Perturbing the tensor trick preserves the coalgebra structure

The tensor trick (TauCeti.LinearSpecialContraction.reducedTensorWords) turns a special contraction of (M, d) onto (N, d') into a special contraction of the reduced tensor coalgebras Tᶜ(M) onto Tᶜ(N), with letterwise inclusion i and projection p, and with homotopy H. Homological transfer then perturbs the letterwise differential D of Tᶜ(M) by a graded coderivation δ which lowers tensor length, such as the part of the bar differential of an A∞ algebra collapsing at least two letters, and applies the basic perturbation lemma (TauCeti.LinearSpecialContraction.perturb).

This file shows that the output of the perturbation lemma is again compatible with deconcatenation: the perturbed inclusion i' = i - H X i and the perturbed projection p' = p - p X H are morphisms of reduced tensor coalgebras, and the perturbed differential D' = D + p X i of Tᶜ(N) is a graded coderivation. Consequently D' is determined by its letter component, which carries the transferred operations on N, and i' and p' are determined by their Taylor components.

The key input is that H is a coderivation homotopy: with τ the letterwise Koszul twist,

Δ H = (H ⊗ i p + τ ⊗ H) Δ

(TauCeti.LinearSpecialContraction.deconcatenation_comp_reducedTensorWordsHomotopy).

Main results

• TauCeti.LinearSpecialContraction.isCoalgHom_reducedTensorWords_perturb_incl: the perturbed inclusion is a coalgebra morphism.
• TauCeti.LinearSpecialContraction.isCoalgHom_reducedTensorWords_perturb_proj: the perturbed projection is a coalgebra morphism.
• TauCeti.LinearSpecialContraction.isGradedCoderivation_reducedTensorWords_perturbedDifferential: the perturbed differential is a graded coderivation.

References

• V. K. A. M. Gugenheim, L. A. Lambe, and J. D. Stasheff, Perturbation theory in differential homological algebra II, Illinois Journal of Mathematics 35 (1991), 357--373.
• J. Huebschmann and T. Kadeishvili, Small models for chain algebras, Mathematische Zeitschrift 207 (1991), 245--280.

Local nilpotence on pairs of words

The perturbed tensor-trick contraction

theorem TauCeti.LinearSpecialContraction.isCoalgHom_reducedTensorWords_perturb_incl {R : Type uR} {M : Type uM} {N : Type uN} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {dM : Module.End R M} {dN : Module.End R N} (c : LinearSpecialContraction dM dN) {G : InternalGrading R M} {H : InternalGrading R N} (hdM : LinearMap.IsHomogeneous dM G.piece G.piece 1) (hh : LinearMap.IsHomogeneous c.homotopy G.piece G.piece (-1)) (hincl : LinearMap.IsHomogeneous c.incl H.piece G.piece 0) (hproj : LinearMap.IsHomogeneous c.proj G.piece H.piece 0) {δ : Module.End R (ReducedTensorWords R M)} (hsq : (ReducedTensorWords.gradedCoderiv G (dM ∘ₗ ReducedTensorWords.letter R M) 1 + δ) ∘ₗ (ReducedTensorWords.gradedCoderiv G (dM ∘ₗ ReducedTensorWords.letter R M) 1 + δ) = ReducedTensorWords.gradedCoderiv G (dM ∘ₗ ReducedTensorWords.letter R M) 1 ∘ₗ ReducedTensorWords.gradedCoderiv G (dM ∘ₗ ReducedTensorWords.letter R M) 1) (hU : IsUnit (1 + δ * (c.reducedTensorWords G H hdM hh hincl hproj).homotopy)) (hδ : ReducedTensorWords.IsGradedCoderivation G 1 δ) (hδfil : ∀ (n : ℕ), Submodule.map δ (ReducedTensorWords.filtration R M (n + 1)) ≤ ReducedTensorWords.filtration R M n) : ReducedTensorWords.IsCoalgHom R ((c.reducedTensorWords G H hdM hh hincl hproj).perturb δ hsq hU).incl

The coalgebra perturbation lemma for the tensor trick. Perturbing the tensor-trick contraction by a graded coderivation δ which lowers tensor length, the perturbed inclusion i' = i - H X i is again a morphism of reduced tensor coalgebras. The unit hypothesis hU of the perturbation lemma holds automatically for such δ, by TauCeti.ReducedTensorWords.exists_pow_comp_apply_eq_zero_of_filtration_lowering and Module.End.isUnit_one_add_of_forall_exists_pow_apply_eq_zero.

theorem TauCeti.LinearSpecialContraction.isCoalgHom_reducedTensorWords_perturb_proj {R : Type uR} {M : Type uM} {N : Type uN} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {dM : Module.End R M} {dN : Module.End R N} (c : LinearSpecialContraction dM dN) {G : InternalGrading R M} {H : InternalGrading R N} (hdM : LinearMap.IsHomogeneous dM G.piece G.piece 1) (hh : LinearMap.IsHomogeneous c.homotopy G.piece G.piece (-1)) (hincl : LinearMap.IsHomogeneous c.incl H.piece G.piece 0) (hproj : LinearMap.IsHomogeneous c.proj G.piece H.piece 0) {δ : Module.End R (ReducedTensorWords R M)} (hsq : (ReducedTensorWords.gradedCoderiv G (dM ∘ₗ ReducedTensorWords.letter R M) 1 + δ) ∘ₗ (ReducedTensorWords.gradedCoderiv G (dM ∘ₗ ReducedTensorWords.letter R M) 1 + δ) = ReducedTensorWords.gradedCoderiv G (dM ∘ₗ ReducedTensorWords.letter R M) 1 ∘ₗ ReducedTensorWords.gradedCoderiv G (dM ∘ₗ ReducedTensorWords.letter R M) 1) (hU : IsUnit (1 + δ * (c.reducedTensorWords G H hdM hh hincl hproj).homotopy)) (hδ : ReducedTensorWords.IsGradedCoderivation G 1 δ) : ReducedTensorWords.IsCoalgHom R ((c.reducedTensorWords G H hdM hh hincl hproj).perturb δ hsq hU).proj

The perturbed projection is a coalgebra morphism. Perturbing the tensor-trick contraction by a graded coderivation δ, the perturbed projection p' = p - p X H is again a morphism of reduced tensor coalgebras. Unlike the perturbed inclusion, no filtration hypothesis on δ is needed beyond the invertibility of 1 + δ H.

theorem TauCeti.LinearSpecialContraction.isGradedCoderivation_reducedTensorWords_perturbedDifferential {R : Type uR} {M : Type uM} {N : Type uN} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {dM : Module.End R M} {dN : Module.End R N} (c : LinearSpecialContraction dM dN) {G : InternalGrading R M} {H : InternalGrading R N} (hdM : LinearMap.IsHomogeneous dM G.piece G.piece 1) (hh : LinearMap.IsHomogeneous c.homotopy G.piece G.piece (-1)) (hincl : LinearMap.IsHomogeneous c.incl H.piece G.piece 0) (hproj : LinearMap.IsHomogeneous c.proj G.piece H.piece 0) {δ : Module.End R (ReducedTensorWords R M)} (hsq : (ReducedTensorWords.gradedCoderiv G (dM ∘ₗ ReducedTensorWords.letter R M) 1 + δ) ∘ₗ (ReducedTensorWords.gradedCoderiv G (dM ∘ₗ ReducedTensorWords.letter R M) 1 + δ) = ReducedTensorWords.gradedCoderiv G (dM ∘ₗ ReducedTensorWords.letter R M) 1 ∘ₗ ReducedTensorWords.gradedCoderiv G (dM ∘ₗ ReducedTensorWords.letter R M) 1) (hU : IsUnit (1 + δ * (c.reducedTensorWords G H hdM hh hincl hproj).homotopy)) (hδ : ReducedTensorWords.IsGradedCoderivation G 1 δ) (hδdeg : LinearMap.IsHomogeneous δ (ReducedTensorWords.gradedPiece G) (ReducedTensorWords.gradedPiece G) 1) (hδfil : ∀ (n : ℕ), Submodule.map δ (ReducedTensorWords.filtration R M (n + 1)) ≤ ReducedTensorWords.filtration R M n) : ReducedTensorWords.IsGradedCoderivation H 1 ((c.reducedTensorWords G H hdM hh hincl hproj).perturbedDifferential δ)

The perturbed tensor-trick differential is a coderivation. Perturbing the tensor-trick contraction by an odd graded coderivation δ which lowers tensor length, the perturbed differential D' = D + p X i of Tᶜ(N) is again a graded coderivation for the Koszul signs of the grading of N.