Filtrations in hyperhomology
For typical complexes, hyperhomology and its two filtrations an intrinsic description is given independent of the hyperhomology apparatus. Let \({\mathcal N}_*\) be a complex and \({\mathcal I}^*{\mathcal N}_*\) an injective Cartan-Eilenberg resolution. The filtrations of the hyperhomology of \({\mathcal N}_*\) for a functor \(F\) are determined by two filtrations of the bigraded object \({\mathcal I}^* {\mathcal N}_*\), viz., by subobjects \(K_p\) of the form \(\oplus_{i,j} {\mathcal I}^i {\mathcal N}_j\) with \(j\leq p\) (which is the first spectral sequence in the terminology of \textit{H. Cartan} and \textit{S. Eilenberg} [``Homological algebra'', Princeton Univ. Press, Princeton (1956; Zbl 0075.24305), Chap. XVII, \S 2]) and similar subobjects \(K^q\) with \(q\leq i\), the second sequence in the op. cit. Then there exists the complex \(F({\mathcal I}^*{\mathcal N}_*)\) with the standard total grading \(\oplus_{j-i=n} F({\mathcal I}^i {\mathcal N}_j)\) with respect to the total differential, \(d=d' +(-1)^j d''\), where \(d'\) and \(d''\) are differentials induced by \({\mathcal N}_*\) and \({\mathcal I}^*\) respectively. Let \(\Phi_p \mathbb{H}_m\) and \(\Phi^q \mathbb{H}_m\) be the images of the homology of its subcomplexes \(F(K_p)\) and \(F(K^q) \), respectively, in the hyperhomology \(H_m ({\mathcal N}_*;F) =H_m(F( {\mathcal I}^* {\mathcal N}_*))\) of the complex \({\mathcal N}_*\) for the functor \(F\). They induce the filtrations \[ \mathbb{H}_m({\mathcal N}_*;F) \dots \supset \Phi_{p+1} \mathbb{H}_m \supset \Phi_p \mathbb{H}_m \supset \cdots \supset \Phi_m \mathbb{H}_m \supset \Phi_{p-1} \mathbb{H}_m=0 \] and \[ \mathbb{H}_m ({\mathcal N}_*; F)= \Phi^0 \mathbb{H}_m \supset \Phi^1 \mathbb{H}_m \supset\cdots \supset \Phi^q \mathbb{H}_m \supset \dots \] in the groups \(\mathbb{H}_m ({\mathcal N}_*;F)\). Any \(F\)-acyclic Cartan-Eilenberg resolution \(C^*{\mathcal N}_*\) of the complex \({\mathcal N}_*\) admits a resolution transformation \(\alpha: C^*{\mathcal N}_* \to {\mathcal I}^* {\mathcal N}_* \), which is unique up to homotopy in the sense of Cartan-Eilenberg [op. cit., Chap. IV, \S 4]. The transformation \(\alpha\) induces isomorphisms of spectral sequences of the first and second kind (i.e., the sequences whose second terms \(E_2^{pq}\) are \(H_{-p} (F^q ({\mathcal N}_*))\) and \(F^p(H_{-q} ({\mathcal N}_*))\), respectively). Since the filtrations of the first kind are regular, \(\alpha\) induces an isomorphism \(\alpha_*: H_*(F(C^* {\mathcal N}_*)) \to\mathbb{H}_* ({\mathcal N}_*;F)\) of the filtered groups of the first kind. The transformation \(\alpha\) also induces the inclusion \(\overline \Phi^q \mathbb{H}_m \subset \Phi^q \mathbb{H}_m\), where the subgroups of the second kind \(\widetilde \Phi^q \mathbb{H}_m\) that filter \(\mathbb{H}_m ({\mathcal N}_*; F)\) are determined by \(C^*{\mathcal N}_*\). The filtration of the second kind does not depend of the injective resolution \({\mathcal I}^* {\mathcal N}_*\). The following proposition 1 gives an intrinsic description of the hyperhomology \(\mathbb{H}_m ({\mathcal N}_*;F)\) and the filtration of the first kind: If \(H_*({\mathcal N}_*)=0\), then the group \(\mathbb{H}_m ({\mathcal N}_*;F)\) is the direct limit of the sequence formed by the groups \(F^k(Z_{m+k})\) and their homomorphisms \((-1)^{m+k_i k_{m+k}}\). The subgroups \(\Phi_p\mathbb{H}_m\) coincide with the images of the groups \(F^p(Z_{m+p})\) in the direct limit. \((Z_m\) is the kernel of the boundary operator and \(i_{m,k}:F^k(Z_m)\to F^{k+1} (Z_{m+1})\) the homomorphism induced by the exact sequence \(0\to Z_{m+1} \to {\mathcal N}_{m+1} \to Z_m\to 0\) for \({\mathcal N}_m\) the images of the boundary operator of \({\mathcal N}_*\). For the second typical special case when the complex is formed of \(F\)-acyclic objects, the author proves proposition 2: If \({\mathcal N}_*\) consists of \(F\)-acyclic objects, then the second filtration in \(\mathbb{H}_m({\mathcal N}_*;F) =H_m (F({\mathcal N}_*))\) is determined by the images of the groups \(F^q (Z_{m+q})\) under iterations of the homomorphisms \((-1)^{m+ k_ik_{m+k}}\) with \(k\leq q\).
scientific article; zbMATH DE number 1237998