Connected Components of the Stability Space

Openness of connected components, seminorm equivalence on components, topological basis, and membership of central charges in the finite seminorm subgroup.

theorem CategoryTheory.Triangulated.exists_basisNhd_subset_connectedComponent (C : Type u) [Category.{v, u} C] [Limits.HasZeroObject C] [HasShift C ℤ] [Preadditive C] [∀ (n : ℤ), (shiftFunctor C n).Additive] [Pretriangulated C] [IsTriangulated C] {Λ : Type u'} [AddCommGroup Λ] {v : K₀ C →+ Λ} (σ : StabilityCondition.WithClassMap C v) : ∃ (ε : ℝ), 0 < ε ∧ ε < 1 / 8 ∧ WeakStabilityCondition.StabilityCondition.Deformation.basisNhd C σ ε ⊆ {τ : StabilityCondition.WithClassMap C v | ConnectedComponents.mk τ = ConnectedComponents.mk σ}

Every stability condition admits a small Bridgeland basis neighbourhood contained in its topological connected component. This is the local connectedness input actually needed later.

theorem CategoryTheory.Triangulated.stabilityCondition_isOpen_connectedComponent (C : Type u) [Category.{v, u} C] [Limits.HasZeroObject C] [HasShift C ℤ] [Preadditive C] [∀ (n : ℤ), (shiftFunctor C n).Additive] [Pretriangulated C] [IsTriangulated C] {Λ : Type u'} [AddCommGroup Λ] {v : K₀ C →+ Λ} (σ : StabilityCondition.WithClassMap C v) : IsOpen (connectedComponent σ)

Connected components in StabilityCondition C are open, because small Bridgeland basis neighbourhoods stay inside the connected component of their centre.

theorem CategoryTheory.Triangulated.stabilitySeminorm_dominated_of_connected (C : Type u) [Category.{v, u} C] [Limits.HasZeroObject C] [HasShift C ℤ] [Preadditive C] [∀ (n : ℤ), (shiftFunctor C n).Additive] [Pretriangulated C] [IsTriangulated C] {Λ : Type u'} [AddCommGroup Λ] {v : K₀ C →+ Λ} (σ τ : StabilityCondition.WithClassMap C v) (h : ConnectedComponents.mk σ = ConnectedComponents.mk τ) : ∃ (K : ENNReal), K ≠ ⊤ ∧ ∀ (f : Λ →+ ℂ), WeakStabilityCondition.StabilityCondition.Deformation.stabilitySeminorm C σ f ≤ K * WeakStabilityCondition.StabilityCondition.Deformation.stabilitySeminorm C τ f

Lemma 6.2: On a connected component, seminorms are equivalent (domination).

theorem CategoryTheory.Triangulated.finiteSeminormSubgroup_eq_of_connected (C : Type u) [Category.{v, u} C] [Limits.HasZeroObject C] [HasShift C ℤ] [Preadditive C] [∀ (n : ℤ), (shiftFunctor C n).Additive] [Pretriangulated C] [IsTriangulated C] {Λ : Type u'} [AddCommGroup Λ] {v : K₀ C →+ Λ} (σ τ : StabilityCondition.WithClassMap C v) (h : ConnectedComponents.mk σ = ConnectedComponents.mk τ) : finiteSeminormSubgroup C σ = finiteSeminormSubgroup C τ

Lemma 6.2: On a connected component, the finite-seminorm subgroups agree.

def CategoryTheory.Triangulated.basisNhdFamily (C : Type u) [Category.{v, u} C] [Limits.HasZeroObject C] [HasShift C ℤ] [Preadditive C] [∀ (n : ℤ), (shiftFunctor C n).Additive] [Pretriangulated C] {Λ : Type u'} [AddCommGroup Λ] {v : K₀ C →+ Λ} : Set (Set (StabilityCondition.WithClassMap C v))

The generating family of Bridgeland basis neighborhoods on Stab(D).

Equations

• One or more equations did not get rendered due to their size.

theorem CategoryTheory.Triangulated.exists_basisNhd_subset_of_mem_open (C : Type u) [Category.{v, u} C] [Limits.HasZeroObject C] [HasShift C ℤ] [Preadditive C] [∀ (n : ℤ), (shiftFunctor C n).Additive] [Pretriangulated C] {Λ : Type u'} [AddCommGroup Λ] {v : K₀ C →+ Λ} (σ : StabilityCondition.WithClassMap C v) {s : Set (StabilityCondition.WithClassMap C v)} (hs : IsOpen s) (hσ : σ ∈ s) : ∃ (ε : ℝ), 0 < ε ∧ ε < 1 / 8 ∧ WeakStabilityCondition.StabilityCondition.Deformation.basisNhd C σ ε ⊆ s

Every open neighborhood of σ contains a centered Bridgeland basis neighborhood.

theorem CategoryTheory.Triangulated.isTopologicalBasis_basisNhd (C : Type u) [Category.{v, u} C] [Limits.HasZeroObject C] [HasShift C ℤ] [Preadditive C] [∀ (n : ℤ), (shiftFunctor C n).Additive] [Pretriangulated C] {Λ : Type u'} [AddCommGroup Λ] (v : K₀ C →+ Λ) : TopologicalSpace.IsTopologicalBasis (basisNhdFamily C)

The Bridgeland neighborhoods form a topological basis for Stab(D).

theorem CategoryTheory.Triangulated.exists_basisNhd_subset_of_mem_nhds (C : Type u) [Category.{v, u} C] [Limits.HasZeroObject C] [HasShift C ℤ] [Preadditive C] [∀ (n : ℤ), (shiftFunctor C n).Additive] [Pretriangulated C] {Λ : Type u'} [AddCommGroup Λ] {v : K₀ C →+ Λ} (σ : StabilityCondition.WithClassMap C v) {s : Set (StabilityCondition.WithClassMap C v)} (hs : s ∈ nhds σ) : ∃ (ε : ℝ), 0 < ε ∧ ε < 1 / 8 ∧ WeakStabilityCondition.StabilityCondition.Deformation.basisNhd C σ ε ⊆ s

Neighborhood-form extraction of a centered Bridgeland basis neighborhood.

theorem CategoryTheory.Triangulated.eq_zero_of_vanishes_on_cl (C : Type u) [Category.{v, u} C] [Limits.HasZeroObject C] [HasShift C ℤ] [Preadditive C] [∀ (n : ℤ), (shiftFunctor C n).Additive] [Pretriangulated C] {Λ : Type u'} [AddCommGroup Λ] {v : K₀ C →+ Λ} (U : Λ →+ ℂ) (hv : Function.Surjective ⇑v) (hU : ∀ (E : C), U (classOf C v E) = 0) : U = 0

An additive homomorphism on Λ is zero if it vanishes on all class images and v is surjective. The surjectivity ensures classOf C v generates all of Λ.

theorem CategoryTheory.Triangulated.eq_zero_of_stabilitySeminorm_toReal_eq_zero (C : Type u) [Category.{v, u} C] [Limits.HasZeroObject C] [HasShift C ℤ] [Preadditive C] [∀ (n : ℤ), (shiftFunctor C n).Additive] [Pretriangulated C] {Λ : Type u'} [AddCommGroup Λ] {v : K₀ C →+ Λ} (σ : StabilityCondition.WithClassMap C v) (hv : Function.Surjective ⇑v) {U : Λ →+ ℂ} (hfin : WeakStabilityCondition.StabilityCondition.Deformation.stabilitySeminorm C σ U < ⊤) (hzero : (WeakStabilityCondition.StabilityCondition.Deformation.stabilitySeminorm C σ U).toReal = 0) : U = 0

If the Bridgeland seminorm of U is finite and equal to zero, then U = 0. Requires surjectivity of v so that vanishing on classOf C v implies vanishing on all of Λ.

theorem CategoryTheory.Triangulated.Z_mem_finiteSeminormSubgroup (C : Type u) [Category.{v, u} C] [Limits.HasZeroObject C] [HasShift C ℤ] [Preadditive C] [∀ (n : ℤ), (shiftFunctor C n).Additive] [Pretriangulated C] {Λ : Type u'} [AddCommGroup Λ] {v : K₀ C →+ Λ} (σ : StabilityCondition.WithClassMap C v) : σ.Z ∈ finiteSeminormSubgroup C σ

Z(σ) has finite σ-seminorm: ‖Z(σ)‖_σ ≤ 1, hence Z(σ) ∈ V(σ).