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

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

for all ${\displaystyle q\geq 0}$, where ${\displaystyle {\check {H}}^{q}({\mathcal {U}},{\mathcal {F}})}$ is the ${\displaystyle q}$-th Čech cohomology group of ${\displaystyle {\mathcal {F}}}$ with respect to the open cover ${\displaystyle {\mathcal {U}}}$, and ${\displaystyle H^{q}(X,{\mathcal {F}})}$ 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 ${\displaystyle {\mathcal {F}}}$ be a sheaf on ${\displaystyle X}$ and ${\displaystyle {\mathcal {U}}}$ an open cover. There exists a spectral sequence relating Čech cohomology to abstract sheaf cohomology, whose second page is given by:

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

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

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

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

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

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

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

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

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

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.

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