Reducing matrix factorizations to two-periodic complexes #
If a ring homomorphism sends the potential of a matrix factorization to zero, extension of scalars turns its two curved differential equations into square-zero equations. In particular, reduction modulo the ideal generated by the potential gives a two-periodic complex over the hypersurface quotient. The construction acts on closed even morphisms and hence is a functor.
Scalar extension first gives a finite-projective matrix factorization of the image potential,
and it commutes with the parity shift. When that image is zero, forgetting finite projectivity
gives a square-zero duplex and a two-periodic complex. The reduction modulo the potential is the
periodic complex used in Eisenbud, Homological algebra on a complete intersection, with an
application to group representations, Trans. Amer. Math. Soc. 260 (1980), Section 5.
When the potential is a non-zero-divisor, the reduction of a finite-projective factorization
modulo the potential is exact in both parities (exact_quotient_d₁_d₀, exact_quotient_d₀_d₁):
if d₀ x = w y, then w x = d₁ d₀ x = w d₁ y, so x = d₁ y. Hence the reduced two-periodic
complex is acyclic (quotientPeriodicComplex_acyclic).
Extension of scalars carries a finite-projective matrix factorization of w to one of
f w, without requiring the potential to vanish.
Instances For
The even object of the scalar-extended factorization is the scalar extension of the original even module.
The odd object of the scalar-extended factorization is the scalar extension of the original odd module.
The even-to-odd differential of the scalar-extended factorization is the scalar
extension of d₀.
The odd-to-even differential of the scalar-extended factorization is the scalar
extension of d₁.
The even component of a map in the scalar-extended factorization is the scalar
extension of f₀.
The odd component of a map in the scalar-extended factorization is the scalar extension
of f₁.
Scalar extension preserves addition of closed even maps.
The even differential of the base change of a parity shift is the even differential of the
parity shift of the base change: both are -(T ⊗ d₁).
The odd differential of the base change of a parity shift is the odd differential of the
parity shift of the base change: both are -(T ⊗ d₀).
Base change commutes with the parity shift. Both sides have the same components, and the negated differentials agree because extension of scalars is additive.
Instances For
The even component of parityShiftCompBaseChangeFunctorIso is an identity.
The odd component of parityShiftCompBaseChangeFunctorIso is an identity.
The even component of the inverse of parityShiftCompBaseChangeFunctorIso is an identity.
The odd component of the inverse of parityShiftCompBaseChangeFunctorIso is an identity.
Extension of scalars along a map killing the potential sends a finite-projective matrix factorization to a square-zero duplex. The target components are the tensor products of the original components with the target ring.
Instances For
The even object of the square-zero duplex is the scalar extension of the original even module.
The odd object of the square-zero duplex is the scalar extension of the original odd module.
The even-to-odd differential of the square-zero duplex is the scalar extension of d₀.
The odd-to-even differential of the square-zero duplex is the scalar extension of d₁.
The even component of a map in the square-zero duplex is the scalar extension of f₀.
The odd component of a map in the square-zero duplex is the scalar extension of f₁.
Extension of scalars is additive on closed even maps of matrix factorizations.
The two-periodic complex obtained by reducing a matrix factorization along a ring map that kills its potential.
Instances For
The zero component of the two-periodic complex is the even component of the square-zero duplex.
The one component of the two-periodic complex is the odd component of the square-zero duplex.
The differential from zero to one is the even differential of the square-zero duplex.
The differential from one to zero is the odd differential of the square-zero duplex.
The zero component of a periodic map is the even component of the duplex map.
The one component of a periodic map is the odd component of the duplex map.
The two-periodic complex over S/(w) associated to a matrix factorization of w.
Instances For
The even object of the hypersurface quotient complex is the scalar extension of the original even module.
The odd object of the hypersurface quotient complex is the scalar extension of the original odd module.
The zero-to-one differential of the hypersurface quotient complex is the scalar
extension of d₀.
The one-to-zero differential of the hypersurface quotient complex is the scalar
extension of d₁.
The zero component of a map in the hypersurface quotient complex is the scalar
extension of f₀.
The one component of a map in the hypersurface quotient complex is the scalar extension
of f₁.
The reduction modulo a non-zero-divisor potential w of a finite-projective matrix
factorization is exact at its even component: the kernel of d₀' is the image of d₁'.
The reduction modulo a non-zero-divisor potential w of a finite-projective matrix
factorization is exact at its odd component: the kernel of d₁' is the image of d₀'.
The two-periodic complex over S/(w) obtained by reducing a finite-projective matrix
factorization modulo a non-zero-divisor potential w is exact in both parities.