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

The comment(s) below were originally left at Talk:Sheaf cohomology/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

References needed. Geometry guy 23:37, 20 May 2007 (UTC) ...as well as clear explanation of sheaf cohomology vs cohomology of topological spaces (both in historical context and in mathematical theory). Also, seems biased towards algebraic geometry too much. Arcfrk 11:32, 26 May 2007 (UTC)

Last edited at 11:32, 26 May 2007 (UTC). 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