Grothendieck spectral sequence

If F \colon\mathcal{A}\to\mathcal{B} and G \colon \mathcal{B}\to\mathcal{C} are two additive and left exact functors between abelian categories such that both \mathcal{A} and \mathcal{B} have enough injectives and F takes injective objects to G-acyclic objects, then for each object A of \mathcal{A} there is a spectral sequence: :E_2^{pq} = ({\rm R}^p G \circ{\rm R}^q F)(A) \Longrightarrow {\rm R}^{p+q} (G\circ F)(A), where {\rm R}^p G denotes the p-th right-derived functor of G, etc., and where the arrow '\Longrightarrow' means convergence of spectral sequences. Five term exact sequence The exact sequence of low degrees reads :0\to {\rm R}^1G(FA)\to {\rm R}^1(GF)(A) \to G({\rm R}^1F(A)) \to {\rm R}^2G(FA) \to {\rm R}^2(GF)(A). == Examples ==

The Leray spectral sequence If X and Y are topological spaces, let \mathcal{A} = \mathbf{Ab}(X) and \mathcal{B} = \mathbf{Ab}(Y) be the category of sheaves of abelian groups on X and Y, respectively. For a continuous map f \colon X \to Y there is the (left-exact) direct image functor f_* \colon \mathbf{Ab}(X) \to \mathbf{Ab}(Y). We also have the global section functors :\Gamma_X \colon \mathbf{Ab}(X)\to \mathbf{Ab} and \Gamma_Y \colon \mathbf{Ab}(Y) \to \mathbf {Ab}. Then since \Gamma_Y \circ f_* = \Gamma_X and the functors f_* and \Gamma_Y satisfy the hypotheses (since the direct image functor has an exact left adjoint f^{-1}, pushforwards of injectives are injective and in particular acyclic for the global section functor), the sequence in this case becomes: :H^p(Y,{\rm R}^q f_*\mathcal{F})\implies H^{p+q}(X,\mathcal{F}) for a sheaf \mathcal{F} of abelian groups on X. Local-to-global Ext spectral sequence There is a spectral sequence relating the global Ext and the sheaf Ext: let F, G be sheaves of modules over a ringed space (X, \mathcal{O}); e.g., a scheme. Then :E^{p,q}_2 = \operatorname{H}^p(X; \mathcal{E}xt^q_{\mathcal{O}}(F, G)) \Rightarrow \operatorname{Ext}^{p+q}_{\mathcal{O}}(F, G). This is an instance of the Grothendieck spectral sequence: indeed, :R^p \Gamma(X, -) = \operatorname{H}^p(X, -), R^q \mathcal{H}om_{\mathcal{O}}(F, -) = \mathcal{E}xt^q_{\mathcal{O}}(F, -) and R^n \Gamma(X, \mathcal{H}om_{\mathcal{O}}(F, -)) = \operatorname{Ext}^n_{\mathcal{O}}(F, -). Moreover, \mathcal{H}om_{\mathcal{O}}(F, -) sends injective \mathcal{O}-modules to flasque sheaves, which are \Gamma(X, -)-acyclic. Hence, the hypothesis is satisfied. == Derivation ==

We shall use the following lemma: {{math_theorem|name=Lemma|If K is an injective complex in an abelian category C such that the kernels of the differentials are injective objects, then for each n, :H^n(K^{\bullet}) is an injective object and for any left-exact additive functor G on C, :H^n(G(K^{\bullet})) = G(H^n(K^{\bullet})).}} Proof: Let Z^n, B^{n+1} be the kernel and the image of d: K^n \to K^{n+1}. We have :0 \to Z^n \to K^n \overset{d}\to B^{n+1} \to 0, which splits. This implies each B^{n+1} is injective. Next we look at :0 \to B^n \to Z^n \to H^n(K^{\bullet}) \to 0. It splits, which implies the first part of the lemma, as well as the exactness of :0 \to G(B^n) \to G(Z^n) \to G(H^n(K^{\bullet})) \to 0. Similarly we have (using the earlier splitting): :0 \to G(Z^n) \to G(K^n) \overset{G(d)} \to G(B^{n+1}) \to 0. The second part now follows. \square We now construct a spectral sequence. Let A^0 \to A^1 \to \cdots be an injective resolution of A. Writing \phi^p for F(A^p) \to F(A^{p+1}), we have: :0 \to \operatorname{ker} \phi^p \to F(A^p) \overset{\phi^p}\to \operatorname{im} \phi^p \to 0. Take injective resolutions J^0 \to J^1 \to \cdots and K^0 \to K^1 \to \cdots of the first and the third nonzero terms. By the horseshoe lemma, their direct sum I^{p, \bullet} = J \oplus K is an injective resolution of F(A^p). Hence, we found an injective resolution of the complex: :0 \to F(A^{\bullet}) \to I^{\bullet, 0} \to I^{\bullet, 1} \to \cdots. such that each row I^{0, q} \to I^{1, q} \to \cdots satisfies the hypothesis of the lemma (cf. the Cartan–Eilenberg resolution.) Now, the double complex E_0^{p, q} = G(I^{p, q}) gives rise to two spectral sequences, horizontal and vertical, which we are now going to examine. On the one hand, by definition, :{}^{\prime \prime} E_1^{p, q} = H^q(G(I^{p, \bullet})) = R^q G(F(A^p)), which is always zero unless q = 0 since F(A^p) is G-acyclic by hypothesis. Hence, {}^{\prime \prime} E_{2}^n = R^n (G \circ F) (A) and {}^{\prime \prime} E_2 = {}^{\prime \prime} E_{\infty}. On the other hand, by the definition and the lemma, :{}^{\prime} E^{p, q}_1 = H^q(G(I^{\bullet, p})) = G(H^q(I^{\bullet, p})). Since H^q(I^{\bullet, 0}) \to H^q(I^{\bullet, 1}) \to \cdots is an injective resolution of H^q(F(A^{\bullet})) = R^q F(A) (it is a resolution since its cohomology is trivial), :{}^{\prime} E^{p, q}_2 = R^p G(R^qF(A)). Since {}^{\prime} E_r and {}^{\prime \prime} E_r have the same limiting term, the proof is complete. \square ==Notes==