Harder--Narasimhan reduction after phase tilting #
This file isolates the cohomological quotient step used in Proposition 19.5
and in the weak-HN part of Proposition 14.16. If the last original HN factor
of F is U, then quotienting an extension of F[1] by the shifted prefix
first produces an extension of U[1] by a zero-charge object. Saturating
that extension gives the last tilted semistable quotient. The kernel of the
composite quotient is again an extension of the shifted prefix by a
zero-charge object, which is the recursive state for the shorter original
HN filtration.
Boundary-phase saturation is performed inside the quotient step: no right-orthogonality hypothesis on the shifted last factor is required.
The H⁻¹ last-factor reduction.
Suppose G ⟶ F ⟶ U is the last step of an original HN filtration,
and F[1] ⟶ E ⟶ V is an extension by a tilted zero-charge object.
Then E has a
semistable quotient B with the charge of U[1]. Its kernel K fits into
an extension G[1] ⟶ K ⟶ A with A zero-charge.
Thus the first triangle removes the last original HN factor, while the second triangle is exactly the recursive input attached to the prefix.
Iterating the strengthened last-factor reduction constructs a weak HN
filtration for every extension of F[1] by a tilted zero-charge object, when
F belongs to the phase-cut torsion-free class.
The induction discards zero terminal factors in the chosen slicing HN tower.
For every nonzero factor it applies phaseTilt_hnLastQuotient; the recursive
kernel is the same kind of extension for the prefix. The returned last-slope
identity records the last surviving original factor and turns strict phase
separation into the strict tilted-slope inequality required by
append_hn_filtration_of_mono.
Every free-shift/zero-charge extension has a weak HN filtration.
The H⁰ last-factor kernel calculation.
Given short exact sequences F[1] ⟶ E ⟶ N and G ⟶ N ⟶ U in the
tilted heart, the kernel of the composite quotient E ⟶ U fits into
F[1] ⟶ K ⟶ G. This is the dual kernel-of-composite diagram used to
iterate the original H⁰ HN filtration.
Iterating the original H⁰ HN filtration, with the H⁻¹ induction as
its base case, proves the weak HN property for the phase-tilted heart.
Charged terminal H⁰ factors below phase 1 are directly of phase-tilt type
one. At the boundary phase, the original zero-charge torsion pair first
saturates the factor; its zero-charge kernel is absorbed by the already
formalized H⁻¹ induction.