Sheaf cohomology - Leray spectral sequence (general form)

Intuition Beginner

The Leray spectral sequence is a way to compute cohomology on a space (X) by using a map from (X) to another space (Y).

First push the sheaf from (X) down to (Y). This produces an ordinary pushforward sheaf and, in higher degrees, higher pushforward sheaves. Then compute cohomology on (Y) with those sheaves.

The spectral sequence says that this two-stage information assembles into the cohomology of the original sheaf on (X). In practice, this lets a difficult cohomology problem on (X) be replaced by data over the often simpler base (Y).

Visual Beginner

The vertical direction records higher pushforwards. The horizontal direction records cohomology on the target.

Worked example Beginner

Suppose a family of spaces sits over a base space (Y). A sheaf on the total space (X) can be examined fiber by fiber and then organized over (Y).

The higher pushforward sheaves store how cohomology varies along the fibers. Taking cohomology of those sheaves on (Y) accounts for how the fiberwise information glues across the base. Leray's spectral sequence combines these two layers into cohomology on (X).

This is why Leray is central in algebraic geometry: many spaces are studied through maps to simpler spaces.

Check your understanding Beginner

Formal definition Intermediate+

Let $$ f\to Y $$ be a continuous map, and let (\mathcal F) be a sheaf of abelian groups on (X). The direct image functor $$ f_*:\operatorname{Sh}(X)\to\operatorname{Sh}(Y) $$ is left exact, and its right-derived functors are the higher direct images (R^qf_*\mathcal F).

The global-sections functor on (Y) is $$ \Gamma(Y,-):\operatorname{Sh}(Y)\to\mathbf{Ab}. $$ Since $$ \Gamma(Y,f_*\mathcal F)=\Gamma(X,\mathcal F), $$ the global-sections functor on (X) factors as $$ \Gamma(X,-)=\Gamma(Y,-)\circ f_*. $$

Applying the Grothendieck spectral sequence to this composite gives the Leray spectral sequence: $$ E_2^{p,q}=H^p\bigl(Y,R^qf_*\mathcal F\bigr) \Rightarrow H^{p+q}(X,\mathcal F). $$

The abutment carries a finite filtration in each total degree under the usual boundedness hypotheses: $$ E_\infty^{p,q}\cong F^pH^{p+q}(X,\mathcal F)/F^{p+1}H^{p+q}(X,\mathcal F). $$

Key theorem with proof Intermediate+

Theorem (Leray spectral sequence). For every continuous map (f\to Y) and every sheaf of abelian groups (\mathcal F) on (X), there is a natural first-quadrant spectral sequence $$ E_2^{p,q}=H^p(Y,R^qf_*\mathcal F) \Rightarrow H^{p+q}(X,\mathcal F). $$

Proof. Work in the abelian categories of sheaves of abelian groups on (X) and (Y). The functors (f_*) and (\Gamma(Y,-)) are left exact, and their composite is (\Gamma(X,-)).

We verify the acyclicity hypothesis in Grothendieck's spectral sequence. If (I) is an injective sheaf on (X), then (I) is flabby. The direct image of a flabby sheaf is flabby: for open sets (V\subseteq U) in (Y), the restriction $$ (f_*I)(U)=I(f^{-1}U)\to I(f^{-1}V)=(f_*I)(V) $$ is surjective because (f^{-1}V\subseteq f^{-1}U). Flabby sheaves are acyclic for global sections, so (f_*I) is (\Gamma(Y,-))-acyclic.

Thus (f_*) sends injective sheaves on (X) to (\Gamma(Y,-))-acyclic sheaves on (Y). The Grothendieck spectral sequence for $$ \operatorname{Sh}(X)\xrightarrow{f_*}\operatorname{Sh}(Y) \xrightarrow{\Gamma(Y,-)}\mathbf{Ab} $$ therefore gives $$ E_2^{p,q} =R^p\Gamma(Y,-)(R^qf_*\mathcal F) =H^p(Y,R^qf_*\mathcal F) \Rightarrow R^{p+q}\Gamma(X,\mathcal F). $$ The target is (H^{p+q}(X,\mathcal F)), proving the theorem.

Basic consequences Intermediate+

If (Y) is a point, then (f_*\mathcal F) is the group of global sections on (X), and the spectral sequence recovers ordinary sheaf cohomology.

If (R^qf_*\mathcal F=0) for all (q>0), the spectral sequence collapses to $$ H^p(Y,f_*\mathcal F)\cong H^p(X,\mathcal F). $$ This is the common vanishing shortcut.

If (i\hookrightarrow X) is a closed inclusion and (i_*) is exact on the sheaf category under discussion, then higher direct images vanish and Leray identifies the cohomology of (Z) with the cohomology of (X) with coefficients in (i_*\mathcal F): $$ H^p(X,i_*\mathcal F)\cong H^p(Z,\mathcal F). $$

Relation to higher direct images Intermediate+

The sheaf (R^qf_*\mathcal F) is local on (Y): for an open set (U\subseteq Y), it records the (q)-th derived pushforward over (U). Under favorable hypotheses its stalk at a point (y\in Y) compares to the cohomology of the fiber over (y).

Thus Leray has two readings:

• geometrically, it computes global cohomology by first organizing fiberwise or relative cohomology over the base;
• categorically, it is the Grothendieck spectral sequence for the factorization (\Gamma_X=\Gamma_Y\circ f_*).

Both readings are needed in algebraic geometry. The geometric reading drives examples; the categorical reading supplies functoriality and the proof.

Edge maps and low-degree terms Master

The five-term exact sequence attached to Leray begins $$ 0\to H^1(Y,f_*\mathcal F)\to H^1(X,\mathcal F) \to H^0(Y,R^1f_*\mathcal F) \to H^2(Y,f_*\mathcal F) \to H^2(X,\mathcal F). $$

The map $$ H^1(X,\mathcal F)\to H^0(Y,R^1f_*\mathcal F) $$ restricts a global cohomology class to its relative degree-one behavior over the base. The connecting map after it measures whether that relative class glues to an absolute class on (X).

These edge maps are often more useful than the whole spectral sequence in low-dimensional problems.

Collapse criteria Master

The Leray spectral sequence collapses when degree reasons or vanishing force all possible differentials to be zero. Common cases include:

• (R^qf_*\mathcal F=0) for (q>0);
• (H^p(Y,R^qf_*\mathcal F)=0) for (p>0);
• the (E_2)-page has nonzero entries in too few rows or columns for a nonzero differential to start and land.

Even after collapse, extension problems may remain: (E_\infty) gives the associated graded pieces of a filtration on (H^n(X,\mathcal F)). Additional structure, such as cup products, weights, dimensions, or module decompositions, may be needed to identify the cohomology group itself.

Historical note Master

Leray introduced the spectral sequence in 1946 while developing sheaf methods for maps of spaces. Grothendieck's 1957 Tohoku paper recast the construction as an instance of the composite-functor spectral sequence in abelian categories. Godement and later algebraic-geometry texts made this the standard proof of the sheaf-theoretic Leray spectral sequence.

The result remains central because it converts a map (X\to Y) into a cohomological computation over (Y), where geometry, vanishing theorems, and base-change statements are often easier to apply.

Connections Master

• 04.03.13 supplies the Grothendieck spectral sequence used in the proof.
• 04.03.14 supplies the filtered-complex convergence language behind the pages and abutment.
• 04.03.07 studies higher direct images and base-change refinements.
• 03.13.02 is the Leray-Serre fibration form, distinct from this sheaf-theoretic form for a general continuous map.

Exercises Intermediate+

Bibliography Master

• J. Leray, "L'anneau d'homologie d'une représentation," Comptes Rendus de l'Académie des Sciences de Paris 222 (1946), 1366-1368.
• A. Grothendieck, "Sur quelques points d'algèbre homologique," Tohoku Mathematical Journal 9 (1957).
• R. Godement, Topologie algébrique et théorie des faisceaux, Hermann, 1958.
• R. Hartshorne, Algebraic Geometry, Springer, Ch. III.
• B. Iversen, Cohomology of Sheaves, Springer, 1986.