Plaything sheaf cohomology calculation

I asked this question a while back on MO :

One point that actually aided in finding out the Serre SS was doing certain calculations (like $H^*(CP^{\infty})$)

I wonder, as a type of followup if any person can recommend:

  1. A reference where tiny calculations are executed? or
  2. A details calculation to do with a tiny adequate sheaf an some straightforward topological room that would certainly have the ability to offer one a feeling for sheaf cohomology. So this room that we are persuading need not be a system, actually it would possibly be ideal if it were not a system given that I do not recognize them fairly yet. And also exist methods of the profession to calculating these points? or do individuals simply hammer away consumed injective resolutions?

Basically, please recommend a room and also a sheaf on it that I need to work with calculating the sheaf cohomology of.

PS: I certainly welcome any kind of various other pointers for recognizing just how to calculate sheaf cohomology.

2019-05-07 00:17:39
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Answers: 3

Rotman does some really primary specific calculations of Cech cohomology in his publication Homological Algebra.

If I bear in mind appropriately he does these calculations making use of resolutions, spooky series, and also by simply beginning with some series.

As a full newbie to this product, I had the ability to recognize his therapy and also calculate some details instances on my very own.

I wish this aids :)

P.S. I simply considered the first version, and also it appears to be various a little. For your details I made use of the 2nd version.

2019-05-09 07:02:01

This is instead system - y, yet there's an actually wonderful paper by Kempf (with any luck you have institutional accessibility : () that offers a really standard and also primary evidence that the greater cohomology of a seemingly - systematic sheaf on an affine system is unimportant. The first component of the paper makes use of absolutely nothing greater than the standard buildings (as an example lengthy specific series) of cohomology, and also could be enjoyable. I assumed it was enjoyable, anyhow ; it's additionally wonderful due to the fact that it reveals that Hartshorne is needlessly limiting in adhering to noetherian affine systems in phase III (also if one intends to stay clear of anything expensive).

OK, upgrade : below is the evidence explained (unquestionably by a newbie :) ).

2019-05-09 05:16:20

Any de Rham cohomology (or Dolbeault cohomology) calculation is a calculation in sheaf cohomology. In fact - - - any kind of calculation in single cohomology is a calculation in sheaf cohomology!! ; -) We're simply taking various resolutions of the ideal constant sheaf.

IIRC, there are some excellent Cech cohomology calculations and also instances in Bott - Tu. Additionally, have you read area 3.H of Hatcher's algebraic topology publication, on "neighborhood coefficients"?

For a straightforward instance from algebraic geometry, calculate the cohomology of the framework sheaf of $\mathbb{A}^2$ minus a factor.

I appear to remember a workout or an instance in Hartshorne in which the category of a level $d$ contour in $\mathbb{P}^2$ is calculated making use of Cech cohomology.

The area in Hartshorne on the cohomology of $\mathbb{P}^n$ makes use of Cech cohomology, and also I bear in mind locating it rather instructional.

Eisenbud's commutative algebra publication possibly has great deals of examples.

2019-05-08 19:46:29