Graded complexes of free modules over a polynomial ring modulo the variables #
Let S = R[V_v : v ∈ σ] be a polynomial ring and let d be a square-zero S-linear
endomorphism of the free module ι →₀ S. Setting every variable to zero, that is, applying
MvPolynomial.constantCoeff to every coordinate, turns d into an R-linear endomorphism d₀
of ι →₀ R, determined by d₀ ∘ ρ = ρ ∘ d for the reduction ρ. This file proves a graded
Nakayama lemma for such complexes: if each generator i carries an integer degree g i, the
degrees g i are bounded above, every variable V_v has negative degree w v, and d is
homogeneous of some degree r, then d is exact as soon as d₀ is
(LinearMap.ker_le_range_of_mapRange_constantCoeff). Applied to mapping cones, a homogeneous
chain map f between two such complexes induces a bijection on homology as soon as its reduction
f₀ does (LinearMap.homologyMap_bijective_of_mapRange_constantCoeff).
This is the algebraic input for deducing statements about the unblocked grid complexes GC⁻
over 𝔽₂[V₀, …, V_{n-1}] from the fully blocked complexes, in which every variable is set to
zero: there the generators are the finitely many grid states, the variables lower the Maslov
grading by two, and the differentials lower it by one.
The proof filters a cycle z by the powers of the ideal J = (V_v : v ∈ σ). If z lies in
J ^ k • (ι →₀ S), its coefficients at the monomials V ^ e of total degree k form cycles of
d₀, since the higher terms of the matrix coefficients of d only contribute to monomials of
larger degree. Choosing primitives of these cycles under d₀ corrects z by a boundary into
J ^ (k + 1) • (ι →₀ S). The grading makes this process stop: the primitives can be chosen of
degree at least a fixed bound, while an element of J ^ k • (ι →₀ S) has degree at most
max g - k.
Without the grading the statement fails: on S = R[V], multiplication by 1 + V is a chain map
from S to itself, both with zero differential, which becomes the identity after setting V = 0
but is not surjective.
Main results #
LinearMap.ker_le_range_of_mapRange_constantCoeff: a graded square-zero endomorphism of a free module over a polynomial ring is exact if its reduction modulo the variables is.LinearMap.homologyMap_bijective_of_mapRange_constantCoeff: a graded chain map between such complexes induces a bijection on homology if its reduction modulo the variables does.
References #
The graded homological algebra over 𝔽[V₁, …, Vₙ] in which this reduction is used is that of
P. Ozsváth, A. Stipsicz, Z. Szabó, Grid Homology for Knots and Links, AMS Mathematical Surveys
and Monographs 208, 2015, Appendix A.
Graded Nakayama lemma for free complexes over a polynomial ring. Let d be a square-zero
endomorphism of the free module ι →₀ R[V_v : v ∈ σ] which is homogeneous of degree r when the
generator i has degree g i and the variable V_v has negative degree w v, with the degrees
g i bounded above. If the reduction d₀ of d modulo the variables, the endomorphism of
ι →₀ R with d₀ ∘ ρ = ρ ∘ d for ρ the coordinatewise constant coefficient, is exact, then so
is d.
Quasi-isomorphisms of graded free complexes over a polynomial ring are detected modulo the
variables. Let d and e be square-zero endomorphisms of the free modules ι →₀ S and
κ →₀ S over S = R[V_v : v ∈ σ], and f a chain map between them. Suppose the generators
carry degrees g i and g' j, bounded above, the variables V_v have negative degrees w v, the
maps d and e are homogeneous of the same degree r, and f is homogeneous of degree δ. If
the reduction f₀ of f modulo the variables induces a bijection from the homology of the
reduction d₀ of d to that of the reduction e₀ of e, then f induces a bijection on
homology.