Sheaf cohomology

Intuition [Beginner]

Sheaf cohomology measures the global obstructions to assembling local data into global objects. A sheaf gives consistent local data; if you can always glue locally-compatible data into global sections, the sheaf has zero cohomology in degree 1 and above. When gluing fails — when locally-compatible data exists but no global section assembles — you have a nonzero cohomology class.

The simplest example: on the punctured plane ${\mathbb{R}}^2\setminus \{0\}$, the sheaf of "potential functions for the angular 1-form $d\theta$" has local sections everywhere (locally, $\theta$ exists) but no global section (the angle $\theta$ wraps around the origin, so it can't be a global function). This is a nonzero class in $H^1$.

Sheaf cohomology unifies de Rham cohomology, singular cohomology, group cohomology, and Galois cohomology into a single framework. It is the central computational tool of algebraic geometry, where computing cohomology of coherent sheaves on a scheme drives Riemann-Roch, Hodge theory, and much more.

Visual [Beginner]

A topological space with sheaf data on opens; the cohomology measures the "global twisting" of the sheaf, distinct from any single point's data.

Worked example [Beginner]

The classic example: the Mittag-Leffler problem on a Riemann surface. Given specified poles and Laurent tails on a Riemann surface, when can you find a global meromorphic function with exactly that prescribed singular behaviour?

For genus 0 (the sphere): always possible — no obstruction. For higher genus: there are obstructions, captured by $H^1$ of an appropriate sheaf. This higher-genus obstruction is exactly the Riemann-Roch defect.

A concrete numerical example: on the genus-1 torus, you cannot have a meromorphic function with a single simple pole — Riemann-Roch / cohomology says the obstruction lives in $H^1(\text{torus},\mathcal{O})$, which is nonzero (one-dimensional).

The sheaf cohomology framework makes such obstructions computable through systematic methods (Čech, derived functors, spectral sequences).

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $X$ be a topological space 02.01.01 and $\mathcal{F}$ a sheaf of abelian groups on $X$ 04.01.01. The sheaf cohomology groups $H^i(X;\mathcal{F})$ for $i\ge 0$ are defined as the right-derived functors of the global sections functor $\Gamma (X,-):{\mathbf{Sh}}_{\mathbf{Ab}}(X)\to \mathbf{Ab}$:

$$H^i(X;\mathcal{F}):=R^i\Gamma (X,\mathcal{F}).$$

Concretely: choose an injective resolution $\mathcal{F}\to {\mathcal{I}}^{\bullet }$ in ${\mathbf{Sh}}_{\mathbf{Ab}}(X)$, apply $\Gamma$ to get the cochain complex $\Gamma (X,{\mathcal{I}}^{\bullet })$, and take cohomology:

$$H^i(X;\mathcal{F}):=H^i(\Gamma (X,{\mathcal{I}}^{\bullet }))=\frac{\ker (d^i)}{\mathrm{im}(d^{i-1})}.$$

The sheaf-cohomology groups are independent of the choice of injective resolution (standard derived-functor argument).

Properties:

• $H^0(X;\mathcal{F})=\Gamma (X,\mathcal{F})=\mathcal{F}(X)$.
• $H^i(X;-)$ is a covariant functor for each $i$.
• A short exact sequence of sheaves $0\to \mathcal{F}\to \mathcal{G}\to \mathcal{H}\to 0$ induces a long exact sequence in cohomology:

$$0\to H^0(\mathcal{F})\to H^0(\mathcal{G})\to H^0(\mathcal{H})\to H^1(\mathcal{F})\to H^1(\mathcal{G})\to H^1(\mathcal{H})\to H^2(\mathcal{F})\to \cdots$$

This is the central computational tool: short exact sequences of sheaves give long exact sequences in cohomology.

Čech cohomology. A more concrete definition for "nice" sheaves: for an open cover $\mathcal{U}=\{U_i\}$ of $X$, the Čech complex of $\mathcal{F}$ with respect to $\mathcal{U}$ is

$${\check{C}}^p(\mathcal{U};\mathcal{F}):=\prod _{i_0<i_1<\cdots <i_p}\mathcal{F}(U_{i_0}\cap \cdots \cap U_{i_p})$$

with Čech differential ${\delta }^p:{\check{C}}^p\to {\check{C}}^{p+1}$ the alternating-sum-of-restriction map. The Čech cohomology ${\check{H}}^p(\mathcal{U};\mathcal{F})$ is the cohomology of this complex; Čech cohomology of $\mathcal{F}$ on $X$ is the colimit over refining covers, ${\check{H}}^p(X;\mathcal{F})$.

Čech vs de Rham differentials — typographic discipline

When sheaf cohomology is computed via Čech cohomology of a covering and the sheaf in question is itself a complex of sheaves (e.g., the de Rham complex ${\Omega }^{\bullet }$), two different differentials appear: the Čech differential $\delta$ on ${\check{C}}^p$, and the internal differential of the sheaf complex (here the exterior derivative $d$ on ${\Omega }^q$). Bott-Tu §8 fixes this typographic discipline:

• $\delta$ for the Čech differential — alternating-sum restriction on ${\check{C}}^{\ast }$.
• $d$ for the exterior derivative on ${\Omega }^{\ast }$.

The total differential of the Čech-de Rham double complex is $D=d+(-1{)}^p\delta$ on bidegree $(p,q)$, with sign convention chosen so that $D^2=0$. Codex pre-Pass-4 collapsed both into a generic $d^p$; the present unit (D2) and Codex's Čech-de Rham double-complex unit 03.04.11 adopt the $\delta$ vs $d$ convention throughout, both because it is Bott-Tu's originator-text discipline and because it is load-bearing for the tic-tac-toe pedagogy of §9.

Comparison. For "nice" $X$ (paracompact Hausdorff) and any sheaf $\mathcal{F}$, ${\check{H}}^p(X;\mathcal{F})\cong H^p(X;\mathcal{F})$. So Čech cohomology computes derived-functor sheaf cohomology in the cases of interest.

Acyclic sheaves. A sheaf $\mathcal{F}$ is acyclic if $H^i(X;\mathcal{F})=0$ for $i>0$. Important examples: flabby sheaves (every section extends globally), soft sheaves, fine sheaves (admit partitions of unity). On smooth manifolds, sheaves of $C^{\infty }$-functions and their tensor-power variants are fine; this implies de Rham cohomology computes singular cohomology.

Key theorem with proof [Intermediate+]

Theorem (Long exact sequence in sheaf cohomology). Given a short exact sequence of sheaves of abelian groups $0\to \mathcal{F}\to \mathcal{G}\to \mathcal{H}\to 0$ on a topological space $X$, there is a long exact sequence

$$0\to H^0(X;\mathcal{F})\to H^0(X;\mathcal{G})\to H^0(X;\mathcal{H})\stackrel{\delta }{\to }{H}^1(X;\mathcal{F})\to H^1(X;\mathcal{G})\to \cdots$$

Proof sketch. This is the standard derived-functor long exact sequence applied to $\Gamma$.

The global sections functor $\Gamma$ is left exact: if $0\to \mathcal{F}\to \mathcal{G}\to \mathcal{H}\to 0$ is short exact, then $0\to \Gamma (\mathcal{F})\to \Gamma (\mathcal{G})\to \Gamma (\mathcal{H})$ is exact (but the rightmost map need not be surjective).

To extend to a long exact sequence: pick injective resolutions $\mathcal{F}\to {\mathcal{I}}^{\bullet }$ and $\mathcal{H}\to {\mathcal{J}}^{\bullet }$. The horseshoe lemma constructs an injective resolution $\mathcal{G}\to {\mathcal{K}}^{\bullet }$ fitting into a short exact sequence of complexes $0\to {\mathcal{I}}^{\bullet }\to {\mathcal{K}}^{\bullet }\to {\mathcal{J}}^{\bullet }\to 0$. Apply $\Gamma$: a short exact sequence of complexes of abelian groups (since each ${\mathcal{I}}^p,{\mathcal{K}}^p,{\mathcal{J}}^p$ is injective and $\Gamma$ preserves products of injectives in each degree). The snake lemma then produces the long exact sequence connecting cohomologies. $\square$

The long exact sequence is the workhorse of sheaf-cohomology computation: it converts a "vertical" exact sequence of sheaves into a "horizontal" exact sequence of cohomology groups, allowing iterative computation by attacking simpler sheaves.

Bridge. The construction here builds toward 04.04.01 (riemann-roch theorem for curves), where the same data is developed in the next layer of the strand. The defining pattern appears again in those units in a sharpened form, where the local data is glued or quotiented. Putting these together, the foundational insight is that the data of this unit gives the structural signature that the rest of the strand reads off.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

lean_status: partial — Mathlib has Sheaf.derivedCategory and basic derived-functor cohomology in the categorical setting.

[object Promise]

Advanced results [Master]

Theorem (Leray on good covers). Let $X$ be a paracompact Hausdorff space and $\mathcal{F}$ a sheaf of abelian groups on $X$. If $\mathcal{U}=\{U_i\}$ is an open cover such that every finite intersection $U_{i_0}\cap \cdots \cap U_{i_p}$ is $\mathcal{F}$-acyclic — meaning $H^q(U_{i_0}\cap \cdots \cap U_{i_p};\mathcal{F})=0$ for all $q\ge 1$ — then the canonical map

$${\check{H}}^p(\mathcal{U};\mathcal{F})\stackrel{\sim }{\to }{H}^p(X;\mathcal{F})$$

is an isomorphism for every $p\ge 0$.

Proof sketch. The Čech-to-derived-functor spectral sequence has $E_2^{p,q}={\check{H}}^p(\mathcal{U};{\mathcal{H}}^q(\mathcal{F}))$, where ${\mathcal{H}}^q(\mathcal{F})$ is the presheaf $U\mapsto H^q(U;\mathcal{F})$. The acyclic-cover hypothesis forces ${\mathcal{H}}^q(\mathcal{F})=0$ for $q\ge 1$ on every finite intersection, so the spectral sequence collapses on the $q=0$ row, yielding ${\check{H}}^p(\mathcal{U};\mathcal{F})=H^p(X;\mathcal{F})$.

This is the Leray cover theorem: when intersections are acyclic, Čech computes derived-functor cohomology on a single cover, no refinement needed. The corollary on smooth manifolds: a good cover (every finite intersection contractible) is acyclic for the constant sheaf $\underline{\mathbb{R}}$, so de Rham cohomology and Čech cohomology of $\underline{\mathbb{R}}$ on a good cover coincide.

Grothendieck's vanishing theorem. For a Noetherian topological space $X$ of dimension $n$ and any sheaf of abelian groups $\mathcal{F}$, $H^i(X;\mathcal{F})=0$ for $i>n$.

Leray spectral sequence. For a continuous map $f:X\to Y$ and a sheaf $\mathcal{F}$ on $X$, there is a spectral sequence $E_2^{p,q}=H^p(Y;R^q{f}_{\ast }\mathcal{F})\Rightarrow H^{p+q}(X;\mathcal{F})$. This decomposes the cohomology of $X$ in terms of cohomology on $Y$ with coefficients in the higher direct images.

Hodge decomposition. For a compact Kähler manifold (in particular, a smooth complex projective variety) $X$, the singular cohomology decomposes:

$$H^n(X;\mathbb{C})=\underset{p+q=n}{⨁}{H}^q(X;{\Omega }_X^p)$$

where ${\Omega }_X^p$ is the sheaf of holomorphic $p$-forms. This Hodge decomposition is one of the deepest results linking sheaf cohomology to topology.

Tic-tac-toe — alternative proof of Leray on good covers (constant sheaf $\underline{\mathbb{R}}$)

For the case of the constant sheaf $\underline{\mathbb{R}}$ on a smooth manifold $M$ with a finite good cover $\mathcal{U}=\{U_{\alpha }\}$, the Leray theorem admits a second proof via the Čech-de Rham double complex 03.04.11. The proof avoids the spectral-sequence machinery in favour of a direct diagonal-ascent argument — Bott-Tu's tic-tac-toe pedagogy.

Form the double complex

$$K^{p,q}=\prod _{{\alpha }_0<\cdots <{\alpha }_p}{\Omega }^q(U_{{\alpha }_0}\cap \cdots \cap U_{{\alpha }_p}),$$

with vertical $d$ (de Rham) and horizontal $\delta$ (Čech). The total differential $D=d+(-1{)}^p\delta$ satisfies $D^2=0$.

First filtration (rows first). Take cohomology of the vertical $d$ first. By the Poincaré lemma on each contractible intersection, every column collapses: only the degree-zero rows of constant functions survive. The remaining horizontal complex is the Čech complex of the constant sheaf $\underline{\mathbb{R}}$:

$${}_I{E}_2^{p,q}=\{\begin{array}{ll}{\check{H}}^p(\mathcal{U};\underline{\mathbb{R}})& q=0\\ 0& q\ge 1.\end{array}$$

The total cohomology of the double complex is therefore ${\check{H}}^{\ast }(\mathcal{U};\underline{\mathbb{R}})$.

Second filtration (columns first). Take cohomology of the horizontal $\delta$ first. Each sheaf ${\Omega }^q$ is fine, hence acyclic, so the Čech complex of ${\Omega }^q$ on the cover collapses to the global sections in degree zero:

$${}_{II}{E}_2^{p,q}=\{\begin{array}{ll}{H}_{\mathrm{dR}}^q(M)& p=0\\ 0& p\ge 1.\end{array}$$

The total cohomology is therefore $H_{\mathrm{dR}}^{\ast }(M)$.

The two filtrations of the same total complex give the same total cohomology, so ${\check{H}}^{\ast }(\mathcal{U};\underline{\mathbb{R}})\cong H_{\mathrm{dR}}^{\ast }(M)$. Combined with sheaf cohomology agreeing with de Rham cohomology of the constant sheaf via the fine resolution ${\Omega }^{\bullet }$ (Exercise 5), this gives Leray's theorem for $\underline{\mathbb{R}}$ on good covers.

The tic-tac-toe pedagogy is the same machinery as the spectral-sequence proof, with the bookkeeping done graphically on a 2D grid rather than symbolically as $E_r$ pages. Bott-Tu §9 carries the full diagonal-ascent diagram. The dual-proof discipline — Leray's theorem proved twice, once by spectral sequence and once by tic-tac-toe — is the same pattern that recurs in the Künneth formula 03.04.12 and in Poincaré duality.

Coherent cohomology of projective space. For projective space ${\mathbb{P}}_k^n$ over a field $k$ and the line bundle $\mathcal{O}(d)$:

$$H^i({\mathbb{P}}_k^n;\mathcal{O}(d))=\{\begin{array}{ll}\left(\genfrac{}{}{0ex}{}{n+d}{n}\right)& i=0,d\ge 0\\ \left(\genfrac{}{}{0ex}{}{-d-1}{-d-n-1}\right)& i=n,d\le -n-1\\ 0& \text{otherwise.}\end{array}$$

This is the foundational computation in projective algebraic geometry — every coherent-cohomology calculation eventually reduces to this and Riemann-Roch.

Synthesis. This construction generalises the pattern fixed in 04.01.01 (sheaf), with the symmetric data replaced by its skew or twisted analogue. Read in the opposite direction, the construction is dual to the metric story: complements and orthogonality are taken with respect to the bilinear datum of this unit, not a metric, and the resulting category of subobjects is the one the rest of the strand classifies. The central insight is that this datum identifies algebra with geometry: functions become vector fields, subspaces become quotients, and invariants become cohomology classes — and that identification is the engine driving every theorem downstream.

Full proof set [Master]

Detailed proofs (with all spectral-sequence arguments) of the long exact sequence theorem, Grothendieck vanishing, and the comparison between Čech and derived-functor cohomology are deferred to a worked supplement; the key derived-functor machinery is summarised in the formal-definition section above.

Connections [Master]

• De Rham cohomology 03.04.06 — sheaf cohomology of the constant sheaf $\underline{\mathbb{R}}$ on a smooth manifold; computed by the fine resolution ${\Omega }^{\bullet }$.

• Čech-de Rham double complex 03.04.11 — the tic-tac-toe alternative proof of Leray on good covers proceeds via the double complex of $\underline{\mathbb{R}}$ on a finite good cover. Connection type: foundation-of.

• Singular cohomology and de Rham theorem 03.04.13 — sheaf cohomology of the constant sheaf $\underline{\mathbb{R}}$ on a paracompact Hausdorff space agrees with singular cohomology with $\mathbb{R}$ coefficients; this is route 3 of the de Rham theorem. Connection type: bridging-theorem.

• Local systems and monodromy 04.03.02 — local systems are special locally constant sheaves; twisted cohomology is sheaf cohomology of these. Connection type: specialisation.

• Singular cohomology — for paracompact Hausdorff $X$ and constant sheaf $\underline{A}$, $H_{\mathrm{sing}}^i(X;A)\cong H^i(X;\underline{A})$.

• Riemann-Roch theorem for curves 04.04.01 — direct application of sheaf cohomology of line bundles on a curve, computing $\dim H^0(C;\mathcal{L})-\dim H^1(C;\mathcal{L})$.

• Riemann-Roch for compact Riemann surfaces 06.04.01 — analytic version, equivalent to the algebraic statement via GAGA.

• Étale cohomology — Grothendieck-topology refinement of sheaf cohomology, the foundation of arithmetic geometry and the Weil conjectures.

• Galois cohomology — sheaf cohomology in the étale topology of $\mathrm{Spec}(k)$ for a field $k$.

• Group cohomology — sheaf cohomology on the classifying space $BG$ of a discrete group $G$, with constant coefficients.

Throughlines and forward promises. Sheaf cohomology is the foundational organising language for cohomology with coefficients. We will see local systems generalise the constant sheaf in 04.03.02; we will see Leray's theorem for Čech cohomology on a good cover identify Čech and sheaf cohomology in 03.04.11; we will later see Hodge theory and Dolbeault cohomology specialise to holomorphic-coefficient sheaf cohomology. This pattern recurs throughout algebraic and complex geometry. The foundational reason sheaf cohomology vanishes on affines is exactly Serre's vanishing theorem; this is precisely the statement that affine geometry is module theory in disguise. Putting these together: sheaf cohomology is an instance of derived-functor cohomology, the generalisation of singular cohomology to non-constant coefficients, and the bridge between local algebraic data and global topological invariants. This is exactly the Grothendieck reformulation that organises modern algebraic geometry.

Historical & philosophical context [Master]

Sheaf cohomology emerged in the 1940s–50s through the work of Jean Leray (in a Nazi prison camp), Henri Cartan, Jean-Pierre Serre, and Alexander Grothendieck. Leray invented the concept while imprisoned to hide his expertise in fluid dynamics; he developed sheaves and their cohomology as part of a self-imposed program to "do harmless mathematics."

Cartan and Serre formalised the theory in the Cartan seminar (1948–64), and Serre's foundational paper Faisceaux Algébriques Cohérents (FAC, 1955) introduced sheaf cohomology to algebraic geometry. The framework was completed by Grothendieck's Tôhoku paper (1957), which defined sheaf cohomology in any abelian category with enough injectives via right-derived functors.

The conceptual revolution was the realisation that cohomology is local-to-global — every classical cohomology theory is sheaf cohomology of an appropriate sheaf. De Rham cohomology, singular cohomology, group cohomology, Galois cohomology — all become unified examples of derived functor cohomology of sheaves on appropriately chosen spaces (or sites).

This unification is one of the great conceptual achievements of 20th-century mathematics: a single framework explains the deep similarities across topology, complex analysis, algebraic geometry, and number theory.

Bibliography [Master]

• Hartshorne, Algebraic Geometry — Chapter III is the standard introduction to sheaf cohomology in algebraic geometry.
• Bott & Tu, Differential Forms in Algebraic Topology — masterful introduction covering Čech cohomology, spectral sequences, and de Rham theory.
• Iversen, Cohomology of Sheaves — comprehensive treatment of derived-functor sheaf cohomology.
• Godement, Topologie algébrique et théorie des faisceaux — the original derived-functor formulation, still a standard reference.
• Grothendieck, "Sur quelques points d'algèbre homologique" (Tôhoku 1957) — the foundational paper defining derived-functor cohomology in abelian categories.
• Serre, Faisceaux Algébriques Cohérents (FAC, 1955) — the introduction of sheaf cohomology to algebraic geometry.