Subsection 4.5. (0NHE) · ScalingStacks

0NGT

Definition 4.18. Let $\mathcal{C}$ be an additive category. The split Grothendieck group of $\mathcal{C}$ is defined as:

K_{0}(\mathcal{C}):=\frac{\mathrm{Span}_{\mathbb{Z}}\{\text{isomorphism classes }[x]\text{ of objects in }\mathcal{C}\}}{([x\oplus y]=[x]+[y]\mid x,y\in\mathrm{Ob}(\mathcal{C}))}

0NGU

Definition 4.19. A $K$ -linear additive category $\mathcal{C}$ is called Krull–Schmidt if every object decomposes uniquely into a finite direct sum of indecomposable objects with local endomorphism rings.

0NGV

Proposition 4.20. For a Krull-Schmidt category, the split Grothendieck group is a free abelian group on the isomorphism classes of indecomposable objects in $\mathcal{C}$ .

0NGW

Definition 4.21. For a $K$ -linear additive category $\mathcal{C}$ , the Chern character is the $K$ -linear map

h_{\mathcal{C}}\colon K_{0}(\mathcal{C})\otimes_{\mathbb{Z}}K\to\mathrm{HH}_{0}(\mathcal{C}),\quad[x]\otimes 1\mapsto[\operatorname{Id}_{x}\colon x\to x]

0NGX

Proposition 4.22 (Proposition 2.4 in [5]). If $K=\mathbbm{k}$ is a perfect field and $\mathcal{C}$ is Krull-Schmidt with a finite-dimensional endomorphism algebra for each indecomposable object, then the Chern character $h_{\mathcal{C}}$ is injective.

0NGY

Theorem 4.23. Let $p\geq 2$ . Then $\mathcal{S}_{0}^{2}(S^{1}\times B^{3};S^{1}\times P_{p},\mathbbm{k})$ is infinite-dimensional in bidegree $(0,0)$ .

0NGZ

Proof. We let $s=t=p\;\mathrm{points}$ and again have isomorphisms

\mathcal{S}_{0}^{2}(S^{1}\times B^{3};S^{1}\times P_{p},\mathbbm{k})\cong\mathrm{HH}_{0}(\mathcal{S}_{0}^{2}(B^{3};P_{p},\mathbbm{k}))\cong\mathrm{HH}_{0}(\boldsymbol{\mathrm{T}}_{2}(s,t))

and we consider the category $\boldsymbol{\mathrm{T}}_{2}(s,t)$ as a full subcategory of the enriched morphism category $H^{\bullet}(\boldsymbol{\mathrm{Foam}}_{2}^{\mathrm{dg}})^{*}(s,t)$ .

The $\mathbbm{k}$ -linear, additive category $H^{\bullet}(\boldsymbol{\mathrm{Foam}}_{2}^{\mathrm{dg}})^{*}(s,t)$ is Krull-Schmidt and hence idempotent complete; see e.g. the discussion in [30, Sections 4.5, 4.8] based on Bar-Natan’s category, which is equivalent to $\boldsymbol{\mathrm{Foam}}_{2}$ by [6].

Now $\mathrm{Kar}(\boldsymbol{\mathrm{T}}_{2}(s,t))^{\oplus}$ may be considered as an additive, idempotent complete full subcategory of $H^{\bullet}(\boldsymbol{\mathrm{Foam}}_{2}^{\mathrm{dg}})^{*}$ ; it is thus itself Krull–Schmidt. We have $\mathrm{HH}_{0}(\boldsymbol{\mathrm{T}}_{2}(s,t))\cong\mathrm{HH}_{0}(\mathrm{Kar}(\boldsymbol{\mathrm{T}}_{2}(s,t))^{\oplus})$ by Fact 4.12. Therefore, it suffices to compute its zeroth Hochschild homology of $\mathrm{Kar}(\boldsymbol{\mathrm{T}}_{2}(s,t))^{\oplus}$ .

It is straightforward to check that the objects of $\mathrm{Kar}(\boldsymbol{\mathrm{T}}_{2}(s,t))^{\oplus}$ have finite-dimensional endomorphism algebras, and since $\mathbbm{k}$ is perfect, the Chern character

h\colon K_{0}(\mathrm{Kar}(\boldsymbol{\mathrm{T}}_{2}(s,t))^{\oplus})\otimes_{\mathbb{Z}}\mathbbm{k}\to\mathrm{HH}_{0}(\mathrm{Kar}(\boldsymbol{\mathrm{T}}_{2}(s,t))^{\oplus})

is injective; see Proposition 4.22. To prove that $\mathcal{S}_{0}^{2}(S^{1}\times B^{3};S^{1}\times P_{p},\mathbbm{k})$ is infinite-dimensional in bidegree $(0,0)$ , it is thus sufficient to show that $K_{0}(\mathrm{Kar}(\boldsymbol{\mathrm{T}}_{2}(s,t))^{\oplus})\otimes_{\mathbb{Z}}\mathbbm{k}$ is infinite-dimensional.

Moreover, $K_{0}(\mathrm{Kar}(\boldsymbol{\mathrm{T}}_{2}(s,t))^{\oplus})$ is free abelian on the isomorphism classes of its indecomposable objects; cf. Proposition 4.20. Thus, we will be done once we can exhibit infinitely many indecomposable and pairwise non-isomorphic complexes appearing as (direct summands in) tangle complexes.

We will see that such complexes can be constructed as invariants of braids. Clearly, for $p\geq 2$ there are infinitely many braids on $p$ strands. Moreover, the braid complexes are invertible under tensoring with the complex for the respective inverse braid. Since the complex of the trivial braid is indecomposable (its endomorphism algebra $\left(\mathbbm{k}[X]/(X^{2})\right)^{\otimes p}$ is local), so are the complexes for all other braids. It is also known that all braid complexes are pairwise non-isomorphic. This can e.g. be deduced from the faithfulness of the braid group action of Khovanov–Seidel [22]. For us, however, it is enough to consider infinitely many braids that are powers of a single Artin braid generator. For these complexes it is straightforward to check by hand that they are pairwise non-isomorphic. ∎

4.5. The 3-ball category with four or more points