Complex Analytic and Differential Geometry 2025, 12 - Spectral Sequences of Filtered Complexes

A filtration of an R-module M for a commutative ring R of submodules M_p = F_p(M) in M, p ∈ ℤ, with M_p+1 ⊂ M_p ∀ p, and ⋓M_p = M and ⋒M_p = {0}. The associated graed module G(M) = ⊕_p∈ℤG_p(M), G_p(M) = M_p/M_p+1. A differential module (K, d) with filtration (K_p) by differential submodules, and any finite real number r defines Z_r^p = K_p∩d^-1K_p+r mod K_p+1, Z_∞^p = K_p∩d^-1{0} mod K_p+1 B_r^p = K_p∩dK_p-r+1 mod K_p+1, B_∞^p = K_p∩dK mod K_p+1 ... ⊂ B_r^p ⊂ B_r+1^p ⊂... ⊂ B_∞^p ⊂ Z_∞^p ⊂ ... ⊂ Z_r+1^p ⊂ Z_r^p ⊂ ... The differential d induces an isomorphism. A canonical isomorphism E^˙_r+1 ≡ H^˙(E^˙_r) exists, and the sequence of differential complexes (E, d) is called the spectral sequence of the filtered differential module (K, d). The filtration of the homology module H(K) defined by F_p(H(K)) = Im(H(K_p) → H(K)), then there exists a canonical isomorphism E^p_∞ = G_p(H(K)).

A filtration (K˙p) of a complex K˙ is regular if for each degree l there are indices ν(l) ≤ N(l) with Klp = Kl for p < ν(l) and 0 for p > N(l). A spectral sequence collapses in E˙r if all terms Z, B, E are constant for k ≥ r or if dk = 0 in all bidegrees for k ≥ r. Assume an integer r ≥ 2 and index q0 with Erp,q = 0 for q ≠ q0, then this remains true for larger r and dr = 0. The spectral sequence collapses in E˙r, and Hl(K˙) = El-q0, q0r.

There is an exact sequence 0→E₂^1,0→H¹(K^˙)→E₂^0,1→^d₂E₂^2,0→H²(K^˙) where the non-indicated arrows are edge homomorphism. Let K^p,q → L^{p, q} a morphism of double complexes, then they induce morphisms _KE_r^{˙, ˙} → _LE_r^{˙, ˙} , r ≥ 0 of the associated spectral sequences. If one of them is isomorphic for some r, then H^l(K^˙) → H^l(L^˙) is isomorphic.

The groupd H^q(K^˙_L) are "hypercohomology groups" of L^˙, denoted ℍ^q(K^˙, L^˙). They are connected via natural morphism of double complexes. If L^q = 0 for q ≠ 0, then ℍ^q(K^˙, L^˙) = H^q(K⁰, L^˙). f L^˙[s] denotes the complex L^˙ shifted of s indices to the right, then ℍ^q(K^˙, L^˙[s]) = ℍ^q-s(K^˙, L^˙). Hypercohomology groups retain exact sequences of sheaf complexes. A quasi-isomorphism φ^˙: L^˙ → M^˙ implies an isomorphism ℍ^l(φ^˙): ℍ^l(X, L^˙) → ℍ^l(X, M^˙).