Proposition 12.2 in its general form #
deformation-refinements.tex (prop:characteristic-two-normal-form): "Let
C = kQ/(r_v : v ∈ V) be a finite-dimensional graded algebra over any field, with Q finite and
the r_v homogeneous quadratic relations. For each q ≥ 3, choose homogeneous path tuples p_i
of degree d_i = q whose classes form a basis of coker D_{q-1}. […] Every algebra E with a
finite filtration and a specified identification gr E = C has a presentation
E ≅ kQ/(𝔪^{N+1}, r_v − Σ_i λ_i (p_i)_v) for some scalars λ_i. The isomorphism induces the
specified identification on the associated grading."
In the setting of the general Proposition 5.4 (NormalFormGeneral.lean: E a finite filtered
k-linear category on the vertices with symbols realizing gr E = C), with retained tuples
p_j of degrees ≥ 3 whose classes together with im D_{q-1} span (the basis hypothesis is
stronger), there are scalars λ and arrow lifts satisfying r_v = Σ λ_j (p_j)_v exactly
(exists_retained); the realization kQ → E is onto every Hom space, its kernel is the ideal
generated by r_v − Σ λ_j (p_j)_v and the paths of length > N for any N with F^N = 0, and
it induces the identification on the associated grading (normalFormRetained). The proof is the
source's: in increasing degree q, split the degree-q errors as D_{q-1}(f) + Σ c_j [p_j],
retain the c_j and change the arrows by lifts of f; a retained word of degree d_j ≥ 3
changes only in degree ≥ q + d_j − 2 > q, and the two-correction terms have degree 2q − 2.
Changing the arrows #
The arrow change a ↦ L a − ft a by maps of filtration degree q' ≥ 2.
Instances For
A path changes by maps of filtration degree length − 1 + q' under an arrow change of
degree q'.
The change of a quadratic relation under the arrow change L.sub (as in
exists_correction).
Retained tuples #
The perturbed relations r_v − Σ_j λ_j (p_j)_v.
Instances For
One correction step with retained tuples: if the errors of the perturbed relations lie in
F^{q'+1} and their symbols are D_{q'}(f) + Σ_j c_j [p_j] with c supported in degree
q'+1, retaining c and changing the arrows by lifts of f pushes the errors into F^{q'+2}.
Exact relations with retained tuples: if the classes of the retained tuples together with
im D_{q-1} span the degree-q relation families for every q ≥ 3, there are scalars λ and
arrow lifts satisfying r_v = Σ λ_j (p_j)_v.
The presentation #
The relations of the presentation: the perturbed relations and the paths of length > N.
Instances For
Morphisms with all components of degree > N lie in the ideal.
The realization kills the presentation ideal.
Replacing the quadratic relations by the perturbed ones in a homogeneous component.
The kernel of the realization is the presentation ideal (𝔪^{N+1}, r_v − Σ λ_j (p_j)_v).
Proposition 12.2 (prop:characteristic-two-normal-form), general form: for quadratic
relations r_v, retained tuples p_j of degrees ≥ 3 whose classes together with
im D_{q-1} span the degree-q relation families (q ≥ 3; a basis of the cokernel in the
source), and any finite filtered category E on the vertices with gr E = C, there are scalars
λ and arrow lifts satisfying r_v = Σ λ_j (p_j)_v; the realization kQ → E is onto every Hom
space, its kernel is the ideal (𝔪^{N+1}, r_v − Σ λ_j (p_j)_v) for every N with F^N = 0, so
E ≅ kQ/(𝔪^{N+1}, r_v − Σ λ_j (p_j)_v) (full and faithful Φ), and it induces the given
identification gr E = C.