Duality for matrix factorizations #
The dual of a finite-projective matrix factorization P of w is the factorization
(Pᵛ)₀ = P₀ᵛ (Pᵛ)₁ = P₁ᵛ
d₀ᵛ = (d₁)ᵗ : P₀ᵛ ⟶ P₁ᵛ d₁ᵛ = -(d₀)ᵗ : P₁ᵛ ⟶ P₀ᵛ
of -w: the two composites are multiplication by -w, because a crossed pair of transposes is
the transpose of the original composite and the single minus sign turns w into -w.
TauCeti.Algebra.Homology.Curved.Dual proves the same statement for curved duplexes of
projective modules, and the crossed transposes are exactly the convention used there.
Duality is contravariant: a morphism f : X ⟶ Y of matrix factorizations dualizes to a
morphism dualMap f : Yᵛ ⟶ Xᵛ, whose two components are the transposes of the components of
f, and whose commutativity conditions are the commutativity conditions of f with the two
differentials exchanged. The two are packaged together as the contravariant functor
MatrixFactorization.dualFunctor from the opposite category of matrix factorizations of w to
the category of matrix factorizations of -w, so duality is available through the category
API.
The object-level double dual is also here, and is isomorphic to the original: dualizing twice
lands back at the potential w and at an isomorphic factorization.
Main definitions #
MatrixFactorization.dual: the dual of a matrix factorization ofw, of potential-w.MatrixFactorization.dualMap: the dual of a morphism, contravariantly.MatrixFactorization.dualFunctor: duality as the contravariant functor(MatrixFactorization S w)ᵒᵖ ⥤ MatrixFactorization S (-w).MatrixFactorization.doubleDual,MatrixFactorization.doubleDualIso: the double dual, of the original potential, and its isomorphism with the original factorization.
Main results #
MatrixFactorization.dualMap_f₀,MatrixFactorization.dualMap_f₁: the components of a dual morphism.MatrixFactorization.dualMap_id,MatrixFactorization.dualMap_comp,MatrixFactorization.dualMap_zero,MatrixFactorization.dualMap_add: the dual of a morphism reverses identity and composition and is additive, which is what makes duality contravariant.MatrixFactorization.dualFunctor_obj,MatrixFactorization.dualFunctor_map: the object and morphism maps of the contravariant duality functor, and the instance saying that it is an additive functor.MatrixFactorization.doubleDualIso,MatrixFactorization.doubleDualIso_f₀,MatrixFactorization.doubleDualIso_f₁,MatrixFactorization.doubleDualIso_inv_f₀,MatrixFactorization.doubleDualIso_inv_f₁: a double dual is isomorphic to the original matrix factorization, by the evaluation pairing with the single minus sign on the even component, and its two components and the two components of its inverse are the evaluation isomorphisms.
References #
- D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980), Section 5, for duality of matrix factorizations of finite free modules.
- The finite-projective formulation and the ambient curved-duplex convention follow
TauCeti.CommutativeAlgebra.MatrixFactorization.BasicandTauCeti.Algebra.Homology.Curved.Duplex.
The dual of a finite-projective matrix factorization of w is a finite-projective matrix
factorization of -w whose differentials are the crossed transposes.
Equations
Instances For
The underlying curved duplex of a dual matrix factorization is the dual curved duplex.
The even component of a dual matrix factorization is the dual of the even component.
The odd component of a dual matrix factorization is the dual of the odd component.
The even differential of a dual matrix factorization is the transpose of the odd differential of the original.
The odd differential of a dual matrix factorization is the negated transpose of the even
differential of the original, the sign that turns the potential w into -w.
The dual of a morphism of matrix factorizations is a morphism from the dual of the target to the dual of the source, with transposed components.
Equations
Instances For
The underlying curved-duplex morphism of a dual morphism of matrix factorizations.
The even component of the dual of a morphism of matrix factorizations is the transpose of the even component of the morphism.
The odd component of the dual of a morphism of matrix factorizations is the transpose of the odd component of the morphism.
Dualization is contravariant on morphisms: the dual of the identity of a matrix factorization is the identity of its dual.
Dualization is contravariant on morphisms: the dual of a composite is the composite of the duals in the opposite order.
Dualization is contravariant on morphisms and additive: the dual of the zero morphism of a matrix factorization is the zero morphism of its dual.
Dualization is contravariant on morphisms and additive: the dual of a sum of morphisms of matrix factorizations is the sum of the duals.
Duality of matrix factorizations is contravariant: a morphism of matrix factorizations of
w dualizes to a morphism of matrix factorizations of -w from the dual of its target to the
dual of its source, and duality reverses identity and composition. The morphism map is
MatrixFactorization.dualMap and the two functor laws are MatrixFactorization.dualMap_id and
MatrixFactorization.dualMap_comp, so the dual is available through the category API rather than
only as a pair of separate functions.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The object map of the contravariant duality functor is MatrixFactorization.dual, applied
to the matrix factorization underlying the object of the opposite category.
The morphism part of dualFunctor is the transpose MatrixFactorization.dualMap of the
morphism of the opposite category, carried along the two canonical identifications that
MatrixFactorization.dualFunctor_obj exhibits between the duals of the two objects and the
matrix factorizations the morphism map is stated at.
Duality of matrix factorizations is an additive functor, as the transpose of a linear
map is: the morphism map of the contravariant duality functor MatrixFactorization.dualFunctor
preserves addition, by MatrixFactorization.dualMap_add, and so is a morphism of abelian groups,
which also sends the zero morphism to the zero morphism.
The double dual of a matrix factorization is a matrix factorization of the same potential
w: each of its differentials is the negated double transpose of the corresponding differential
of the original, and the two minus signs cancel when the differentials are composed.
Equations
Instances For
The underlying curved duplex of a double dual matrix factorization is the double dual curved duplex.
The even component of a double dual matrix factorization is the double dual of the even component.
The odd component of a double dual matrix factorization is the double dual of the odd component.
The even differential of a double dual matrix factorization is the negated double transpose of the even differential of the original.
The odd differential of a double dual matrix factorization is the negated double transpose of the odd differential of the original.
A double dual of a matrix factorization is isomorphic to the original, by the evaluation pairing with the single minus sign on the even component.
Equations
Instances For
The underlying morphism of the double dual isomorphism is the double dual isomorphism of the underlying curved duplex.
The underlying morphism of the inverse of the double dual isomorphism is the inverse double dual isomorphism of the underlying curved duplex.
The even component of the double dual isomorphism is the negated evaluation isomorphism on the even component, the placement of the sign which makes the two commutativity conditions hold against the negated double transposes.
The odd component of the double dual isomorphism is the evaluation isomorphism on the odd component.
The even component of the inverse of the double dual isomorphism is the negated inverse evaluation isomorphism.
The odd component of the inverse of the double dual isomorphism is the inverse evaluation isomorphism.