The homotopy category of matrix factorizations is triangulated

Let S be a commutative ring and w : S. Finite-projective matrix factorizations of w with the componentwise split exact structure form a Frobenius exact category, and its stable category is identified with HMF(S,w), here MatrixFactorization.HomotopyCategory, by MatrixFactorization.stableToHomotopy, compatibly with the shifts by ℤ (stable suspension on one side, the parity shift on the other). This file transports Happel's triangulation of the stable category along this equivalence, which makes HMF(S,w) a triangulated category whose shift by 1 is the parity shift. No regularity hypothesis on S or w is needed.

The distinguished triangles are described concretely. Let X₁ ⟶ X₂ ⟶ X₃ be a short complex of matrix factorizations which splits in both components. Since the disk sum diskSum X₁ is contractible, the inflation X₁ ⟶ diskSum X₁ extends along X₁ ⟶ X₂ to a map a : X₂ ⟶ diskSum X₁, which induces δ : X₃ ⟶ X₁[1] on cokernels. Then

X₁ ⟶ X₂ ⟶ X₃ ⟶ X₁[1]

with third map the homotopy class of δ is distinguished, and every distinguished triangle is isomorphic to one of these.

Main definitions

• TauCeti.MatrixFactorization.HomotopyCategory.instPretriangulated: the pretriangulated structure on HMF(S,w).

Main results

• TauCeti.MatrixFactorization.HomotopyCategory.instIsTriangulated: HMF(S,w) is triangulated.
• TauCeti.MatrixFactorization.stableToHomotopy_isTriangulated: the comparison from the componentwise split stable category is a triangle functor.
• TauCeti.MatrixFactorization.HomotopyCategory.mk_distinguished_of_conflation: componentwise split short complexes give distinguished triangles.
• TauCeti.MatrixFactorization.HomotopyCategory.mem_distTriang_iff: every distinguished triangle is isomorphic to the triangle of a componentwise split short complex.

References

• Dieter Happel, Triangulated Categories in the Representation Theory of Finite Dimensional Algebras, Chapter I, Section 2, Theorem 2.6: the stable category of a Frobenius category is triangulated.
• D. Orlov, Triangulated categories of singularities and D-branes in Landau–Ginzburg models, Proc. Steklov Inst. Math. 246 (2004), Section 3, for the triangulated category of finite-projective matrix factorizations.

The argument follows the curved-duplex version in TauCeti.Algebra.Homology.Curved.Triangulated, restricted to finite-projective components.

@[instance_reducible]

noncomputable instance TauCeti.MatrixFactorization.HomotopyCategory.instPretriangulated (S : Type u) [CommRing S] (w : S) : CategoryTheory.Pretriangulated HomotopyCategory

The homotopy category of matrix factorizations is pretriangulated: its distinguished triangles are the images of Happel's distinguished triangles under the equivalence with the componentwise split stable category.

Equations

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

theorem TauCeti.MatrixFactorization.stableToHomotopy_isTriangulated (S : Type u) [CommRing S] (w : S) : (stableToHomotopy S w).IsTriangulated

The comparison from the componentwise split stable category of matrix factorizations to HMF(S,w) is a triangle functor.

instance TauCeti.MatrixFactorization.HomotopyCategory.instIsTriangulated (S : Type u) [CommRing S] (w : S) : CategoryTheory.IsTriangulated HomotopyCategory

The homotopy category of matrix factorizations is triangulated, with the shift by 1 given by the parity shift.

theorem TauCeti.MatrixFactorization.HomotopyCategory.mk_distinguished_of_conflation {S : Type u} [CommRing S] {w : S} {T : CategoryTheory.ShortComplex (MatrixFactorization S w)} (hT : (splitExact S w).Conflation T) (a : T.X₂ ⟶ T.X₁.diskSum) (δ : T.X₃ ⟶ parityShift.obj T.X₁) (ha : CategoryTheory.CategoryStruct.comp T.f a = T.X₁.toDiskSum) (hδ : CategoryTheory.CategoryStruct.comp T.g δ = CategoryTheory.CategoryStruct.comp a T.X₁.diskSumToParityShift) : CategoryTheory.Pretriangulated.Triangle.mk (nullHomotopic.quotientFunctor.map T.f) (nullHomotopic.quotientFunctor.map T.g) (CategoryTheory.CategoryStruct.comp (nullHomotopic.quotientFunctor.map δ) ((parityShiftCompQuotientFunctorIso S w).hom.app T.X₁)) ∈ CategoryTheory.Pretriangulated.distinguishedTriangles

Componentwise split conflations give distinguished triangles. Let X₁ ⟶ X₂ ⟶ X₃ be a short complex of matrix factorizations which splits in both components, let a : X₂ ⟶ diskSum X₁ extend the inclusion of X₁ into its disk sum, and let δ : X₃ ⟶ X₁[1] be the map induced by a on cokernels. Then the triangle X₁ ⟶ X₂ ⟶ X₃ ⟶ X₁⟦1⟧ of homotopy classes, with third map the class of δ, is distinguished.

theorem TauCeti.MatrixFactorization.HomotopyCategory.mem_distTriang_iff (S : Type u) [CommRing S] (w : S) (D : CategoryTheory.Pretriangulated.Triangle HomotopyCategory) : D ∈ CategoryTheory.Pretriangulated.distinguishedTriangles ↔ ∃ (T : CategoryTheory.ShortComplex (MatrixFactorization S w)) (_ : (splitExact S w).Conflation T) (a : T.X₂ ⟶ T.X₁.diskSum) (δ : T.X₃ ⟶ parityShift.obj T.X₁) (_ : CategoryTheory.CategoryStruct.comp T.f a = T.X₁.toDiskSum) (_ : CategoryTheory.CategoryStruct.comp T.g δ = CategoryTheory.CategoryStruct.comp a T.X₁.diskSumToParityShift), Nonempty (D ≅ CategoryTheory.Pretriangulated.Triangle.mk (nullHomotopic.quotientFunctor.map T.f) (nullHomotopic.quotientFunctor.map T.g) (CategoryTheory.CategoryStruct.comp (nullHomotopic.quotientFunctor.map δ) ((parityShiftCompQuotientFunctorIso S w).hom.app T.X₁)))

The distinguished triangles of the homotopy category of matrix factorizations are exactly the triangles isomorphic to the triangle of homotopy classes X₁ ⟶ X₂ ⟶ X₃ ⟶ X₁⟦1⟧ of a componentwise split short complex, with third map induced by an extension X₂ ⟶ diskSum X₁ of the inclusion of X₁ into its disk sum.