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The opposite untwisted Hall algebra of a hereditary heart embeds in the derived Hall algebra

Statement

Let $T$ be a locally finite dg-enhancement over $F_{q}$ , and let $D = Perf (T)$ . Suppose $D$ has a bounded t-structure whose heart $A$ is a small finitary hereditary abelian category, and suppose the inclusion induces an exact equivalence $D^{b} (A) ≃ D$ under which short exact sequences in $A$ correspond to distinguished triangles with all three terms in the heart. Here hereditary means cohomological dimension at most one: ${Hom}_{D^{b} (A)} (M, N [i]) = 0$ for all $M, N \in A$ and $i > 1$ .

The untwisted Hall product The untwisted Hall algebra of a finitary abelian category uses the convention that $[M] *_{A} [N]$ counts subobjects isomorphic to $N$ with quotient $M$ . With this convention, the basis map

Φ : H_{A, Q} ⟶ D H (T), [M] ⟼ χ_{M}

is an injective algebra anti-homomorphism:

Φ ([M] *_{A} [N]) = Φ ([N]) *_{D H} Φ ([M]) .

Equivalently, it is an algebra embedding $H_{A, Q}^{o p} ↪ D H (T)$ . Its image is the subalgebra spanned by the characteristic functions of heart objects.

Facts & Assumptions

Given: The dg-enhancement, the bounded t-structure and the hereditary heart specified in the statement.

[F1]

A t-structure has $Hom (D^{\leq 0}, D^{\geq 1}) = 0$ ; its heart is $D^{\leq 0} \cap D^{\geq 0}$ , and boundedness is defined using the truncation subcategories (Canonical t structure on a derived category, Truncation functors, t-cohomology, amplitude and bounded t-structures).

[F2]

The derived Hall number $g_{x, y}^{z}$ counts arrows $x \to z$ with homotopy cofiber $y$ , with the higher-Ext factor in Toën derived Hall numbers have the higher-Ext orbit formula.

[F3]

The ordinary product is $[M] *_{A} [N] = \sum_{L} g_{M, N}^{L} [L]$ , where $g_{M, N}^{L}$ counts subobjects $U \subseteq L$ with $U ≅ N$ and $L / U ≅ M$ (The untwisted Hall algebra of a finitary abelian category).

[F4]

This ordinary product is well-defined and associative with unit $[0]$ (Small finitary abelian categories for Hall multiplication, Untwisted Hall structure constants and products are finite, The untwisted Hall product is associative and unital).

[A1]

The heart inclusion has the exact realization stated in the hypotheses: a triangle $x \to z \to y \to x [1]$ with $x, y \in A$ has $z \in A$ and is the triangle of a short exact sequence $0 \to x \to z \to y \to 0$ .

Proof

1.1F2A1

If $x, y$ lie in $A$ and $g_{x, y}^{z} \neq 0$ , then some arrow $x \to z$ has cofiber $y$ , giving a distinguished triangle $x \to z \to y \to x [1]$ . By [A1], $z$ lies in $A$ and the triangle is a short exact sequence $0 \to x \to z \to y \to 0$ . Thus the span of heart classes is closed under the derived Hall product.

1.2F1F2

For $x, z \in A$ , $x \in D^{\leq 0}$ and $z [- i] \in D^{\geq i} \subseteq D^{\geq 1}$ for every $i > 0$ . Orthogonality in [F1] gives ${Ext}_{D}^{- i} (x, z) = {Hom}_{D} (x, z [- i]) = 0$ ; the same holds with $z = x$ . The coefficient formula [F2] therefore reduces to $g_{x, y}^{z} = ∣ [x, z]_{y} ∣ / ∣ Aut (x) ∣$ .

2.1F3A1step 1.2

Each subobject of $z$ with domain isomorphic to $x$ and quotient isomorphic to $y$ gives, after identifying its domain with $x$ , a map in $[x, z]_{y}$ . Two such identifications give maps in the same $Aut (x)$ -orbit. Conversely every map in $[x, z]_{y}$ is the monomorphism in the short exact sequence of [A1]. The action is free because a monomorphism $f$ with $f \circ a = f$ forces $a = {i d}_{x}$ . Thus the orbit count $∣ [x, z]_{y} ∣ / ∣ Aut (x) ∣$ equals $g_{y, x}^{z}$ in the convention of [F3].

3.1F2F4step 1.1step 2.1∎

For all $M, N$ , step 2.1 gives $Φ ([M] *_{A} [N]) = \sum_{L} g_{M, N}^{L} χ_{L} = χ_{N} *_{D H} χ_{M}$ . Hence $Φ$ is an anti-homomorphism, or equivalently a homomorphism from the opposite algebra. It is injective because distinct heart isomorphism classes are distinct basis vectors of $D H (T)$ ; closure follows from step 1.1. The algebra and unit claims are those of [F4] and Toën derived Hall multiplication is associative and unital.

Convention note

Toën's product counts the source as the subobject and the cofiber as the quotient. The current finitary Hall supplier uses the opposite naming order: the second product input is the subobject and the first is the quotient. The opposite-algebra formulation above makes this convention difference explicit.

Depends on

Used by

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Sources

Bertrand Toën, Derived Hall algebras, arXiv:math/0501343v4, §6, Proposition 6.1 (standard reference, not scraped)
Olivier Schiffmann, Lectures on Hall algebras, arXiv:math/0611617v2, §5.6 (standard reference, not scraped)