Leray spectral sequence for sheaf cohomology

By [F1] fix the supplied injective resolution datum $\mathcal{F}\mapsto I^{\bullet }(\mathcal{F})$ in $\mathrm{A}\mathrm{b}(X)$ and use it throughout; the complex $f_{\ast }{I}^{\bullet }$ is then a complex of sheaves on $Y$, and the higher direct images are the sheaves $R^q{f}_{\ast }\mathcal{F}$ defined as the cohomology sheaves of $f_{\ast }{I}^{\bullet }$ in the in-run definition def-higher-direct-image-sheaf, which agrees with the abelian-sheaf construction used here.

The inverse-image functor $f^{-1}$ is exact: by [F3] the stalk of $f^{-1}$ at $x$ is the stalk of the original sheaf at $f(x)$, so $f^{-1}$ is stalkwise the exact functor of taking stalks at a point, and [F4] upgrades stalkwise exactness to exactness of the sequence of sheaves. Since $f^{-1}$ is exact and left adjoint to $f_{\ast }$ by [F5], the right adjoint $f_{\ast }$ preserves injective objects: if $I$ is injective and $M\rightarrowtail N$ is a monomorphism, every map $M\to f_{\ast }I$ corresponds to $f^{-1}M\to I$, extends along the monomorphism $f^{-1}M\rightarrowtail f^{-1}N$ by injectivity of $I$, and transposes back to an extension $N\to f_{\ast }I$.

Hence $f_{\ast }$ carries injective abelian sheaves on $X$ to injective sheaves on $Y$, and in particular to $\mathrm{\Gamma }(Y,-)$-acyclic sheaves, since injective objects are acyclic for any additive left-exact functor whose derived functors are computed on injective resolutions.

Apply the Grothendieck spectral sequence [F6] to the composite of the additive left-exact functors $f_{\ast }:\mathrm{A}\mathrm{b}(X)\to \mathrm{A}\mathrm{b}(Y)$ and $\mathrm{\Gamma }(Y,-):\mathrm{A}\mathrm{b}(Y)\to \mathbf{A}\mathbf{b}$, whose composite is $\mathrm{\Gamma }(Y,f_{\ast }(-))=\mathrm{\Gamma }(X,-)$ by the definition of the direct image; both categories have enough injectives by [F1] applied to $X$ and to $Y$, and the acyclicity hypothesis is exactly step 2.1.

The resulting spectral sequence has $E_2^{p,q}=R^p\mathrm{\Gamma }(Y,-)(R^q{f}_{\ast }(-))(\mathcal{F})=H^p(Y,R^q{f}_{\ast }\mathcal{F})$ by [F2] applied on $Y$, and abutment $R^{p+q}\mathrm{\Gamma }(X,-)(\mathcal{F})=H^{p+q}(X,\mathcal{F})$; this is the displayed spectral sequence, with differentials of bidegree $(r,1-r)$ by [F6].

Since $R^q{f}_{\ast }\mathcal{F}=0$ for $q<0$ and $H^p(Y,-)=0$ for $p<0$, the spectral sequence of step 4.1 is first-quadrant, and [F6] supplies a finite decreasing filtration of $H^n(X,\mathcal{F})$ with ${\mathrm{gr}}^p{H}^n\cong E_{\mathrm{\infty }}^{p,n-p}$; in particular only finitely many terms contribute to $H^n$ in each total degree.

Naturality and the edge maps: the Grothendieck spectral sequence of [F6] is natural in the object $\mathcal{F}$ and in the pair of functors, so a morphism $\mathcal{F}\to \mathcal{G}$ of abelian sheaves induces a morphism of spectral sequences compatible with the filtrations; its edge maps are the natural maps $H^n(Y,f_{\ast }\mathcal{F})\to H^n(X,\mathcal{F})$ arising from the canonical identity of global-section functors $\mathrm{\Gamma }(Y,f_{\ast }(-))=\mathrm{\Gamma }(X,-)$: on an injective resolution the two complexes $\mathrm{\Gamma }(Y,f_{\ast }{I}^{\bullet })$ and $\mathrm{\Gamma }(X,I^{\bullet })$ agree degreewise, and $H^n(X,\mathcal{F})\to H^0(Y,R^n{f}_{\ast }\mathcal{F})$, which are the standard edge homomorphisms of a first-quadrant spectral sequence.

For an arbitrary morphism of schemes, let $\mathcal{F}\to J^{\bullet }$ be a module-injective resolution. Each $J^r$ is flasque as an abelian sheaf by [F7], hence acyclic for sections on every open. A flasque abelian sheaf $J$ is also $f_{\ast }$-acyclic: take an abelian injective resolution $J\to I^{\bullet }$; over each $f^{-1}U$, its terms are flasque and compute $H^q(f^{-1}U,J)=0$ for $q>0$. Thus the complex $f_{\ast }{I}^{\bullet }$ is exact in positive degrees on sections over all opens $U$, and hence as a complex of sheaves. The acyclic-resolution theorem [F7] now identifies $H^q(f_{\ast }{J}^{\bullet })$ with the abelian higher direct images and identifies $H^q(\mathrm{\Gamma }(X,J^{\bullet }))$ with abelian cohomology. The same comparison on $Y$ identifies module and abelian cohomology there. Alternatively, apply [F6] directly to module direct image and global sections: $f_{\ast }{J}^r$ is flasque, since its restrictions are restrictions of $J^r$, and is therefore global-sections-acyclic by the comparison just proved. This yields the claimed module spectral sequence, edges and convergence, with no characteristic assumption. The flatness observation in the statement is a sufficient shortcut over $\mathbb{C}$, not a hypothesis needed for this general argument.

The Axiom of Choice is used exactly in step 1.1, where the supplied injective-resolution datum of [F1] is chosen, and in step 5.3 through the $\mathcal{O}$-module injective supply and flasque acyclicity; AC also supplies the DC required in [F6] and [F7]; the comparison theorems for injective resolutions and the identification of cohomology across resolutions are the published AC-qualified data of [F2]. The statement claims the spectral sequence, its convergence and its edge maps, and nothing about degeneration or about splitting of the filtration.