Cancelling two blocks of a four-term cochain complex #
Suppose a complex has terms Q, A ⊞ (U ⊞ Q), U ⊞ B, and Z in degrees
-3, -2, -1, and 0. In these coordinates its differentials are
(α, β, -id), ((f,g), (-id,δ), (qU,qB)),
and (zU, ε).
The square-zero identity forces qU = α ≫ f - β, qB = α ≫ g + β ≫ δ,
and zU = δ ≫ ε; these are conclusions rather than additional hypotheses.
In TauCeti.Contraction.FourTermCancellationData the components α, β, f, g, δ,
qU, qB, zU and ε are the fields alpha, beta, first, second, delta,
qFirst, qSecond, zFirst and epsilon, and the three identities are its theorems
qFirst_eq, qSecond_eq and zFirst_eq.
The two negative identity blocks cancel, leaving the three-term complex
A ⟶ B ⟶ Z, with differentials f ≫ δ + g and ε. The corrected projection
is (id,0,α) in degree -2 and (δ,id) in degree -1; the corrected inclusion
is (id,f,0) in degree -2 and the second inclusion in degree -1. The homotopy
is minus the projection to Q and minus the map from the second U (degree -1)
to the first U (degree -2).
The off-diagonal maps need not vanish. This is the two-block form of algebraic
Gaussian elimination, and its result is a Contraction K L. When the two blocks
are invertible, the caller supplies coordinate isomorphisms bringing them into
the displayed negative-identity form.
References #
- D. Bar-Natan, Fast Khovanov homology computations, J. Knot Theory Ramifications 16 (2007), 243--255, Lemma 4.2, https://arxiv.org/abs/math/0606318.
Four-term coordinates with two negative identity blocks and a specified surviving complex.
The lowest source term.
The source term containing the surviving and both cancellable summands.
The next source term, containing the second copy of
U.The highest source term.
The lowest surviving target term.
The middle surviving target term.
The highest surviving target term.
The source is supported in these four consecutive degrees.
The target is supported in the three surviving degrees.
The surviving component of the lowest differential.
The component of the lowest differential into the cancellable
U.The first component of the middle differential on
A.The second component of the middle differential on
A.The first component of the middle differential on
Q.The second component of the middle differential on
Q.The second component of the middle differential on
U.The differential out of the cancellable
U.The differential out of the surviving
B.- d_minusThree : CategoryTheory.CategoryStruct.comp (K.d (-3) (-2)) self.minusTwoIso.hom = CategoryTheory.CategoryStruct.comp self.minusThreeIso.hom (CategoryTheory.Limits.biprod.lift self.alpha (CategoryTheory.Limits.biprod.lift self.beta (-CategoryTheory.CategoryStruct.id Q)))
The lowest differential has a negative identity in its
Qcomponent. - d_minusTwo : CategoryTheory.CategoryStruct.comp (K.d (-2) (-1)) self.minusOneIso.hom = CategoryTheory.CategoryStruct.comp self.minusTwoIso.hom (CategoryTheory.Limits.biprod.desc (CategoryTheory.Limits.biprod.lift self.first self.second) (CategoryTheory.Limits.biprod.desc (CategoryTheory.Limits.biprod.lift (-CategoryTheory.CategoryStruct.id U) self.delta) (CategoryTheory.Limits.biprod.lift self.qFirst self.qSecond)))
The middle differential has a negative identity in its
Ucomponent. - d_minusOne : CategoryTheory.CategoryStruct.comp (K.d (-1) 0) self.zeroIso.hom = CategoryTheory.CategoryStruct.comp self.minusOneIso.hom (CategoryTheory.Limits.biprod.desc self.zFirst self.epsilon)
The highest source differential in the specified coordinates.
- target_d_minusTwo : CategoryTheory.CategoryStruct.comp (L.d (-2) (-1)) self.targetMinusOneIso.hom = CategoryTheory.CategoryStruct.comp self.targetMinusTwoIso.hom (CategoryTheory.CategoryStruct.comp self.first self.delta + self.second)
The surviving differential includes the correction through the cancellable
U. - target_d_minusOne : CategoryTheory.CategoryStruct.comp (L.d (-1) 0) self.targetZeroIso.hom = CategoryTheory.CategoryStruct.comp self.targetMinusOneIso.hom self.epsilon
The last surviving differential.
Instances For
Square-zero determines the component from Q to the next U.
Square-zero determines the component from Q to the next U.
Square-zero determines the component from Q to the surviving B.
Square-zero determines the component from Q to the surviving B.
Square-zero determines the last differential on the cancellable U.
Square-zero determines the last differential on the cancellable U.
Cancel the two negative identity blocks and retract onto the surviving three-term complex.
Instances For
The lowest projection includes the correction through the eliminated Q.
The lowest projection includes the correction through the eliminated Q.
The middle projection includes the correction through the eliminated U.
The middle projection includes the correction through the eliminated U.
The lowest inclusion includes the correction into the eliminated U.
The lowest inclusion includes the correction into the eliminated U.
The middle inclusion is the second biproduct inclusion in source coordinates.
The middle inclusion is the second biproduct inclusion in source coordinates.
The degree-zero projection is the specified coordinate identification.
The degree-zero projection is the specified coordinate identification.
The degree-zero inclusion is the inverse coordinate identification.
The degree-zero inclusion is the inverse coordinate identification.
The lower homotopy is minus the projection onto the eliminated Q.
The lower homotopy is minus the projection onto the eliminated Q.
The upper homotopy sends the second U to minus its first copy.
The upper homotopy sends the second U to minus its first copy.
All homotopy components outside the two cancelled pairs vanish.