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sheaf cohomology∗
mathcam†
2013-03-21 16:26:50
Let X be a topological space. The category of sheaves of abelian groups
on X has enough injectives. So we can define the sheaf cohomology H i (X, F)
of a sheaf F to be the right derived functors of the global sections functor
F → Γ(X, F).
Usually we are interested in the case where X is a scheme, and F is a
coherent sheaf. In this case, it does not matter if we take the derived functors
in the category of sheaves of abelian groups or coherent sheaves.
Sheaf cohomology can be explicitly calculated using Čech cohomology. Choose
an open cover {Ui } of X. We define
Y
C i (F) =
F(Uj0 ···ji )
where the product is over i + 1 element subsets of {1, . . . , n} and Uj0 ···ji =
Uj0 ∩ · · · ∩ Uji . If s ∈ F(Uj0 ···ji ) is thought of as an element of C i (F), then the
differential


Y j`+1
Y−1

∂(s) =
(−1)` s|Uj0 ···j` kj`+1 ···ji 
`
k=j` +1
∗
makes C (F) into a chain complex. The cohomology of this complex is denoted
Ȟ i (X, F) and called the Čech cohomology of F with respect to the cover {Ui }.
There is a natural map H i (X, F) → Ȟ i (X, F) which is an isomorphism for
sufficiently fine covers. (A cover is sufficiently fine if H i (Uj , F) = 0 for all
i > 0, for every j and for every sheaf F). In the category of schemes, for
example, any cover by open affine schemes has this property. What this means
is that if one can find a finite fine enough cover of X, sheaf cohomology becomes
computable by a finite process. In fact in [?], this is how the cohomology of
projective space is explicitly calculated.
∗ hSheafCohomologyi created: h2013-03-21i by: hmathcami version: h34587i Privacy
setting: h1i hDefinitioni h14F25i
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References
[1] Grothendieck, A. Sur quelques points d’algèbre homologique, Tôhoku Math.
J., Second Series, 9 (1957), 119–221.
[2] Hartshorne, R. Algebraic Geometry, Springer-Verlag Graduate Texts in
Mathematics 52, 1977
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