Talk:Sheaf cohomology

Untitled
The following two sentences seem to conflict. They are even in the same paragraph! Could someone please explain? Thanks!
 * "The problem with the Čech theory manifests itself in the failure of the long exact sequence..."
 * "Jean-Pierre Serre showed that the Čech theory worked..."

Yzarc314 (talk) 02:33, 20 November 2009 (UTC)


 * The problem was that in general Čech theory fails to have a long exact sequence for non-Hausdorff spaces (meaning you can't really compute well with it). Serre was interested in coherent cohomology, i.e. a special type of sheaf, for the Zariski topology of algebraic varieties, i.e. a particular kind of non-Hausdorff space. There something better did happen. Charles Matthews (talk) 08:42, 20 November 2009 (UTC)

Assessment comment
Substituted at 02:35, 5 May 2016 (UTC)

Todo Theorems
There should be some basic theorems for sheaf cohomology on this page. This should include
 * cohomology and base change for constructible sheaves
 * cohomology and base change for coherent sheaves - https://stacks.math.columbia.edu/tag/07VJ
 * Degeneration theorems for spectral sequences

Todo Computations

 * Cohomology of line bundles on projective spaces: https://stacks.math.columbia.edu/tag/01XS
 * Cohomology of hypersurfaces: take the short exact sequence

0 \to \mathcal{O}(-d) \to \mathcal{O} \to \mathcal{O}_X \to 0 $$ then compute the long exact sequence. The only non-trivial terms will be $H^0$ and $H^{n-1}$. We know that

H^0(X;\mathcal{O}_X) \cong R $$ where $$R$$ is the base ring for $$\mathbb{P}^n$$. For $H^{n-1}$ use the long exact sequence to get the isomorphism

H^{n-1}(X;\mathcal{O}_X) \cong H^n(\mathbb{P}^n;\mathcal{O}_X(-d)) $$
 * Hodge decomposition of hypersurface, include plane curves. These can be computed using the Euler sequence:

0 \to \frac{\mathcal{I}_X}{\mathcal{I}_X^2} \to \Omega_{\mathbb{P}^n} \to \Omega_{X} \to 0 $$ Since

\frac{\mathcal{I}_X}{\mathcal{I}_X^2} \cong \mathcal{I}_X \otimes_{\mathcal{O}} \mathcal{O}_X $$ this can be easily computed in many cases. For smooth plane curves of degree $$d$$ the long exact sequence can be used to compute $$H^{0,1}$$.
 * cohomology of hypersurfaces in products of projective spaces (e.g. hyperelliptic curves)
 * https://mathoverflow.net/questions/157325/cohomology-of-the-tangent-sheaf-of-mathbbp1-2-3?rq=1

Confusing sentence
The section Definition contains this passage:

"The essential point is to fix a topological space X and think of cohomology as a functor from sheaves of abelian groups on X to abelian groups. In more detail, start with the functor E ↦ E(X) from sheaves of abelian groups on X to abelian groups."

I am confused by the phrase "the functor E ↦ E(X)". Does this mean any functor (from sheaves of abelian groups on X to abelian groups)? Or does the use of the word "the" imply that it is a specific functor of this type ... and if so, which one is it?

Logicdavid (talk), on 28 September 2023, says: They mean a specific functor, the one assigning to E the group of global sections of E. I agree it is confusing!

Possible mistake
The section Sheaf cohomology with constant coefficients contains the assertion: "For a continuous map f: X → Y and an abelian group A, the pullback sheaf f*(AY) is isomorphic to AX". Can this be true? What if Y consists of two isolated points, and f is the inclusion of the first point? Then a sheaf can attach any two groups to these points in Y, and the sheaf over X will only have one of them. Right? Logicdavid (talk) 28 September 2023