the cofiber to $y$ . For an arrow $x^{'} \to z$ in that restriction, choose a source equivalence $x ≃ x^{'}$ to express its derived morphism as $f : x \to z$ with cofiber isomorphic to $y$ . A different source identification precomposes $f$ by an element of $Aut (x)$ . Two such representatives lie in the same component precisely when there is a source equivalence over $z$ up to homotopy: after the identifications this says $f^{'} = f \circ a$ in the derived category for some $a \in Aut (x)$ (using its inverse if the equivalence is oriented oppositely). Conversely that equality gives a homotopy between the maps and hence an equivalence of arrows in the target homotopy fiber. The cofiber condition restricts components and adds no framing. Thus the component set is exactly $[x, z]_{y} / Aut (x)$ . This identifies only components; their higher homotopy and stabilizer weights are retained in the following weight calculation.

1.2F1F2F4

By [F4], the weight of the component represented by $f$ is its stabilizer reciprocal times the higher-Ext factor displayed in the first formula. This uses the homotopy exact sequence in [F4], including its low-degree stabilizer tail; in particular the stabilizer is not the component set of the mapping-space fiber.

2.1F1step 1.1step 1.2

Summing the component weights from step 1.2 over the component set in step 1.1 gives the first displayed formula by [F1].

3.1F1F3step 2.1algebra∎

For a finite group $G$ acting on a finite set $S$ , each orbit of $s$ has size $∣ G ∣ / ∣ Stab (s) ∣$ , so $∣ S ∣ / ∣ G ∣ = \sum_{[s] \in S / G} 1 / ∣ Stab (s) ∣$ . Apply this with $G = Aut (x)$ and $S = [x, z]_{y}$ , then collect the common Ext factor in step 2.1. This gives the second formula.

Depends on

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Sources

Bertrand Toën, Derived Hall algebras, arXiv:math/0501343v4, §5, Proposition 5.1 (standard reference, not scraped)
Olivier Schiffmann, Lectures on Hall algebras, arXiv:math/0611617v2, §5.6 (standard reference, not scraped)