(also nonabelian homological algebra)
Context
Basic definitions
Stable homotopy theory notions
Constructions
Lemmas
Homology theories
Theorems
group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic cohomology?
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The Leray spectral sequence is the special case of the Grothendieck spectral sequence for the case where the two functors being composed are a push-forward of sheaves of abelian groups along a continuous map $f : X \to Y$ between topological spaces or more generally the direct image of a morphism of sites, followed by the push-forward $Y \to *$ to the point – the global section functor. This yields a spectral sequence that computes the abelian sheaf cohomology on $X$ in terms of the abelian sheaf cohomology on $Y$.
Let $X, Y$ be suitable sites let and $f : X \to Y$ be a morphism of sites. Let $\mathcal{C} = Ch_\bullet(Sh(X,Ab))$ and $\mathcal{D} = Ch_\bullet(Sh(Y,Ab))$ be the model categories of complexes of sheaves of abelian groups. The direct image $f_*$ and global section functor $\Gamma_Y$ compose to $\Gamma_X$:
Then for $A \in Sh(X,Ab)$ a sheaf of abelian groups on $X$ there is a cohomology spectral sequence
that converges as
and hence computes the abelian sheaf cohomology of $X$ with coefficients in $A$ in terms of the cohomology of $Y$ with coefficients in the derived direct image of $A$.
Lecture notes include
Dan Petersen, Leray spectral sequence, November 2010 (pdf)
Greg Friedman, Some extremely brief notes on the Leray spectral sequence (pdf)
Textbook accounts with an eye specifically towards étale cohomology
Günter Tamme, section I 3.7 of Introduction to Étale Cohomology
James Milne, section 11 of Lectures on Étale Cohomology
Last revised on May 6, 2015 at 19:25:57. See the history of this page for a list of all contributions to it.