In mathematics, the Leray spectral sequence was a pioneering example in homological algebra, introduced in 1946 by Jean Leray. It is usually seen nowadays as a special case of the Grothendieck spectral sequence.
Definition Let f : X → Y {\displaystyle f:X\to Y} be a continuous map of topological spaces, which in particular gives a functor f ∗ {\displaystyle f_{*}} from sheaves of abelian groups on X {\displaystyle X} to sheaves of abelian groups on Y {\displaystyle Y} . Composing this with the functor Γ {\displaystyle \Gamma } of taking sections on Sh Ab ( Y ) {\displaystyle {\text{Sh}}_{\text{Ab}}(Y)} is the same as taking sections on Sh Ab ( X ) {\displaystyle {\text{Sh}}_{\text{Ab}}(X)} , by the definition of the direct image functor f ∗ {\displaystyle f_{*}} :
S h A b ( X ) → f ∗ S h A b ( Y ) → Γ A b . {\displaystyle \mathrm {Sh_{Ab}} (X)\xrightarrow {f_{*}} \mathrm {Sh_{Ab}} (Y)\xrightarrow {\Gamma } \mathrm {Ab} .}
Thus the derived functors of Γ ∘ f ∗ {\displaystyle \Gamma \circ f_{*}} compute the sheaf cohomology for X {\displaystyle X} :
R i ( Γ ⋅ f ∗ ) ( F ) = H i ( X , F ) . {\displaystyle R^{i}(\Gamma \cdot f_{*})({\mathcal {F}})=H^{i}(X,{\mathcal {F}}).}
But because f ∗ {\displaystyle f_{*}} and Γ {\displaystyle \Gamma } send injective objects in Sh Ab ( X ) {\displaystyle {\text{Sh}}_{\text{Ab}}(X)} to Γ {\displaystyle \Gamma } -acyclic objects in Sh Ab ( Y ) {\displaystyle {\text{Sh}}_{\text{Ab}}(Y)} , there is a spectral sequencepg 33,19 whose second page is
E 2 p q = ( R p Γ ⋅ R q f ∗ ) ( F ) = H p ( Y , R q f ∗ ( F ) ) , {\displaystyle E_{2}^{pq}=(R^{p}\Gamma \cdot R^{q}f_{*})({\mathcal {F}})=H^{p}(Y,R^{q}f_{*}({\mathcal {F}})),}
and which converges to
E p + q = R p + q ( Γ ∘ f ∗ ) ( F ) = H p + q ( X , F ) . {\displaystyle E^{p+q}=R^{p+q}(\Gamma \circ f_{*})({\mathcal {F}})=H^{p+q}(X,{\mathcal {F}}).}
This is called the Leray spectral sequence.
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