A coboundary morphism for the Grothendieck spectral sequence (Q2015829)

Recall the content of the Grothendieck spectral sequence [\textit{A. Grothendieck}, Sur quelques points d'algèbre homologique, Tohoku Math. J., II. Ser. 9, 119--221 (1957; Zbl 0118.26104)]. Let \(\mathcal{A}, \mathcal{B}, \mathcal{C}\) be abelian categories and let \(F: \mathcal{A}\to \mathcal{B}\), \(G: \mathcal{B}\to \mathcal{C}\) be left exact functors. Suppose \(\mathcal{A}\), \(\mathcal{B}\) have enough injectives and \(F\) sends injective objects to \(G\)-acyclic objects. Given an object \(A\in \mathcal{A}\) there is a spectral sequence \(\{E_r^{p,q}(A), d^r\}\) consisting of objects in \(\mathcal{C}\) and filtration \(F^p R^n(G\circ F)(A)\) on \(R^n(G\circ F)(A)\) such that the spectral sequence converges to the assiciated graded objects and such that \(E_2^{p,q}(A)=R^p G(R^q F(A))\). This spectral sequence is functorial in \(A\). Consider a short exact sequence \(0\to A \to B\to C\to 0\) in \(\mathcal{A}\). The paper under review gives an answer to the question how the spectral sequences and filtrations associated to \(A\) and \(C\) are related. Theorem 1. Let \(0\to A \to B\to C\to 0\) be a short exact sequence in \(\mathcal{A}\). There are morphisms \(\delta_r: E_r^{p,q}(C)\to E_r^{p,q+1}(A)\) for \(r\geq 2\) between the Grothendieck spectral sequences for \(C\) and \(A\) with the following properties: \begin{itemize} \item- \(\delta_r\) commutes with the differentials \(d_r\) and the induced map at the \((r+1)\)-stage is \(\delta_{r+1}\). \item - \(\delta_2: R^p G(R^q F (C)) \to R^p G(R^{q+1} F (C))\) is the map induced by the coboundary morphism \(R^q F(C)\to R^{q+1}F(A)\) in the long exact sequence of derived functors of \(F\) associated to \(0\to A \to B\to C\to 0\). \item - The coboundaries \(R^n(G\circ F)(C)\to R^{n+1}(G\circ F)(A)\) for the long exact sequence associated to \(G\circ F\) send \(F^p R^n(G\circ F)(C)\to F^p R^{n+1}(G\circ F)(A)\) and thus induce maps \(E_{\infty}^{p,q}(C)\to E_{\infty}^{p,{q+1}}(A)\). These maps coincide with \(\delta_{\infty}\) where \(\delta_{\infty}\) denotes the limit of the \(\delta_r\). The Leray spectral sequence is the Grothendieck spectral sequence for functors \(f_*: \mathcal{A}^X \to \mathcal{A}^Y\), \(\Gamma_Y: \mathcal{A}^Y\to \mathcal{A}\) where \(\mathcal{A}^X\) denotes the category of sheaves on \(X\), \(\Gamma_Y\) is defined by \(\Gamma_Y(A)= A(Y)\), and \(f_*\) is the push-down functor. Furthermore, assume that \(X\), \(Y\) are paracompact topological spaces, and moreover that every subspace of \(X\) is paracompact. Let \(\underline{\mathbb{C}}\), \(\underline{\mathbb{C}}^*\) be the sheaves of continuous functions with values in \(\mathbb{C}\), \(\mathbb{C}^*\), where \(\mathbb{C}^*\) is non-zero complex numbers. Theorem 6. Let \(f: X\to Y\) be a map between spaces X,Y, let \(E_r^{p,q}(\mathbb{Z})\), \(E_r^{p,q}(\underline{\mathbb{C}}^*)\) be the Leray spectral sequence associated to the sheaves \(\mathbb{Z}\), \(\underline{\mathbb{C}}^*\) and let \(F^{p,n}(\mathbb{Z})\), \(F^{p,n}(\underline{\mathbb{C}}^*)\) be the associated filtrations on \(H^n(X, \mathbb{Z})\), \(H^n(X, \underline{\mathbb{C}}^*)\). Then \item - The coboundary \(\delta: H^n(X, \underline{\mathbb{C}}^*)\to H^{n+1}(X, \mathbb{Z})\) restricts to morphisms \( \delta: F^{p,p+q}(\underline{\mathbb{C}}^*) \to F^{p,p+q+1}(\mathbb{Z}) \) which are isomorphisms whenever \(p+q\geq 1\) and surjective for \(p=q=0\). \item - The induced quotient maps \(E_{\infty}^{p,q}(\underline{\mathbb{C}}^*)\to E_{\infty}^{p,{q+1}}(\mathbb{Z})\) are isomorphisms for \(q\geq 1\) and surjections for \(q=0\). In conclusion, the author presents an application of Theorem 6 to the topological \(\mathbf{T}\)-duality.\end{itemize}