Leray's theorem

In algebraic topology and algebraic geometry, Leray's theorem (so named after Jean Leray) relates abstract sheaf cohomology with Čech cohomology.

Let ${\mathcal {F}}$ be a sheaf on a topological space $X$ and ${\mathcal {U}}=\{U_{i}\}_{i\in I}$ an open cover of $X$. If ${\mathcal {F}}$ is acyclic on every finite intersection of elements of ${\mathcal {U}}$ (meaning that $H^{k}(U_{i_{0}}\cap \dots \cap U_{i_{n}},{\mathcal {F}})=0$ for all $k\geq 1$ and all finite intersections $U_{i_{0}}\cap \dots \cap U_{i_{n}}$), then there is an isomorphism

${\check {H}}^{q}({\mathcal {U}},{\mathcal {F}})\cong H^{q}(X,{\mathcal {F}})$

for all $q\geq 0$, where ${\check {H}}^{q}({\mathcal {U}},{\mathcal {F}})$ is the $q$-th Čech cohomology group of ${\mathcal {F}}$ with respect to the open cover ${\mathcal {U}}$, and $H^{q}(X,{\mathcal {F}})$ is the abstract sheaf cohomology group.

Proof

The standard modern proof of Leray's theorem relies on spectral sequences, specifically the Čech-to-derived functor spectral sequence.

Let ${\mathcal {F}}$ be a sheaf on $X$ and ${\mathcal {U}}$ an open cover. There exists a spectral sequence relating Čech cohomology to abstract sheaf cohomology, whose second page is given by:

$E_{2}^{p,q}={\check {H}}^{p}({\mathcal {U}},{\mathcal {H}}^{q}({\mathcal {F}}))\Rightarrow H^{p+q}(X,{\mathcal {F}})$

where ${\mathcal {H}}^{q}({\mathcal {F}})$ denotes the presheaf defined by $V\mapsto H^{q}(V,{\mathcal {F}})$.

The acyclicity hypothesis states that for any finite intersection $V=U_{i_{0}}\cap \dots \cap U_{i_{p}}$ of sets in ${\mathcal {U}}$, we have $H^{q}(V,{\mathcal {F}})=0$ for all $q\geq 1$. Consequently, the presheaf ${\mathcal {H}}^{q}({\mathcal {F}})$ evaluates to $0$ on all finite intersections of the cover.

This implies that the Čech complex computing ${\check {H}}^{p}({\mathcal {U}},{\mathcal {H}}^{q}({\mathcal {F}}))$ is identically zero for $q\geq 1$. Therefore, the spectral sequence degenerates at the $E_{2}$ page:

$E_{2}^{p,q}=0\quad {\text{for }}q>0$

Because the spectral sequence degenerates, the terms on the $p$-axis of the second page are isomorphic to the target of the spectral sequence:

$E_{2}^{p,0}\cong H^{p}(X,{\mathcal {F}})$

For $q=0$, the presheaf ${\mathcal {H}}^{0}({\mathcal {F}})$ assigns to each open set $V$ the module of sections $H^{0}(V,{\mathcal {F}})={\mathcal {F}}(V)$. Thus, ${\mathcal {H}}^{0}({\mathcal {F}})$ is exactly the sheaf ${\mathcal {F}}$ itself, which means $E_{2}^{p,0}={\check {H}}^{p}({\mathcal {U}},{\mathcal {F}})$. This collapse yields the desired isomorphism:

${\check {H}}^{p}({\mathcal {U}},{\mathcal {F}})\cong H^{p}(X,{\mathcal {F}})$

References

• Bonavero, Laurent. Cohomology of Line Bundles on Toric Varieties, Vanishing Theorems. Lectures 16-17 from "Summer School 2000: Geometry of Toric Varieties."

This article incorporates material from Leray's theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.