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Equal actions on K_0 do not imply isomorphic derived autoequivalences

Statement refuted

If an exact autoequivalence of a triangulated category acts as the identity on the Grothendieck group, then it is isomorphic to the identity functor.

Facts & Assumptions

Given: AC; the integer $m = 4$ , so that the braid group is $B_{5}$ , the category $C_{4} = K^{b} ({proj}^{g r} A_{4})$ , its graded Grothendieck group $G (A_{4})$ , and the braid action of $B_{5}$ on $C_{4}$ by the complexes $R_{σ}$ .

[L1]

There is a nontrivial braid $ψ \in B_{5}$ with $ρ_{5}^{m a t} (ψ) = I_{5}$ , the identity matrix in the unreduced Burau representation; the element is the commutator of the half twist about a regular neighbourhood of an arc $α$ with the full twist about a regular neighbourhood of $β \cup \partial D$ (A nontrivial five-strand braid lies in the Burau kernel, The unreduced Burau matrices).

[L2]

The weak action of $B_{5}$ on $C_{4}$ is faithful: $R_{σ} ≅ {Id}_{C_{4}}$ implies $σ = 1$ (The Khovanov-Seidel weak braid action is faithful).

[L3]

The induced action on $G (A_{4})$ is the unreduced Burau action: after the explicit invertible change of basis $C$ and the parameter identification $q = t$ , the operator $[R_{σ}]$ equals $ρ_{5}^{m a t} (σ)$ for every $σ$ (Decategorification is the unreduced Burau action, The graded Grothendieck group of A_m).

Counterexample

technique · direct

1.1L1L2

The witness braid and its categorical action. Take $m = 4$ and let $ψ \in B_{5}$ be the nontrivial braid of [L1], so $ψ \neq 1$ and $ρ_{5}^{m a t} (ψ) = I_{5}$ . By [L2] applied to the nontrivial braid $ψ$ , the endofunctor $R_{ψ}$ is not isomorphic to the identity functor of $C_{4}$ .

2.1step 1.1L1L3

Its action on $K_{0}$ is trivial. By [L3] the operator $[R_{ψ}]$ on $G (A_{4})$ corresponds, in the explicit basis of the decategorification proposition, to the matrix $ρ_{5}^{m a t} (ψ) = I_{5}$ ; hence $[R_{ψ}]$ is the identity operator on $G (A_{4})$ . Thus the exact autoequivalence $R_{ψ}$ acts as the identity on the Grothendieck group while, by step 1.1, it is not isomorphic to the identity functor.

3.1step 2.1∎

Conclusion. The map from derived autoequivalences of $C_{4}$ to operators on $K_{0}$ has a nontrivial kernel, containing the class of $R_{ψ}$ ; the refuted statement is false. AC is inherited from the kernel lemma and the faithfulness theorem; no additional choice is made.

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Sources

Mikhail Khovanov and Paul Seidel, Quivers, Floer Cohomology, and Braid Group Actions, J. Amer. Math. Soc. 15 (2002) 203-271, Corollary 1.2 and Section 2e.1 (standard reference, not scraped)
Stephen J. Bigelow, The Burau representation is not faithful for n=5, Geometry & Topology 3 (1999) 397-404 (standard reference, not scraped)